HKR CHARACTERS AND HIGHER TWISTED SECTORS JACK MORAVA Abstract.This is an expository talk, presented at the ChengDu (Sichuan) ICM Satellite conference on stringy orbifolds. It is intended as an intr* *oduction to the work of Hopkins, Kuhn, and Ravenel on generalized group character* *s, which seems to fit very well with the theory of what physicists call hig* *her twisted sectors in the theory of orbifolds. I would like to acknowledge many conversations with Matthew Ando about the contents of this paper. In a better world, he would be its coauthor. 1.Basic definitions 1.0 Since this paper is intended to be expository, I will work in a convenient * *ad hoc category of orbispaces. For our purposes, an orbispace is a (topological) categ* *ory X := [X=G] defined by an action of a compact Lie group G on a topological space X, subject to the restriction that the isotropy group Gx of any point x 2 X be finite. Morphisms of orbispaces are to be equivalence classes, up to natural transformations, of (continuous) functors between categories. This class of objects is rich enough to contain some interesting examples: Ex 1 If M is a reduced d-dimensional orbifold, then its principal orthogonal fr* *ame `bundle' O(M) is a smooth manifold upon which the orthogonal group O(d) acts with finite isotropy. By a fundamental lemma of Kawasaki (conceivably known to Satake?) the category (or groupoid) [O(M)=O(d)] is equivalent to the category defined by the original orbifold M. Ex 2 If G is a finite group, then the category [*=G] with one object, and the s* *et G of morphisms, is an interesting unreduced orbifold. Remarks: Useful topological constructions take us out of the category of smooth objects, so it is convenient to work with a class slightly larger than the usual orbifolds. In general, I will use the mathcal typeface for an orbispace, and th* *e usual mathematical typeface for its underlying space of objects; thus X := [X=G] has objects X and underlying quotient space X=G. However, there will be exceptions: 1.1 If G is a group, and X 2 (G - spaces), then I(X) := {(g, x) 2 G x X | gx = x} is itself a G-space, with action defined by h(g, x) = (hgh-1, hx) . ____________ Date: 20 August 2002. 1991 Mathematics Subject Classification. 19Lxx, 55Nxx, 57Rxx. The author was supported in part by the NSF. 1 2 JACK MORAVA I is thus a functor from the category of G-spaces to itself. The isotropy group* * of (g, x) 2 I(X) is {h 2 G | h(g, x) = (hgh-1, hx) = (g, x)} ; being a subset of Gx, it is finite if the latter is. It follows that if X = [X=* *G] is an orbispace, in the sense above, then I(X) := [I(X)=G] is again such an orbispace; following the terminology of algebraic geometers, i* *t is now called the inertia stack of X. [It is also the fixed-point orbispace [10] o* *f the circle group, acting on the free loops in X.] The description above makes it cl* *ear that I is an endofunctor of the category of orbispaces. 1.2 These constructions define some useful invariants. I will call the Borel co* *ho- mology H*(X, Q) := H*G(X, Q) := H*(EG xG X, Q) (in this paper all coefficients will be Q-vectorspaces) the ordinary cohomology* * of the orbispace X: its Leray spectral sequence has as E2-term, the cohomology H*(X=G, H*(Gx, Q)) of the quotient with coefficients in a sheaf whose stalk at x is the group coho* *mology of Gx. Since these groups are by hypothesis finite, this sheaf is concentrated* * in degree zero, and the spectral sequence degenerates to an isomorphism H*(X, Q) ~=H*(X=G, Q) = H0(G, H*(X, Q)) with the cohomology of the quotient space. This is interesting enough, but it i* *s not very subtle. A more powerful invariant is defined by the equivariant K-theory K*(X) := K*G(X) of the orbispace. 1.3 Theorem: There is a natural multiplicative transformation K*(X) = KG (X)* ! H*G(I(X), Q) = H*(I(X), Q) which becomes an isomorphism after tensoring with Q on the left. Remarks: Nowadays this (rational) invariant is usually called the Adem-Ruan [2], or classical, orbifold cohomology; it is to be distinguished from the Chen-Ruan* * [4] orbifold cohomology, which has a different multiplicative structure. [I will ig* *nore some deep questions about the gradings of these theories, since I have nothing * *to say about them.] I should note that the existence of some such generalized Chern character was also known to Baum and Brylinski [3]. The cohomology groups on the right have a natural decomposition as g2G^H*C(g)(Xg, Q) where ^Gdenotes the set of conjugacy classes in G, Xg is the set of g-fixed poi* *nts in X, and C(g) is the centralizer of g in G. [More precisely: for any choice of* * g in HKR CHARACTERS AND HIGHER TWISTED SECTORS 3 the appropriate conjugacy class, the cohomology group in question is well-defin* *ed under conjugation by elements of G.] The contributions to this sum, indexed by conjugacy classes other than the iden- tity are now called the twisted sectors of the cohomology. 