L1 -algebra connections and applications to String- and Chern-Simons n-transport Hisham Sati*, Urs Schreiberyand Jim Stasheffz January 24, 2008 Abstract We give a generalization of the notion of a Cartan-Ehresmann connection * *from Lie algebras to L1 - algebras and use it to study the obstruction theory of lifts through highe* *r String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals thi* *s way and describe aspects of their parallel transport and quantization. It is known that over a D-brane the Kalb-Ramond background field of the * *string restricts to a 2- bundle with connection (a gerbe) which can be seen as the obstruction to l* *ifting the PU(H)-bundle on the D-brane to a U(H)-bundle. We discuss how this phenomenon generalizes f* *rom the ordinary central extension U(1) ! U(H) ! PU(H) to higher categorical central extensions, li* *ke the String-extension BU(1) ! String(G) ! G. Here the obstruction to the lift is a 3-bundle with* * connection (a 2-gerbe): the Chern-Simons 3-bundle classified by the first Pontrjagin class. For G* * = Spin(n) this obstructs the existence of a String-structure. We discuss how to describe this obstr* *uction problem in terms of Lie n-algebras and their corresponding categorified Cartan-Ehresmann conne* *ctions. Generalizations even beyond String-extensions are then straightforward. For G = Spin(n) th* *e next step is "Fivebrane structures" whose existence is obstructed by certain generalized Chern-Sim* *ons 7-bundles classified by the second Pontrjagin class. ________________________________* hisham.sati@yale.edu yschreiber@math.uni-hamburg.de zjds@math.upenn.edu Contents 1 Introduction * * 4 2 The Setting and Plan * * 5 2.1 L1 -algebras and their String-like central extensions . . . . . . . . .* * . . . . . . . . . . . . . . 6 2.1.1L1 -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . . . . 6 2.1.2L1 -algebras from cocycles: String-like extensions . . . . . . . . * *. . . . . . . . . . . . . 6 2.1.3 L1 -algebra differential forms . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 7 2.2 L1 -algebra Cartan-Ehresmann connections . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . 8 2.2.1g-Bundle Descent data . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . . . 8 2.2.2Connections on n-bundles: the extension problem . . . . . . . . . .* * . . . . . . . . . . . 9 2.3 Higher String and Chern-Simons n-transport: the lifting problem . . . .* * . . . . . . . . . . . . 10 3 Physical applications: String-, Fivebrane- and p-Brane structures * * 12 4 Statement of the main results * * 14 5 Differential graded-commutative algebra * * 15 5.1 Differential forms on smooth spaces . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . . . . 16 5.1.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 18 5.2 Homotopies and inner derivations . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 19 5.2.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 20 5.3 Vertical flows and basic forms . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . . . . . 21 5.3.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 22 6 L1 -algebras and their String-like extensions * * 25 6.1 L1 -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . . . 25 6.1.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 29 6.2 L1 -algebra homotopy and concordance . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . . 32 6.2.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 36 6.3 L1 -algebra cohomology . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . . . . 37 6.3.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 41 6.4 L1 -algebras from cocycles: String-like extensions . . . . . . . . . . * *. . . . . . . . . . . . . . . 45 6.4.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 48 6.5 L1 -algebra valued forms . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . . 49 6.5.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 50 6.6 L1 -algebra characteristic forms . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . . . . 52 6.6.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 53 7 L1 -algebra Cartan-Ehresmann connections * * 54 7.1 g-Bundle descent data . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . . 55 7.1.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 55 7.2 Connections on g-bundles: the extension problem . . . . . . . . . . . .* * . . . . . . . . . . . . . 59 7.2.1Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . . . . 60 7.3 Characteristic classes . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . . . 61 7.3.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 62 7.4 Universal and generalized g-connections . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 62 7.4.1Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . . . . 63 2 8 Higher String- and Chern-Simons n-bundles: the lifting problem * * 64 8.1 Weak cokernels of L1 -morphisms . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . . . 64 8.1.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 68 8.2 Lifts of g-descent objects through String-like extensions . . . . . . .* * . . . . . . . . . . . . . . 70 8.2.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 71 8.3 Lifts of g-connections through String-like extensions . . . . . . . . .* * . . . . . . . . . . . . . . 72 8.3.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 75 9 L1 -algebra parallel transport * * 78 9.1 L1 -parallel transport . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . . . . . 78 9.1.1Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . . . . 79 9.2 Transgression of L1 -transport . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . . . . 80 9.2.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 81 9.3 Configuration spaces of L1 -transport . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 81 9.3.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . . . 82 A Appendix: Explicit formulas for 2-morphisms of L1 -algebras * * 86 3 1 Introduction The study of extended n-dimensional relativistic objects which arise in string * *theory has shown that these couple to background fields which can be naturally thought of as n-fold categor* *ified generalizations of fiber bundles with connection. There are two popular alternative viewpoints on studyi* *ng such higher structures geometrically. The first is using the language of gerbes and the second using t* *he language of Cheeger-Simons differential characters or Deligne cohomology. fundamental | background _____object____||___field____ | n-particle |n-bundle | (n - 1)-brane (n|- 1)-gerbe Table 1: The two schools of counting higher dimensional structures. Here n is* * in N = {0, 1, 2, . .}.. The first departure from bundles with connections occurs with the fundamenta* *l (super)string which couples to the Neveu-Schwarz (NS) B-field. Locally, the B-field is just an R-va* *lued two-form. However, the study of the path integral, which amounts to `exponentiation', reveals that the* * B-field can be thought of as an abelian gerbe with connection whose curving corresponds to the H-field H3* * or as a Cheeger-Simons differential character, whose holonomy [24] can be described [14] in the langua* *ge of bundle gerbes [46]. The next step up occurs with the M-theory (super)membrane which couples to t* *he C-field [7]. In supergravity, this is viewed locally as an R-valued differential three-form. Ho* *wever, the study of the path integral has shown that this field is quantized in a rather nontrivial way [58]* *. This makes the C-field not precisely a 2-gerbe or degree 3 Cheeger-Simons differential character but rathe* *r a shifted version [21] that can also be modeled using the Hopkins-Singer description of differential charac* *ters [34]. Some aspects of the description in terms of Deligne cohomology is given in [19]. From a purely formal point of view, the need of higher connections for the d* *escription of higher di- mensional branes is not a surprise: n-fold categorified bundles with connection* * should be precisely those objects that allow us to define a consistent assignment of "phases" to n-dimens* *ional paths in their base space. We address such an assignment as parallel n-transport. This is in fact e* *ssentially the definition of Cheeger-Simons differential characters [18] as these are consistent assignments* * of phases to chains. However, abelian bundle gerbes, Deligne cohomology and Cheeger-Simons differential chara* *cters all have one major restriction: they only know about assignments of elements in U(1). While the group of phases that enter the path integral is usually abelian, m* *ore general n-transport is important nevertheless. For instance, the latter plays a role at intermediate s* *tages. This is well understood for n = 2: over a D-brane the abelian bundle gerbe corresponding to the NS fiel* *d has the special property that it measures the obstruction to lifting a PU(H)-bundle to a U(H)-bundle, i.* *e. lifting a bundle with structure group the infinite projective unitary group on a Hilbert space H to t* *he corresponding unitary group [11] [12]. Hence, while itself an abelian 2-structure, it is crucially re* *lated to a nonabelian 1-structure. That this phenomenon deserves special attention becomes clear when we move u* *p the dimensional ladder: The Green-Schwarz anomaly cancelation [29] in the heterotic string leads to a 3* *-structure with the special property that, over the target space, it measures the obstruction to lifting an* * E8xSpin(n)-bundle to a certain nonabelian principal 2-bundle, called a String 2-bundle. Such a 3-structure is * *also known as a Chern-Simons 2-gerbe [15]. By itself this is abelian, but its structure is constrained by ce* *rtain nonabelian data. Namely this string 2-bundle with connection, from which the Chern-Simons 3-bundle aris* *es, is itself an instance of a structure that yields parallel 2-transport. It can be described neither by ab* *elian bundle gerbes, nor by 4 Cheeger-Simons differential characters, nor by Deligne cohomology. In anticipation of such situations, previous works have considered nonabelia* *n gerbes and nonabelian bundle gerbes with connection. However, it turns out that care is needed in ord* *er to find the right setup. For instance, the kinds of nonabelian gerbes with connection studied in [13] [2* *], although very interesting, are not sufficiently general to capture String 2-bundles. Moreover, it is not easy * *to see how to obtain the parallel 2-transport assignment from these structures. For the application to string phy* *sics, it would be much more suitable to have a nonabelian generalization of the notion of a Cheeger-Simons * *differential character, and thus a structure which, by definition, knows how to assign generalized phases t* *o n-dimensional paths. The obvious generalization that is needed is that of a parallel transport n-* *functor. Such a notion was described in [5] [52]: a structure defined by the very fact that it labels n-pa* *ths by algebraic objects that allow composition in n different directions, such that this composition is compatible* * with the gluing of n-paths. One can show that such transport n-functors encompass abelian and nonabelian ge* *rbes with connection as special cases [52]. However, these n-functors are more general. For instance, S* *tring 2-bundles with connection are given by parallel transport 2-functors. Ironically, the strength of the lat* *ter - namely their knowledge about general phase assignments to higher dimensional paths - is to some degree* * also a drawback: for many computations, a description entirely in terms of differential form data would b* *e more tractable. However, the passage from parallel n-transport to the corresponding differential structure i* *s more or less straightforward: a parallel transport n-functor is essentially a morphism of Lie n-groupoids. As* * such, it can be sent, by a procedure generalizing the passage from Lie groups to Lie algebras, to a morphi* *sm of Lie n-algebroids. The aim of this paper is to describe two topics: First, to set up a formalis* *m for higher bundles with connections entirely in terms of L1 -algebras, which may be thought of as a cat* *egorification of the theory of Cartan-Ehresmann connections. This is supposed to be the differential versio* *n of the theory of parallel transport n-functors, but an exhaustive discussion of the differentiation proce* *dure is not given here. Instead we discuss a couple of examples and then show how the lifting problem has a nic* *e description in this language. To do so, we present a family of L1 -algebras that govern the gauge s* *tructure of p-branes, as above, and discuss the lifting problem for them. By doing so, we characterize C* *hern-Simons 3-forms as local connection data on 3-bundles with connection which arise as the obstruction to * *lifts of ordinary bundles to the corresponding String 2-bundles, governed by the String Lie 2-algebra. The formalism immediately allows the generalization of this situation to hig* *her degrees. Indeed we indicate how certain 7-dimensional generalizations of Chern-Simons 3-bundles ob* *struct the lift of ordinary bundles to certain 6-bundles governed by the Fivebrane Lie 6-algebra. The latte* *r correspond to what we define as the fivebrane structure, for which the degree seven NS field H7 plays* * the role that the degree three dual NS field H3 plays for the n = 2 case. The paper is organized in such a way that section 2 serves more or less as a* * self-contained description of the basic ideas and construction, with the rest of the document having all the * *details and all the proofs. In this paper we make use of the homotopy algebras usually referred to as L1* * -algebras. These algebras also go by other names such as sh-Lie algebras [41]. In our context we may als* *o call such algebras Lie 1-algebras which we think of as the abstract concept of an 1-vector space with * *an antisymmetric and coherently Jacobi bracket 1-functor on it, whereas "L1 -algebra" is concretely * *a codifferential coalgebra of sorts. In this paper we will nevertheless follow the standard notation of L1 -a* *lgebra. 2 The Setting and Plan We set up a useful framework for describing higher order bundles with connectio* *n entirely in terms of Lie n- algebras, which can be thought of as arising from a categorification of the con* *cept of an Ehresmann connection on a principal bundle. Then we apply this to the study of Chern-Simons n-bundl* *es with connection as obstructions to lifts of principal G-bundles through higher String-like extensi* *ons of their structure Lie algebra. 5 2.1 L1 -algebras and their String-like central extensions A Lie group has all the right properties to locally describe the phase change o* *f a charged particle as it traces out a worldline. A Lie n-group is a higher structure with precisely all the rig* *ht properties to describe locally the phase change of a charged (n - 1)-brane as it traces out an n-dimensional w* *orldvolume. 2.1.1 L1 -algebras Just as ordinary Lie groups have Lie algebras, Lie n-groups have Lie n-algebras* *. If the Lie n-algebra is what is called semistrict, these are [3] precisely L1 -algebras [41] which have come* * to play a significant role in cohomological physics. A ("semistrict" and finite dimensional) Lie n-algebra is* * any of the following three equivalent structures: o an L1 -algebra structure on a graded vector space g concentrated in the fi* *rst n degrees (0, ..., n - 1); o a quasi-free differential graded-commutative algebra ("qDGCA": free as a g* *raded-commutative) algebra on the dual of that vector space: this is the Chevalley-Eilenberg algebra * *CE(g) of g; o an n-category internal to the category of graded vector spaces and equippe* *d with a skew-symmetric linear bracket functor which satisfies a Jacobi identity up to higher cohe* *rent equivalence. For every L1 -algebra g, we have the following three qDGCAs: o the Chevalley-Eilenberg algebra CE(g) o the Weil algebra W(g) o the algebra of invariant polynomials or basic forms inv(g). These sit in a sequence CE (g)oooo_____W(g)oo________inv(g)?,` (* *1) where all morphisms are morphisms of dg-algebras. This sequence plays the role * *of the sequence of differential forms on the "universal g-bundle". 2.1.2 L1 -algebras from cocycles: String-like extensions A simple but important source of examples for higher Lie n-algebras comes from * *the abelian Lie algebra u(1) which may be shifted into higher categorical degrees. We write bn-1u(1) fo* *r the Lie n-algebra which is entirely trivial except in its nth degree, where it looks like u(1). Just as u(* *1) corresponds to the Lie group U(1) , so bn-1u(1) corresponds to the iterated classifying space Bn-1U(1), real* *izable as the topological group given by the Eilenber-MacLane space K(Z, n). Thus an important source for inter* *esting Lie n-algebras comes from extensions 0 ! bn-1u(1) ! ^g! g ! 0 (* *3) of an ordinary Lie algebra g by such a shifted abelian Lie n-algebra bn-1u(1). * *We find that, for each (n + 1)- cocycle ~ in the Lie algebra cohomology of g, we do obtain such a central exten* *sion, which we describe by 0 ! bn-1u(1) ! g~ ! g ! 0 . (* *4) Since, for the case when ~ = <., [., .]> is the canonical 3-cocycle on a semisi* *mple Lie algebra g, this g~ is known ([4] and [32]) to be the Lie 2-algebra of the String 2-group, we call the* *se central extensions String-like central extensions. (We also refer to these as Lie n-algebras "of Baez-Crans ty* *pe" [3].) Moreover, whenever the cocycle ~ is related by transgression to an invariant polynomial P on the L* *ie algebra, we find that g~ fits into a short homotopy exact sequence of Lie (n + 1)-algebras 0 ! g~ ! csP(~) ! chP(~) ! 0 . (* *5) 6 (pointed) dg-algebras topological (* *2) spaces Chevalley- Eilenberg CE (g)OO G" ` structure group algebra OO| || i*|| |i| | | | fflffl| universal Weil algebra W(g) EG OO | G-bundle | | p*|| |p| algebra of ?O|| fflfflfflffl||classifying space invariant inv(g) BG polynomials for G Figure 1: The universal G-bundle and its analog in the world of dg-al* *gebras. Here csP(g) is a Lie (n + 1)-algebra governed by the Chern-Simons term correspo* *nding to the transgression element interpolating between ~ and P. In a similar fashion chP(g) knows about * *the characteristic (Chern) class associated with P. In summary, from elements of W(g)-cohomology we obtain the String-like exten* *sions of Lie algebras to Lie 2n-algebras and the associated Chern- and Chern-Simons Lie (2n - 1)-algebra* *s: ____________________________________________________ Lie algebra cocycle~ |Baez-Crans Lie n-algebrag~ invariant polynomialP |Chern Lie n-algebra chP(g) _transgression_elementcs|Chern-Simons_Lie_n-algebracsP(g)_ 2.1.3 L1 -algebra differential forms For g an ordinary Lie algebra and Y some manifold, one finds that dg-algebra mo* *rphisms CE(g) ! o(Y ) from the Chevally-Eilenberg algebra of g to the DGCA of differential forms on Y* * are in bijection with g-valued 1-forms A 2 1(Y, g) whose ordinary curvature 2-form FA = dA + [A ^ A] (* *6) vanishes. Without the flatness, the correspondence is with algebra morphisms no* *t respecting the differentials. But dg-algebra morphisms A : W(g) ! o(Y ) are in bijection with arbitrary g-va* *lued 1-forms. These are flat precisely if A factors through CE(g). This situation is depicted in the fo* *llowing diagram: CE(g)oooo______W(g) ____ | (A,FA=0)_______ (A,FA)| fflffl___= fflffl|. (* *7) o(Y )__________ o(Y ) This has an obvious generalization for g an arbitrary L1 -algebra. For g any L1* * -algebra, we write o(Y, g) = Homdg-Alg(W(g), o(X)) (* *8) 7 for the collection of g-valued differential forms and oflat(Y, g) = Homdg-Alg(CE (g), o(X)) (* *9) for the collection of flat g-valued differential forms. 2.2 L1 -algebra Cartan-Ehresmann connections 2.2.1 g-Bundle Descent data A descent object for an ordinary principal G-bundle on X is a surjective submer* *sion ss : Y ! X together with a functor g : Y xX Y ! BG from the groupoid whose morphisms are pairs of p* *oints in the same fiber of Y , to the groupoid BG which is the one-object groupoid corresponding to the* * group G. Notice that the groupoid BG is not itself the classifying space BG of G, but the geometric real* *ization of its nerve, |BG|, is: |BG| = BG. We may take Y to be the disjoint union of some open subsets {Ui} of X that f* *orm a good open cover of X. Then g is the familiar concept of a transition function decribing a bundle t* *hat has been locally trivialized over the Ui. But one can also use more general surjective submersions. For inst* *ance, for P ! X any principal G-bundle, it is sometimes useful to take Y = P. In this case one obtains a cano* *nical choice for the cocycle g : Y xX Y = P xX P ! BG (1* *0) since P being principal means that P xX P 'diffeoP x G . (1* *1) This reflects the fact that every principal bundle canonically trivializes when* * pulled back to its own total space. The choice Y = P differs from that of a good cover crucially in the foll* *owing aspect: if the group G is connected, then also the fibers of Y = P are connected. Cocycles over surj* *ective submersions with connected fibers have special properties, which we will utilize: When the fiber* *s of Y are connected, we may think of the assignment of group elements to pairs of points in one fiber as ar* *ising from the parallel transport with respect to a flat vertical 1-form Avert2 1vert(Y, g), flat along the fibe* *rs. As we shall see, this can be thought of as the vertical part of a Cartan-Ehresmann connection 1-form. This p* *rovides a morphism overt(YA)veCE(g)rtoo_ (1* *2) of differential graded algebras from the Chevalley-Eilenberg algebra of g to th* *e vertical differential forms on Y . Unless otherwise specified, morphism will always mean homomorphism of differ* *ential graded algebra. Averthas an obvious generalization: for g any Lie n-algebra, we say that a g-bu* *ndle descent object for a g-n-bundle on X is a surjective submersion ss : Y ! X together with a morphism * * overt(YA)veCE(g)rtoo_. Now Avert2 overt(Y, g) encodes a collection of vertical p-forms on Y , each ta* *king values in the degree p-part of g and all together satisfying a certain flatness condition, controlled by th* *e nature of the differential on CE (g). 8 2.2.2 Connections on n-bundles: the extension problem Given a descent object overt(Yo)Averto_CE(g) as above, a flat connection on it* * is an extension of the morphism Avertto a morphism Aflatthat factors through differential forms on* * Y overt(Yo)Averto_CE(g)_OO. (1* *3) OO| _____ | ______ | ______ i* ______ || ___Aflat_______ | ______ | --______ o(Y ) In general, such an extension does not exist. A general connection on a g-desce* *nt object Avertis a morphism (A,FA) o(Y )oo_______W(g) (1* *4) from the Weil algebra of g to the differential forms on Y together with a morph* *ism {Ki} o(Y )oo______inv(g)_ (1* *5) from the invariant polynomials on g, as in 2.1.1, to the differential forms on * *X, such that the following two squares commute: overt(Yo)AvertoCE_(g) OOOO| OOOO| | | i*|| || . (1* *6) | | | | | (A,FA) | o(YO)oo________W(g)_OOO | | | | ss*|| || | | | | ?O| ?O| o(X) oo_{Ki}___inv(g) Whenever we have such two commuting squares, we say o Avert2 overt(Y, g) is a g-bundle descent object (playing the role of a tr* *ansition function); o A 2 o(Y, g) is a (Cartan-Ehresmann) connection with values in the L1 -alg* *ebra g on the total space of the surjective submersion; o FA 2 o+1(Y, g) are the corresponding curvature forms; o and the set {Ki 2 o(X)} are the corresponding characteristic forms, whose* * classes {[Ki]} in deRham cohomology {Ki} o(X)oo____________inv(g) (1* *7) {[Ki]} HodeRham(X)oo______Ho(inv(g)) are the corresponding characteristic classes of the given descent object A* *vert. 9 overt(Yo)Averto__CE (g) descentdata OOOO| OOOO| | | || || first |i* | Cartan-Ehresmann || || condition | | | (A,FA) | connection o(YO)oo__________W(g)OOO data | | | | || || second |ss* | Cartan-Ehresmann || || condition | | ?O| ?O| characteristic o(X)oo__{Ki}_____inv(g)_ forms HodR(X)oo_{[K____Ho(inv(g)) Chern-Weil i]} homomorphism Figure 2: A g-connection descent object and its interpretation. For g-any L1 -* *algebra and X a smooth space, a g-connection on X is an equivalence class of pairs (Y, (A, FA))* * consisting of a surjective submersion ss : Y ! X and dg-algebra morphisms forming the above commuting diag* *ram. The equivalence relation is concordance of such diagrams. Hence we realize the curvature of a g-connection as the obstruction to exten* *ding a g-descent object to a flat g-connection. 2.3 Higher String and Chern-Simons n-transport: the lifting problem Given a g-descent object CE (g), (1* *8) t ttt yytAverttttt overt(Y ) and given an extension of g by a String-like L1 -algebra CE(bn-1u(1))ooioo___CE(g~)oo_______?CE(g)`_, (1* *9) we ask if it is possible to lift the descent object through this extension, i.e* *. to find a dotted arrow in CE(bn-1u(1))oooCE(g~)o___oo____CE?(g)`_. (2* *0) ______ ---- ______ -A- _""_____vert""-- overt(Y ) In general this is not possible. We seek a straightforward way to compute the o* *bstruction to the existence of the lift. The strategy is to form the weak (homotopy) kernel of CE(bn-1u(1))iooCE(g~)oo_ (2* *1) 10 which we denote by CE(bn-1u(1) ,! g~) and realize as a mapping cone of qDGCAs. * *This comes canonically with a morphism f from CE(g) which happens to have a weak inverse CE(bn-1u(1) ,! g~). (2* *2) hhnn7_______________________7n hhhhhhf__________________________* *________nn hhhhhhnnnn-1______________________ i tthhhhhh`qq___f____________________________* *___________________________________n CE (bn-1u(1))ooCEo(g~)o_oCE?(g)o_ Then we see that, while the lift to a g~-cocycle may not always exist, the lift* * to a (bn-1u(1) ,! g~)-cocycle does always exist. We form AvertO f-1: CE(bn-1u(1) ,! g~). (2* *3) fffffnnnn fffffff f-1 fffffff nnnn n-1 i ssfffffff ` wwn CE(b u(1))oooCE(g~)o___oo____?CE(g)_ ______ ---- ______ -A- _""_____vert""-- overt(Y ) The failure of this lift to be a true lift to g~ is measured by the component o* *f AvertO f-1 on bn-1u(1)[1] ' bnu(1). Formally this is the composite A0vert:= AvertO f-1 O j in j CE(bn-1u(1) ,! g~)oCE?(bnu(1))`o_.* *(24) fffffnnnn jjjjj fffffff f-1 jjjjj fffffff nnnn jjjjj n-1 i ssfffffff ` wwn A0 jjj CE(b u(1))oooCE(g~)o___oo____CE?(g)_ jjjjvert ________ --- jjjj ______Avert-jjjjj _""___"uujjjjj"-- overt(Y ) The nontriviality of the bnu(1)-descent object A0vertis the obstruction to cons* *tructing the desired lift. We thus find the following results, for any g-cocycle ~ which is in transgre* *ssion with the the invariant polynomial P on g, o The characteristic classes (in deRham cohomology) of g~-bundles are those * *of the corresponding g- bundles modulo those coming from the invariant polynomial P. o The lift of a g-valued connection to a g~-valued connection is obstructed * *by a bnu(1)-valued (n + 1)- connection whose (n+1)-form curvature is P(FA), i.e. the image under the C* *hern-Weil homomorphism of the invariant polynomial corresponding to ~. o Accordingly, the (n + 1)-form connection of the obstructing bnu(1) (n + 1)* *-bundle is a Chern-Simons form for this characteristic class. We call the obstructing bnu(1) (n + 1)-descent object the corresponding Cher* *n-Simons (n + 1)-bundle. For the case when ~ = <., [., .]