2.Higher inertia stacks The first main point of this note is that iterating the inertia stack construct* *ion defines a simplicial object Io(X), which is a convenient device for organizing * *the `higher twisted sectors' defined by the orbifold X. When n = 2 these higher twisted sectors are crucially important in elliptic coh* *o- mology ([6]; cf. also [13]), but for larger n their relevance to physics is no* *t yet clear; but they are certainly interesting invariants, and my second main point * *is that these things already have a deep literature in mathematics. 2.1 Definition: If X = [X=G] as above, let In(X) = [In(X)=G] ; note that In(X) = {(g1, . .,.gn; x) 2 Gn x X | gi2 Gx, 8i, k [gi, gk] = 1} . Proof: See the argument in x1.1, and induct. For example: If X = * is a single point, In[*=G] = Hom (Zn, G)=G is the set of conjugacy classes of commuting n-tuples of elements in G. When n * *= 1, this is just the classical set of conjugacy classes in G. The construction of the inertia stack is essentially local, so more generally [ In[X=G] = [( In[x=Gx] x {x})=G] . x2X 2.2 Recall now that a simplicial object in a category C can be defined as a fun* *ctor C from the category of finite ordered sets to C. We can think of such a functor* * as defined by its sets C[n] of n-simplices, together with various face and degener* *acy maps between them. A simplicial object in the category of spaces (for example, a simplicial set) h* *as a geometric realization a |C| = (C[n] x n)=(face & degeneracy relations) . n 0 For example: a category C can be regarded, following Grothendieck and Segal, as a simplicial set with objects as zero-simplices, morphisms as one-simplices, and chains of n composable morphisms as its n-simplices. The face maps are defined 4 JACK MORAVA by composing maps, and degeneracies are defined by inserting identities. The ge* *o- metric realization of this simplicial set is sometimes called the classifying s* *pace for the category; in particular, |[*=G]| = BG is the classifying space for the (finite) group G, and more generally the geome* *tric realization |[X=G]| ~ EG xG X of a transformation group is homotopy equivalent to its associated Borel constr* *uc- tion. The map EG xG X ! * xG X = X=G which collapses (the free contractible G-space) EG to a point is sometimes call* *ed the `homotopy-to-geometric' quotient map; the arguments of x1.2 above show that for our class of orbispaces, this map induces an isomorphism on rational cohomology. 2.3 The simplicial set n 7! Zn defining [*=Z] is in fact a simplicial object in* * the category of abelian groups: group composition is a homomorphism when the group is abelian. It follows that the covariant functor n 7! (~Z)n := Hom (Zn, Z) is, in a natural sense, a cosimplicial abelian group. Definition: The functor n 7! Hom ((~Z)n, G)=G defines the simplicial set Io[*=G] of commuting tuples of elements in the group* * G, cf. [11 x4]; more generally, [ Io[X=G] := [( Io[x=Gx] x {x})=G] x2X is the simplicial inertia stack of X. We can use this construction to elaborate Adem and Ruan's construction for orb- ifold cohomology: n 7! H*(In(X), Q) is a cosimplicial object in the category of graded-commutative algebras, which * *keeps simultaneous track of the higher inertia stacks of X. 2.5 Theorem: There is a natural transformation |Io[X=G]| ! |[X=G]| which is an equivalence if G is abelian. The proof is by construction; it is easiest to begin in the special case when X* * is a point. Then Io[*=G] is a simplicial set with one zero-simplex; a one-simplex * *is a conjugacy class, a two-simplex is a conjugacy class of commuting elements, etc.