> is the canonical 3-cocycle on a semisimple Li* *e algebra g, this structure (corresponding to a 2-gerbe) has a 3-connection given by the ordinary Chern-Sim* *ons 3-form and has a curvature 4-form given by the (image in deRham cohomology of) the first Pontrja* *gin class of the underlying g-bundle. 11 3 Physical applications: String-, Fivebrane- and p-Brane struc- tures We can now discuss physical applications of the formalism that we have develope* *d. What we describe is a useful way to handle obstructing n-bundles of various kinds that appear in stri* *ng theory. In particular, we can describe generalizations of string structure in string theory. In the context o* *f p-branes, such generalizations have been suggested based on p-loop spaces [23] [6] [48] and, more generally, o* *n the space of maps Map(M, X) from the brane worldvolume M to spacetime X [45]. The statements in this sectio* *n will be established in detail in [60]. From the point of view of supergravity, all branes, called p-branes in that * *setting, are a priori treated in a unified way. In tracing back to string theory, however, there is a distinc* *tion in the form-fields between the Ramond-Ramond (RR) and the Neveu-Schwarz (NS) forms. The former live in gen* *eralized cohomology and the latter play two roles: they act as twist fields for the RR fields and t* *hey are also connected to the geometry and topology of spacetime. The H-field H3 plays the role of a twist in* * K-theory for the RR fields [37] [11] [44]. The twist for the degree seven dual field H7 is observed in [51* *] at the rational level. The ability to define fields and their corresponding partition functions put* *s constraints on the topology of the underlying spacetime. The most commonly understood example is that of fermi* *ons where the ability to define them requires spacetime to be spin, and the ability to describe theories* * with chiral fermions requires certain restrictions coming from the index theorem. In the context of heterotic* * string theory, the Green- Schwarz anomaly cancelation leads to the condition that the difference between * *the Pontrjagin classes of the tangent bundle and that of the gauge bundle be zero. This is called the str* *ing structure, which can be thought of as a spin structure on the loop space of spacetime [38] [20]. In M-t* *heory, the ability to define the partition function leads to an anomaly given by the integral seventh-integr* *al Steifel-Whitney class of spacetime [22] whose cancelation requires spacetime to be orientable with respe* *ct to generalized cohomology theories beyond K-theory [40] . In all cases, the corresponding structure is related to the homotopy groups * *of the orthogonal group: the spin structure amounts to killing the first homotopy group, the string structur* *e and - to some extent- the W7 condition to killing the third homotopy group. Note that when we say that th* *e n-th homotopy group is killed, we really mean that all homotopy groups up to and including the n-th on* *e are killed. For instance, a String structure requires killing everything up to and including the third, h* *ence everything through the sixth, since there are no homotopy groups in degrees four, five or six. The Green-Schwarz anomaly cancelation condition for the heterotic string can* * be translated to the language of n-bundles as follows. We have two bundles, the spin bundle with str* *ucture group G = Spin(10), and the gauge bundle with structure group G0being either SO(32)=Z2 or E8x E8. C* *onsidering the latter, we have one copy of E8 on each ten-dimensional boundary component, which can be* * viewed as an end-of- the-world nine-brane, or M9-brane [35]. The structure of the four-form on the b* *oundary which we write as G4|@ = dH3 (2* *5) implies that the 3-bundle (2-gerbe) becomes the trivializable lifting 2-gerbe o* *f a String(Spin(10)xE8) bundle over the M9-brane. As the four-form contains the difference of the Pontrjagin c* *lasses of the bundles with structure groups G and G0, the corresponding three-form will be a difference of* * Chern-Simons forms. The bundle aspect of this has been studied in [8] and will be revisited in the curr* *ent context in [60]. The NS fields play a special role in relation to the homtopy groups of the o* *rthogonal group. The degree three class [H3] plays the role of a twist for a spin structure. Likewise, the * *degree seven class plays a role of a twist for a higher structure related to BO<9>, the 8-connected cover of BO* *, which we might call a Fivebrane-structure on spacetime. We can talk about such a structure once the s* *pacetime already has a string structure. The obstructions are given in the following table, where A is* * the connection on the G0 12 bundle and ! is a connection on the G bundle. n || 2 6 __________________|_____=_4_._0_+_2________=_4_._1_+_2______ fundamental object|| (n - 1)-brane | string 5-brane ______n-particle___||_______________________________________ target space | string structure fivebrane structure structure ||ch2(A) - p1(!) = 0ch4(A) - 1_48p2(!) = 0 Table 2: Higher dimensional extended objects and the corresponding topologic* *al structures. In the above we alluded to how the brane structures are related to obstructi* *ons to having spacetimes with connected covers of the orthogonal groups as structures. The obstructing classe* *s here may be regarded as classifying the corresponding obstructing n-bundles, after we apply the general* * formalism that we outlined earlier. The main example of this general mechanism that will be of interest to* * us here is the case where g is an ordinary semisimple Lie algebra. In particular, we consider g = spin(n)* *. For g = spin(n) and ~ a (2n + 1)-cocycle on spin(n), we call spin(n)~ the (skeletal version of the) (2n* * - 1)-brane Lie (2n) - algebra. Thus, the case of String structure and Fivebrane structure occurring in the fun* *damental string and NS fivebrane correspond to the cases n = 1 and n = 3 respectively. Now applying ou* *r formalism for g = spin(n), and ~3, ~7 the canonical 3- and 7-cocycle, respectively, we have: o the obstruction to lifting a g-bundle descent object to a String 2-bundle * *(a g~3-bundle descent object) is the first Pontryagin class of the original g-bundle cocycle; o the obstruction to lifting a String 2-bundle descent object to a Fivebrane* * 6-bundle cocycle (a g~7-bundle descent object) is the second Pontryagin class of the original g-bundle co* *cycle. The cocyles and invariant polynomials corresponding to the two structures ar* *e given in the following table _p-brane_______cocycle____________invariant_polynomial______________ p = 1 = 4 . 0 +~13= <., [., .]> P1= <., .> first Pontrjagin p = 5 = 4 . 1 +~17= <., [., .], [.,P.],2[.,=.]><., .,s.,e.>cond Pontrjagin Table 3: Lie algebra cohomology governing NS p-branes. In case of the fundamental string, the obstruction to lifting the PU(H) bund* *les to U(H) bundles is measured by a gerbe or a line 2-bundle. In the language of E8 bundles this corr* *esponds to lifting the loop group LE8 bundles to the central extension ^LE8 bundles [44]. The obstruction f* *or the case of the String structure is a 2-gerbe and that of a Fivebrane structure is a 6-gerbe. The stru* *ctures are summarized in the following table _______obstruction_________________________G-bundle______________^G-bundle______ 1-gerbes / line 2-bundles PU(H)-bundles U(H)-bundles 2-gerbes / line 3-bundlesobstruct the liftSofpin(n)-bundlestoString(n)-2-bund* *les 6-gerbes / line 7-bundles Spin(n)-bundles FiveBrane(n)-6-bund* *les Table 4: Obstructing line n-bundles appearing in string theory. A description can also be given in terms of (higher) loop spaces, generalizi* *ng the known case where a String structure on a space X can be viewed as a Spin structure on the loop spa* *ce LX. A fuller discussion of the ideas of this section will be given in [60]. 13 4 Statement of the main results We define, for any L1 -algebra g and any smooth space X, a notion of o g-descent objects over X; and an extension of these to o g-connection descent objects over X . These descent objects are to be thought of as the data obtained from locally tr* *ivializing an n-bundle (with connection) whose structure n-group has the Lie n-algebra g. Being differentia* *l versions of n-functorial descent data of such n-bundles, they consist of morphisms of quasi free differe* *ntial graded-commutative algebras (qDGCAs). We define for each L1 -algebra g a dg-algebra inv(g) of invariant polynomial* *s on g. We show that every g-connection descent object gives rise to a collection of deRham classes on X: * *its characteristic classes. These are images of the elements of inv(g). Two descent objects are taken to b* *e equivalent if they are concordant in a natural sense. Our first main result is Theorem 1 (characteristic classes)Characteristic classes are indeed characteris* *tic of g-descent objects (but do not necessarily fully characterize them) in the following sense: o Concordant g-connection descent objects have the same characteristic class* *es. This is our proposition 33. Remark. We expect that this result can be strengthened. Currently our characte* *ristic classes are just in deRham cohomology. One would expect that these are images of classes in integra* *l cohomology. While we do not attempt here to discuss integral characteristic classes in general, we d* *iscuss some aspects of this for the case of abelian Lie n-algebras g = bn-1u(1) in 7.1.1 by relating g-descent * *objects to Deligne cohomology. We define String-like extensions g~ of L1 -algebras coming from any L1 -alge* *bra cocycle ~: a closed element in the Chevalley-Eilenberg dg-algebra CE(g) corresponding to g: ~ 2 CE(* *g). These generalize the String Lie 2-algebra which governs the dynamics of (heterotic) superstrings. Our second main results is Theorem 2 (string-like extensions and their properties)Every degree (n + 1)-coc* *ycle ~ on an L1 - algebra g we obtain the string-like extension g~ which sits in an exact seqeuen* *ce 0 ! bn-1u(1) ! g~ ! g ! 0 . When ~ is in transgression with an invariant polynomial P we furthermore obtain* * a weakly exact sequence 0 ! g~ ! csP(g) ! chP(~) ! 0 of L1 -algebras, where csP(g) ' inn(g~) is trivializable (equivalent to the tri* *vial L1 -algebra). There is an algebra of invariant polynomials on g associated with csP(g) and we show that i* *t is that of g modulo the polynomial P. This is proposition 21, proposition 22 and proposition 25. Our third main result is 14 Theorem 3 (obstructions to lifts through String-like extensions)For ~ 2 CE(g) a* *ny degree n + 1 g-cocycle that transgresses to an invariant polynomial P 2 inv(g), the obstruct* *ion to lifting a g-descent object to a g~-descent object is a (bnu(1))-descent object whose single characteristic* * class is the class corresponding to P of the original g-descent object. This is reflected by the fact that the cohomology of the basic forms on the * *Chevalley-Eilenberg algebra of the corresponding Chern-Simons L1 -algebra csP(g) are those of g modulo the ide* *al generated by P. This is our proposition 25 and proposition 45. We discuss the following applications. o For g an ordinary semisimple Lie algebra and ~ its canonical 3-cocycle, th* *e obstruction to lifting a g-bundle to a String 2-bundle is a Chern-Simons 3-bundle. This is a specia* *l case of our proposition 45 which is spelled out in detail in in 8.3.1. The vanishing of this obstruction is known as a String structure [47]. In * *categorical language, this issue was first discussed in [56]. o This result generalizes to all String-like extensions. Using the 7-cocycle* * on so(n) we obtain lifts through extensions by a Lie 6-algebra, which we call the Fivebrane Lie 6-algebra. * * Accordingly, fivebrane structures on string structures are obstructed by the second Pontrjagin cl* *ass. This pattern continues and one would expect our obstruction theory for lif* *ts through string-like exten- sions with respect to the 11-cocycle on so(n) to correspond to Ninebrane s* *tructure. The issue of p-brane structures for higher p was discussed before in [45].* * In contrast to the discussion there, we here see p-brane structures only for p = 4n + 1, corresponding t* *o the list of invariant polynomials and cocycles for so(n). While our entire obstruction theory ap* *plies to all cocycles on all Lie 1-algebras, it is only for those on so(n) and maybe e8 for which the p* *hysical interpretation in the sense of p-brane structures is understood. o We discuss how the action functional of the topological field theory known* * as BF-theory arises from an invariant polynomial on a strict Lie 2-algebra, in a generalization of the* * integrated Pontrjagin 4-form of the topological term in Yang-Mills theory. See proposition 19 and the e* *xample in 6.6.1. This is similar to but different from the Lie 2-algebraic interpretation o* *f BF theory indicated in [27, 28], where the "cosmological" bilinear in the connection 2-form is not consider* *ed and a constraint on the admissable strict Lie 2-algebras is imposed. o We indicate the parallel transport induced by a g-connection, relate it to* * the n-functorial parallel transport of [52, 53, 54] and point out how this leads to oe-model actions* * in terms of dg-algebra morphisms. See section 9. o We indicate how by forming configuration spaces by something close to an i* *nternal hom in DGCAs, we obtain for every g-connection descent object configuration spaces of ma* *ps equipped with an action functional induced by the transgressed g-connection. We show that the alge* *bra of differential forms on these configuration spaces naturally supports the structure of the corresp* *onding BRST-BV complex, with the iterated ghost-of-ghost structure inherited from the higher degre* *e symmetries induced by g. This construction is similar in spirit to the one given in [1], reviewed i* *n [50], but at least superficially different. 5 Differential graded-commutative algebra Differential N-graded commutative algebras (DGCAs) play a prominent role in our* * discussion. One way to understand what is special about DGCAs is to realize that every DGCA can be reg* *arded, essentially, as the algebra of differential forms on some generalized smooth space. 15 We explain what this means precisely in 5.1. The underlying phenomenon is e* *ssentially the familiar governing principle of Sullivan models in rational homotopy theory [33, 57], bu* *t instead of working with simplicial spaces, we here consider presheaf categories. This will not become * *relevant, though, until the discussion of configuration spaces, parallel transport and action functionals i* *n 9. 5.1 Differential forms on smooth spaces We can think of every differential graded commutative algebra essentially as be* *ing the algebra of differential forms on some space, possibly a generalized space. Definition 1Let S be the category whose objects are the open subsets of R [ R2 * *[ R3 [ . .a.nd whose morphisms are smooth maps between these. We write op S1 := SetS (2* *6) for the category of set-valued presheaves on S. So an object X in S1 is an assignment of sets U 7! X(U) to each open subset* * U, together with an assignment OE*X ( U _OE_//_V) 7! ( X(U)oo___X(V )) (2* *7) of maps of sets to maps of smooth subsets which respects composition. A morphism f : X ! Y (2* *8) of smooth spaces is an assignment U 7! ( X(U)__fU_//Y (U)) of maps of sets to o* *pen subsets, such that for all smooth maps of subsets U __OE//_Vwe have that the square X(V )_fV_//_Y (V ) (2* *9) |OE*X| OE*Y|| |fflfflfU fflffl| X(U) ____//_Y (U) commutes. We think of the objects of S1 smooth spaces. The set X(U) that such a* * smooth space X assigns to an open subset U is to be thought of as the set of smooth maps from U into X* *. As opposed to manifolds which are locally isomorphic to an object in S, smooth spaces can hence be thou* *ght of as being objects which are just required to have the property that they may be probed by objects* * of S. Every open subset V becomes a smooth space by setting V : U 7! HomS1 (U, V ) . (3* *0) This are the representable presheaves. Similarly, every ordinary manifold X bec* *omes a smooth space by setting X : U 7! Hommanifolds(U, X) . (3* *1) The special property of smooth spaces which we need here is that they form a (c* *artesian) closed category: o for any two smooth spaces X and Y there is a cartesian product X xY , whic* *h is again a smooth space, given by the assignment X x Y : U 7! X(U) x Y (U) ; (3* *2) where the cartesian product on the right is that of sets; 16 o the collection hom(X, Y ) of morphisms from one smooth space X to another * *smooth space Y is again a smooth space, given by the assignment hom(X, Y ) : U 7! HomS1 (X x U, Y ) . (3* *3) A very special smooth space is the smooth space of differential forms. Definition 2We write o for the smooth space which assigns to each open subset * *the set of differential forms on it o : U 7! o(U) . (3* *4) Using this object we define the DGCA of differential forms on any smooth space * *X to be the set o(X) := HomS1 (X, o) (3* *5) equipped with the obvious DGCA structure induced by the local DGCA structure of* * each o(U). Therefore the object o is in a way both a smooth space as well as a differe* *ntial graded commutative algebra. In fact, it provides an adjunction between these categories. Definition 3There are contravariant functors from smooth spaces to DGCAs given * *by o : S1! DGCA s X 7! o(X) (3* *6) and Hom(-, o(-)) : DGCA s! S1 A 7! XA (3* *7) These form an adjunction of categories. The unit _____Id_____________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *___ ________________________________________________* *_____________________________________________________________________________* *________________________________________u _________________________''________________________* *__________________________ff'|| DGCAsHom(-,_o(-))//_S1___o__//_DGCAs (3* *8) of this adjunction is a natural transformation whose component map embeds each * *DGCA A into the algebra of differential forms on the smooth space it defines " o AO_________// (XA) (3* *9) by sending every a 2 A to the map of presheaves (f 2 HomDGCAs(A, o(U))) 7! (f(a) 2 o(U)) . (4* *0) This way of obtaining forms on XA from elements of A will be crucial for our co* *nstruction on differential forms on spaces of maps, hom(X, Y ), used in 9.3. Definition 4 (spaces of maps)Given any two DGCAs A and B, we define the smooth * *space of "maps" from B to A maps(B, A) (4* *1) by the assignment maps(B, A) : U 7! HomDGCAs(B, A o(U)) . (4* *2) This extends to a functor maps: DGCAs opx DGCAs ! S1 . (4* *3) 17 Hence we obtain the algebra of differential forms o(maps(B, A)) (4* *4) on the space of maps from B to A. Definition 5 (forms on maps functor)This construction yields a functor o(maps(-, -)) : DGCAs x DGCAsop! DGCAs . (4* *5) 5.1.1 Examples Diffeological spaces Particularly useful are smooth spaces X which, while not * *quite manifolds, have the property that there is a set Xs such that X : U 7! X(U) HomSet(U, Xs) (4* *6) for all U 2 S. These are the Chen-smooth or diffeological spaces used in [5, 53* *, 54]. In particular, all spaces of maps homS1(X, Y ) for X and Y manifolds are of this form. Then this includes* * for instance loop spaces. Using the fact that for any two smooth spaces X and Y , DGCA homomorphisms o(X* *) ! o(Y ) are in bijection with smooth maps X ! Y we obtain: Proposition 1For any two smooth spaces X and Y , we have maps( o(Y ), o(X)) ' homS1(X, Y ) . (4* *7) Forms on spaces of maps. When we discuss parallel transport and its transgress* *ion to configuration spaces in 9.3, we need the following construction of differential forms on spac* *es of maps. Definition 6 (currents)For A any DGCA, we say that a current on A is a smooth l* *inear map c : A ! R . (4* *8) For A = o(X) this reduces to the ordinary notion of currents. Proposition 2For each element b 2 B and current c on A, we get an element in o* *(Hom DGCAs(B, A o(-))) by mapping, for each U 2 S Hom DGCAs(B, A o(U))! o(U) f* 7! c(f*(b)) . (4* *9) If b is in degree n and c in degree m n, then this differential form is in* * degree n - m. The superpoint. Most of the DGCAs we shall consider here are non-negatively gr* *aded or even positively graded. These can be thought of as Chevalley-Eilenberg algebras of Lie n-algebr* *oids and Lie n-algebras, respectively, as discussed in more detail in 6. However, DGCAs of arbitrary deg* *ree do play an important role, too. Notice that a DGCA of non-positive degree is in particular a cochain* * complex of non-positive degree. But that is the same as a chain complex of non-negative degree. The following is a very simple but important example of a DGCA in non-positi* *ve degree. Definition 7 (superpoint)The "algebra of functions on the superpoint" is the DG* *CA C(pt) := (R R[-1], dpt) (5* *0) where the product on R R[-1] is the tensor product over R, and where the diff* *erential dpt: R[-1] ! R is the canonical isomorphism. 18 The smooth space associated to this algebra according to definition 3 is jus* *t the ordinary point, because for any test domain U the set Hom DGCAs(C(pt), o(U)) (5* *1) contains only the morphism which sends 1 2 R to the constant unit function on U* *, and which sends R[-1] to 0. However, as is well known from the theory of supermanifolds, the algebra * *C(pt) is important in that morphisms from any other DGCA A into it compute the (shifted) tangent space cor* *responding to A. From our point of view here this manifests itself in particular by the fact that for* * X any manifold, we have a canonical injection o(TX) ,! o(maps(C1 (X), C(pt))) (5* *2) of the differential forms on the tangent bundle of X into the differential form* *s on the smooth space of algebra homomorphisms of C1 (X) to C(pt): for every test domain U an element in HomDGCAs(C1 (X), C(pt o(U))) comes f* *rom a pair consisting of a smooth map f : U ! X and a vector field v 2 (TX). Together this constitu* *tes a smooth map f^: U ! TX and hence for every form ! 2 o(TX) we obtain a form on maps(C1 (X),* * C(pt)) by the assignment ((f, v) 2 HomDGCAs(C1 (X), C(pt o(U)))) 7! (f^*! 2 o(U)) (5* *3) over each test domain U. In 6.1.1 we discuss how in the analogous fashion we obtain the Weil algebra * *W(g) of any L1 -algebra g from its Chevalley-Eilenberg algebra CE(g) by mapping that to C(pt). This says * *that the Weil alghebra is like the space of functions on the shifted tangent bundle of the "space" that t* *he Chevalley-Eilenberg algebra is the space of functions on. See also figure 3. 5.2 Homotopies and inner derivations When we forget the algebra structure of DGCAs, they are simply cochain complexe* *s. As such they naturally live in a 2-category Chowhose objects are cochain complexes (V o, dV), whose mo* *rphisms * (V o, dV)oof_____(Wo, dW ) (5* *4) * are degree preserving linear maps V oofo__Wo that do respect the differentials, [d, f*] := dV O f* - f* O dW = 0 , (5* *5) and whose 2-morphisms __f*______________________________________* *___________________________________________________ _____________________________________________* *_____________________________________________________________________________* *_______________________________________________________________ ______________________________________________* *___________________________________|| (V o,bdV)b |ae| (Wo,_dW_)______ (5* *6) ______________________________________________* *_________________________________________|| _____________________________________________* *_____________________________________________________________________________* *_________________________________ff'|| ___________________________________________* *______________________________________________________________ g* are cochain homotopies, namely linear degree -1 maps ae : Wo ! V owith the prop* *erty that g* = f* + [d, ae] = f* + dV O ae + ae O dW . (5* *7) Later in 6.2 we will also look at morphisms that do preserve the algebra struct* *ure, and homotopies of these. Notice that we can compose a 2-morphism from left and right with 1-morph* *isms, to obtain another 19 2-morphism f* ____________________________________________* *_____________________________________________________________________________* *______________________________________________________________ ______________________________________________* *_____________________________________________________________________________ * ______________________________________________* *_____j*|| (Uo, dU)ooh_____(V_o,bdV)b ae|| (Wo,_dW_)______oo(Xo,_dX ) (5* *8) ______________________________________________* *_________________________________________|| _____________________________________________* *_____________________________________________________________________________* *_________________________________ff'|| ___________________________________________* *______________________________________________________________ g* whose component map now is * ae h* h*O ae O j* : Xo __j_//_Wo___//_V_o_//_Uo. (5* *9) This will be important for the interpretation of the diagrams we discuss, of th* *e type 68 and 74 below. Of special importance are linear endomorphisms V oooae_V oof DGCAs which are* * algebra derivations. Among them, the inner derivations in turn play a special role: Definition 8 (inner derivations)On any DGCA (V o, dV), a degree 0 endomorphism (V o, dV)ooL____(V_o, dV) (6* *0) is called an inner derivation if o it is an algebra derivation of degree 0; o it is connected to the 0-derivation, i.e. there is a 2-morphims ___0____________________________________* *____________________________________________________ ___________________________________________* *_____________________________________________________________________________* *___________________________________________ ____________________________________________* *_____________________________________|| (V o,bdV)b |ae| (V_o,_dV)______, (6* *1) ____________________________________________* *___________________________________________|| ___________________________________________* *_____________________________________________________________________________* *________________________ff'|| _________________________________________* *_____________________________________________________ L=[dV,ae] where ae comes from an algebra derivation of degree -1. Remark. Inner derivations generalize the notion of a Lie derivative on differe* *ntial forms, and hence they encode the notion of vector fields in the context of DGCAs. 5.2.1 Examples Lie derivatives on ordinary differential forms. The formula somtetimes known a* *s "Cartan's magic formula", which says that on a smooth space Y the Lie derivative Lv! of a diffe* *rential form ! 2 o(Y ) along a vector field v 2 (TY ) is given by Lv! = [d, 'v] , (6* *2) where 'v : o(Y ) ! o(Y ), says that Lie derivatives on differential forms are* * inner derivations, in our sense. When Y is equipped with a smooth projection ss : Y ! X, it is of importance to * *distinguish the vector fields vertical with respect to ss. The abstract formulation of this, applicable to ar* *bitrary DGCAs, is given in 5.3 below. 20 5.3 Vertical flows and basic forms We will prominently be dealing with surjections * A ooiooB_ (6* *3) of differential graded commutative algebras that play the role of the dual of a* *n injection "i F O___//_P (6* *4) of a fiber into a bundle. We need a way to identitfy in the context of DGCAs wh* *ich inner derivations of P are vertical with respect to i. Then we can find the algebra corresponding to t* *he basis of P as those elements of B which are annihilated by all vertical derivations. Definition 9 (vertical derivations)Given any surjection of differential graded * *algebras * F ooiooP_ (6* *5) we say that the vertical inner derivations __0______________________________________* *___________________________________________________________ ___________________________________________* *________________________________________ ____________________________________________* *_________|| P""__ |ae| P____________________ (6* *6) ____________________________________________* *_________________|| ___________________________________________* *_____________________________________________________________________________* *________________________________ff'|| [dP,ae] (this diagram is in the category of cochain complexes, compare the beginning of* * 6.2) on P with respect to i* are those inner derivations, for which there exists an inner derivation of F __0______________________________________* *___________________________________________________________ ___________________________________________* *________________________________________ ____________________________________________* *_________|| F""__ |ae0|F____________________ (6* *7) ____________________________________________* *_________________|| ___________________________________________* *_____________________________________________________________________________* *________________________________ff'|| [dP,ae0] such that _0______________________________________* *__________ __________________________________________* *___________________________________________________________ ""_________________________________________* *______________|| FOaa_O|ae0__F________________OO| O___________________________________________* *____________________________________________________OOOff'|| | _________________________________________* *__________________________________________________| || [d,ae0]|| | | (6* *8) | 0_____|________ | _________|________________________________* *_____________________________________________________________________________* *____________________________________ |""________|______________________________|| P aa_ |ae|__P________________ ___________________________________________* *____________________________________________________ff'|| _________________________________________* *__________________________________________________ [d,ae] Definition 10 (basic elements)Given any surjection of differential graded algeb* *ras * F ooiooP_ (6* *9) 21 we say that the algebra " Pbasic= ker(ae) \ ker(ae O dp) (7* *0) aevertical of basic elements of P (with respect to the surjection i*) is the subalgebra of* * P of all those elements a 2 P which are annihilated by all i*-vertical derivations ae, in that ae(a)= 0 (7* *1) ae(dPa)= 0 . (7* *2) We have a canonical inclusion p* Poo___Pbasic?.` (7* *3) Diagrammatically the above condition says that __0______________________________________* *_______________________ ____________________________________________* *_____________________________________________________ zz___________________________________________* *____________________________|| FOdd__________F________________________________* *_OOOae0||.(74) O_____________________________________________* *____________________________________________________OOOff'|| | ___________________________________________* *________________________________________________| || [d,ae0] || i*| |i* | 0______|_______ | ____________|_______________________________* *_____________________________________________________________________________* *__________________________________ |zz___________|_______________________________* *______________________|| POdd__________P________________________________* *_OOOae|| _____________________________________________* *____________________________________________________ff'|| | ___________________________________________* *________________________________________________| || [d,ae] || p*| |p* | | | | ?O| 0 ?O| Pbasicoo______Pbasic 5.3.1 Examples Basic forms on a bundle As a special case of the above general defintion, we r* *eobtain the standard notion of basic differential forms on a smooth surjective submersion ss : Y ! X* * with connected fibers. Definition 11Let ss : Y ! X be a smooth map. The vertical deRham complex , ove* *rt(Y ), with respect to Y is the deRham complex of Y modulo those forms that vanish when restricted * *in all arguments to vector fields in the kernel of ss* : (TY ) ! (TX), namely to vertical vector fields. The induced differential on overt(Y ) sends !vert= i*! to dvert: i*! 7! i*d! . (7* *5) Proposition 3This is well defined. The quotient overt(Y ) with the differenti* *al induced from o(Y ) is indeed a dg-algebra, and the projection * overt(Yo)io____ o(Y ) (7* *6) is a homomorphism of dg-algebras (in that it does respect the differential). Proof. Notice that if ! 2 o(Y ) vanishes when evalutated on vertical vector fi* *elds then obviously so does ff ^ !, for any ff 2 o(Y ). Moreover, due to the formula X X d!(v1, . .,.vn+1) = voe1!(voe2, . .,.voen+1) + !([voe1, vo* *e2],(voe3,7.7.,.voen+1)) oe2Sh(1,n+1) oe2Sh(2,n+1) 22 and the fact that for v, w vertical so is [v, w] and hence d! is also vertical.* * This gives that vertical differential forms on Y form a dg-subalgebra of the algebra of all forms on Y . Therefore if* * i*! = i*!0then di*!0= i*d!0= i*d(! + (!0- !)) = i*d! + 0 = di*! . (7* *8) Hence the differential is well defined and i* is then, by construction, a morph* *ism of dg-algebras. Recall the following standard definition of basic differential forms. Definition 12 (basic forms)Given a surjective submersion ss : Y ! X, the basic * *forms on Y are those with the property that they and their differentials are annihilated by all vert* *ical vector fields ! 2 o(Y )basic, 8v 2 ker(ss) : 'v! = 'vd! = 0 . (7* *9) It is a standard result that Proposition 4If ss : Y ! X is locally trivial and has connected fibers, then th* *e basic forms are precisely those comping from pullback along ss o(Y )basic' o(X) . (8* *0) Remark. The reader should compare this situation with the definition of invari* *ant polynomials in 6.3. The next proposition asserts that these statements about ordinary basic diff* *erential forms are indeed a special case of the general definition of basic elements with respect to a surj* *ection of DGCAs, definition 10. Proposition 5Given a surjective submersion ss : Y ! X with connected fibers, th* *en * * * o the inner derivations of o(Y ) which are vertical with respect to overt(* *Yi)oooo(Yo)_according to the general definition 9, come precisely from contractions 'v with vertical ve* *ctor fields v 2 ker(ss*) (TY ); o the basic differential forms on Y according to definition 12 conincide wit* *h the basic elements of o(Y ) relative to the above surjection * overt(Yi)oooo(Yo)_ (8* *1) according to the general definition 10. Proof. We first show that if o(Y )oaeo_o(Y )is a vertical algebra derivation, * *then ae has to annihilate all forms in the image of ss*. Let ff 2 o(Y ) be any 1-form and ! = ss*fi for * *fi 2 1(X). Then the wedge product ff ^ ! is annihilated by the projection to overt(Y ) and we find ae(!)O^offo_________0O_OOO | || | | | | |i*| i*| | || . (8* *2) `| | ae(!) ^ ffoaeo____O_`| +ae(ff) ^ ! ff ^ ! We see that ae(!) ^ ff has to vanish for all ff. Therefore ae(!) has to vanish * *for all ! pulled back along ss*. Hence ae must be contraction with a vertical vector field. It then follows from* * the condition 68 that a basic form is one annihilated by all such ae and all such ae O d. Possibly the most familiar kinds of surjective submersions are 23 o Fiber bundles. Indeed, the standard Cartan-Ehresmann theory of connections of principal b* *undles is obtained in our context by fixing a Lie group G and a principal G-bundle p : P ! X and the* *n using Y = P itself as the surjective submersion. The definition of a connection on P in terms of a g* *-valued 1-form on P can be understood as the descent data for a connection on P obtained with respect* * to canonical trivialization of the pullback of P to Y = P. Using for the surjective submersion Y a pri* *ncipal G-bundle P ! X is also most convenient for studying all kinds of higher n-bundles obstructin* *g lifts of the given G-bundle. This is why we will often make use of this choice in the following. o Covers by open subsets. The disjoint union of all sets in a cover of X by open subsets of X forms * *a surjective submersion ss : Y ! X. In large parts of the literature on descent (locally trivializ* *ed bundles), these are the only kinds of surjective submersions that are considered. We will find here, th* *at in order to characterize principal n-bundles entirely in terms of L1 -algebraic data, open covers a* *re too restrictive and the full generality of surjective submersions is needed. The reason is that, for ss* * : Y ! X a cover by open subsets, there are no nontrivial vertical vector fields ker(ss) = 0, (8* *3) hence overt(Y ) = 0 . (8* *4) With the definition of g-descent objects in 7.1 this implies that all g-de* *scent objects over a cover by open subsets are trivial. There are two important subclasses of surjective submersions ss : Y ! X: o those for which Y is (smoothly) contractible; o those for which the fibers of Y are connected. We say Y is (smoothly) contractible if the identity map Id: Y ! Y is (smooth* *ly) homotopic to a map Y ! Y which is constant on each connected component. Hence Y is a disjoint unio* *n of spaces that are each (smoothly) contractible to a point. In this case the Poincar'e lemma says * *that the dg-algebra o(Y ) of differential forms on Y is contractible; each closed form is exact: ___0______________________________________* *________________________________________________ ____________________________________________* *_______________________________________________________________________ ""___________________________________________* *____________________________________|| o(Ya)a o|| o(Y_)_______. (8* *5) _____________________________________________* *__________________________________________|| ____________________________________________* *______________________________________________________________ff'|| __________________________________________* *________________________________________ [d,o] Here o is the familiar homotopy operator that appears in the proof of the Poinc* *ar'e lemma. In practice, we often make use of the best of both worlds: surjective submersions that are (* *smoothly) contractible to a discrete set but still have a sufficiently rich collection of vertical vector* * fields. The way to obtain these is by refinement: starting with any surjective submersion ss : Y ! X which has * *good vertical vector fields but might not be contractible, we can cover Y itself with open balls, whose dis* *joint union, Y 0, then forms a surjective submersion Y 0! Y over Y . The composite ss0 Y 0A_________//Y AAA """" (8* *6) ssAA__AA"""""" X 24 is then a contractible surjective submersion of X. We will see that all our des* *cent objects can be pulled back along refinements of surjective submersions this way, so that it is possible, w* *ithout restriction of generality, to always work on contractible surjective submersions. Notice that for these th* *e structure of overt(Yo)ooo____ o(Y )oo______?_o(X)` (8* *7) is rather similar to that of CE (g)oooo_____W(g)oo________inv(g)?,` (8* *8) since W(g) is also contractible, according to proposition 7. Vertical derivations on universal g-bundles. The other important exmaple of ve* *rtical flows, those on DGCAs modelling universal g-bundle for g an L1 -algebra, is discussed at the be* *ginning of 6.3. 6 L1 -algebras and their String-like extensions L1 -algebras are a generalization of Lie algebras, where the Jacobi identity is* * demanded to hold only up to higher coherent equivalence, as the category theorist would say, or "strongly h* *omotopic", as the homotopy theorist would say. 6.1 L1 -algebras L Definition 13Given a graded vector space V , the tensor space To(V ) := n=0V * * nwith V 0being the ground field. We will denote by Ta(V ) the tensor algebra with the concatenatio* *n product on To(V ): O x1 x2 . . .xp xp+1 . . .xn 7! x1 x2 . . .xn (8* *9) and by Tc(V ) the tensor coalgebra with the deconcatenation product on To(V ): X O x1 x2 . . .xn 7! x1 x2 . . .xp xp+1 . . .xn. (9* *0) p+q=n The graded symmetric algebra ^o(V ) is the quotient of the tensor algebra Ta(V * *) by the graded action of the symmetric groups Sn on the components V n. The graded symmetric coalgebra _o(V* * ) is the sub-coalgebra of the tensor coalgebra Tc(V ) fixed by the graded action of the symmetric grou* *ps Sn on the components V n. Remark. _o(V ) is spanned by graded symmetric tensors x1_ x2_ . ._.xp (9* *1) for xi2 V and p 0, where we use _ rather than ^ to emphasize the coalgebra as* *pect, e.g. x _ y = x y y x. (9* *2) In characteristic zero, the graded symmetric algebra can be identified with * *a sub-algebra of Ta(V ) but that is unnatural and we will try to avoid doing so. The coproduct on _o(V ) is* * given by X X (x1_ x2. ._.xn) = ffl(oe)(xoe(1)_ xoe(2).x.o.e(p)) (xoe(p* *+1)_(.9.x.oe(n))3.) p+q=noe2Sh(p,q) On notation 25 o Sh(p, q) is the subset of all those bijections (the "unshuffles") of {1, 2* *, . .,.p+q} that have the property that oe(i) < oe(i + 1) whenever i 6= p; o ffl(oe), which is shorthand for ffl(oe, x1_ x2, . .x.p+q), the Koszul sign* *, defined by x1_ . ._.xn = ffl(oe)xoe(1)_ . .x.oe(n). (9* *4) Definition 14 (L1 -algebra)An L1 -algebra g = (g, D) is a N+-graded vector spac* *e g equipped with a degree -1 differential coderivation D : _og ! _og (9* *5) on the graded co-commutative coalgebra generated by g, such that D2 = 0. This i* *nduces a differential dCE(g): Symo(g) ! Symo+1(g) (9* *6) on graded-symmetric multilinear functions on g. When g is finite dimensional th* *is yields a degree +1 differ- ential dCE(g): ^og* ! ^og* (9* *7) on the graded-commutative algebra generated from g*. This is the Chevalley-Eile* *nberg dg-algebra correspond- ing to the L1 -algebra g. Remark. That the original definition of L1 -algebras in terms of multibrackets* * yields a codifferential coalgebra as above was shown in [41]. That every such codifferential comes from* * a collection of multibrackets this way is due to [42]. Example For (g[-1], [., .]) an ordinary Lie algebra (meaning that we regard th* *e vector space g to be in degree 1), the corresponding Chevalley-Eilenberg qDGCA is CE(g) = (^og*, dCE(g)) (9* *8) with [.,.]* * * dCE(g): g*____//_g ^ g. (9* *9) If we let {ta} be a basis of g and {Cabc} the corresponding structure constants* * of the Lie bracket [., .], and if we denote by {ta} the corresponding basis of g*, then we get dCE(g)ta = -1_2Cabctb^ tc. (10* *0) If g is concentrated in degree 1, . .,.n, we also say that g is a Lie n-algebra* *. Notice that built in we have a shift of degree for convenience, which makes ordinary Lie 1-algebras be in degr* *ee 1 already. In much of the literature a Lie n-algebra would be based on a vector space concentratred in de* *grees 0 to n - 1. An ordinary Lie algebra is a Lie 1-algebra. Here the coderivation differential D = [., .] i* *s just the Lie bracket, extended as a coderivation to _og, with g regarded as being in degree 1. In the rest of the paper we assume, just for simplicity and since it is suff* *icient for our applications, all g to be finite-dimensional. Then, by the above, these L1 -algebras are equivale* *ntly conceived of in terms of their dual Chevalley-Eilenberg algebras, CE(g), as indeed every quasi-free diff* *erential graded commutative algebra ("qDGCA", meaning that it is free as a graded commutative algebra) corr* *esponds to an L1 -algebra. We will find it convenient to work entirely in terms of qDGCAs, which we will u* *sually denote as CE(g). While not very interesting in themselves, truly free differential algebras a* *re a useful tool for handling quasi-free differential algebras. 26 Definition 15We say a qDGCA is free (even as a differential algebra) if it is o* *f the form F(V ) := (^o(V * V *[1]), dF(V)) (10* *1) with dF(V|)V *= oe : V *! V *[1] (10* *2) the canonical isomorphism and dF(V|)V *[1]= 0 . (10* *3) Remark. Such algebras are indeed free in that they satisfy theVuniversal prope* *rty: given any linear map V ! W, it uniquely extends to a morphism of qDGCAs F(V ) ! ( o(W*), d) for any* * choice of differential d. Example. The free qDGCA on a 1-dimensional vector space in degree 0 is the gra* *ded commutative algebra freely generated by two generators, t of degree 0 and dt of degree 1, with the * *differential acting as d : t 7! dt and d : dt 7! 0. In rational homotopy theory, this models the interval I = [0, * *1]. The fact that the qDGCA is free corresponds to the fact that the interval is homotopy equivalent to the* * point. We will be interested in qDGCAs that arise as mapping cones of morphisms of * *L1 -algebras. Definition 16 ("mapping cone" of qDGCAs)Let * CE(h)oot__CE(g) (10* *4) be a morphism of qDGCAs. The mapping cone of t*, which we write CE(h !tg), is t* *he qDGCA whose underlying graded algebra is ^o(g* h*[1]) (10* *5) and whose differential dt*is such that it acts as ` ' dt*= dgt*0d . (10* *6) h We postpone a more detailed definition and discussion to 8.1; see definition 39* * and proposition 37. Strictly speaking, the more usual notion of mapping cones of chain complexes applies to * *t : h ! g, but then is extended as a derivation differential to the entire qDGCA. Definition 17 (Weil algebra of an L1 -algebra)The mapping cone of the identity * *on CE(g) is the Weil algebra W(g) := CE(g Id!g) (10* *7) of g. Proposition 6For g an ordinary Lie algebra this does coincide with the ordinary* * Weil algebra of g. Proof. See the example in 6.1.1. The Weil algebra has two important properties. Proposition 7The Weil algebra W(g) of any L1 -algebra g o is isomorphic to a free differential algebra W(g) ' F(g) , (10* *8) and hence is contractible; 27 o has a canonical surjection * CE (g)oiooW(g)o_. (10* *9) Proof. Define a morphism f : F(g) ! W(g) (11* *0) by setting f : a 7! a (11* *1) f : (dF(Va)= oea) 7! (dW(g)a = dCE(g)a + oea) (11* *2) for all a 2 g* and extend as an algebra homomorphism. This clearly respects the* * differentials: for all a 2 V * we find dF(g) OdF(g) aO_______//`oea` oea_____//_`0` |f| |f| and |f| |f|. (11* *3) fflffl|O fflffl| fflffl| fflffl| ad___//_dCE(g)a + oea dW(g)aO__//_0 W(g) dW(g) One checks that the strict inverse exists and is given by f-1|g*: a 7! a (11* *4) f-1|g*[1]:oea 7! dF(g)a - dCE(g)a . (11* *5) * * * Here oe : g* ! g*[1] is the canonical isomorphism that shifts the degree. The s* *urjection CE (g)oiooW(g)o_ simply projects out all elements in the shifted copy of g: i*|^og*= id (11* *6) i*|g*[1]=0 . (11* *7) This is an algebra homomorphism that respects the differential. As a corollary we obtain Corollary 1For g any L1 -algebra, the cohomology of W(g) is trivial. Proposition 8The step from a Chevalley-Eilenberg algebra to the corresponding W* *eil algebra is functorial: for any morphism * CE(h)oo__f____CE_(g) (11* *8) we obtain a morphism ^f* W(h) oo_______W(g)_ (11* *9) and this respects composition. Proof. The morphism ^f*acts as for all generators a 2 g* as ^f*: a 7! f*(a) (12* *0) and f^*: oea 7! oef*(a) . (12* *1) 28 We check that this does repect the differentials dW(g) O dW(g) a`O_______//dCE(g)a + oea oea`____//-oe(dCE(g)a) | ` | ` , (12* *2) |^f*| |^f*| |^f*| |^f*| fflffl|dW(h) fflffl| fflffl|dW(h)fflffl| f*(a)____//_dCE(h)f*(a) + oef*(a)oef*(a)//_-oe(dCE(h)a) Remark. As we will shortly see, W(g) plays the role of the algebra of differen* *tial forms on the universal * g-bundle. The surjection CE(g) oiooW(g)o_plays the role of the restriction to t* *he differential forms on the fiber of the universal g-bundle. 6.1.1 Examples In section 6.4 we construct large families of examples of L1 -algebras, based o* *n the first two of the following examples: 1. Ordinary Weil algebras as Lie 2-algebras. What is ordinarily called the Weil* * algebra W(g) of a Lie algebra (g[-1], [., .]) can, since it is again a DGCA, also be interpreted as t* *he Chevalley-Eilenberg algebra of a Lie 2-algebra. This Lie 2-algebra we call inn(g). It corresponds to the Li* *e 2-group INN(G) discussed in [49]: W(g) = CE(inn(g)) . (12* *3) We have W(g) = (^o(g* g*[1]), dW(g)) . (12* *4) Denoting by oe : g* ! g*[1] the canonical isomorphism, extended as a derivation* * to all of W(g), we have [.,.]*+oe * * * dW(g): g*__________//g ^ g g [1] (12* *5) and -oeOdCE(g)Ooe-1 dW(g): g*[1]________//_g* g*[1]. (12* *6) With {ta} a basis for g* as above, and {oeta} the corresponding basis of g*[1] * *we find dW(g): ta 7! -1_2Cabctb^ tc+ oeta (12* *7) and dW(g): oeta 7! -Cabctboetc. (12* *8) The Lie 2-algebra inn(g) is, in turn, nothing but the strict Lie 2-algebra as i* *n the third example below, which comes from the infinitesimal crossed module (g Id!g ad!der(g)). 2. Shifted u(1). By the above, the qDGCA corresponding to the Lie algebra u(1) * *is simply CE(u(1)) = (^oR[1], dCE(u(1))= 0) . (12* *9) We write CE(bn-1u(1)) = (^oR[n], dCE(bnu(1))= 0) (13* *0) for the Chevalley-Eilenberg algebras corresponding to the Lie n-algebras bn-1u(* *1). 29 3. Infinitesimal crossed modules and strict Lie 2-algebras. An infinitesimal c* *rossed module is a diagram ( h__t_//_g_ff//_der(h)) (13* *1) of Lie algebras where t and ff satisfy two compatibility conditions. These cond* *itions are equivalent to the nilpotency of the differential on CE(h t!g) := (^o(g* h*[1]), dt) (13* *2) defined by dt|g* = [., .]*g+ t* (13* *3) dt|h*[1]= ff*, (13* *4) where we consider the vector spaces underlying both g and h to be in degree 1. * *Here in the last line we regard ff as a linear map ff : g h ! h. The Lie 2-algebras (h !tg) thus defi* *ned are called strict Lie 2-algebras: these are precisely those Lie 2-algebras whose Chevalley-Eilenberg * *differential contains at most co-binary components. 4. Inner derivation L1 -algebras. In straightforward generalization of the firs* *t exmaple we find: for g any L1 -algebra, its Weil algebra W(g) is again a DGCA, hence the Chevalley-Eil* *enberg algebra of some other L1 -algbera. This we address as the L1 -algebra of inner derivations and * *write CE (inn(g)) := W(g) . (13* *5) This identification is actually useful for identifying the Lie 1-groups that co* *rrespond to an integrated picture underlying our differential discussion. In [49] the Lie 3-group corresp* *onding to inn(g) for g any strict 2-group is discussed. This 3-group is in particular the right codomain for inco* *rporating the the non-fake flat nonabelian gerbes with connection considered in [13] into the integrated v* *ersion of the picture discussed here. This is indicated in [54] and should be discussed elsewhere. tangent categoryinner(automorphismnL+i1)-groupnneriderivationeW(ne+i1)-algebr* *alsalgebrahiftedtangent bundle CE(Lie(TBG))_____CE(Lie(INN(G)))_____CE(inn(g))_______W(g)________C1 (T[1]g) Figure 3: A remarkable coincidence of concepts relates the notion of tangency t* *o the notion of universal bundles. The leftmost equality is discussed in [49]. The second one from the ri* *ght is the identification 135. The rightmost equality is equation 147. Proposition 9For g any finite dimensional L1 -algebra, the differential forms o* *n the smooth space of morphisms from the Chevalley-Eilenberg algebra CE (g) to the algebra of "functi* *ons on the superpoint", definition 7, i.e. the elements in o(maps(CE (g), C(pt))), which come from cu* *rrents as in definition 6, form the Weil algebra W(g) of g: W(g) o(maps(CE (g), C(pt))) . (13* *6) Proof. For any test domain U, an element in HomDGCAs(CE (g), C(pt) o(U)) is * *specified by a degree 0 algebra homomorphism ~ : CE(g) ! o(U) (13* *7) 30 and a degree +1 algebra morphism ~ : CE(g) ! o(U) (13* *8) by a 7! ~(a) + c ^ !(a) (13* *9) for all a 2 g* and for c denoting the canonical degree -1 generator of C(pt); s* *uch that the equality in the bottom right corner of the diagram dCE(g) a`O___________________//dCE(g)a (14* *0) | ` | | | | | | | | | | | | fflffl| dpt+dU~(d afflffl|)+c^!(d a) ~(a) + c ^ !(a)O_____//_=CE(g)d(~(a))+!(a)-cCE(g)^d(!(a)) holds. Under the two canonical currents on C(pt) of degree 0 and degree 1, resp* *ectively, this gives rise for each a 2 g* of degree |a| to an |a|-form and an (|a|+1) form on maps(CE (g), C(* *pt)) whose values on a given plot are ~(a) and !(a), respectively. By the above diagram, the differential of these forms satisfies d~(a) = ~(dCE(g)a) + !(a) (14* *1) and d!(a) = -!(dCE(g)a) . (14* *2) But this is precisely the structure of W(g). To see the last step, it may be helpful to consider this for a simple case in t* *erms of a basis: let g be an ordinary Lie algebra, {ta} a basis of g* and {Cabc} the correspo* *nding structure constants. Then, using the fact that, since we are dealing with algebra homomorphisms, we * *have ~(ta^ tb) = ~(ta) ^ ~(tb) (14* *3) and !(ta^ tb) = c ^ (!(ta) ^ ~(tb) - ~(ta) ^ !(tb)) (14* *4) we find d~(ta) = -1_2Cabc~(tb) ^ ~(tc) + !(ta) (14* *5) and d!(ta) = -Cabc~(tb) ^ !(tc) . (14* *6) This is clearly just the structure of W(g). Remark. As usual, we may think of the superpoint as an "infinitesimal interval* *". The above says that the algebra of inner derivations of an L1 -algebra consists of the maps from th* *e infinitesimal interval to the supermanifold on which CE(g) is the "algebra of functions". On the one hand thi* *s tells us that W(g) = C1 (T[1]g) (14* *7) in supermanifold language. On the other hand, this construction is clearly anal* *ogous to the corresponding discussion for Lie n-groups given in [49]: there the 3-group INN(G) of inner au* *tomorphisms of a strict 2- group G was obtained by mapping the "fat point" groupoid pt= { o____//_O} into * *G. As indicated there, this is a special case of a construction of "tangent categories" which mimics t* *he relation between inn(g) and the shifted tangent bundle T[1]g in the integrated world of Lie 1-groups. T* *his relation between these concepts is summarized in figure 3. 