* * If is an n-simplex, then its faces are the maps 7! HKR CHARACTERS AND HIGHER TWISTED SECTORS 5 and its degeneracies are the maps which insert identity elements. These are exa* *ctly the maps defining the classifying space of G; but we are working now not with group elements, but conjugacy classes. It may clarify the situation to observe that conjugation by a group element def* *ines a functor from the category [*=G] to itself; thus G acts on BG. However, the endofunctor defined by conjugation with a group element is naturally equivalent to the identity endofunctor: the element itself defines the transformation. Nat* *ural transformations of functors become homotopies under geometric realization, so t* *his action of G on BG is homotopically trivial, and the quotient map BG ! BG=G is an equivalence. The promised map is then the quotient of the obvious equivariant inclusion Hom ((~Z)o, G) ! BG by G. Because Io is a local construction, this definition now extends directly* * to [X=G]; alternately, we can display the simplicial object Io[X=G] (with most of * *its maps supressed) as a a . .!. [(Xg \ Xh)=C(g, h)] ! [Xg=C(g)] ! [X=G] , where the nth coproduct is indexed by conjugacy classes of commuting n-tuples, and C(g1, . .,.gn) is the centralizer of the commuting tuple. Remarks: It is tempting to think of this construction as a kind of blowup or resolution of the Borel construction; it seems analogous in some ways to Segal'* *s [14] reconstruction of a manifold, up to homotopy, from the category defined by the * *sets of an atlas with inclusions as morphisms. In our case, the charts are reminisce* *nt of the complete sets of commuting observable of classical quantum mechanics. Kuhn [8 x7] remarks that |Io[*=G]| is in fact a -space [though not, in general, a s* *pecial -space] in the sense of Segal. I am reluctant to admit that I don't know how a single example works out. Symme* *t- ric groups and finite subgroups of Sl2(C) are of course very interesting candid* *ates. This construction may also be related to Kontsevich's theory of motivic integra* *tion: if X is an algebraic variety, say over the complexes, the n-simplices of Io are* * roughly deformations of the scheme over fields of transcendence degree n. To make this precise would require a better understanding of the degree-shifting numbers [4;* * 9 x8; 11 x2], which do not appear in the formalism above. When n = 1, these are locally constant Q-valued functions w on the fixed-point * *set Xg, which are slightly more sophisticated than the function which assigns to g,* * the number logdet(g| ) , where g| represents the action of g on the normal bundle of Xg in X. In genera* *l, the normal bundle to the fixed point set of a commuting n-tuple h* *as a flag decomposition as the sum of the normal bundles Xg1\ . .\.Xgi-1 Xg1\ . .\.Xgi, 6 JACK MORAVA and it seems reasonable to expect that the degree-shifting number of this n-tup* *le will be the sum of the degree-shifting numbers of these subbundles. 3.HKR characters 3.0 A homomorphism from a free abelian group to a finite group G factors through some finite abelian quotient group, so Y * Hom (Zn, G)=G = Hom (^Zn, G)=G = pHom (Znp, G)=G decomposes as the restricted product (with only finitely many nontrivial entrie* *s) of p-local contributions, indexed by primes p. This uses the fact that Y ^Z= Zp , p where Zp = limZ=pnZ the p-adic integers. Since products of simplicial sets (and spaces) are defined coordinate-wise, Io[* *X=G] can be expressed as a restricted infinite fiber product (over [X=G]) of p-local* * objects Iop[X=G] built like I but with Zp replacing Z. I will ignore questions about in* *finite restricted products by assuming that [X=G] is `ramified' at a finite set of pri* *mes (dividing #G, say, when the group is finite); the cohomology of the simplicial * *inertia stack can then be calculated from the local contributions, one prime at a time. 3.1 In this context, Hopkins, Kuhn, and Ravenel [7] provide, for each n 1, an interpretation of H*(Inp[X=G], Q) which reduces when n = 1 to the theorem of Adem and Ruan in x1.3 above. To state these results, however, requires a short digression about cobordism. Very briefly, then: cobordism is to homology as smooth manifolds are to simplic* *es. In this theory, a d-dimensional chain is not some sum of nasty singular simplic* *es, but a map, say f : M ! X, of a nice smooth d-manifold M to the space X of interest. Instead of boundaries of simplices, we take boundaries of manifolds; * *thus @f : @M ! X is the boundary of f, which is said to be closed if @M = 0. Similar* *ly, f = @F if 9F : W ! M such that @W = M and F |@W = f. The analog of the homology of X is the quotient of the abelian semigroup of closed objects (cycle* *s) by the subsemigroup of boundaries; this is well-defined, since of course @ O @ * *= 0. It is more usual to say that these