31 6.2 L1 -algebra homotopy and concordance Like cochain complexes, differental graded algebras can be thought of as being * *objects in a higher categorical structure, which manifests itself in the fact that there are not only morphisms* * between DGCAs, but also higher morphisms between these morphisms. It turns out that we need to conside* *r a couple of slightly differening notions of morphisms and higher morphisms for these. While differin* *g, these concepts are closely related among each other, as we shall discuss. In 5.2 we had already considered 2-morphisms of DGCAs obtained after forgett* *ing their algebra structure and just remembering their differential structure. The 2-morphisms we present n* *ow crucially do know about the algebra structure. ____________|_______name______________________nature________________| | chain homotopy | f* | _________________________* *_____________________________________________________________________________* *_______________________________________________________________ infinitesimal ||transformation ""____________________j|| | CE(g)aa___CE(h)______________* *___________|| | _________________________* *_____________________________________________________________________________* *_______________________________________________________ff' | g* _______________|____________________________________________________| | extension over interval | g* | ___________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *________________________ | ________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________ finite ||homotopy/concordance wooId_s*w______o_oo_*____________* *__________ | CE(g)oo__CE*(g)_gg__(I)j_CE_(h)______* *__________________ | __Id_t__________________________* *_____________________________________________________________________________* *______________________ | _____________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *___ | f* | Table 5: The two different notions of higher morphisms of qDGCAs. Infinitesimal homotopies between dg-algebra homomorphisms. When we restrict at* *tention to cochain maps between qDGCAs which respect not only the differentials but also t* *he free graded commutative algebra structure, i.e. to qDGCA homomorphisms, it becomes of interest to expre* *ss the cochain homotopies in terms of their action on generators of the algebra. We now define transforma* *tions (2-morphisms) between morphisms of qDGCAs by first defining them for the case when the domain is a We* *il algebra, and then extending the definition to arbitrary qDGCAs. Definition 18 (transformation of morphisms of L1 -algebras)We define transforma* *tions between qDGCA morphisms in two steps o A 2-morphism _f*___________________________________* *_________________________________________ _________________________________________* *____________________________________________________________________ ""________________________________________* *_______________________________________|| CE (g)`` |j| F(h)________ (14* *8) __________________________________________* *___________________________________________|| _________________________________________* *___________________________________________________ff'|| _______________________________________* *_________________________________________ g* is defined by a degree -1 map j : h* h*[1] ! CE(g) which is extended to * *a linear degree -1 map j : ^o(h* h*[1]) ! CE(g) by defining it on all monomials of generators by* * the formula j : x1^ . .^.xn 7! 32 k-1P 1_X ffl(oe) nX(-1)i=1|xoe(i)|g*(x ^ . .^.x ) ^ j(x ) ^ f*(x (14* *9^). .^.x ) n!oe k=1 oe(1) oe(k-1) oe(k) oe(k* *+1) oe(n) for all x1, . .,.xn 2 h* h*[1], such that this is a chain homotopy from f* ** to g*: g* = f* + [d, j] . (15* *0) o A general 2-morphism __f*___________________________________* *________________________________________ __________________________________________* *_____________________________________________________________________________* *______ ""_________________________________________* *______________________________________|| CE(g)aa |j| CE_(h)_______ (15* *1) ___________________________________________* *____________________________________________|| __________________________________________* *______________________________________________________________ff'|| ________________________________________* *__________________________________________ g* is a 2-morphism g*___CE_(h)_____________________________________* *___________________________ ________________________________________ccccHH ________________________________i*HHH __________________HHH _________________Hxxxx'xx (15* *2) CE(g)WW xxxxxxxxW(h) oo__F(h)_ _______________vv_____________________xxxxxx ____________vvv_______________________xxxxx _________---i*-vvvv____________________________* *____ *______________________________________________* *_________________ f CE (g) of the above kind that vanishes on the shifted generators, i.e. such that g*___CE_(h)_____________________________________* *___________________________ ________________________________________ccccHH ________________________________i*HHH __________________HHH _________________Hxxxxxx (15* *3) CE(g)WW xxxxxxxx W(h) oo__h*[1]?_` _______________vv_____________________xxxxxx ____________vvv_______________________xxxxx _________---i*-vvvv____________________________* *____ *______________________________________________* *_________________ f CE (g) vanishes. Proposition 10Formula 149 is consistent in that g*|h* h*[1]= (f* + [d, j])|h* h* **[1]implies that g* = f* + [d, j] on all elements of F(h). Remark. Definition 18, which may look ad hoc at this point, has a practical an* *d a deep conceptual motivation. o Practical motivation. While it is clear that 2-morphisms of qDGCAs should * *be chain homotopies, it is not straightforward, in general, to characterize these by their action * *on generators. Except when the domain qDGCA is free, in which case our formula 18 makes sense. The prescr* *iption 152 then provides a systematic algorithm for extending this to arbitrary qDGCAs. In particular, using the isomorphism W(g) ' F(g) from proposition 7, the a* *bove yields the usual explicit description of the homotopy operator o : W(g) ! W(g) with IdW(g)=* * [dW(g), o]. Among other things, this computes for us the transgression elements ("Chern-Simons ele* *ments") for L1 -algbras in 6.3. 33 o Conceptual motivation. As we will see in 6.3 and 6.5, the qDGCA W(g) plays* * an important twofold role: it is both the algebra of differential forms on the total space of t* *he universal g-bundle - while CE(g) is that of forms on the fiber -, as well as the domain for g-valued * *differential forms, where the shifted component, that in h*[1], is the home of the corresponding curvatu* *re. In the light of this, the above restriction 153 can be understood as sayin* *g either that - vertical transformations induce transformations on the fibers; or - gauge transformations of g-valued forms are transformations under which* * the curvatures transform covariantly. Finite transformations between qDGCA morphisms: concordances. We now consider * *the finite transformations of morphisms of DGCAs. What we called 2-morphisms or transform* *ations for qDGCAs above would in other contexts possibly be called a homotopy. Also the followin* *g concept is a kind of homotopy, and appears as such in [55] which goes back to [10]. Here we wish to * *clearly distinguish these different kinds of homotopies and address the following concept as concordance * *- a finite notion of 2-morphism between dg-algebra morphisms. Remark. In the following the algebra of forms o(I) on the interval I := [0, 1] plays an important role. Essentially everything would also go through by instea* *d using F(R), the DGCA on a single degree 0 generator, which is the algebra of polynomial forms on the* * interval. This is the model used in [55]. Definition 19 (concordance)We say that two qDGCA morphisms * CE(g)oo__g____CE_(h) (15* *4) and * CE(g)oo_h_____CE_(h) (15* *5) are concordant, if there exists a dg-algebra homomorphism * CE (g) o(I)joo____CE_(h) (15* *6) from the source CE(h) to the the target CE(g) tensored with forms on the interv* *al, which restricts to the two given homomorphisms when pulled back along the two boundary inclusions _s__//_ {o}____//_I, (15* *7) t so that the diagram of dg-algebra morphisms g* ______________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *____________________________________________________ _________________________________________________* *_________________________________________________________________________ ooId_s*ww_________j*_____________________________* *_______ CE (g)oo__CE(g) o(I)oo_CE(h) (15* *8) ggId_t*__________________________________________* *_____________________________________________________________________ _______________________________________________* *_____________________________________________________________________________* *___________________________________________________________________ ____________________________________________* *________________________________________________________________________ f* commutes. 34 See also table 5. Notice that the above diagram is shorthand for two separat* *e commuting diagrams, one involving g* and s*, the other involving f* and t*. We can make precise the statement that definition 18 is the infinitesimal ve* *rsion of definition 19, as follows. Proposition 11Concordances * CE (g) o(I)joo____CE_(h) (15* *9) are in bijection with 1-parameter families ff : [0, 1] ! Homdg-Alg(CE (h), CE(g)) (16* *0) of morphisms whose derivatives with respect to the parameter is a chain homotop* *y, i.e. a 2-morphism __0____________________________* *___________________________________________________________ __________________________________* *_____________________________________________________________________________* *__ ""________________________________* *_______________________________________________|| 8t 2 [0, 1] : CE(g)aa ae|| CE(h)________ (16* *1) __________________________________* *_____________________________________________________|| _________________________________* *_______________________________________________________________________ff'|| _______________________________* *___________________________________________________ d_dtff(t)=[d,ae] in the 2-category of cochain complexes. For any such ff, the morphisms f* and g* ** between which it defines a concordance are defined by the value of ff on the boundary of the interval. Proof. Writing t : [0, 1] ! R for the canonical coordinate function on the i* *nterval I = [0, 1] we can decompose the dg-algebra homomorphism j* as j* : ! 7! (t 7! ff(!)(t) + dt ^ ae(!)(t)) . (16* *2) ff is itself a degree 0 dg-algebra homomorphism, while ae is degree -1 map. The* *n the fact that j* respects the differentials implies that for all ! 2 CE(h) we have dh !O_______________________________//`|dh!` . (16* *3) | | | | | | |j*| |j*| | | | fflffl| | fflffl| dg+dt (t 7! (ff(dh!)(t) + dt ^ ae(dh!)* *(t)) (t 7! (ff(!)(t) + dt ^ ae(!)(t)))O________//_= (t 7! (dg(ff(!))(t) +dt ^ (_d_dtff(!) - dgae(!))(t)) The equality in the bottom right corner says that ff O dh- dgO ff = 0 (16* *4) and 8! 2 CE(g) : d_dtff(!) = ae(dh!) + dg(ae(!)) . (16* *5) But this means that ff is a chain homomorphism whose derivative is given by a c* *hain homotopy. 35 6.2.1 Examples Transformations between DGCA morphisms. We demonstrate two examples for the ap* *plication of the notion of transformations of DGCA morphisms from definition 18 which are re* *levant for us. Computation of transgression forms. As an example for the transformation in* * definition 18, we show how the usual Chern-Simons transgression form is computed using formula 14* *9. The reader may wish to first skip to our discussion of Lie 1-algebra cohomology in 6.3 for more backgr* *ound. So let g be an ordinary Lie algebra with invariant bilinear form P, which we regard as a dW(g)-closed e* *lement P 2 ^2g*[1] W(g). We would like to compute oP, where o is the contracting homotopy of W(g), such * *that [d, o] = IdW(g), (16* *6) which according to proposition 7 is given on generators by o : a 7! 0 (16* *7) o : dW(g)a 7! a (16* *8) for all a 2 g*. Let {ta} be a chosen basis of g* and let {Pab} be the component* *s of P in that basis, then P = Pab(oeta) ^ (oetb) . (16* *9) In order to apply formula 149 we need to first rewrite this in terms of monomia* *ls in {ta} and {dW(g)ta}. Hence, using oeta = dW(g)ta+ 1_2Cabctb^ tc, we get ` * * ' oP = o Pab(dW(g)ta) ^ (dW(g)ta) - Pab(dW(g)ta) ^ Cbcdtc^ td+ 1_4PabCacdCbeft* *c^.td^(tc^1td70) Now equation 149 can be applied to each term. Noticing the combinatorial prefac* *tor 1_n!, which depends on the number of factors in the above terms, and noticing the sum over all permuta* *tions, we find a a a b o Pab(dW(g)t ) ^ (dW(g)t=)Pab(dW(g)t ) ^ t a b c d1 b b c d 1 a b * * c o -Pab(dW(g)t ) ^ C cdt ^=t__3!. 2 PabCcdt ^ t ^ t = _3Cabct ^ t(^1* *t7,1) i j where we write Cabc:= PadCdbcas usual. Finally o 1_4PabCacdCbeftc^ td^ tc^=td0* * . In total this yields oP = Pab(dW(g)ta) ^ tb+ 1_3Cabcta^ tb^ tc. (17* *2) By again using dW(g)ta = -1_2Cabctb^ tc+ oeta together with the invariance of P* * (hence the dW(g)-closedness of P which implies that the constants Cabcare skew symmetric in all three indic* *es), one checks that this does indeed satisfy dW(g)oP = P . (17* *3) In 6.5 we will see that after choosing a g-valued connection on space space Y t* *he generators ta here will get sent to components of a g-valued 1-form A, while the dW(g)ta will get sent to t* *he components of dA. Under this map the element oP 2 W(g) maps to the familiar Chern-Simons 3-form CSP(A) := P(A ^ dA) + 1_3P(A ^ [A ^ A]) (17* *4) whose differential is the characteristic form of A with respect to P: dCSP (A) = P(FA ^ FA) . (17* *5) Characteristic forms, for arbitrary Lie 1-algebra valued forms, is discussed fu* *rther in 6.6. 36 2-Morphisms of Lie 2-algebras Proposition 12For the special case that g is any Lie 2-algebra (any L1 -algebra* * concentrated in the first two degrees) the 2-morphisms defined by definition 18 reproduce the 2-morphisms* * of Lie 2-algebras as stated in [3] and used in [4]. Proof. The proof is given in the appendix. This implies in particular that with the 1- and 2-morphisms as defined above* *, Lie 2-algebras do form a 2-category. There is an rather straightforward generalization of definition 18 * *to higher morphisms, which one would expect yields correspondingly n-categories of Lie n-algebras. But thi* *s we shall not try to discuss here. 6.3 L1 -algebra cohomology The study of ordinary Lie algebra cohomology and of invariant polynomials on th* *e Lie algebra has a simple formulation in terms of the qDGCAs CE(g) and W(g). Furthermore, this has a stra* *ightforward generalization to arbitrary L1 -algebras which we now state. * For CE (g)oooio____W(g) the canonical morphism from proposition 7, notice th* *at CE(g) ' W(g)=ker(i*) (17* *6) and that ker(i*) = W(g), (17* *7) the ideal in W(g) generated by g*[1]. Algebra derivations 'X : W(g) ! W(g) (17* *8) for X 2 g are like (contractions with) vector fields on the space on which W(g)* * is like differential forms. In the case of an ordinary Lie algebra g, the corresponding inner derivations [dW(* *g), 'X ] for X 2 g are of degree -1 and are known as the Lie derivative LX . They generate flows exp([dW(g), 'X * *]) : W(g) ! W(g) along these vector fields. Definition 20 (vertical derivations)We say an algebra derivation o : W(g) ! W(g* *) is vertical if it vanishes on the shifted copy g*[1] of g* inside W(g), o|g*[1]= 0 . (17* *9) Proposition 13The vertical derivations are precisely those that come from contr* *actions 'X : g* 7! R (18* *0) for all X 2 g, extended to 0 on g*[1] and extended as algebra derivations to al* *l of ^o(g* g*[1]). The reader should compare this and the following definitions to the theory o* *f vertical Lie derivatives and basic differential forms with respect to any surjective submersion ss : Y ! X. * *This is discussed in 5.3.1. Definition 21 (basic forms and invariant polynomials)The algebra W(g)basicof ba* *sic forms in W(g) is the intersection of the kernels of all vertical derivations and Lie derivati* *ves. i.e. all the contractions 'X and Lie derivatives LX for X 2 g. Since LX = [dW(g), 'X ], it follows that in t* *he kernel of 'X , the Lie derivative vanishes only if 'X dW(g)vanishes. 37 As will be discussed in a moment, basic forms in W(g) play the role of invar* *iant polynomials on the L1 -algebra g. Therefore we often write inv(g) for W(g)basic: inv(g) := W(g)basic. (18* *1) * Using the obvious inclusion W(g) opo_inv(g)?_`we obtain the sequence * p* ` CE(g)oooio____W(g)_oo_______inv(g)?_ (18* *2) of dg-algebras that plays a major role in our analysis: it can be interpreted a* *s coming from the universal bundle for the Lie 1-algebra g. As shown in figure 4, we can regard vertical d* *erivations on W(g) as derivations along the fibers of the corresponding dual sequence. ___0_________________________________________________* *_________________________________________ ww_____________________________________________________* *_________________________________(co)adjoint action CE (g)gg 'X|| CE(g)_______ . OO_______________________________________________________* *_____________________________________________________________________________* *________________________________________ofOgOon|itselfOOOO|ff'|| | [d,'X] | | | | ____0______|__________________________________________* *_______________________________________________________________________ | ww__________|__________________________________________* *____vertical derivation W(g)gg 'X|| W(g)_______ OO_______________________________________________________* *_____________________________________________________________________________* *________________________________________on|W(g)OO|ff'|| | [d,'X] | | | | ____0______|__________________________________________* *_____________________________________________________________________________* *______________ ?O|ww__________?O|__________________________leaves basic f* *orms inv(g)gg 0|| inv(g)_______ _______________________________________________________* *_____________________________________________________________________________* *________________________________________invariantff'|| 0 Figure 4: Interpretation of vertical derivations on W(g). The algebra CE(g) pla* *ys the role of the algebra of differential forms on the Lie 1-group that integrates the Lie 1-alge* *bra g. The coadjoint action of g on these forms corresponds to Lie derivatives along the fibers of the univ* *ersal bundle. These vertical derivatives leave the forms on the base of this universal bundle invariant. The* * diagram displayed is in the 2-category Cho of cochain complexes, as described in the beginning of 6.2. Definition 22 (cocycles, invariant polynomials and transgression elements)Let g* * be an L1 -algebra. Then o An L1 -algebra cocycle on g is a dCE(g)-closed element of CE(g). ~ 2 CE(g) , dCE(g)~ = 0 . (18* *3) o An L1 -algebra invariant polynomial on g is an element P 2 inv(g) := W(g)b* *asic. o An L1 -algebra g-transgression element for a given cocycle ~ and an invari* *ant polynomial P is an element cs2 W(g) such that dW(g)cs= p*P (18* *4) i*cs= ~ . (18* *5) If a transgression element for ~ and P exists, we say that ~ transgresses to* * P and that P suspends to ~. If ~ = 0 we say that P suspends to 0. The situation is illustrated diagra* *mmatically in figure 5 and figure 6. 38 Definition 23 (suspension toA0)n element P 2 inv(g) is said to suspend to 0 if * *under the inclusion * ker(i*)oop____?W(g)`_ (18* *6) it becomes a coboundary: p*P = dker(i*)ff (18* *7) for some ff 2 ker(i*). Remark. We will see that it is the intersection of inv(g) with the cohomology * *of ker(i*) that is a candidate, in general, for an algebraic model of the classifying space of the object that * *integrates the L1 -algebra g. But at the moment we do not entirely clarify this relation to the integrated th* *eory, unfortunately. cocycle transgression elementinv. polynomial " i p GO_____________//EG______________////_BG 0OO d|| `| O 0OO p*POoo____p*________PO |d| d|| `| O`| ~oo_____________cs i* Figure 5: Lie algebra cocycles, invariant polynomials and transgression forms i* *n terms of coho- mology of the universal G-bundle. Let G be a simply connected compact Lie grou* *p with Lie algebra g. Then invariant polynomials P on g correspond to elements in the cohomology Ho(B* *G) of the classifying space of G. When pulled back to the total space of the universal G-bundle EG ! * *BG, these classes become trivial, due to the contractability of EG: p*P = d(cs). Lie algebra cocycles, o* *n the other hand, correspond to elements in the cohomology Ho(G) of G itself. A cocycle ~ 2 Ho(G) is in tran* *sgression with an invariant polynomial P 2 Ho(BG) if ~ = i*cs. Proposition 14For the case that g is an ordinary Lie algebra, the above definit* *ion reproduces the ordinary definitions of Lie algebra cocycles, invariant polynomials, and transgression e* *lements. Moreover, all elements in inv(g) are closed. Proof. That the definitions of Lie algebra cocycles and transgression elements * *coincides is clear. It remains to be checked that inv(g) really contains the invariant polynomials. In the ord* *inary definition a g-invariant polynomial is a dW(g)-closed element in ^o(g*[1]). Hence one only needs to che* *ck that all elements in ^o(g*[1]) with the property that their image under dW(g)is again in ^o(g*[1]) a* *re in fact already closed. This can be seen for instance in components, using the description of W(g) give* *n in 6.1.1. Remark. For ordinary Lie algebras g corresponding to a simply connected compac* *t Lie group G, the situation is often discussed in terms of the cohomology of the universal G-bund* *le. This is recalled in figure 5 and in 6.3.1. The general definition above is a precise analog of that famili* *ar situation: W(g) plays the role of the algebra of (left invariant) differential forms on the universal g-b* *undle and CE(g) plays the role of 39 cocycle transgression elementinv. polynomial * p* ` CE(g)oooo_____i__W(g)oo____________inv(g)? 0OO dW(g)|| `| O 0OO p*PG________oo_____*P_ __________________pOO |dCE(g)| o ______________________________________dW(g)* *|| ~`| i* _OEOE____________________O`| oo_____________cs Figure 6: The homotopy operator o is a contraction homotopy for W(g). Acting wi* *th it on a closed invariant polynomial P 2 inv(g) ^og[1] W(g) produces an element cs2 W(g) wh* *ose "restriction to the fiber" ~ := i*csis necessarily closed and hence a cocycle. We say that csinduce* *s the transgression from ~ to P, or that P suspends to ~. the algebra of (left invariant) differential forms on its fiber. Then inv(g) pl* *ays the role of differential forms on the base, BG = EG=G. In fact, for G a compact and simply connected Lie group* * and g its Lie algebra, we have Ho(inv(g)) ' Ho(BG, R) . (18* *8) In summary, the situation we thus obtain is that depicted in figure 1. Compa* *re this to the following fact. Proposition 15For p : P ! X a principal G-bundle, let vert(P) (TP) be the ve* *rtical vector fields on P. The horizontal differential forms on P which are invariant under vert(P) are* * precisely those that are pulled back along p from X. These are called the basic differential forms in [30]. Remark. We will see that, contrary to the situation for ordinary Lie algebras,* * in general invariant poly- nomials of L1 algebras are not dW(g)-closed (the dW(g)-differential of them is * *just horizontal). We will also see that those indecomposable invariant polynomials in inv(g), i.e. those * *that become exact in ker(i*), are not characteristic for the corresponding g-bundles. This probably means tha* *t the real cohomology of the classifying space of the Lie 1-group integrating g is spanned by invariant * *polynomials modulo those suspending to 0. But here we do not attempt to discuss this further. Proposition 16For every invariant polynomial P 2 ^og[1] W(g) on an L1 -algebr* *a g such that dW(g)p*P = 0, there exists an L1 -algebra cocycle ~ 2 CS(g) that transgresses to P. Proof. This is a consequence of proposition 7 and proposition 1. Let P 2 W(g* *) be an invariant polyno- * * * mial. By proposition 7, p*P is in the kernel of the restriction homomorphism CE* * (g)oiooW(g)o_: i*P = 0. By proposition 1, p*P is the image under dW(g)of an element cs:= o(p*P) and by * *the algebra homomorphism property of i* we know that its restriction, ~ := i*cs, to the fiber is closed,* * because dCE(g)i*cs= i*dW(g)cs= i*p*P = 0 . (18* *9) Therefore ~ is an L1 -algebra cocycle for g that transgresses to the invariant * *polynomial P. 40 Remark. Notice that this statement is useful only for indecomposable invariant* * polynomials. All others trivially suspend to the 0 cocycle. Proposition 17An invariant polynomial which suspends to a Lie 1-algebra cocycle* * that is a coboundary also suspends to 0. Proof. Let P be an invariant polynomial, csthe corresponding transgression elem* *ent and ~ = i*csthe corresponding cocycle, which is assumed to be a coboundary in that ~ = dCE(g)b * *for some b 2 CE(g). Then by the definition of dW(g)it follows that ~ = i*(dW(g)b). Now notice that cs0:= cs- dW(g)b (19* *0) is another transgression element for P, since dW(g)cs0= p*P . (19* *1) But now i*(cs0) = i*(cs- dW(g)b) = 0 . (19* *2) Hence P suspends to 0. 6.3.1 Examples The cohomologies of G and of BG in terms of qDGCAs. To put our general conside* *rations for L1 -algebras into perspective, it is useful to keep the following classical res* *ults for ordinary Lie algebras in mind. A classical result of E. Cartan [16] [17] (see also [36]) says that for a co* *nnected finite dimensional Lie group G, the cohomology Ho(G) of the group is isomorphic to that of the Chevall* *ey-Eilenberg algebra CE(g) of its Lie algebra g: Ho(G) ' Ho(CE (g)) , (19* *3) namely to the algebra of Lie algebra cocycles on g. If we denote by QG the spac* *e of indecomposable such cocycles, and form the qDGCA ^oQG = Ho(^oQG) with trivial differential, the abo* *ve says that we have an isomorphism in cohomology Ho(G) ' Ho(^oQG) = ^oQG (19* *4) which is realized by the canonical inclusion " i : ^oQG O___//_CE(g) (19* *5) of all cocycles into the Chevalley-Eilenberg algebra. Subsequently, we have the classical result of Borel [9]: For a connected fin* *ite dimensional Lie group G, the cohomology of its classifying space BG is a finitely generated polynomial a* *lgebra on even dimensional generators: Ho(BG) ' ^oPG . (19* *6) Here PG is the space of indecomposable invariant polynomials on g, hence Ho(BG) ' Ho(inv(g)) . (19* *7) In fact, PG and QG are isomorphic after a shift: PG ' QG[1] (19* *8) and this isomorphism is induced by transgression between indecomposable cocycle* *s ~ 2 CE(g) and indecom- posable invariant polynomials P 2 inv(g) via a transgression element cs= oP 2 W* *(g). 41 Cohomology and invariant polynomials of bn-1u(1) Proposition 18For every integer n 1, the Lie n-algebra bn-1u(1) (the (n - 1)-* *folded shifted version of ordinary u(1)) from 6.1.1) we have the following: o there is, up to a scalar multiple, a single indecomposable Lie 1-algebra c* *ocycle which is of degree n and linear, ~bn-1u(1)2 R[n] CE(bn-1u(1)) , (19* *9) o there is, up to a scalar multiple, a single indecomposable Lie 1-algebra i* *nvariant polynomial, which is of degree (n + 1) Pbn-1u(1)2 R[n + 1] inv(bn-1u(1)) . (20* *0) o The cocycle ~bn-1u(1)is in transgression with Pbn-1u(1). These statements are an obvious consequence of the definitions involved, but th* *ey are important. The fact that bn-1u(1) has a single invariant polynomial of degree (n+1) will immediatel* *y imply, in 7, that bn-1u(1)- bundles have a single characteristic class of degree (n+1): known (at least for* * n = 2, as the Dixmier-Douady class). Such a bn-1u(1)-bundle classes appear in 8 as the obstruction classes f* *or lifts of n-bundles through string-like extensions of their structure Lie n-algebra. Cohomology and invariant polynomials of strict Lie 2-algebras. Let g(2)= (h t!* *g ff!der(h)) be a strict Lie 2-algebra as described in section 6.1. Notice that there is a canoni* *cal projection homomorphism * CE(g)oooo_j____CE(h t!g) (20* *1) which, of course, extends to the Weil algebras * W(g) ooooj____W(h_t!g). (20* *2) Here j* is simply the identity on g* and on g*[1] and vanishes on h*[1] and h*[* *2]. Proposition 19Every invariant polynomial P 2 inv(g) of the ordinary Lie algebra* * g lifts to an invariant polynomial on the Lie 2-algebra (h t!g): W(h t!g)ccG | G | GG i*|| GG . (20* *3) | GG fflfflfflffl||1GQ W(g)oo_________?inv(g)` However, a closed invariant polynomial will not necessarily lift to a closed on* *e. Proof. Recall that dt:= dCE(h!tg)acts on g* as dt|g*= [., .]