groups are defined by classes of maps of smo* *oth manifolds to X under the equivalence relation defined by cobordism: which is to say that two maps of closed manifolds to X are related if they are the boundary values of maps defined on a smooth manifold of one higher dimension. These groups are obviously homotopy-invariant (use the cobordism defined by a cylinder) and covariant: a map OE : X ! Y pushes the class [f] to the class [OE* * O f]. Atiyah's convention is to call this (graded-abelian-group-valued, homological) * *func- tor the bordism of X; there is a corresponding cohomological theory (contravari- ant under pullback or fiber product, using Thom's theory of transversality), now usually called the cobordism of X. One advantage of the latter theory is a nice multiplicative structure, defined by the obvious Cartesian product, without need for any Eilenberg-Zilber foolishness. HKR CHARACTERS AND HIGHER TWISTED SECTORS 7 Cobordism theory has very natural connections with the theory of group actions on manifolds: the Borel construction EG xG M ! EG xG * = BG associated to a G-manifold M is a kind of relative manifold, which defines a (-* *d)- dimensional class in the cobordism of BG. This is the beginnings of a rich subj* *ect; a more sophisticated approach can be found in [5]. HKR theory is a natural gene* *r- alization of the classical theory of characters of representations of groups on* * vector spaces to a theory of characters for actions on manifolds. The advantages of cobordism (geometric naturality, etc.) are recognized in the Russian literature, where it is usually called `intrinsic homology'. Its disadv* *antages include the fact that there are many cobordism theories, depending on one's fav* *orite choice of manifold: oriented, spin, symplectic, framed . .e.ach with its own sp* *ecial features. A more substantial issue is that the ground ring of such a theory (ie* *, the value of the cohomology theory on a point) tends to be quite large. It is argua* *bly the cobordism theory of stably almost complex manifolds (with a complex structu* *re on the sum of the tangent bundle with some trivial bundle) which is technically most accessible; that theory, called complex cobordism, has a polynomial ground ring MU* := MU*(pt) ~=MU* ~=Z[xi| i 1] with one generators of each even degree. [Frank Adams's convention is to write ML* for the cobordism theory of manifolds with structure group reduced to the Lie group L, eg U for weakly almost complex manifolds]. Over the rationals, MU*(pt) Q = Q[CPn | n 1] is the polynomial ring generated by the complex projective spaces; but these cl* *asses do not generate over the integers. More generally, an old argument of Dold shows that there is a natural multiplic* *ative transformation MU*(X) ! H*(X, MU* Q) which factors through an isomorphism of the rationalization of the left-hand si* *de. Over the rationals, then, there is in some sense little difference between cobo* *rdism and ordinary cohomology. The advantage of the former theory lies in its geometr* *ic naturality: its cycles are geometric objects, which carry characteristic class * *data (for example, of the sort familiar to physicists in the theory of `gravitationa* *l de- scendents'). 3.2 These cohomology theories are often too big to be technically convenient - * *for example, their ground rings are not Noetherian, so topologists have developed an arsenal of techniques to make them more useful. One useful ruse is to work p-lo* *cally, at some fixed prime. It turns out that to understand MU in general, it suffices* * to understand a hierarchy of cohomology theories with ground rings E^*n= Zp[v1, . .,.vn-1]((v-1n)) indexed by integers n 1, defined as truncations (in a suitable sense) of the * *p- completion of MU; here vk can be taken to be the cobordism class of a degree p hypersurface in CP (pk), and A((x)) is the formal Laurent series extension of a* * ring 8 JACK MORAVA A which allows only finitely many negative powers of x. When n = 1, this theory is a version of p-adically completed complex K-theory. The study of these theories tends to involve some quite subtle number theory, a* *nd one of the main technical advances in [7] is the construction of a certain fait* *hfully flat ring extension E^n D^n, which is most naturally interpreted as a kind of generalized Galois extension, with Galois group Gln(Zp). Theorem: There is a natural multiplicative transformation E^*n(|[X=G]|) ! H*(Inp[X=G], ^Dn Q)Gln(^Zp)-inv which factors through an isomorphism with the rationalization of the group on t* *he left. The term on the right is the subring of invariants under the action of the Galo* *is group Gln(Zp), but that action requires some clarification. The point is that t* *his group acts on the coefficient ring ^Dn, but it also acts on Inp, through its co* *nstruction in terms of conjugacy classes of homomorphisms from Znpto G. The relevant action on the right is the (diagonal) product of these two natural actions. 