*g+ t*. (20* *4) By definition 16 and definition 17 it follows that dW(h!tg)acts on g*[1] as dW(h!tg)|g*[1]= -oe O [., .]*g- oe O t* (20* *5) 42 and on h*[1] as dW(h!tg)|h*[1]= -oe O ff*. (20* *6) Then notice that (oe O t*) : g*[1] ! h*[2] . (20* *7) But this means that dW(h!tg)differs from dW(g)on ^o(g*[1]) only by elements tha* *t are annihilated by vertical 'X . This proves the claim. It may be easier to appreciate this proof by looking at what it does in term* *s of a chosen basis. Same discussion in terms of a basis. Let {ta} be a basis of g* and {bi} be * *a basis of h*[1]. Let {Cabc}, {ffiaj}, and {tai}, respectively, be the components of [., .]g, ff and * *t in that basis. Then corresponding to CE(g), W(g), CE(h t!g), and W(h t!g), respectively, we have the differentials dCE(g): ta 7! -1_2Cabctb^ tc, (20* *8) dW(g): ta 7! -1_2Cabctb^ tc+ oeta, (20* *9) dCE(h!tg): ta 7! -1_2Cabctb^ tc+ taibi, (21* *0) and dW(h!tg): ta 7! -1_2Cabctb^ tc+ taibi+ oeta. (21* *1) Hence we get dW(g): oeta 7! -oe(-1_2Cabctb^ tc) = Cabc(oetb) ^ tc (21* *2) as well as dW(h!tg): oeta 7! -oe(-1_2Cabctb^ tc+ taibi) = Cabc(oetb) ^ tc+(t* *aioebi.213) Then if P = Pa1...an(oeta1) ^ . .^.(oetan) (21* *4) is dW(g)-closed, i.e. an invariant polynomial on g, it follows that dW(h!tg)P = nPa1,a2,...an(ta1ioebi) ^ (oeta2) ^ . .^.(oetan)* *(.215) (all terms appearing are in the image of the shifting isomorphism oe), hence P * *is also an invariant polynomial on (h t!g). We will see a physical application of this fact in 6.6. Remark. Notice that the invariant polynomials P lifted from g to (h t!g) this * *way are no longer closed, in general. This is a new phenomenon we encounter for higher L1 -algebras. Whil* *e, according to proposition 14, for g an ordinary Lie algebra all elements in inv(g) are closed, this is no* * longer the case here: the lifted elements P above vanish only after we hit with them with both dW(h!tg)and a ver* *tical o. Proposition 20Let P be any invariant polynomial on the ordinary Lie algebra g i* *n transgression with the cocycle ~ on g. Regarded both as elements of W(h t!g) and CE(h t!g) respectivel* *y. Notice that dCE(h!tg)~ in in general non-vanishing but is of course now an exact cocycle on (h t!g). We have : the (h t!g)-cocycle dCE(h!tg)~ transgresses to dinv(h!tg)P. The situation is illustrated by the diagram in figure 7. 43 0OO . | | dW(h!tg)|| | | `| p*W(h!tg) 0OO p*dinv(h!tg)PK_oo___dinv(h!tg)PO | ________________________OOOO | ________________________|| | ____________________| || d t| oW(h!tg)________________________________________* *____d|d| t t CE(h!g)| ________________________|| * * W(h!g)inv(h!g) | ________________________|| | ______________| | `| * _ssss_____________`| `| dCE(h!tg)~oo_i___Oop*dinv(h!tg)P_ P Figure 7: Cocycles and invariant polynomials on strict Lie 2-algebras (h t!g), * *induced from cocycles and invariant polynomials on g. An invariant polynomial P on g in transgression* * with a cocycle ~ on g lifts to a generally non-closed invariant polynomial on (h t!g). The diagram says tha* *t its closure, dinv(h!tg)P, suspends to the dCE(h!tg)-closure of the cocycle ~. Since this (h t!g)-cocycle * *d(h!tg)~ is hence a coboundary, it follows from proposition 17 that dinv(h!tg)P suspends also to 0. Nevertheles* *s the situation is of interest, in that it governs the topological field theory known as BF theory. This is dis* *cussed in section 6.6.1. Concrete Example: su(5) ! sp(5). It is known that the cohomology of the Cheval* *ley-Eilenberg algebras for su(5) and sp(5) are generated, respectively, by four and five indecomposabl* *e cocycles, Ho(CE (su(5))) = ^o[a, b, c, d] (21* *6) and Ho(CE (sp(5))) = ^o[v, w, x, y, z] , (21* *7) |generator * *degree | a * * 3 | b * * 5 HoCE (su(5))| c * * 7 _____________|____d_____* *___9____ which have degree as indicated in the following table: | * * . | v * * 3 | w * * 7 Ho(CE (sp(5))) | x * * 11 | y * * 15 | z * * 19 As discussed for instance in [30], the inclusion of groups SU(5) ,! Sp(5) (21* *8) is reflected in the morphism of DGCAs * CE(su(5))oooto___CE_(sp(5)) (21* *9) which acts, in cohomology, on v and w as aoo__Ov_ (22* *0) coo__Ow_ 44 and which sends x, y and z to wedge products of generators. We would like to apply the above reasoning to this situation. Now, su(5) is * *not normal in sp(5) hence (su(5) ,! sp(5)) does not give a Lie 2-algebra. But we can regard the cohomolog* *y complexes Ho(CE (su(5))) and Ho(CE (sp(5))) as Chevalley-Eilenberg algebras of abelian L1 -algebras in t* *heir own right. Their inclu- sion is normal, in the sense to be made precise below in definition 38. By usef* *ul abuse of notation, we write now CE(su(5) ,! sp(5)) for this inclusion at the level of cohomology. Recalling from 133 that this means that in CE(su(5) ,! sp(5)) we have dCE(su(5),!sp(5))v := oea (22* *1) and dCE(su(5),!sp(5)))w := oec (22* *2) we see that the generators oea and oeb drop out of the cohomology of the Cheval* *ley-Eilenberg algebra V o * * CE(su(5) ,! sp(5)) = ( (sp(5) su(5) [1]), dt) (22* *3) of the strict Lie 2-algebra coming from the infinitesimal crossed module (t : s* *u(5) ,! sp(5)). A simple spectral sequence argument shows that products are not killed in Ho* *(CE (su(5) ,! sp(5))), but they may no longer be decomposable. Hence Ho(CE (su(5) ,! sp(5))) (22* *4) is generated by classes in degrees 6 and 10 by oeb and oed, and in degrees 21 a* *nd 25, which are represented by products in 223 involving oea and oec, with the only no zero product being 6 ^ 25 = 10 ^ 21 , (22* *5) where 31 is the dimension of the manifold Sp(5)=SU(5). Thus the strict Lie 2-al* *gebra (t : su(5) ,! sp(5)) plays the role of the quotient Lie 1-algebra sp(5)=su(5). We will discuss the g* *eneral mechanism behind this phenomenon in 8.1: the Lie 2-algebra (su(5) ,! sp(5)) is the weak cokernel, i.e* *. the homotopy cokernel of the inclusion su(5) ,! sp(5). The Weil algebra of (su(5) ,! sp(5)) is W(su(5) ,! sp(5)) = (^o(sp(5)* su(5)*[1] sp(5)*[1] su(5)*[2]),(dW* *(su(5),!sp(5)))2.26) Recall the formula 128 for the action of dW(su(5),!sp(5))on generators in sp(5)* **[1] su(5)*[2]. By that formula, oev and oew are invariant polynomials on sp(5) which lift to non-close* *d invariant polynomials on su(5) ,! sp(5)): dW(su(5),!sp(5))) : oev 7! -oe(dCE(su(5),!sp(5))v) = -oeoea* *(227) by equation 221; and dW(su(5),!sp(5))) : oew 7! -oe(dCE(su(5),!sp(5))w) = -oeoec* *(228) by equation 222. Hence oev and oew are not closed in CE(su(5) ,! sp(5)), but t* *hey are still invariant polynomials according to definition 21, since their differential sits entirely * *in the shifted copy (sp(5)* su(5)*[1])[1]. On the other hand, notice that we do also have closed invariant polynomials * *on (su(5) ,! sp(5)), for instance oeoeb and oeoed. 6.4 L1 -algebras from cocycles: String-like extensions We now consider the main object of interest here: families of L1 -algebras that* * are induced from L1 -cocycles and invariant polynomials. First we need the following Definition 24 (String-like extensions of L1 -algebras)Let g be an L1 -algebra. 45 o For each degree (n + 1)-cocycle ~ on g, let g~ be the L1 -algebra defined * *by CE(g~) = (^o(g* R[n]), dCE(g~)) (22* *9) with differential given by dCE(g~)|g*:= dCE(g), (23* *0) and dCE(g~))|R[n]: b 7! -~ , (23* *1) where {b} denotes the canonical basis of R[n]. This we call the String-l* *ike extension of g with respecto to ~, because, as described below in 6.4.1, it generalizes the co* *nstruction of the String Lie 2-algebra. o For each degree n invariant polynomial P on g, let chP(g) be the L1 -algeb* *ra defined by CE (chP(g)) = (^o(g* g*[1] R[2n - 1]), dCE(chP(g)))(23* *2) with the differential given by dCE(chP(g))|g* g*[1]:= dW(g) (23* *3) and dCE(chP(g)))|R[2n-1]: c 7! P , (23* *4) where {c} denotes the canonical basis of R[2n-1]. This we call the Chern * *L1 -algebra corresponding to the invariant polynomial P, because, as described below in 6.5.1, conne* *ctions with values in it pick out the Chern-form corresponding to P. o For each degree 2n - 1 transgression element cs, let csP(g) be the L1 -alg* *ebra defined by CE(csP(g)) = (^o(g* g*[1] R[2n - 2] R[2n - 1]), dCE(chP(* *g)))(235) with dCE(csP(g))|^o(g* g*[1])= dW(g) (23* *6) dCE(csP(g))|R[2n-2]: b 7! -cs+ c (23* *7) dCE(chp(g))|R[2n-1]: c 7! P , (23* *8) where {b} and {c} denote the canonical bases of R[2n - 2] and R[2n - 1], r* *espectively. This we call the Chern-Simons L1 -algebra with respect to the transgression element cs,* * because, as described below in 6.5.1, connections with values in these come from (generalized) C* *hern-Simons forms. The nilpotency of these differentials follows directly from the very definit* *ion of L1 -algebra cocoycles and invariant polynomials. Proposition 21 (the string-like extensions)For each L1 -cocycle ~ 2 ^n(g*) of d* *egree n, the corre- sponding String-like extension sits in an exact sequence 0oo___CE(bn-1u(1))oooo____CE_(g~)oo______?CE(g)`_oo_0 (23* *9) Proof. The morphisms are the canonical inclusion and projection. 46 Proposition 22For cs2 W(g) any transgression element interpolating between the * *cocycle ~ 2 CE(g) and the invariant polynomial P 2 ^o(g[1]) W(g), we obtain a homotopy-exact sequen* *ce CE (g~)oooo_____CE(csP(g))oo______?CE(chP(g))`_. (24* *0) |'| | W(g~) Here the isomorphism f : W(g~) __'_//_CE(csP(g)) (24* *1) is the identity on g* g*[1] R[n] f|g* g*[1] R[n]= Id (24* *2) and acts as f|R[n+1]: b 7! c + ~ - cs (24* *3) for b the canonical basis of R[n] and c that of R[n + 1]. We check that this do* *es respect the differentials dW(g~) O dW(g~) bO________//_`-~`+ c c_____________//`|oe~` | | | | | | | | |f| f|| |f| |f|. (24* *4) | | | | | | | | fflffl|OdCE(csfflffl|P(g))fflffl|OdCE(csPfflffl|(g)) b_________//_-cs+ c c + ~ -_cs_______//oe~ Recall from definition 39 that oe is the canonical isomorphism oe : g* ! g*[1] * *extended by 0 to g*[1] and then as a derivation to all of ^o(g* g*[1]). In the above the morphism between the * *Weil algebra of g~ and the Chevalley-Eilenberg algebra of csP(g) is indeed an isomorphism (not just an equ* *ivalence). This isomorphism exhibits one of the main points to be made here: it makes manifest that the inv* *ariant polynomial P that is related by transgression to the cocycle ~ which induces g~ becomes exact wit* *h respect to g~. This is the statement of proposition 24 below. L1 -algebra cohomology and invariant polynomials of String-like extensions. Th* *e L1 -algebra g~ obtained from an L1 -algebra g with an L1 -algebra cocycle ~ 2 Ho(CE (g)) ca* *n be thought of as being obtained from g by "killing" a cocycle ~. This is familiar from Sullivan models* * in rational homotopy theory. Proposition 23Let g be an ordinary semisimple Lie algebra and ~ a cocycle on it* *. Then Ho(CE (g~)) = Ho(CE (g))=<~>. (24* *5) Accordingly, one finds that in cohomology the invariant polynomials on g~ ar* *e those of g except that the polynomial in transgression with ~ now suspends to 0. Proposition 24Let g be an ordinary semisimple Lie algebra and ~ 2 Ho(CE (g)) a * *class which is necessarily of odd degree, so that ~ ^ ~ = 0 automatically. Let ~ be in transgression with* * the invariant polynomial P 2 inv(g). Then with respect to g~ the polynomial P suspends to 0. Proof. Since ~ is a coboundary in CE(g~), this is a corollary of 17. 47 Remark. We will see in 7 that those invariant polynomials which suspend to 0 d* *o actually not contribute to the characteristic classes. As we will also see there, this can be understo* *od in terms of the invari- ant polynomials not with respect to the projection CE (g)ooooW(g)_but with resp* *ect to the projection CE(g)ooooCE_(csP(g~))r,ecalling from 22 that W(g) is isomorphic to csP(g). Proposition 25For g any L1 -algebra with cocycle ~ of degree 2n + 1 in transgre* *ssion with the invariant polynomial P, denote by csP(g)basicthe DGCA of basic forms with respect to the * *canonical projection CE (g)ooooCE(csP(g~))_, (24* *6) according to the general definition 10. Then the cohomology of csP(g)basicis that of inv(g) modulo P: Ho(csP(g)basic) ' Ho(inv(g))=

. (24* *7) Proof. One finds that the vertical derivations on CE (csP(g)) = ^o(g* g*[1] * * R[n] R[n + 1]) are those that vanish on everything except the unshifted copy of g*. Therefore the * *basic forms are those in ^o(g*[1] R[n] R[n + 1]) such that also their dcsP(g)-differential is in tha* *t space. Hence all invariant g-polynomials are among them. But one of them now becomes exact, namely P. Remark. The first example below, definition 25, introduces the String Lie 2-al* *gebra of an ordinary semisim- ple Lie algebra g, which gave all our String-like extensions its name. It is kn* *own that the real cohomology of the classifying space of the 2-group integrating it is that of G = exp(g), m* *odulo the ideal generated by the class corresponding to P. Hence CE(csP(g)) is an algebraic model for this s* *pace. 6.4.1 Examples The String Lie 2-algebra. Definition 25Let g be a semisiple Lie algebra and ~ = <., [., .]> the canonical* * 3-cocycle on it. Then string(g) (24* *8) is defined to be the strict Lie 2-algebra coming from the crossed module (^g ! Pg) , (24* *9) where Pg is the Lie algebra of based paths in g and ^g the Lie algebra of based* * loops in g, with central extension induced by ~. Details are in [4]. Proposition 26 ([4])The Lie 2-algebra g~ obtained from g and ~ as in definition* * 24 is equivalent to the strict string Lie 2-algebra g~ ' string(g) . (25* *0) This means there are morphisms g~ ! string(g) and string(g) ! g~ whose composit* *e is the identity only up to homotopy g~____//_______str;;ing(g)//_g~string(g)//_g~//_string(g) (25* *1) _____________________________________________________________* *____________________;;____________________________________________ ___________________________________________________________* *_____________________________________________________________________________* *__=ff'||_____________________________________________________________________* *___________________________________________________jff'|| ________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *__________________________________________@ Id Id We call g~ the skeletal and string(g) the strict version of the String Lie 2* *-algebra. 48 The Fivebrane Lie 6-algebra Definition 26Let g = so(n) and ~ the canonical 7-cocycle on it. Then fivebrane(g) (25* *2) is defined to be the strict Lie 7-algebra which is equivalent to g~ g~ ' fivebrane(g) . (25* *3) A Lie n-algebra is strict if it corresponds to a differential graded Lie alg* *ebra on a vector space in degree 1 to n. (Recall our grading conventions from 6.1.) Remark. It is a major open problem to identify the strict fivebrane(g). Propos* *ition 26 suggests that it might involve hyperbolic Kac-Moody algebras and/or the torus algebra of g, sinc* *e these would seem to be what comes beyond the affine Kac-Moody algebras relevant for string(n). The BF-theory Lie 3-algebra. Definition 27For g any ordinary Lie algebra with bilinear invariant symmetric f* *orm <., .> 2 inv(g) in transgression with the 3-cocycle ~, and for h t!g a strict Lie 2-algebra based * *on g, denote by ~^:= dCE(h!tg)~ (25* *4) the corresponding exact 4-cocycle on (h t!g) discussed in 6.3.1. Then we call t* *he string-like extended Lie 3-algebra bf(h t!g) := (h t!g)^~ (25* *5) the corresponding BF-theory Lie 3-algebra. The terminology here will become clear once we describe in 8.3.1 and 9.1.1 h* *ow the BF-theory action functional discussed in 6.6.1 arises as the parallel 4-transport given by the b* *3u(1)-4-bundle which arises as the obstruction to lifting (h t!g)-2-descent objects to bf(h t!g)-3-descent obj* *ects. 6.5 L1 -algebra valued forms Consider an ordinary Lie algebra g valued connection form A regarded as a linea* *r map g* ! 1(Y ). Since CE(g) is free as a graded commutative algebra, this linear map extends un* *iquely to a morphism of graded commutative algebras, though not in general of differential graded commu* *tative algebra. In fact, the deviation is measured by the curvature FA of the connection. However, the diffe* *rential in W(g) is precisely such that the connection does extend to a morphism of differential graded-commu* *tative algebras W(g) ! o(Y ) . (25* *6) This implies that a good notion of a g-valued differential form on a smooth spa* *ce Y , for g any L1 -algebra, is a morphism of differential graded-commutative algebras from the Weil algebra* * of g to the algebra of differential forms on Y . Definition 28 (g-valued forms)For Y a smooth space and g an L1 -algebra, we call o(Y, g) := Homdgc-Alg(W(g), o(Y )) (25* *7) the space of g-valued differential forms on X. 49 Definition 29 (curvature)We write g-valued differential forms as (A,FA) o ( o(Y )oo_______W(g)) 2 (Y, g) , (25* *8) where FA denotes the restriction to the shifted copy g*[1] given by FA_______________________* *__________________________________________________ _____________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_________________ _______________________________* *_____________________________________________________________________________* *________ (A,FA) o -(A,FA)-___ __*`_____________* *____________ curv: ( o(Y )oo__W(g)) 7! ( (Y )oo__W(g)oo__?g_[1]) . (25* *9) FA we call the curvature of A. (A,FA) Proposition 27The g-valued differential form o(Y )oo_______W(g)_factors throug* *h CE(g) precisely when its curvature FA vanishes. CE(g)_oooo_____W(g) . (26* *0) ___ | ____ | ____ | _(A,FA=0)_____ |(A,FA) ___ | ____ | fflffl____= fflffl| o(Y )_________ o(Y ) In this case we say that A is flat. Hence the space of flat g-valued forms is oflat(Y, g) ' Homdgc-Alg(CE (g), o(Y )) . (26* *1) Bianchi identity. Recall from 6.1 that the Weil algebra W(g) of an L1 -algebra* * g is the same as the Chevalley-Eilenberg algebra CE(inn(g)) of the L1 -algebra of inner derivation o* *f g. It follows that g-valued differential forms on Y are the same as flat inn(g)-valued differential forms o* *n Y : o(Y, g) = oflat(inn(g)) . (26* *2) By the above definition of curvature, this says that the curvature FA of a g-va* *lued connection (A, FA) is itself a flat inn(g)-valued connection. This is the generalization of the ordinary Bia* *nchi identity to L1 -algebra valued forms. Definition 30Two g-valued forms A, A02 o(Y, g) are called (gauge) equivalent p* *recisely if they are related by a vertical concordance, i.e. by a concordance, such that the corres* *ponding derivation ae from proposition 11 is vertical, in the sense of definition 9. 6.5.1 Examples 1. Ordinary Lie-algebra valued 1-forms. We have already mentioned ordinary Li* *e algebra valued 1-forms in this general context in 2.1.3. 2. Forms with values in shifted bn-1u(1) A bn-1u(1)-valued form is nothing but an ordinary n-form A 2 n(Y ): o(bn-1u(1), Y ) ' n(Y ) . A flat bn-1u(1)-valued form is precisely a closed n-form. 50 CE (bn-1u(1))oooo____W(bn-1u(1)) __ | ____ | ____ | (A)_______ (A,FA)| dA=0___ A=dA| ___ | fflffl____= fflffl| o(Y )______________// o(Y ) 3. Crossed module valued forms. Let g(2)= (h t!g) be a strict Lie 2-algebra com* *ing from a crossed module. Then a g(2)-valued form is an ordinary g-valued 1-form A and an ordinar* *y h-valued 2-form B. The corresponding curvature is an ordinary g-valued 2-form fi = FA + t(B) and an or* *dinary h-valued 3-form H = dAB. This is denoted by the right vertical arrow in the following diagram. CE (h_t!g)oooo_____W(h t!g) ____ | ____ | (A,B)_____ (A,B,fi,H)| . (26* *3) ___ fi=F|+t(B) FA+t(B)=0___ H=dAAB ___ fflffl____= fflffl| o(Y )____________ o(Y ) Precisely if the curvature components fi and H vanish, does this morphism on th* *e right factor through CE (h t!g), which is indicated by the left vertical arrow of the above diagram. 4. String Lie n-algebra valued forms. For g an ordinary Lie algebra and ~ a deg* *ree (2n + 1)-cocycle on g the situation is captured by the following diagram String-like Chern-Simons Chern . (26* *4) 1 2n 2n + 1 2n + 1 " ` CE(g)O____CE/(g~)/_oooo_CE(csP(g))oo_?_CE(chP(g))_ | _____ | | | ____ | | (A)| (A,B)______ (A,B,C)| (A,C)| FA=0| dB+FA=0CS(A)=0_C=dB+CSP(A)| dC=k((FA)n+1)| | _k____ | | fflffl| fflffl_=__ fflffl|= fflffl| o(Y )_=___ o(Y )________ o(Y )________ o(Y ) Here CSP(A) denotes the Chern-Simons form such that dCSP (A) = P(FA), given * *by the specific con- tracting homotopy. 51 The standard example is that corresponding to the ordinary String-extension. " CE(g)O___/CE(string(g))/_oooo_W(stringk(g)) || || || || '|| |'| || " || || CE(g)O______CE/(g~)/oooo__CE_(csk(g))oo__`?CE(chP(g)) (A)| (A,B)______ (A,B,C)| (A,C)| | FA=0_ | | FfflfflA=0|dB+fflfflC=SP(A)=0fflfflC==dB+CSP(A)|fflffldC=| o(Y )__=___ o(Y )_________ o(Y )________ o(Y ) (26* *5) In the above, g is semisimple with invariant bilinear form P = <., .> related b* *y transgression to the 3-cocycle ~ = <., [., .]>. Then the Chern-Simons 3-form for any g-valued 1-form A is CS<.,.>(A) = + 1_3 . (26* *6) 6.6 L1 -algebra characteristic forms Definition 31For g any L1 algebra and (A,FA) o(Y )oo__W(g)_ (26* *7) any g-valued differential form, we call the composite {P(FA)}___________________________________ ____________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *________________________________________ ______________________________________________* *______________________________________________________________________ (A,FA)zz___ ________`___________________ o(Y )oo_W(g)_oo__inv(g)?_ (26* *8) the collection of invariant forms of the g-valued form A. We call the deRham c* *lasses [P(FA)] of the characteristic forms arising as the image of closed invariant polynomials {Pi(FA)}___________________________________ ______________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *______________________________________ ________________________________________________* *____________________________________________________________________ y(A,FA)y___ _________`__________________ o(Y )oo__W(g) oo____inv(g)? (26* *9) {[P(FA)]} HodR(Y )oo__________Ho(inv(g))_ the collection of characteristic classes of the g-valued form A. Recall from 6.3 that for ordinary Lie algebras all invariant polynomials are* * closed, while for general L1 - algebras it is only true that their dW(g)-differential is horizontal. Notice th* *at Y will play the role of a cover of some space X soon, and that characteristic forms really live down on X. We w* *ill see shortly a constraint imposed which makes the characteristic forms descend down from the Y here to su* *ch an X. Proposition 28Under gauge transformations as in definition 30, characteristic c* *lasses are invariant. 52 Proof. This follows from proposition 11: By that proposition, the derivative of* * the concordance form ^A along the interval I = [0, 1] is a chain homotopy _d_^A(P) = [d, ' ]P = do(P) + ' (d P) . (27* *0) dt X X W(g) By definition of gauge-transformations, 'X is vertical. By definition of basic * *forms, P is both in the kernel of 'X as well as in the kernel of 'X O d. Hence the right hand vanishes. 6.6.1 Examples Characteristic forms of bn-1u(1)-valued forms. Proposition 29A bn-1u(1)-valued form o(Y )oo___A___W(bn-1u(1))_is precisely an* * n-form on Y : o(Y, bn-1u(1)) ' n(Y ) . (27* *1) If two such bn-1u(1)-valued forms are gauge equivalent according to definition * *30, then their curvatures coincide 0 ( o(Y )ooA_W(bn-1u(1))) ~ ( o(Y )oAo_W(bn-1u(1))) ) dA = dA0. (27* *2) BF-theory. We demonstrate that the expression known in the literature as the a* *ction functional for BF- theory with cosmological term is the integral of an invariant polynomial for g-* *valued differential forms where g is a Lie 2-algebra. Namely, let g(2)= (h t!g) be any strict Lie 2-algebra as * *in 6.1. Let P = <., .> (27* *3) be an invariant bilinear form on g, hence a degree 2 invariant polynomial on g.* * According to proposition 19, P therefore also is an invariant polynomial on g(2). Now for (A, B) a g(2)-valued differential form on X, as in the example in 6.* *5, ((A,B),(fi,H)) o(Y )oo______W(g(2))_, (27* *4) one finds ((A,B),(fi,H)) ` o(Y )oo_______W(g(2))oo______?inv(g(2))_hh____ (27* *5) ____________________________________________________* *___________________________________ __________________________________________________* *_____________________________________________________________________________* *_________________________________ ______________________________________________* *_____________________________________________________________________________* *_________________________________________ P7! so that the corresponding characteristic form is the 4-form P(fi, H) = = <(FA + t(B)) ^ (FA + t(B))> . (27* *6) Collecting terms as P(fi, H) = _-z___"+2 _-z___"+ _-z____"(277) Pontryagin term BF-term "cosmological* * constant" we recognize the Lagrangian for topological Yang-Mills theory and BF theory wit* *h cosmological term. For X a compact 4-manifold, the corresponding action functional S : o(X, g(2)) ! R (27* *8) 53 sends g(2)-valued 2-forms to the intgral of this 4-form Z (A, B) 7! ( + 2 + ) (27* *9) X The first term here is usually not considered an intrinsic part of BF-theory, b* *ut its presence does not affect the critical points of S. The critical points of S, i.e. the g(2)-valued differential forms on X that * *satisfy the equations of motion defined by the action S, are given by the equation fi := FA + t(B) = 0 . (28* *0) Notice that this implies dAt(B) = 0 (28* *1) but does not constrain the full 3-curvature H = dAB (28* *2) to vanish. In other words, the critical points of S are precisely the fake flat* * g(2)-valued forms which precisely integrate to strict parallel transport 2-functors [27, 53, 5]. While the 4-form looks similar to the Pontrjagin 4-form * *for an ordinary connection 1-form A, one striking difference is that is, in general, not closed.* * Instead, according to equation 215, we have d = 2 . (28* *3) Remark. Under the equivalence [4] of the skeletal String Lie 2-algebra to its * *strict version, recalled in proposition 26, the characteristic forms for strict Lie 2-algebras apply also t* *o one of our central objects of interest here, the String 2-connections. But a little care needs to be exercise* *d here, because the strict version of the String Lie 2-algebra is no longer finite dimensional. Remark. Our interpretation above of BF-theory as a gauge theory for Lie 2-alge* *bras is not unrelated to, but different from the one considered in [27, 28]. There only the Lie 2-algebra* * coming from the infinitesimal crossed module (|g| !0 g ad!der(g)) (for g any ordinary Lie algebra and |g| its* * underlying vectorRspace, regarded as an abelian Lie algebra) is considered, and the action is restricted* * to the term . We can regard the above discussion as a generalization of this approach to arbitra* *ry Lie 2-algebras. Standard BF-theory (with "cosmological" term) is reproduced with the above Lagrangian by* * using the Lie 2-algebra inn(g) corresponding to the infinitesimal crossed module (g Id!g ad!der(g)) dis* *cussed in 6.1.1. 7 L1 -algebra Cartan-Ehresmann connections We will now combine all of the above ingredients to produce a definition of g-v* *alued connections. As we shall explain, the construction we give may be thought of as a generalization of the * *notion of a Cartan-Ehresmann connection, which is given by a Lie algebra-valued 1-form on the total space of* * a bundle over base space satisfying two conditions: o first Cartan-Ehresmann condition: on the fibers the connection form restri* *cts to a flat canonical form o second Cartan-Ehresmann condition: under vertical flows the connections tr* *ansforms nicely, in such a way that its characteristic forms descend down to base space. We will essentially interpret these two conditions as a pullback of the univ* *ersal g-bundle, in its DGC- algebraic incarnation as given in equation 182. The definition we give can also be seen as the Lie algebraic image of a simi* *lar construction involving locally trivializable transport n-functors [5, 54], but this shall not be furth* *er discussed here. 54 7.1 g-Bundle descent data Definition 32 (g-bundle descent data)Given a Lie n-algebra g, a g-bundle descen* *t object on X is a pair (Y, Avert) consisting of a choice of surjective submersion ss : Y ! X with conn* *ected fibers (this condition will be dropped when we extend to g-connection descent objects in 7.2) together with* * a morphism of dg-algebras overt(Yo)Averto_CE(g). (28* *4) Two such descent objects are taken to be equivalent A0vert ( overt(YA)vCEe(g)rtoo_) ~ ( overt(Yo0)oCE(g)_) (28* *5) precisely if their pullbacks ss*1Avertand ss*2A0vertto the common refinement ss1 Y xX Y 0____//Y (28* *6) ss2|| ss|| fflffl|ssfflffl|0 Y 0______//X are concordant in the sense of definition 19. Thus two such descent objects Avert, A0verton the same Y are equivalent if t* *here is j*vertsuch that _______Avert___________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *________________________ ___________________________________________________* *_____________________________________________________________ vv_____ j*vert_____________________________ overt(Yo)s*_overt(Yox_I)t*_oo_CE(g)oo________. (28* *7) gg_________________________________________________* *___________________________ __________________________________________________* *_____________________________________________________________________________* *____________ _____A0vert__________________________________* *_____________________________________________________________________________* *____________________________________ Recall from the discussion in 2.2.1 that the surjective submersions here pla* *y the role of open covers of X. 7.1.1 Examples Example: ordinary G-bundles. The following example is meant to illustrate how * *the notion of descent data with respect to a Lie algebra g as defined here can be related to the ordi* *nary notion of descent data with respect to a Lie group G. Consider the case where g is an ordinary Lie (1-* *)algebra. A g-cocycle then is a surjective submersion ss : Y ! X together with a g-valued flat vertical 1-* *form Averton Y . Assume the fiber of ss : Y ! X to be simply connected. Then for any two points (y, y0) 2 Y* * xX Y in the same fiber we obtain an element g(y, y0) 2 G, where G is the simply connected Lie group integ* *rating g, by choosing any path y __fl//_y0in the fiber connecting y with y0and forming the parallel trans* *port determined by Avert along this path Z g(y, y0) := P exp( Avert) . (28* *8) fl By the flatness of Avertand the assumption that the fibers of Y are simply conn* *ected o g : Y xX Y ! G is well defined (does not depend on the choice of paths), a* *nd o satisfies the cocycle condition for G-bundles y0 ?o@? @@?? g(y,y0)""@@g(y0,y00) g : ???? 7! """" @@@@ . (28* *9) OO? __________OO//_" y __________//_y00 o g(y,y00)o 55 Any such cocycle g defines a G-principal bundle. Conversely, every G-principal * *bundle P ! X gives rise to a structure like this by choosing Y := P and letting Avertbe the canonical i* *nvariant g-valued vertical 1-form on Y = P. Then suppose (Y, Avert) and (Y, A0vert) are two such cocycles * *defined on the same Y , and let (Y^:= Y x I, ^Avert) be a concordance between them. Then, for every path y x {0}___fl___//_y x {1} (29* *0) connecting the two copies of a point y 2 Y over the endpoints of the interval, * *we again obtain a group element Z h(y) := P exp( A^vert) . (29* *1) fl By the flatness of ^A, this is o well defined in that it is independent of the choice of path; o has the property that for all (y, y0) 2 Y xX Y we have h(y) y x {0}_________//y x {1} o _________//_o | | || || h : || || 7! g(y,y0)| g0(y,y0)|.(29* *2) | | | | | | | | | | | 0| fflffl| fflffl| offlffl|h(y/fflffl|)/_o y0x {0}________//_y0x {1} Therefore h is a gauge transformation between g and g0, as it should be. Note that there is no holonomy since the fibers are assumed to be simply con* *nected in this example. Abelian gerbes, Deligne cohomology and (bn-1u(1))-descent objects For the case* * that the L1 - algebra in question is shifted u(1), i.e. g = bn-1u(1), classes of g-descent ob* *jects on X should coincide with classes of "line n-bundles", i.e. with classes of abelian (n-1)-gerbes on X, he* *nce with elements in Hn(X, Z). In order to understand this, we relate classes of bn-1u(1))-descent objects to * *Deligne cohomology. We recall Deligne cohomology for a fixed surjective submersion ss : Y ! X. For comparison* * with some parts of the literature, the reader should choose Y to be the disjoint union of sets of a go* *od cover of X. More discussion of this point is in 5.3.1. The following definition should be thought of this way: a collection of p-fo* *rms on fiberwise intersections of a surjective submersion Y ! X are given. The 0-form part defines an n-bundle* * (an (n - 1)-gerbe) itself, while the higher forms encode a connection on that n-bundle. Definition 33 (Deligne cohomology)Deligne cohomology can be understood as the c* *ohomology on dif- ferential forms on the simplicial space Y ogiven by a surjective submersion ss * *: Y ! X, where the complex of forms is taken to start as 0 ____//_C1 (Y [n],dR=Z)//_ 1(Y_[n],dR)//_ 2(Yd[n],/R)/_.,. . (29* *3) where the first differential, often denoted dlogin the literature, is evaluated* * by acting with the ordinary differential on any R-valued representative of a U(1) ' R=Z-valued function. More in detail, given a surjective submersion ss : Y ! X, we obtain the augm* *ented simplicial space _ ss ! ___1//_ _ss1//_ ss Y o= . .Y.[3]ss2_//_//_Y/[2]/_Y//_Y [0] (29* *4) ss3 ss2 56 of fiberwise cartesian powers of Y , Y [n]:= Y_xX_Y_xX_.-.x.XYz_______", with Y* * [0]:= X. The double complex of n factors differential forms M M M o(Y o) = n(Y o) = r(Y [s]) (29* *5) n2N n2Nr,s2Nr+s=n on Y ohas the differential d ffi coming from the deRham differential d and the * *alternating pullback operation ffi : r(Y [s])! r(Y [s+1]) ffi : !7!ss*1! - ss*2! + ss*3! + . .-.(-1)s+1. (29* *6) Here we take 0-forms to be valued in R=Z. The map 0(Y )_d_//_ 1(Yt)akes any * *R-valued repre- sentative f of an R=Z-valued form and sends that to the ordinary df. This oper* *ation is often denoted dlog 1 o o [l] 0(Y )___//_ (Y.)Writing k(Y ) for the space of forms that vanish on Y for* * l < k we define (every- thing with respect to Y ): o A Deligne n-cocycle is a closed element in n(Y o); o a flat Deligne n-cocycle is a closed element in n1(Y o); o a Deligne coboundary is an element in (d ffi) o1(Y o) (i.e. no component* * in Y [0]= X); o a shift of connection is an element in (d ffi) o(Y o) (i.e. with possibl* *y a contribution in Y [0]= X). The 0-form part of a Deligne cocycle is like the transition function of a U(* *1)-bundle. Restricting to this part yields a group homomorphism [.] : Hn( o(Y o))___////_Hn(X, Z) (29* *7) to the integral cohomology on X. (Notice that the degree on the right is indeed* * as given, using the total degree on the double comples o(Y o) as given.) Addition of a Deligne coboundary is a gauge transformation. Using the fact [* *46] that the "fundamental complex" r(X)__ffi//_ r(Y_)ffi//_ r(Y [2]) . . . (29* *8) is exact for all r 1, one sees that Deligne cocycles with the same class in H* *n(X, Z) differ by elements in (d ffi) o(Y o). Notice that they do not, in general, differ by an element in * * o1(Y o): two Deligne cochains which differ by an element in (d ffi) o1(Y o) describe equivalent line n-bund* *les with equivalent connections, while those that differ by something in (d ffi) o0(Y o) describe equivalent l* *ine n-bundles with possibly inequivalent connections on them. Let v : o(Y o) ! overt(Y ) (29* *9) be the map which sends each Deligne n-cochain a with respect to Y to the vertic* *al part of its (n - 1)-form on Y [1] : a 7! a| n-1vert(Y.[1]) (30* *0) (Recall the definition 11 of overt(Y ).) Then we have Proposition 30If two Deligne n-cocycles a and b over Y have the same class in H* *n(X, Z), then the classes of (a) and (b) coincide. 57 Proof. As mentioned above, a and b have the same class in Hn(X, Z) if and only * *if they differ by an element in (d ffi)( o(Y o)). This means that on Y [1]they differ by an element of the* * form dff + ffifi = dff + ss*fi . (30* *1) Since ss*fi is horizontal, this is exact in overt(Y [1]). Proposition 31If the (n-1)-form parts B, B02 n-1(Y ) of two Deligne n-cocycles* * differ by a d ffi-exact part, then the two Deligne cocycles have the same class in Hn(X, Z). Proof. If the surjective submersion is not yet contractible, we pull everything* * back to a contractible refine- ment, as described in 5.3.1. So assume without restriction of generality that a* *ll Y [n]are contractible. This implies that HodeRham(Y [n]) = H0(Y [n]), which is a vector space spanned by th* *e connected components of Y [n]. Now assume B - B0= dfi + ffiff (30* *2) on Y . We can immediately see that this implies that the real classes in Hn(X, * *R) coincide: the Deligne cocycle property says d(B - B0) = ffi(H - H0) (30* *3) hence, by the exactness of the deRham complex we have now, ffi(H - H0) = ffi(dff) (30* *4) and by the exactness of ffi we get [H] = [H0]. To see that also the integral classes coincide we use induction over k in Y * *[k]. For instance on Y [2]we have ffi(B - B0) = d(A - A0) (30* *5) and hence ffidfi = d(A - A0) . (30* *6) Now using again the exactness of the deRham differential d this implies A - A0= ffifi + dfl . (30* *7) This way we work our way up to Y [n], where it then follows that the 0-form coc* *ycles are coboundant, hence that they have the same class in Hn(X, Z). Proposition 32bn-1u(1)-descent objects with respect to a given surjective subme* *rsion Y are in bijection with closed vertical n-forms on Y : ae oe overt(Yo)oAvert_CE(bn-1u(1)) $ {Avert2 nvert(Y ) , dAvert=.0}(30* *8) Two such bn-1u(1) descent objects on Y are equivalent precisely if these forms * *represent the same cohomology class (Avert~ A0vert) , [Avert] = [A0vert] 2 Hn( overt(Y )) . (30* *9) Proof. The first statement is a direct consequence of the definition of bn-1u(1* *) in 6.1. The second statement follows from proposition 11 using the reasoning as in proposition 28. Hence two Deligne cocycles with the same class in Hn(X, Z) indeed specify th* *e same class of bn-1u(1)- descent data. 58 7.2 Connections on g-bundles: the extension problem It turns out that a useful way to conceive of the curvature on a non-flat g n-b* *undle is, essentially, as the (n + 1)-bundle with connection obstructing the existence of a flat connection o* *n the original g-bundle. This superficially trivial statement is crucial for our way of coming to grips with * *non-flat higher bundles with connection. Definition 34 (descent object for g-connection)Given g-bundle descent object overt(Yo)AvertoCE_(g) (31* *0) as above, a g-connection on it is a completion of this morphism to a diagram overt(Yo)Averto_CE(g). (31* *1) OOOO| OOOO| | | |i*| || | | | | | (A,FA) | o(YO)oo________W(g)OOO | | | | |ss*| || | | | | ?O| ?O| o(X)oo__{Ki}__ inv(g) As before, two g-connection descent objects are taken to be equivalent, if thei* *r pullbacks to a common refine- ment are concordant. The top square can always be completed: any representative A 2 o(Y ) of Ave* *rt2 overt(Y ) will do. The curvature FA is then uniquely fixed by the dg-algebra homomorphism property* *. The existence of the top square then says that we have a 1-form on a total space which resticts to a* * canonical flat 1-form on the firbers. The commutativity of the lower square means that for all invariant pol* *ynomials P of g, the form P(FA) on Y is a form pulled back from X and is the differential of a form cs th* *at vanishes on vertical vector fields P(FA) = ss*K . (31* *2) The completion of the bottom square is hence an extra condition: it demands tha* *t A has been chosen such that its curvature FA has the property that the form P(FA) 2 o(Y ) for all inv* *ariant polynomials P are lifted from base space, up to that exact part. o The commutativity of the top square generalizes the first Cartan-Ehresmann* * condition: the con- nection form on the total space restricts to a nice form on the fibers. o The commutativity of the lower square generalizes the second Cartan-Ehresm* *ann condition: the connection form on the total space has to behave in such a way that the in* *variant polynomials applied to its curvature descend down to the base space. The pullback f*(Y, (A, FA)) = (Y 0, (f*A, f*FA)) (31* *3) 59 of a g-connection descent object (Y, (A, FA)) on a surjective submersion Y alon* *g a morphism f Y 0AA________//Y" AAA """" (31* *4) ss0A__AA"ss"""" X is the g-connection descent object depicted in figure 8. f* Avert overt(Yo0)o_overt(Yo)o______CE(g)ff_______. OO__________________________________________________* *__________________________OOOOOOOOOO | ________________________________________________* *____________________________________________________________________________| | _____________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________|| | f*Avert | i0*| | * | | |i | | | | | f* | (A,FA) | o(Y 0)oo___fo(Yf)oo________W(g)_______ OO__________________________________________________* *__________________________OOOO | ________________________________________________* *____________________________________________________________________________| | _____________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________|| | (f*A,Ff*A | ss0*| | * | | |ss | | | | ?O| Id ?O| ?O| o(X)foo____fo(X)oo__{Ki}___inv(g)______ __________________________________________________* *__________________________ ________________________________________________* *____________________________________________________________________________ _____________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________ {Ki} Figure 8: Pullback of a g-connection descent object (Y, (A, FA)) along a morphi* *sm f : Y 0! Y of surjective submersions, to f*(Y, (A, FA)) = (Y 0, (f*A, Ff*A)). Notice that the characteristic forms remain unaffected by such a pullback. T* *his way, any two g-connection descent objects may be pulled back to a common surjective submersion. A concor* *dance between two g- connection descent objects on the same surjective submersion is depicted in fig* *ure 9. Suppose (A, FA) and (A0, FA0) are descent data for g-bundles with connection* * over the same Y (possibly after having pulled them back to a common refinement). Then a concordance betwe* *en them is a diagram as in figure 9. 7.2.1 Examples. Example (ordinary Cartan-Ehresmann connection). Let P ! X be a principal G-bun* *dle and con- sider the descent object obtained by setting Y = P and letting Avertbe the cano* *nical invariant vertical flat 1-form on fibers P. Then finding the morphism (A,FA) o(Y )oo_______W(g) (31* *5) such that the top square commutes amounts to finding a 1-form on the total spac* *e of the bundle which restricts to the canonical 1-form on the fibers. This is the first of the two c* *onditions on a Cartan-Ehresmann connection. Then requiring the lower square to commute implies requiring that t* *he 2n-forms Pi(FA), formed from the curvature 2-form FA and the degree n-invariant polynomials Piof g, hav* *e to descend to 2n-forms Kion the base X. But that is precisely the case when Pi(FA) is invariant under * *flows along vertical vector fields. Hence it is true when A satisfies the second condition of a Cartan-Ehre* *smann connection, the one that says that the connection form transforms nicely under vertical flows. 60 Avert______________ _______________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________ __________________________________________________* *___________________________________________________________________ vv_____ j*vert___________________________* *__ overt(Yo)s*_overt(Yox_I)t*_oo_CEo(g)o________ OOgg_________________________________________________* *____________________________________________________OOOOOOOOOO || _______________A0vert___________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *______________________________________________________________________||| | | | | | | _____|____(A,FA)______|______________________* *_____________________________________________________________________________* *________________________________________________________________________ | _________|________________|______________________* *_____________________________________________________________________________* *__________________ | voo_*__v___|______j*________|______________________* *_____________________ o(Y )oost*__o(Y x I)oo________W(g)_______ OOgg_________________________________________________* *___________________________OOOO | _________________________________________________* *_________________| | ____________(A0,FA0)__________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____|| | | | | | | _____|____{Ki}________|______________________* *_____________________________________________________________________________* *______________________________________________________________________ | __________|________________|______________________* *_____________________________________________________________________________* *___________________________________ ?O|voo_*__v__?O|______________?O|______________________ o(X) oost*__o(X x I)oo________inv(g)______ gg_________________________________________________* *___________________________ _________________________________________________* *_____________________________________________________________________________* *_____________ ____{K0i}___________________________________* *_____________________________________________________________________________* *____________________________________ Figure 9: Concordance between g-connection descent objects (Y, (A, FA)) and (Y,* * (A0, FA0)) defined on the same surjective submersion ss : Y ! X. Concordance between descent obje* *cts not on the same surjective submersion is reduced to this case by pulling both back to a common * *refinement, as in figure 8. Further examples appear in 8.3.1. 7.3 Characteristic classes Definition 35For any g-connection descent object (Y, (A, FA)) we say that the d* *eRham classes [Ki] 2 HodeRham(X) in overt(Yo)Averto__CE (g) (31* *6) OOOO OOOO |i*| || | (A,FA) | o(YO)oo__________W(g)OOO |ss*| || ?O| ?O| o(X)oo__{Ki}____inv(g)_ HodR(X)oo_{[K___Ho(inv(g))_ i]} are the characteristic classes of (Y, (A, FA)). The goal is to show that characteristic classes know about equivalence class* *es of g-descent objects. Proposition 33If two g-connection descent objects (Y, (A, FA)) and (Y 0, (A0, F* *A0)) are equivalent, then they have the same characteristic classes: (Y, (A, FA)) ~ (Y 0, (A0, FA0)) ) {[Ki]} = {[K0i]} . (31* *7) 61 Proof. By definition, the two objects are equivalent if their pullbacks to a co* *mmon refinement ss1 Y xX Y 0____//Y ss2|| ss||, (31* *8) fflffl|ssfflffl|0 Y 0_____//_X as in figure 8, are concordant, as in figure 9. We have seen that pullback does* * not change the characteristic forms. It follows from proposition 28 that the characteristic classes are invar* *iant under concordance. 7.3.1 Examples Ordinary characteristic classes of g-bundles Let g be an ordinary Lie algebra * *and (Y, (A, FA)) be a g-descent object corresponding to an ordinary Cartan-Ehresmann connection as * *in 7.2.1. Using the fact 14 we know that inv(g) contains all the ordinary invariant polynomials P on g. * *Hence the characteristic classes [P(FA)] are precisely the standard characteristic classes (in deRham co* *homology) of the G-bundle with connection. 7.4 Universal and generalized g-connections We can generalize the discussion of g-bundles with connection on spaces X, by o allowing all occurrences of the algebra of differential forms to be replac* *ed with more general differential graded algebras; this amounts to admitting generalized smooth spaces as in* * 5.1; o by allowing all Chevalley-Eilenberg and Weil algebras of L1 -algebras to b* *e replaced by DGCAs which may be nontrivial in degree 0. This amounts to allowing not just structure* * 1-groups but also structure 1-groupoids. Definition 36 (generalized g-connection descent objects)Given any L1 -algebra g* *, and given any DGCA A, we say a g-connection descent object for A is * o a surjection F ooiooP_such that A ' Pbasic; o a choice of horizontal morphisms in the diagram Avert FOoo________CE(g)OOO | OOOO | | | | i*|| || | | ; (31* *9) | | | (A,FA) | POoo________W(g)O | OO | | | | | | | | | | | | ?O| ?O| A oo__{Ki}__inv(g) The notion of equivalence of these descent objects is as before. 62 CE(g)oo__Id____CE(g) OOOO| OOOO| | | |i*| || | | (32* *1) | | | Id | W(g)Ooo________W(g)OOO | | | | | | | | | | | | ?O| ?O| inv(g)oo_Id____inv(g) Figure 10: The universal g-connection descent object. Horizontal forms Given any algebra surjection FOO | | i*|| | | | P we know from definition 9 what the "vertical directions" on P are. After we hav* *e chosen a g-connection on P, we obtain also notion of horizontal elements in P: Definition 37 (horizontal elements)Given a g-connection (A, FA) on P, the algeb* *ra of horizontal ele- ments horA(P) P of P with respect to this connection are those elements not in the ideal genera* *ted by the image of A. Notice that horA(P) is in general just a graded-commutative algebra, not a diff* *erential algebra. Accordingly the inclusion horA(P) P is meant just as an inclusion of algebras. 7.4.1 Examples. The universal g-connection. The tautological example is actually of interest: * *for any L1 -algebra g, there is a canonical g-connection descent object on inv(g). This comes from cho* *osing * i* ( F oiooPo_) := ( CE(g)ooooW(g)_) (32* *0) and then taking the horizontal morphisms to be all identities, as shown in figu* *re 10: We can then finally give an intrinsic interpretation of the decomposition of* * the generators of the Weil algebra W(g) of any L1 -algebra into elemenets in g* and elements in the shifte* *d copy g*[1], which is crucial for various of our constructions (for instance for the vanishing condition in 1* *53): Proposition 34The horizontal elements of W(g) with respect to the univeral g-co* *nnection (A, FA) on W(g) are precisely those generated entirely from the shifted copy g*[1]: horA(W(g)) = ^o(g*[1]) W(g) . 63 Line n-bundles on classifying spaces Proposition 35Let g be any L1 -algebra and P 2 inv(g) any closed invariant poly* *nomial on g of degree n + 1. Let cs:= oP be the transgression element and ~ := i*csthe cocycle that P* * transgresses to according to proposition 16. Then we canonically obtain a bn-2u(1)-connection descent obj* *ect in inv(g): CE(g)oo______~______CE (bn-1u(1)) OOOO| OOOO| | | |i*| || | | (32* *2) | | | (cs,P) | W(g)Ooo_____________W(bn-1u(1))_OOO | | | | | | | | | | | | ?O| ?O| inv(g)oo_____P__________inv(bnu(1)) = CE(bn-1u(1)) Remark. For instance for g an ordinary semisimple Lie algebra and ~ its canoni* *cal 3-cocylce, we obtain a descent object for a Lie 3-bundle which plays the role of what is known as th* *e canonical 2-gerbe on the classifying space BG of the simply connected group G integrating g [15]. From t* *he above and using 6.5.1 we read off that its connection 3-form is the canonical Chern-Simons 3-form. We* * will see this again in 9.3.1, where we show that the 3-particle (the 2-brane) coupled to the above g-connecti* *on descent object indeed reproduces Chern-Simons theory. 8 Higher String- and Chern-Simons n-bundles: the lifting prob- lem We discuss the general concept of weak cokernels of morphisms of L1 -algebras. * *Then we apply this to the special problem of lifts of differential g-cocycles through String-like extensi* *ons. 8.1 Weak cokernels of L1 -morphisms After introducing the notion of a mapping cone of qDGCAs, the main point here i* *s proposition 41, which establishes the existence of the weak inverse f-1 that was mentioned in 2.3. It* * will turn out to be that very weak inverse which picks up the information about the existence or non-existenc* *e of the lifts discussed in 8.3. We can define the weak cokernel for normal L1 -subalgebras: Definition 38 (normal L1 -subalgebra)We say a Lie 1-algebra h is a normal sub L* *1 -algebra of the L1 -algebra g if there is a morphism * CE(h)oooot____CE_(g) (32* *3) which the property that t*1 o on g* it restricts to a surjective linear map h* oooog*_; o if a 2 ker(t*) then dCE(g)a 2 ^o(ker(t*1)). 