3.3 This indeed restricts when n = 1 to the theorem of x1.3, plus (a p-adic ver* *sion of) a theorem of Artin: there is a natural multiplicative transformation R(G) ! Fns(G^, Qcyc)Gal(Qcyc=Q)-inv which factors through an isomorphism with the rationalization of the left-hand * *side; where Qcycis the cyclotomic closure of the rationals (defined by adjoining all * *roots of unity). This natural transformation is nothing but the map which assigns to a character* * its representation; this version of the theorem encompasses the fact, also due to A* *rtin, that the values of such characters lie in Qcyc. The Galois group Gal(Qcyc=Q) ~=^Zx of this extension is the multiplicative group of profinite integers, whose p-lo* *cal component is the p-adic unit group Zxp= Gl1(Zp) . The statement above conceals an action of ^Zxon the conjugacy classes, in which k 2 Z sends the class of g to the class of gk (away from the order of g). 3.4 Here are a few closing remarks: i) The rings ^Enclassify (in a suitable sense) one-dimensional formal groups of* * height n over p-adic integer rings, and the rings ^Dnclassify such groups, together wi* *th a level structure: this is a preferred basis for the torsion subgroup. In the theory of algebraic stacks [1], the cyclotomic Galois action plays a dis* *tin- guished role; the level structure is just a choice of isomorphism of Qp=Zp with* * the group of p-power roots of unity. In the case of a stack defined over Q, it is n* *atural to think of the center of Gln(Z^p) as acting through the determinant det: Gln(Zp) ! Zxp HKR CHARACTERS AND HIGHER TWISTED SECTORS 9 on the roots of unity. ii) The E^n's and the D^n's do not fit together naturally as a (co)simplicial r* *ing. In particular, the natural action of the symmetric group n on In gets lost in * *the action of Gln on ^Dn. This suggests that there is lots of room in the transition between chromatic le* *vels for all sorts of gerbish orbifold twisting, and other kinds of noncommutative m* *onkey business . . . References 1. D. Abramovich, T. Graber, A. Vistoli, Algebraic orbifold quantum products, a* *vailable at math.AG/00112004; Madison orbifolds (Contemporary Math, to appear) 2. A. Adem, Y.B. Ruan: Twisted orbifold K-theory, available at math.AT/0107168 3. P. Baum, J.L. Brylinski: Noncommutative topology: talk at AMS Winter meeting* * (2000) 4. W. Chen, Y.B. Ruan, Orbifold quantum cohomology, available at math.AG/0005198 5. J. P. C. Greenlees, N. P. Strickland: Varieties and local cohomology for chr* *omatic group cohomology rings, Topology 38 (1999) 1093 - 1139 6. M. J. Hopkins: Characters and elliptic cohomology, in Advances in homotopy t* *heory (Cortona, 1988), 87 - 104; London Math. Soc. Lecture Notes 139, Cambridge Un* *iv. Press (1989) 7. M. J. Hopkins, N. J. Kuhn, D. C. Ravenel: Generalized group characters and c* *omplex oriented cohomology theories. J. Amer. Math. Soc. 13 (2000) 553 - 594 8. N. J. Kuhn: Character rings in algebraic topology, in Advances in homotopy t* *heory (Cortona, 1988), 111 - 126, London Math. Soc. Lecture Notes 139, Cambridge U* *niv. Press (1989) 9. E. Looijenga: Motivic measures, in Seminaire Bourbaki, Asterisque 276 (2002) 10.E. Lupercio, B. Uribe: Loop groupoids, gerbes, and twisted sectors on orbifo* *lds, available at math.AT/0110207; Madison orbifolds (Contemporary Math, to appear) 11.J. Morava: Some Weil group representations motivated by algebraic topology, * *in Elliptic curves and modular forms in algebraic topology, Springer Lecture Notes in Ma* *the- matics 1326 (1986) 12.M. Reid: La correspondance de McKay, in Seminaire Bourbaki, Asterisque 276 (* *2002) 13.S. Norton: Appendix [on generalized moonshine] to G. Mason, Finite groups an* *d modular functions, in Proc. Sympos. Pure Math., 47: Arcata Conference 181-210 (1987) 14.G. Segal: Classifying spaces and spectral sequences, Publ. Math. IHES 34 (19* *68) Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218 E-mail address: jack@math.jhu.edu