64 Proposition 36For h and g ordinary Lie algebras, the above notion of normal sub* * L1 -algebra coincides with the standard notion of normal sub Lie algebras. Proof. If a 2 ker(t*) then for any x, y 2 g the condition says that (dCE(g)a)(x* *_y) = -a(D[x_y]) = -a([x, y]) vanishes when x or y are in the image of t. But a([x, y]) vanishes when [x, y] * *is in the image of t. Hence the condition says that if at least one of x and y is in the image of t, then their* * bracket is. Definition 39 (mapping cone of qDGCAs; crossed module of normal sub L1 -algebra* *s)Let t : h ,! g be an inclusion of a normal sub L1 -algebra h into g. The mapping cone of t* is* * the qDGCA whose underlying graded algebra is ^o(g* h*[1]) (32* *4) and whose differential dtis such that it acts on generators schematically as ` ' dt= dgt*0d . (32* *5) h In more detail, dt*is defined as follows. We write oet* for the degree +1 deriv* *ation on ^o(g* h*[1]) which acts on g* as t* followed by a shift in degree and which acts on h*[1] as 0. Th* *en, for any a 2 g*, we have dta := dCE(g)a + oet*(a) . (32* *6) and dtoet*(a) := -oet*(dCE(g)a) = -dtdCE(g)a . (32* *7) Notice that the last equation o defines dton all of h*[1] since t* is surjective; o is well defined in that it agrees for a and a0if t*(a) = t*(a0), since t i* *s normal. Proposition 37The differential dtdefined this way indeed satisfies (dt)2= 0. Proof. For a 2 g* we have dtdta = dt(dCE(g)a + oet*(a)) = oet*(dCE(g)a) - oet*(dCE(g)a)(=* *302.8) Hence (dt)2 vanishes on ^o(g*). Since dtdtoet*(a) = -dtdtdCE(g)a (32* *9) and since dCE(g)a 2 ^o(g*) this implies (dt)2= 0. We write CE(h t,!g) := (^o(g* h*[1]),fdt)or the resulting qDGCA and (h t,!g* *) for the corresponding L1 -algebra. The next proposition asserts that CE(h t,!g) is indeed a (weak) kernel of t*. Proposition 38There is a canonical morphism CE (g)oo__CE(h t,!g)with the proper* *ty that * t CE(h)oot__CE(g)oo__CE_(he,!g)e. (33* *0) _________________________________________________* *__________________________________________________KS _______________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *___o|| ____________________________________________* *_________________________________________________________ 0 65 Proof. On components, this morphism is the identity on g* and 0 on h*[1]. One c* *hecks that this respects the differentials. The homotopy to the 0-morphism sends o : oet*(a) 7! t*(a) . (33* *1) Using definition 18 one checks that then indeed [d, o] : a 7! o(dCE(g)a + oet*a) = a (33* *2) and [d, o] : oet*a 7! dCE(g)a + o(-oet*(dCE(g)a)) = 0 . (33* *3) Here the last step makes crucial use of the condition 153 which demands that o(dW(h,t!g)oet*a - dCE(h,t!g)oet*a) = 0 (33* *4) and the formula (149) which induces precisely the right combinatorial factors. Notice that not only is CE(h t,!g) in the kernel of t*, it is indeed the uni* *versal object with this property, hence is the kernel of t* (of course up to equivalence). * u* ` Proposition 39Let CE (h)otooCE(g)o_oo?CE(f)_be a sequence of qDGCAs with t* nor* *mal, as above, and with the property that u* restricts, on the underlying vector spaces of gen* *erators, to the kernel of the linear map underlying t*. Then there is a unique morphism f : CE(f) ! CE(h t,!g* *) such that * t CE(h)oot__CE(g)oo__CE_(hO,!g)O. (33* *5) s99 |u*|ssfs | s CE(f) Proof. The morphism f has to be in components the same as CE(g) CE(f). By the* * assumption that this is in the kernel of t*, the differentials are respected. Remark. There should be a generalization of the entire discussion where u* is * *not restricted to be the kernel of t* on generators. However, for our application here, this simple situ* *ation is all we need. Proposition 40For a string-like extension g~ from definition 24, the morphism * CE (bn-1u(1))otooo___CE(g~) (33* *6) is normal in the sense of definition 38. Proposition 41In the case that the sequence * u* ` CE (h)oooto____CE (g)oo______?CE(f)_ (33* *7) above is a String-like extension * u* ` CE(bn-1u(1))otooo___CE_(g~)oo______?CE(g)_ (33* *8) 66 from proposition 21 or the corresponding Weil-algebra version * u* ` W(bn-1u(1))oooot________W(g~)oo____________W(g)? (33* *9) |=| |=| =|| | t* | u* | CE (inn(bn-1u(1)))oooo__CE_(inn(g~))oo_____?CE(inn(g))`_ the morphisms f : CE(f) ! CE(h t,!g) and ^f: W(f) ! W(h t,!g) have weak inverse* *s f-1 : CE(h t,!g) ! CE (f) and ^f-1: W(h t,!g) ! W(f) , respectively. . Proof. We first construct a morphism f-1 and then show that it is weakly invers* *e to f. The statement for f^the follows from the functoriality of forming the Weil algebra, proposition 8* *. Start by choosing a splitting of the vector space V underlying g* as V = ker(t*) V1. (34* *0) This is the non-canonical choice we need to make. Then take the component map o* *f f-1 to be the identity on ker(t*) and 0 on V1. Moreover, for a 2 V1 set f-1 : oet*(a) 7! -(dCE(g)a)|^oker(t*), (34* *1) where the restriction is again with respect to the chosen splitting of V . We c* *heck that this assignment, extended as an algebra homomorphism, does respect the differentials. For a 2 ker(t*) we have aO__dt//_`dCE(g)a | `| f-1|| f-1| fflffl|Odfflffl|CE(f) (34* *2) a____//_dCE(g)a using the fact that t* is normal. For a 2 V1 we have a`O_______dt__//_dCE(g)a + oet*(a) | `| f-1|| f-1| fflffl|OdCE(f) fflffl| . (34* *3) 0 ____//_(dCE(g)a)|^oker(t*)- (dCE(g)a)|^oker(t*) and oet*(a)O____dt____//`-oet*(dCE(g)a)` f-1|| f-1|| fflffl| dCE(f) fflffl| . (34* *4) -(dCE(g)a)|^oker(t*)O//_-dCE(f)((dCE(g)a)|ker(t*)) This last condition happens to be satisfied for the examples stated in the prop* *osition. The details for that are discussed in 8.1.1 below. By the above, f-1 is indeed a morphism of qDGCAs. Next we check that f-1 is a weak inverse of f. Clearly CE(f)oo__CE(h_t,!g)oo__CE(f) (34* *5) is the identity on CE(f). What remains is to construct a homotopy CE (h t,!g)oo_CE(f)oo__CE_(het,!g)e_______. (34* *6) ________________________________________________* *______________________________ _______________________________________________* *_____________________________________________________________________________* *______off'|| ____________________________________________* *_____________________________________________________________________________* *_______________________________________________________________________ Id 67 ' One checks that this is accomplished by taking o to act on oeV1 as o : oeV1! V* *1 and extended suitably. 8.1.1 Examples Weak cokernel for the String-like extension. Let our sequence * u* ` CE (h)oooto____CE (g)oo______?CE(f)_ (34* *7) be a String-like extension * u* ` CE(bn-1u(1))otooo___CE_(g~)oo______?CE(g)_ (34* *8) from proposition 21. Then the mapping cone Chevalley-Eilenberg algebra CE(bn-1u(1) ,! g~) (34* *9) is ^o(g* R[n] R[n + 1]) (35* *0) with differential given by dt|g*= dCE(g) (35* *1) dt|R[n]= -~ + oe (35* *2) dt|R[n+1]= 0 . (35* *3) (As always, oe is the canonical degree shifting isomorphism on generators exten* *ded as a derivation.) The morphism -1 CE (g)oo____f'_CE_(bn-1u(1) ,! g~) (35* *4) acts as f-1|g*= Id (35* *5) f-1|R[n]= 0 (35* *6) f-1|R[n+1]= ~ . (35* *7) To check the condition in equation 344 explicitly in this case, let b 2 R[n] an* *d write b := t*b for simplicity (since t* is the identity on R[n]). Then dt oebO___//0 (35* *8) |f-1| |f-1| fflffl|fflffl|OdCE(g) ~_____//0 does commute. Weak cokernel for the String-like extension in terms of the Weil algebra. We w* *ill also need the analogous discussion not for the Chevalley-Eilenberg algebras, but for the corr* *esponding Weil algebras. To that end consider now the sequence * u* ` W(bn-1u(1))ootoo____W(g~)_oo_______W(g)?_. (35* *9) 68 This is handled most conveniently by inserting the isomorphism W(g~) ' CE(csP(g)) (36* *0) from proposition 22 as well as the identitfcation W(g) = CE(inn(g)) (36* *1) such that we get * u* ` CE (inn(bn-1u(1)))toooo_CE_(csP(g))oo______CE?(inn(g)). (36* *2) Then we find that the mapping cone algebra CE(bn-1u(1) ,! csP(g)) is ^o(g* g*[1] (R[n] R[n + 1]) (R[n + 1] R[n + 2])) .(3* *63) Write b and c for the canonical basis elements of R[n] R[n + 1], then the dif* *ferential is characterized by dt|g* g*= dW(g) (36* *4) dt : b 7! c - cs+ oeb (36* *5) dt : c 7! P + oec (36* *6) dt : oeb 7! -oec (36* *7) dt : oec 7! 0 . (36* *8) Notice above the relative sign between oeb and oec. This implies that the canon* *ical injection CE (bn-1u(1) ,! csP(g))ioo__W(bnu(1))_ (36* *9) also carries a sign: if we denote the degree n + 1 and n + 2 generators of W(bn* *u(1)) by h and dh, then i : h 7! oeb (37* *0) i : dh 7! -oec . (37* *1) This sign has no profound structural role, but we need to carefully keep track * *of it, for instance in order for our examples in 8.3.1 to come out right. The morphism f-1 CE (bn-1u(1) ,! csP(g))'oo__W(g)_ (37* *2) acts as f-1|g* g*[1]=Id (37* *3) f-1 : oeb7!cs (37* *4) f-1 : oec7!-P . (37* *5) Again, notice the signs, as they follow from the general prescription in propos* *ition 41. We again check explicitly equation (344): dt oebO___//_`-oec` |f-1| f-1||. (37* *6) fflffl|Offlffl|dW(g) cs_____//_P 69 8.2 Lifts of g-descent objects through String-like extensions We need the above general theory for the special case where we have the mapping* * cone CE(bn-1u(1) ,! g~) as the weak kernel of the left morphism in a String-like extension CE(bn-1u(1))oooo____CE_(g~)oo______?CE(g)`_ (37* *7) coming from an (n + 1) cocycle ~ on an ordinary Lie algebra g. In this case CE(* *bn-1u(1) ,! g~) looks like CE (bn-1u(1) ,! g~) = (^o(g* R[n] R[n + 1]), dt) . (37* *8) By chasing this through the above definitions, we find Proposition 42The morphism f-1 : CE(bn-1u(1) ,! g~) ! CE(g) (37* *9) acts as the identity on g* f-1|g*= Id, (38* *0) vanishes on R[n] f-1|R[n]: b 7! 0, (38* *1) and satisfies f-1|R[n+1]: oet*b 7! ~ . (38* *2) Therefore we find the (n + 1)-cocycle A^vert overt(Yo)o_____CE_(bnu(1)) (38* *3) obstructing the lift of a g-cocycle overt(Yo)Averto_CE(g), (38* *4) according to 2.3 given by j CE(bn-1u(1) ,! g~)oCE(bnu(1))?o`_,* *(385) fffffnnnn jjjjj fffffff f-1 jjjjj fffffff nnnn jjjjj i* ssfffffff `wwn ^ jjj CE(bn-1u(1))oooCE(g~)o__oo_____CE?(g)_ jjAvertj _________ --- jjjjj ______Avertjjjjj- _""___uujjjjj""-- overt(Y ) to be the (n + 1)-form ~(Avert) 2 n+1vert(Y ) . (38* *6) Proposition 43Let Avert2 1vert(Y, g) be the cocycle of a G-bundle P ! X for g * *semisimple and let ~ = <., [., .]> be the canonical 3-cocycle. Then g~ is the standard String Lie * *3-algebra and the obstruction to lifting P to a String 2-bundle, i.e. lifitng to a g~-cocycle, is the Chern-Simo* *ns 3-bundle with cocycle given by the vertical 3-form 2 3vert(Y ) . (38* *7) In the following we will express these obstruction in a more familiar way in* * terms of their characteristic classes. In order to do that, we first need to generalize the discussion to dif* *ferential g-cocycle. But that is now straightforward. 70 8.2.1 Examples The continuation of the discussion of 6.3.1 to coset spaces gives a classical i* *llustration of the lifting construc- tion considered here. Cohomology of coset spaces. The above relation between the cohomology of group* *s and that of their Chevalley-Eilenberg qDGCAs generalizes to coset spaces. This also illustrates t* *he constructions which are discussed later in 8. Consider the case of an ordinary extension of (compact co* *nnected) Lie groups: 1 ! H ! G ! G=H ! 1 (38* *8) or even the same sequence in which G=H is only a homogeneous space and not itse* *lf a group. For a closed connected subgroup t : H ,! G, there is the induced map Bt : BH ! BG and a comm* *utative diagram * W(g)____dt____//W(h)OOOO. (38* *9) | | | | | | | | | | | | ?O| * ?O| ^oPG____dt____//^oPH By analyzing the fibration sequence G=H ! EG=H ' BH ! BG, (39* *0) Halperin and Thomas [31] show there is a morphism ^o(PG QH ) ! o(G=K) (39* *1) inducing an isomorphism in cohomology. It is not hard to see that their morphis* *m factors through ^o(g* h*[1]). (39* *2) In general, the homogeneous space G=H itself is not a group, but in case of an * *extension H ! G ! K, we also have BK and the sequences K ! BH ! BG and BH ! BG ! BK. Up to homotopy equ* *ivalence, the fiber of the bundle BH ! BG is K and that of BG ! BK is BH. In particular, * *consider an extension of g by a String-like Lie 1-algebra CE(bn-1u(1))oiooo___CE(g~)oo_______?CE(g)`_. (39* *3) Regard g now as the quotient g~=bn-1u(1) and recognize that corresponding to BH* * we have bnu(1). Thus we have a quasi-isomorphism CE (bn-1u(1) ,! g~) ' CE(g) (39* *4) and hence a morphism CE (bnu(1)) ! CE(g). (39* *5) Given a g-bundle cocycle CE(g) (39* *6) t ttt yyAvertttttt overt(Y ) 71 and given an extension of g by a String-like Lie 1-algebra CE(bn-1u(1))ooioo___CE_(g~)oo______?CE(g)`_ (39* *7) we ask if it is possible to lift the cocycle through this extension, i.e. to fi* *nd a dotted arrow in CE(bn-1u(1))oooCE(g~)o___oo____CE?(g)`_. (39* *8) ______ ---- ______ -A- _""_____vert""-- overt(Y ) In general this is not possible. Indeed, consider the map A0vertgiven by CE(bnu* *(1)) ! CE(g) composed with Avert. The nontriviality of the bnu(1)-cocycle A0vertis the obstruction to cons* *tructing the desired lift. 8.3 Lifts of g-connections through String-like extensions In order to find the obstructing characteristic classes, we would like to exten* *d the above lift 398 of g-descent objects to a lift of g-connection descent objects extending them, according to * *7.2. Hence we would like first to extend Avertto (A, FA) CE (bn-1u(1))oo_OOCE(g~)oo____________CE(g) (39* *9) | OO| ttt OO| | | Avertt | | | ttt | | | yyt | | | overt(Y ) | | | OO| | | | | | | | | | | | | | W(bn-1u(1))oo__W(g~)oo_____ |________W(g) OO| OO| | tt OO| | | | tt | | | | (A,FA)tt| | | | yytt | | | o(Y ) | | | OO| | | | | | | | | | | | | | inv(bn-1u(1))oo_inv(g~)oo____|________inv(g) | tt | tt | t{Ki}t | yytt o(X) 72 and then lift the resulting g-connection descent object (A, FA) to a g~-conn* *ection object (A^, FA^) CE(bn-1u(1))oo__OOCE(g~)oo___________CE (g). * *(400) | OO|KK ttt OO| | | ^A Avertt | | | vertKK ttt | | | %% yyt | | | overt(Y ) | | | OO| | | | | | | | | | | | | | W(bn-1u(1))oo__ W(g~)oo_____ |________W(g) OO| OO|KK | tt OO| | | ^ | tt | | |(Avert,FA^|ve(A,FA)tt|rt)K | | K%%| yytt | | | o(Y ) | | | OO| | | | | | | | | | | | | | inv(bn-1u(1))oo_inv(g~)oo___ |________inv(g) K K | tt ^ | tt {Ki}K | t{Ki}t K%%| yytt o(X) The situation is essentially an obstruction problem as before, only that ins* *tead of single morphisms, we are now lifting an entire sequence of morphisms. As before, we measure the obstr* *uction to the existence of the lift by precomposing everything with the a map from a weak cokernel: CE(bnu(1))OO llll | llll | llll | vvl | CE(bn-1u(1) ! g~) | eeeee nn OO| | eieeeeeeee nnn | | eeeeeeee nn'n | | rreeeeeeeee wwnnn | | CE (bn-1u(1))oo__CE(g~)oo_____________CE (g) | W(bnu(1)) OO| OO|K tt OO| | ll OO| | | KK ttt | | llll | | | K ttt | | llll | | | K%% zztt | | vvlll | | | overt(Y ) | W(bn-1u(1) ! g~) | | | OO| | nn OO| | | | | | nnn | | | | | | nn'n | | | | | | wwnnn | | W(bn-1u(1))oo__W(g~) oo_____ |________W(g) | inv(bnu(1)) OO| OO|KK | tt OO| | lll | | ^ | tt | | llll | | (A,FA^)K| (A,FA)t|t | llll | | K%%| zztt | | vvll | | o(Y ) | inv(bn-1u(1) ! g~) | | OO| | nn | | | | nnn | | | | nn'n | | | | wwnnn inv(bnU(1))oo__inv(g~)oo____ |________inv(g) K | tt KK | tt K | t{Ki}t K%%| yytt o(X) The result is a bnu(1)-connection object. We will call (the class of) thi* *s the generalized Chern-Simons (n + 1)-bundle obstructing the lift. 73 CE(bnu(1))OO llll___|______* *_________________________________ llll ____|______* *____________________________ llll ______|______* *____________________________ vvl _______|______* *____________________________ CE(bn-1u(1) ! g_________|______* *_____________________________~) eeeee nn OO __________|______* *_____________________________ eieeeeeeee nnn | CS(A)vert___ | eeeeeeee nn'n | ______________|______* *_______________________________ eeeeeeee wwnnn _________________|______* *________________________________________________________| n-1 oo___CE(g )oo_____________rre __________________________* *______________________________ n CE (b OOu(1)) OO~K CEO(g)O _____________________________* *_________________________________W(bOu(1))O_____ | | K ttt ________________________________* *______________________________________||__________________________________llll | | K ttt | ____________________________________* *_____________________________________________||______________________________* *_____llll | | KK ttt ________________________________________* *_______________________________________________||||__________________________* *________________________vvlllll || || o%%(Yzqq___________________________________________* *_____________________________________________________________________________* *_______|||___________________________________________________________________* *)ztW(bn-1u(1) ! g ) | | vertOO| | nn OO __________|______* *__________________________________________ * * * * ~ | | | | nnn |(CS(A),P(FA))___|_ | | | | nn'n | ______________|______* *_______________________________ | | | | wwnnn _________________|______* *________________________________________________________| n-1 oo___ oo_____ ________ __________________________* *______________________________ n W(b Ou(1))O W(g~)OOK | W(g)OO _____________________________* *_________________________________inv(b_u(1))___ | | K | ttt ________________________________* *_____________________________________|__________________________________llll | | (A^,F ) | (A,Ft) | ___________________________________* *______________________________________________|______________________________* *_____lll | | KA^K| ttt A ________________________________________* *_____________________________________________________|||_____________________* *_______________________________vvllllll || || %o|(Yqq____________________________________________* *_____________________________________________________________________________* *___________________|_________________________________________________________* *____________)zzt%inv(bn-1u(1) ! g~) | | OO| | nn _________________* *_________________________________________ | | | | nnn {P(FA)}___ | | | | nn'n _____________________* *_______________________________ | | | | wwnnn ________________________* *_______________________________________________________ n oo___inv(g )oo____ |________ ___________________________* *______________________________ inv(b U(1)) ~K inv(g) _____________________________* *__________________________________ K | ttt ________________________________* *_______________________________________ K | {Ki}t ____________________________________* *_______________________________________________ K K%%||yytttt ________________________________________* *_________________________________________________ o(X) qq____________________________________________* *_____________________________________________________________________________* *___________ Figure 11: The generalized Chern-Simons bnu(1)-bundle that obstructs the lif* *t of a given g-bundle to a g~-bundle, or rather the descent object representing it. In order to construct the lift it is convenient, for similar reasons as i* *n the proof of proposition 24, to work with CE(csP(g)) instead of the isomorphic W(g~), using the isomorphism from * *proposition 22. Furthermore, using the identity W(g) = CE(inn(g)) * *(401) mentioned in 6.1, we can hence consider instead of W(bn-1)oooo______W(g~)oo________W(g)? ` * *(402) the sequence CE(inn(bn-1))oooo____CE(csP(g))oo______?CE(inn(g))`_. * *(403) Fortunately, this still satisfies the assumptions of proposition 39. So in * *complete analogy, we find the extension of proposition 42 from g-bundle cocyces to differential g-cocycles: Proposition 44The morphism f-1 : CE(inn(bn-1u(1)) ,! CE(csP(g)) ! CE(inn(g)) * *(404) constructed as in proposition 42 acts as the identity on g* g*[1] f-1|g* g*[1]= Id * *(405) and satisfies f-1|R[n+2]: c 7! P . * *(406) 74 This means that, as an extension of proposition 43, we find the differential* * bnu(1) (n + 1)-cocycle ^A o(Y )oo_______W(bnu(1)) (40* *7) obstructing the lift of a differential g-cocycle (A,FA) o(Y )oo_______W(g), (40* *8) according to the above discussion j CE (inn(bn-1u(1)) ,! inn(g~))W(bnu(1))?o* *`o_,(409) ffffffllllll iiiiiii ffffffff-1 iiiii fffffff lllll iiiii n-1 i*oooo_ ssfffffff?o`ouull_ (A^,Fi^ii W(b u(1)) W(g~)_ W(g) iii A) ______ iiii _____(A,FA) iiiii ______ iiiii ___""ttiiiii o(Y ) to be the connection (n + 1)-form ^A= CS(A) 2 n+1(Y ) (41* *0) with the corresponding curvature (n + 2)-form FA^= P(FA) 2 n+2(Y ) . (41* *1) Then we finally find, in particular, Proposition 45For ~ a cocycle on the ordinary Lie algebra g in transgression wi* *th the invariant polynomial P, the obstruciton to lifting a g-bundle cocycle through the String-like extens* *ion determined by ~ is the characteristic class given by P. Remark. Notice that, so far, all our statements about characteristic classes a* *re in deRham cohomology. Possibly our construction actually obtains for integral cohomology classes, but* * if so, we have not extracted that yet. A more detailed consideration of this will be the subject of [60]. 8.3.1 Examples Chern-Simons 3-bundles obstructing lifts of G-bundles to String(G)-bundles. Co* *nsider, on a base space X for some semisimple Lie group G, with Lie algebra g a principal G-bundl* *e ss : P ! X. Identify our surjective submersion with the total space of this bundle Y := P . (41* *2) Let P be equipped with a connection, (P, r), realized in terms of an Ehresmann * *connection 1-form A 2 1(Y, g) (41* *3) with curvature FA 2 2(Y, g) (41* *4) i.e. a dg-algebra morphism (A,FA) o(Y )oo_______W(g) (41* *5) 75 satisfying the two Ehresmann conditions. By the discussion in 7.2.1 this yields* * a g-connection descent object (Y, (A, FA)) in our sense. We would like to compute the obstruction to lifting this G-bundle to a Strin* *g 2-bundle, i.e. to lift the g-connection descent object to a g~-connection descent object, for 0 ! bu(1) ! g~ ! g ! 0 (41* *6) the ordinary String extension from definition 25. By the above discussion in 8.* *3, the obstruction is the (class of the) b2u(1)-connection descent object (Y, (H(3), G(4))) whose connection and* * curvature are given by the composite W(b2u(1))_____ kk ______________________* *__________ kkkk________________________* *_________ kkkk _________________________* *__________ uukkkk __________________________* *__________,(417) ____________________________* *__________ (W(bu(1)) ! CE(csP(_____________________________* *__________g))) lll _______________________________* *__________ lll _________________________________* *___________ lll' __________________________________* *____________ vvllll _____________________________________* *_____________ W(g) ______________________________________* *___ v (H(3),G(4))________________________________* *________ vv ______________________________________________* *____________________________ (A,FA)v ___________________________________________________* *________________________________________ --vv ______________________________________________________* *__________________________ o(Y _pp___________________________________________________________* *_____________________________________________________________________________* *___) where, as discussed above, we are making use of the isomorphism W(g~) ' CE(csP(* *g)) from proposition 22. The crucial aspect of this composite is the isomorphism -1 W(g)oo___f'___(W(bu(1)) ! CEP(g)) (41* *8) from proposition 41. This is where the obstruction data is picked up. The impor* *tant formula governing this is equation 341, which describes how the shifted elements coming from W(bu* *(1)) in the mapping cone (W(bu(1)) ! CEP(g)) are mapped to W(g). Recall that W(b2u(1)) = F(R[3]) is generated from elements (h, dh) of degree* * 3 and 4, respectively, that W(bu(1)) = F(R[2]) is generated from elements (c, dc) of degree 2 and 3, respec* *tively, and that CE(csP(g)) is generated from g* g*[1] together with elements b and c of degree 2 and 3, r* *espectively, with dCE(csP(g))b = c - cs (41* *9) and dCE(csP(g))c = P , (42* *0) where cs2 ^3(g* g*[1]) is the transgression element interpolating between the c* *ocycle ~ = <., [., .]> 2 ^3(g*) and the invariant polynomial P = <., .> 2 ^2(g*[1]). Hence the map f-1 acts as f-1 : oeb 7! -(dCE(csP(g))b)|^o(g* g*[1])= +cs (42* *1) and f-1 : oec 7! -(dCE(csP(g))c)|^o(g* g*[1])= -P . (42* *2) 76 Therefore the above composite (H(3), G(4)) maps the generators (h, dh) of W(b2u* *(1)) as l_hB________________,l lll_____________________* *____________l llll______________________* *_____________ llll _______________________* *_____________ uull _________________________* *____________(423) __________________________* *_____________ + oeb ___________________________* *______________ kkk _____________________________* *______________ k'kk _______________________________* *_______________ kkkk _________________________________* *________________ uukkkk ___________________________________* *___________________ 5cs ___________________ uuu (H(3),G(4))______________________________* *___________ u ____________________________________________* *____________________________________ zz(A,FA)uuu________________________________________________* *__________________________________________________ CSP(FA)pp_______________________________________________________* *_____________________________________________________________________________* *______________________________________________ and lCdh_________________,l lll______________________* *___________ llll _______________________* *___________ llll ________________________* *___________ uull __________________________* *___________.(424) -oec ___________________________* *___________ , _____________________________* *___________ llll ______________________________* *_____________ ' ll ________________________________* *_____________ lllll __________________________________* *______________ uullll ____________________________________* *________________ 9P ___________________________ yy (H(3),G(4))_______________________________* *_________ yy _____________________________________________* *________________________________ (A,FA)y _________________________________________________* *________________________________________________ __yy ____________________________________________________* *_____________________ P(FA)pp_________________________________________________________* *_________________________________________________________________________ Notice the signs here, as discussed around equation 369. We then have that the * *connection 3-form of the Chern-Simons 3-bundle given by our obstructing b2u(1)-connection descent object* * is the Chern-Simons form H(3)= -CS(A, FA) = - - 1_3 2 3(Y ) (42* *5) of the original Ehresmann connection 1-form A, and its 4-form curvature is ther* *efore the corresponding 4-form G(4)= -P(FA) = 2 4(Y ) . (42* *6) This descends down to X, where it constitutes the characteristic form which cla* *ssifies the obstruction. Indeed, noticing that inv(b2u(1)) = ^o(R[4]), we see that (this works the same * *for all line n-bundles, i.e., for all bn-1u(1)-connection descent objects) the characteristic forms of the obstru* *cting Chern-Simons 3-bundle inv(b2u(1))_____ nn_________________________* *________ nnn__________________________* *________ nnnn ___________________________* *________ vvnnn ____________________________* *________(427) _____________________________* *_________ inv(bu(1) ! g~______________________________* *__________) ppp ________________________________* *__________ pp _________________________________* *___________ pp'p ___________________________________* *____________ wwpp _____________________________________* *_____________ inv(g) ______________________________________ u _{G(4)}___________________________________* *__ uu _____________________________________________* *____________________________ {Ki} ________________________________________________* *_________________________________________ zzuuu ____________________________________________________* *_____________________________ o(X)pp________________________________________________________* *_____________________________________________________________________________ consist only and precisely of this curvature 4-form: the second Chern-form of t* *he original G-bundle P. 77 9 L1 -algebra parallel transport One of the main points about a connection is that it allows to do parallel tran* *sport. Connections on ordinary bundles give rise to a notion of parallel transport along curves, known as holo* *nomy if these curves are closed. Higher connections on n-bundles should yield a way to obtain a notion of par* *allel transport over n- dimensional spaces. In physics, this assignment plays the role of the gauge cou* *pling term in the non-kinetic part of the action functional: the action functional of the charged particle is* * essentially its parallel transport with respect to an ordinary (1-)connection, while the action functional of the * *string contains the parallel transport of a 2-connection (the Kalb-Ramond field). Similarly the action func* *tional of the membrane contains the parallel transport of a 3-connection (the supergravity "C-field"). There should therefore be a way to assign to any one of our g-connection des* *cent objects for g any Lie n-algebra o a prescription for parallel transport over n-dimensional spaces; o a configuration space for the n-particle coupled to that transport; o a way to transgress the transport to an action functional on that configur* *ation space; o a way to obtain the corresponding quantum theory. Each point separately deserves a separate discussion, but in the remainder w* *e shall quickly give an impression for how each of these points is addressed in our context. 9.1 L1 -parallel transport In this section we indicate briefly how our notion of g-connections give rise t* *o a notion of parallel transport over n-dimensional spaces. The abelian case (meaning here that g is an L1 algeb* *ra such that CE(g) has trivial differential) is comparatively easy to discuss. It is in fact the only * *case considered in most of the literature. Nonabelian parallel n-transport in the integrated picture for n up * *to 2 is discussed in [5, 52, 53, 54]. There is a close relation between all differential concepts we develop here and* * the corresponding integrated concepts, but here we will not attempt to give a comprehensive discussion of th* *e translation. Given an (n - 1)-brane ("n-particle") whose n-dimensional worldvolume is mod* *eled on the smooth pa- rameter space (for instance = S1 for the closed string) and which propagate* *s on a target space X in that its configurations are given by maps OE : ! X (42* *8) hence by dg-algebra morphisms * o( )oo__OE____ o(X) (42* *9) we can couple it to a g-descent connection object (Y, (A, FA)) over X pulled ba* *ck to if Y is such that for every map OE : ! X (43* *0) the pulled back surjective submersion has a global section OE*Y== O^E-- --- |ss|. (43* *1) --Id- fflffl| _____//_ Definition 40 (parallel transport)Given a g-descent object (Y, (A, FA)) on a ta* *rget space X and a pa- rameter space such that for all maps OE : ! X the pullback OE*Y has a globa* *l section, we obtain a map tra(A): HomDGCA( o(X), o( )) ! HomDGCA(W(g), o( )) (43* *2) 78 by precomposition with (A,FA) o(Y )oo_______W(g)_. (43* *3) This is essentially the parallel transport of the g-connection object (Y, (A, F* *A)). A full discussion is beyond the scope of this article, but for the special case that our L1 -algebra is (n-* *1)-fold shifted u(1), g = bn-1u(1), the elements in Hom dgca(W(g), o( )) = o( , bn-1u(1)) ' n( ) (43* *4) are in bijection with n-forms on . Therefore they can be integrated over . Th* *en the functional Z traA: Homdgca( o(Y ), o( )) ! R (43* *5) is the full parallel transport of A. Proposition 46The map tra(A)is indeed well defined, in that it depends at most * *on the homotopy class of the choice of global section ^OEof OE. Proof. Let ^OE1and ^OE2be two global sections of OE*Y . Let ^OE: x I ! OE*Y b* *e a homotopy between them, i.e. such that ^OE|0 = ^OE1and ^OE|1 = ^OE2. Then the difference in the paralle* *l transport using ^OE1and ^OE2is the integral of the pullback of the curvature form of the g-descent object over x* * I. But that vanishes, due to the commutativity of (A,FA) o(OE*Yo)o_____________W(g)OOOO (43* *6) """| | """ | || ^OE*"" || | """ || || """""OE* ?O| K ?O| o( x I)oo__?_o(`)oo_iioo___________inv(bn-1u(1))_=_bnu(1)_______* *________________________________________________________ ___________________________________________________________* *_____________________________________________________________________________* *______________________________________ _________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *________________________________________________ ____________________________________________________* *_____________________________________________________________________________* *___________________________________ 0 The composite of the morphisms on the top boundary of this diagram send the sin* *gle degree (n+1)-generator of inv(bn-1u(1)) = CE(bnu(1)) to the curvature form of the g-connection descent* * object pulled back to . It is equal to the composite of the horizontal morphisms along the bottom bound* *ary by the definition of g-descent objects. These vanish, as there is no nontrivial (n + 1)-form on the * *n-dimensional . 9.1.1 Examples. Chern-Simons and higher Chern-Simons action functionals Proposition 47For G simply connected, the parallel transport coming from the Ch* *ern-Simons 3-bundle discussed in 8.3.1 for g = Lie(G) reproduces the familiar Chern-Simons action f* *unctional Z ` ' + 1_ (43* *7) 3 over 3-dimensional . Proof. Recall from 8.3.1 that we can build the connection descent object for th* *e Chern-Simons connection on the surjective submersion Y coming from the total space P of the underlying * *G-bundle P ! X. Then OE*Y = OE*P is simply the pullback of that G-bundle to . For G simply connecte* *d, BG is 3-connected and 79 hence any G-bundle on is trivializable. Therefore the required lift ^OEexists* * and we can construct the above diagram. By equation 425 one sees that the integral which gives the parallel tr* *ansport is indeed precisely the Chern-Simons action functional. Higher Chern-Simons n-bundles, coming from obstructions to fivebrane lifts o* *r still higher lifts, similarly induce higher dimensional generalizations of the Chern-Simons action functional. BF-theoretic functionals From proposition 20 it follows that we can similarly * *obtain the action func- tional of BF theory, discussed in 6.6, as the parallel transport of the 4-conne* *ction descent object which arises as the obstruction to lifting a 2-connection descent object for a strict Lie 2-* *algebra (h !tg) through the string-like extension b2u(1) ! (h t!g)dCE(h!tg)~! (h t!g) (43* *8) for ~ the 3-cocycle on ~ which transgresses to the invariant polynomial P on g * *which appears in the BF-action functional. 9.2 Transgression of L1 -transport An important operation on parallel transport is its transgression to mapping sp* *aces. This is familiar from simple examples, where for instance n-forms on some space transgress to (n - 1)* *-forms on the corresponding loop space. We should think of the n-form here as a bn-1u(1)-connection which t* *ransgresses to an bn-2u(1) connection on loop space. This modification of the structure L1 -algebra under transgression is crucia* *l. In [53] it is shown that for parallel transport n-functors (n = 2 there), the operation of transgression is * *a very natural one, corresponding to acting on the transport functor with an inner hom operation. As shown there,* * this operation automatically induces the familiar pull-back followed by a fiber integration on the correspon* *ding differential form data, and also automatically takes care of the modification of the structure Lie n-gr* *oup. The analogous construction in the differential world of L1 algebras we state* * now, without here going into details about its close relation to [53]. Definition 41 (transgression of g-connections)For any g-connection descent obje* *ct Avert FOoo_________CEO(g)OO (43* *9) | OOOO | | | | i*|| || | | | | | (A,FA) | POoo_________W(g)O|OO | | | | ss*|| || | | | | ?O| ?O| Pbasicoo{Ki}__inv(g) and any smooth space par, we can form the image of the above diagram under the * *functor o(maps(-, o(par))) : DGCAs ! DGCAs (44* *0) 80 from definition 5 to obtain the generalized g-connection descent object (accord* *ing to definition 36) tgpar(Avert) o(maps(F, o(par))oo______ o(maps(CE (g), o(par)) OOOO| OOOO| | | tgpari*|| || | | . (44* *1) | | | tgpar(A,FA) | o(maps(P,OOo(par))oo______ o(maps(W(g),OOo(par)) | | | | tgpar(ss*)|| || | | | | ?O| ?O| o(maps(Pbasic, o(par))tgoo_ o(maps(inv(g), o(par)) par({Ki}) This new maps(CE (g), o(par))-connection descent object we call the transgress* *ion of the original one to par. The operation of transgression is closely related to that of integration. 9.2.1 Examples Transgression of bn-1u(1)-connections. Let g be an L1 -algebra of the form shi* *fted u(1), g = bn-1u(1). By proposition 18 the Weil algebra W(bn-1u(1)) is the free DGCA on a single deg* *ree n-generator b with differential c := db. Recall from 6.5.1 that a DGCA morphism W(bn-1u(1)) ! o(Y* * ) is just an n-form on Y . For every point y 2 parand for every multivector v 2 ^nTyparwe get a 0-form* * on the smooth space maps(W(bn-1u(1)), o(par)) (44* *2) of all n-forms on par, which we denote A(v) 2 (maps(W(bn-1u(1)), o(par))) . (44* *3) This is the 0-form on this space of maps obtained from the element b 2 W(bn-1u(* *1)) and the current ffiy (the ordinary delta-distribution on 0-forms) according to proposition 2. Its va* *lue on any any n-form ! is the value of that form evaluated on v. Since this, and its generalizations which we discuss in 9.3.1, is crucial fo* *r making contact with standard constructions in physics, it may be worthwhile to repeat that statement more ex* *plicitly in terms of compo- nents: Assume that par= Rk and for any point y let v be the unit in ^TnRn ' R. * *Then A(v) is the 0-form on the space of forms which sends any form ! = !~1~2...~ndx~1^ . .^.dx~n to its* * component A(v) : ! 7! !(y)12,...n. (44* *4) This implies that when a bn-1u(1)-connection is transgressed to the space of ma* *ps from an n-dimensional parameter space par, it becomes a map that pulls back functions on the space of* * n-forms on parto the space of functions on maps from parameter space into target space. But such pullbacks* * correspond to functions (0-forms) on the space of maps par! tarwith values in the space of n-forms on t* *ra. 9.3 Configuration spaces of L1 -transport With the notion of g-connections and their parallel transport and transgression* * in hand, we can say what it means to couple an n-particle/(n - 1)-brane to a g-connection. 81 Definition 42 (the charged n-particle/(n - 1)-brane)We say a charged n-particle* */(n - 1)-brane is a tuple (par, (A, FA)) consisting of o parameter space par: a smooth space o a background field (A, FA): a g-connection descent object involving - target space tar: the smooth space that the g-connection (A, FA) lives * *over; - space of phases phas: the smooth space such that o(phas) ' CE(g) From such a tuple we form o configuration space o(conf) := o(maps( o(tar)), o(par)); o the action functional exp(S) := tgpar: the transgression of the background* * field to configuration space. The configuration space thus defined automatically comes equipped with a notion* * of vertical derivations as described in 5.3. ______0______________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________ uu_______________________________________________* *_____________________________________________________________________________* *______________________________________|| o(maps(F, jo(par)))jae0|| o(maps(F,__o(par)))_________. (44* *5) OOO_________________________________________________* *_____________________________________________________________________________* *_____________________OOOOOff'|| | ______________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_______________________| || [d,ae0] || | | | 0 | | _____________________|_________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *____________________ | uu_____________________|_________________________* *______________________________________|| o(maps(P, jo(par)))jae|| o(maps(P,__o(par)))_________ _________________________________________________* *_____________________________________________________________________________* *_____________________ff'|| ______________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_______________________ [d,ae] These form o the gauge symmetries ggauge: an L1 -algebra. These act on the horizontal elements of configuration space, which form o the anti-fields and anti-ghosts in the language of BRST-BV-quantization [43]. We will not go into further details of this here, except for spelling out, a* *s the archetypical example, some details of the computation of the configuration space of ordinary gauge theory. 9.3.1 Examples Configuration space of ordinary gauge theory. We compute here the the configur* *ation space of ordi- nary gauge theory on a manifold parwith respect to an ordinary Lie algebra g. A* * configuration of such a the- ory is a g-valued differential form on par, hence, according to 6.5, an element* * in HomDGCAs(W(g), o(par)). So we are interested in understanding the smooth space maps(W(g), o(par)) =: o(par, g) (44* *6) according to definition 4, and the differential graded-commutative algebra o(maps(W(g), o(par))) =: o( o(par, g)) (44* *7) of differential forms on it. 82 To make contact with the physics literature, we describe everything in compo* *nents. So let par= Rn and let {x~} be the canonical set of coordinate functions on par. Choose a basis {t* *a} of g and let {ta} be the corresponding dual basis of g*. Denote by ffiy'_@_@x~ (44* *8) the delta-current on o(par), according to definition 6, which sends a 1-form !* * to !~(y) := !(_@_@x~)(y) . (44* *9) Summary of the structure of forms on configuration space of ordinary gauge t* *heory. Recall that the Weil algebra W(g) is generated from the {ta} in degree 1 and the oetai* *n degree 2, with the differential defined by dta = -1_2Cabctb^ tc+ oeta (45* *0) d(oeta)= -Cabctb^ (oetc) . (45* *1) We will find that maps(W(g), o(par)) does look pretty much entirely like th* *is, only that all generators are now forms on par. See table 6. fields |Aa~(y), (FA)~ (y) 2 0( (par, g)) | y 2 par, ~, 2 {1, . .,.dim(* *par), a 2 {1, . .,.dim(g)}} | ghosts | ca(y) 2 1( (par, g)) | y 2 par, a 2 {1, . .,.dim(g)}} |n * * o antifields ||'(ffiAa~(y))2 Hom( 1( (par, g)), R) | y 2 par, ~ 2 {1, . .,.dim* *(par), a 2 {1, . .,.dim(g)}} | anti-ghosts |'(fia(y))2 Hom( 2( (par, g)), R) | y 2 par, dim(par), a 2 {1, . .* *,.dim(g)}} Table 6: The BRST-BV field content of gauge theory obtained from our almost int* *ernal hom of dg- algebras, definition 5. The dgc-algebra maps(W(g), o(par)) is the algebra of d* *ifferential forms on a smooth space of maps from parto the smooth space underlying W(g). In the above table f* *i is a certain 2-form that one finds in this algebra of forms on the space of g-valued forms. Remark. Before looking at the details of the computation, recall from from * *5.1 that an n-form ! in n(maps(W(g), o(par))) is an assignment !U o U Hom DGCAs(W(g), o(parx U))______//_ (U) (45* *2) | OO OO | | | | | | OE|| OE*|| |OE*| | | | | | | fflffl| | !V | V Hom DGCAs(W(g), o(parx V_))_____//_ o(V ) of forms on U to g-valued forms on parx U for all plot domains U (subsets of R * *[ R2[ . .f.or us), natural in U. We concentrate on those n-forms ! which arise in the way of proposition 2. 0-Forms. The 0-forms on the space of g-value forms are constructed as in pr* *oposition 2 from an element ta 2 g* and a current ffiy'_@_@x~using taffiy'_@_@x~ (45* *3) 83 and from an element oeta 2 g*[1] and a current ffiy'_@_@x~'_@_@x. (45* *4) This way we obtain the families of functions (0-forms) on the space of g-valued* * forms: Aa~(y) : ( o(parx U) W(g) : A) 7! (u 7! '_@_@x~A(ta)(y, u))* *(455) and Fa~(y) : ( o(parx U) W(g) : FA) 7! (u 7! '_@_@x~'_@_@xFA(oeta)* *(y,(u))456) which pick out the corresponding components of the g-valued 1-form and of its c* *urvature 2-form, respectively. These are the fields of ordinary gauge theory. 1-Forms. A 1-form on the space of g-valued forms is obtained from either st* *arting with a degree 1 element and contracting with a degree 0 delta-current taffiy (45* *7) or starting with a degree 2 element and contracting with a degree 1 delta curre* *nt: (oeta)ffiy_@_@x~. (45* *8) To get started, consider first the case where U = I is the interval. Then a DGC* *A morphism (A, FA) : W(g) ! o(par) o(I) (45* *9) can be split into its components proportional to dt 2 o(I) and those not conta* *ining dt. We can hence write the general g-valued 1-form on parx I as (A, FA) : ta 7! Aa(y, t) + ga(y, t) ^ dt (46* *0) and the corresponding curvature 2-form as (A, FA) : oeta 7! (dpar+ dt)(Aa(y, t) + ga(y, t) ^ dt) + 1_2Cabc(Aa(y, t) + ga(* *y, t) ^ dt) ^ (Ab(y, t) + gb(y, t) ^ dt) = FaA(y, t) + (@tAa(y, t) + dparga(y, t) + [g, A]a) ^ dt(.* *461) By contracting this again with the current ffiy_@_@x~we obtain the 1-forms t 7! ga(y, t)dt (46* *2) and t 7! (@tAa~(y, t) + @~ga(y, t) + [g, A~]a)dt (46* *3) on the interval. We will identify the first one with the component of the 1-for* *ms on the space of g-valued forms on parcalled the ghosts and the second one with the 1-forms which are kil* *led by the objects called the anti-fields. To see more of this structure, consider now U = I2, the unit square. Then a * *DGCA morphism (A, FA) : W(g) ! o(par) o(I2) (46* *4) can be split into its components proportional to dt1, dt22 o(I2). We hence can* * write the general g-valued 1-form on Y x I as (A, FA) : ta 7! Aa(y, t) + gai(y, t) ^ dti, (46* *5) 84 and the corresponding curvature 2-form as (A, FA) : oeta 7! (dY + dI2)(Aa(y, t) + gai(y, t) ^ dti) +1_2Cabc(Aa(y, t) + gai(y, t) ^ dti) ^ (Ab(y, t) + gbi(y, t) * *^ dti) = FaA(y, t) + (@tiAa(y, t) + dYgai(y, t) + [gi, A]a) ^ dti +(@igaj+ [gi, gj]a)dti^ dtj. (46* *6) By contracting this again with the current ffiy_@_@x~we obtain the 1-forms t 7! gai(y, t)dti (46* *7) and t 7! (@tAa~(y, t) + @~gai(y, t) + [gi, A~]a)dti (46* *8) on the unit square. These are again the local values of our ca(y) 2 1( o(par, g)) (46* *9) and ffiAa~(Y ) 2 1( o(par, g)) . (47* *0) The second 1-form vanishes in directions in which the variation of the g-valued* * 1-form A is a pure gauge transformation induced by the function ga which is measured by the first 1-form* *. Notice that it is the sum of the exterior derivative of the 0-form Aa~(y) with another term. ffiAa~(y) = d(Aa~(y)) + ffigAa~(y) . (47* *1) The first term on the right measures the change of the connection, the second s* *ubtracts the contribution to this change due to gauge transformations. So the 1-form ffiAa~(y) on the space * *of g-valued forms vanishes along all directions along which the form A is modfied purely by a gauge transf* *ormation. The ffiAa~(y) are the 1-forms the pairings dual to which will be the antifields. 2-Forms. We have already seen the 2-form appear on the standard square. We * *call this 2-form fia 2 2( o(par, g)) , (47* *2) corresponding on the unit square to the assignment fia : ( o(parx I2) W(g) : A) 7! (@igaj+ [gi, gj]a)dti^ dtj.* *(473) There is also a 2-form coming from (oeta)ffiy. Then one immediately sees that o* *ur forms on the space of g-valued forms satisfy the relations dca(y)= -1_2Cabccb(y) ^ cc(y) + fia(y) (47* *4) dfia(y)= -Cabccb(y) ^ fic(y) . (47* *5) The 2-form fi on the space of g-valued forms is what is being contracted by the* * horizontal pairings called the antighosts. We see, in total, that o( o(par, g)) is the Weil algebra of a * *DGCA, which is obtained from the above formulas by setting fi = 0 and ffiA = 0. This DGCA is the algebra of * *the gauge groupoid, that where the only morphisms present are gauge transformations. The computation we have just performed are over U = I2. However, it should b* *e clear how this extends to the general case. 85 Chern-Simons theory. One can distinguish two ways to set up Chern-Simons theor* *y. In one approach one regards principal G-bundles on abstract 3-manifolds, in the other approach * *one fixes a given principal G-bundle P ! X on some base space X, and pulls it back to 3-manifolds equipped * *with a map into X. Physically, the former case is thought of as Chern-Simons theory proper, while * *the latter case arises as the gauge coupling part of the membrane propagating on X. One tends to want to rega* *rd the first case as a special case of the second, obtained by letting X = BG be the classifying space* * for G-bundles and P the universal G-bundle on that. In our context this is realized by proposition 35, which gives the canonical* * Chern-Simons 3-bundle on BG in terms of a b2u(1)-connection descent object on W(g). Picking some 3-dimen* *sional parameter space manifold par, we can transgress this b2u(1)-connection to the configuration spa* *ce maps(W(g), o(par)), which we learned is the configuration space of ordinary gauge theory. tgpar~ CE (g)oo___~___CE (bn-1u(1)) o(maps(CE (g), o(par))oo(maps(CEo(bn-1u(* *1)),_ o(par))) OOOO| OOOO| OOOO| OOOO| | | | | i*|| || |i*| || | | | | | | | | | (cs,P) | | tgpar(cs,P) | W(g)oo________W(bn-1u(1))OOOO7! o(maps(W(g),OOo(par)))oo(maps(W(bn-1u(1))* *,o_o(par)))OO. | | | | | | | | | | | | | | | | | | | | | | | | ?O| ?O| ?O| ?O| inv(g)oo___P____inv(bnu(1)) o(maps(inv(g), o(par)))to(maps(CEg(bn-1u* *(1)),ooo(par)))_ parP Proposition 47 says that the transgressed connection is the Chern-Simons action* * functional. Further details of this should be discussed elsewhere. A Appendix: Explicit formulas for 2-morphisms of L1 -algebras To the best of our knowledge, the only place in the literature where 2-morphism* *s between 1-morphisms of L1 -algebras have been spelled out in detail is [3], which gives a definition o* *f 2-morphisms for Lie 2-algebras, i.e. for L1 -algebras concentrated in the lowest two degrees. Our definition * *18 provides an algorithm for computing 2-morphisms between morphisms of arbitrary (finite dimensional) L* *1 -algebras. We had already demonstrated in 6.2 one application of that algorithm, showing explicit* *ly how it allows to compute transgression elements (Chern-Simons forms). For completeness, we demonstrate that the formulas given in [3] for the spec* *ial case of Lie 2-algebras also follow as a special case from our general definition 18. This is of releva* *nce to our discussion of the String Lie 2-algebra, since the equivalence of its strict version with its weak* * skeletal version, mentioned in our proposition 26, has been established in [4] using these very formulas. Firs* *t we quickly recall the relevant definitions from [3, 4]: A "2-term" L1 -algebra is an L1 -algebra concentrated * *in the lowest two degrees. A morphism ' : g ! h (47* *6) of 2-term L1 -algebras g and h is a pair of maps OE0 : g1! h1 (47* *7) OE1 : g2! h2 (47* *8) together with a skew-symmetric map OE2: g1 g1! h2 (47* *9) 86 satisfying OE0(d(h)) = d(OE1(h)) (48* *0) as well as d(OE2(x, y))=OE0(l2(x, y)) - l2(OE0(x), OE0(y)) (48* *1) OE2(x, dh)= OE1(l2(x, h)) - l2(OE0(x), OE1(h)) (48* *2) and finally l3(OE0(x), OE0(y), OE0(z)) - OE1(l3(x,=y,Oz))E2(x, l2(y, z)) + OE2(y, l2(z, x* *)) + OE2(z, l2(x, y)) + l2(OE0(x), OE2(y, z)) + l2(OE0(y), OE2(z, x)* *) + l2(OE0(z), OE2(x, y)) . for all x, y, z 2 g1 and h 2 g2. This follows directly from the requirement tha* *t morphisms of L1 -algebras be homomorphisms of the corresponding codifferential coalgebras, according to defi* *nition 14. The not quite so obvious aspect are the analogous formulas for 2-morphisms: Definition 43 (Baez-Crans)A 2-morphism OE _________________________________________* *________________________________________________________________ __________________________________________* *___________________________ __________oo________________________________* *______|| g__________BBho|| (48* *3) __________________________________________|| __________________________________________* *_____________________________ff'|| ________________________________________* *___________________________________________________________ _ of 1-morphisms of 2-term L1 -algebras is a linear map o : g1! h2 (48* *4) such that _0- OE0= tW O o (48* *5) _1- OE1= o O tv (48* *6) and OE2(x, y) - _2(x, y) = l2(OE0(x), o(y)) + l2(o(x), _0(y)) - o(l* *2(x,(y))487) Notice that [d, o] := dhO o + o O dg and that it restricts to dhO o on g1 an* *d to o O dg on g2. Proposition 48For finite dimensional L1 -algebras, definition 43 is equivalent * *to the restriction of our definition 18 to 2-term L1 -algebras. Proof. Let g = g1 g2 and h = h1 h2 be any two 2-term L1 -algebras. Then take _, OE : g ! h (48* *8) to be any two L1 morphisms with *,OE* CE(g)oo_______CE_(h) (48* *9) the corresponding DGCA morphisms. We would like to describe the collection of a* *ll 2-morphisms _OE*_____________________________________* *_______________________________________ ____________________________________________* *_____________________________________________________________________________* *____ ""___________________________________________* *____________________________________|| CE(g)aa |o| CE_(h)_______ (49* *0) _____________________________________________* *__________________________________________|| ____________________________________________* *____________________________________________________________ff'|| __________________________________________* *________________________________________ _* 87 according to definition 18. We do this in terms of a basis. With {ta} a basis f* *or h1 and {bi} a basis for h2, and accordingly {t0a} and {b0i} a basis of g1 and g2, respectively, this comes * *from a map o* : h*1 h*2 h*1[1] h*2[1] ! ^og* (49* *1) of degree -1 which acts on these basis elements as o* : bi7! oiat0a (49* *2) and o* : aa 7! 0 (49* *3) for some coefficients {oia}. Now the crucial requirement 153 of definition 18 * *is that 152 vanishes when restricted OE*___CE(h)__________________________________________* *_______________________(494) ________________________________________ccccHH ________________________________HHH __________________HHH _________________Hxxxxxx CE(g)WW xxxxxxxxW(h) oo__h*[1]?_` _______________vv_____________________xxxxxx ____________vvv____________________________xxxxx *________----vvvv___________________________________* *_______________________________ _ __CE(h)_______________________ to generators in the shifted copy of the Weil algebra. This implies the followi* *ng. For o* to vanish on all oeta we find that its value on dW(h)ta = -1_2Cabcta^ tb- taibi+ oeta is fixed to be o* : dW(g)ta 7! -taioibt0b (49* *5) and on dW(h)bi= -ffiajta^ bj+ cito be o*(dbi) = o*(-ffiajta^ bj) . (49* *6) The last expression needs to be carefully evaluated using formula 149. Doing so* * we get [d, o*] : ta 7! -taioibt0b (49* *7) and [d, o*] : bi7! -1_2oiaC0abct0bt0c- oiat0ajb0j+ ffiaj1_2(OE + _)a* *bojct0bt0c.(498) Then the expression OE*- _* = [d, o*] (49* *9) is equivalent to the following ones (_ab- OEab)t0b=taioibt0b (50* *0) (_ij- OEij)b0j=oiat0ajb0j (50* *1) 1_(OEi - _i )t0at0b=-1_oi C0a t0bt0c+ ffi 1_(OE + _)a oj t0bt0c* *.(502) 2 ab ab 2 a bc aj2 b c The first two equations express the fact that o is a chain homotopy with respec* *t to t and t0. The last equation is equivalent to OE2(x, y) - _2(x,=y)-o([x, y]) + [q(x) + 1_2t(o(x)), o(y)] - [q0(y) - * *1_2t(o(y)), o(x)] = -o([x, y]) + [q(x), o(y)] + [o(x), q0(y)] (50* *3) 88 This is indeed the Baez-Crans condition on a 2-morphism. Acknowledgements We thank Danny Stevenson for many helpful comments. Among other things, the* * discussion of the cohomology of the string Lie 2-algebra has been greatly influenced by conversat* *ion with him. We acknowledge stimulating remarks by John Baez on characteristic classes of string bundles. H* *.S. thanks Matthew Ando and U.S. thanks Stephan Stolz and Peter Teichner for discussions at an early st* *age of this work. U.S. thankfully acknowledges invitations to the University of Oxford by Nige* *l Hitchin; to the conference "Lie algebroids and Lie groupoids in differential geometry" in Sheffield by Kir* *ill Mackenzie and Ieke Mo- erdijk; to the Erwin Schr"odinger institute in the context of the program "Pois* *son oe-models, Lie algebroids, deformations and higher analogues"; and to Yale University which led to collabo* *ration with H.S. and to useful discussions with Mikhail Kapranov. U.S. thanks Todd Trimble for discussi* *on of DGCAs, and their relation to smooth spaces, thanks Mathieu Dupont and Larry Breen for discussion* * about weak cokernels and thanks Simon Willerton and Bruce Bartlett for helpful disucssion of the notion * *of transgression. We thank the contributors to the weblog The n-Category Caf'e for much intere* *sting, often very helpul and sometimes outright invaluable discussion concerning the ideas presented here an* *d plenty of related issues. 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New Haven, CT 06511 BundesstrasseD55-20146 Hamburg 209PSouthh33rdiStreet* *ladelphia, PA 19104-6395 92