A rudimentary theory of topological 4D gravity Jack Morava Department of Mathematicss The Johns Hopkins University Baltimore 21218 Md. jack@math.jhu.edu Abstract A theory of topological gravity is a homotopy-theoretic represen- tation of the Segal-Tillmann topologification of a two-category with cobordisms as morphisms. This note describes some relatively accessi- ble examples of such a thing, suggested by the wall-crossing formulas of Donaldson theory. 1 Gravity categories A cobordism category has manifolds as objects, and cobordisms as mor- phisms. Such categories were introduced by Milnor [22], but following Se- gal's definition of conformal field theory [29] and Atiyah's subsequent ab- straction of the notion of topological quantum field theory [1] they have been studied very widely. Recently, Tillmann [31] has shown the utility in this context of certain closely related two-categories (which generalize the classi- cal notion of category, by admitting morphism-objects which are themselves categories). The following definition is based on her ideas. ___________________________ e-print archive: http://xxx.lanl.gov/math.DG/0007018 Morava 1 Definition A gravity two-category has o (closed) manifolds as objects, o cobordisms as morphisms, and o isomorphisms of these cobordisms, equal to the identity on the boundary, as two-morphisms. There are many possible variations on this theme, and I will not try for maximal generality. If the objects of the category have dimension d (so the cobordisms are (d+1)-dimensional) then I will say that the gravity category is (d+1)-dimensional. I will assume that manifolds are smooth, compact and oriented, but not necessarily connected, and (following Segal) I understand the empty set to be a manifold of any dimension. 1.1 If V and V 0are d-manifolds, a morphism W : V ! V 0 is (the germ of) an orientation-preserving diffeomorphism (Vop[ V 0) x [0, 1) ~= (@W ) of the manifold on the left with a collar neighborhood of the boundary of the (d + 1)-manifold W ; the subscript op signifies reversed orientation. The morphism category Mor(V, V 0) has such cobordisms as its objects; it is a topological category, in which the space of morphisms between two cobordisms W and W~ consists of orientation- and boundary-identification- preserving diffeomorphisms W ~=W~ . Gluing along the boundary defines a continuous composition functor W, W 07! W O W 0: Mor(V, V 0) x Mor(V 0, V 00) ! Mor(V, V 00) , while disjoint union of objects gives this two-category a monoidal structure, with the empty set as identity object. By replacing Mor(V, V 0) with its set ß0Mor(V, V 0) of equivalence classes of objects, we obtain the category employed by Atiyah to define a topo- logical quantum field theory; in other words, we can pass from a gravity two-category, in which the morphism objects are enriched by a categorical structure, to a classical category, in which the morphism objects are simply 2 PRETTY GOOD GRAVITY sets. Tillmann's more perspicacious alternative is to interpret the topological category Mor(V, V 0) as a simplicial topological space and to replace it with its geometric realization Mor (V, V 0). This construction preserves Cartesian products (as does ß0: indeed the set of equivalence classes of objects in Mor is the set of components of the space Mor ), defining a topological grav- ity category (i.e., a category in which the morphism objects are topological spaces, and the composition maps are continuous). A topological quantum field theory in the sense of Atiyah [12 x1.7] is thus a (continous) monoidal functor from a topological gravity category to the (topological) category of modules over a discrete topological ring. However, we can consider monoidal functors to more general categories: for example, the singular chains on the morphism spaces of a gravity category define a monoidal category enriched over chain complexes, whose represen- tations are the (co)homological field theories of physics. In the language of homotopy theory, these are representations in a category of modules over some Eilenberg-MacLane ring-spectrum. In general, I will call any monoidal functor from a topological gravity category to the category of dualizeable objects over a ring-spectrum, a theory of topological gravity. One of the points of this paper is that there is a rich supply of such things. 1.2 This terminology needs some explanation. If W is a manifold with boundary, let Diff+(W ) be the topological group of orientation-preserving diffeomorphisms of W which restrict to the identity in some neighborhood of @W . The components of Mor(V, V 0) are indexed by equivalence classes of cobordisms W : V ! V 0, and the components themselves are the classifying spaces BDiff+(W ). Gluing [20] defines a continuous homomorphism Diff+(W ) x Diff+(W 0) ! Diff+(W O W 0) ; thus the (components of the) composition map in the topological gravity category are the maps these compositions induce on classifying spaces. On the other hand, a fundamental tautology of Riemannian geometry asserts that an isometry of a complete connected Riemannian manifold which fixes a frame at some point is the identity: such a map preserves the geodesics out of the framed point, and any other point in the manifold can be reached by such a geodesic. It follows that group of diffeomorphisms framing some basepoint will act freely on the (contractible) space of Riemannian metrics on a compact connected manifold. The space BDiff+(W ) is the homotopy quotient of the space of metrics [10, 11] on W by the diffeomorphism group Morava 3 and we can think of morphisms in the (d + 1)-dimensional gravity category as cobordisms between d-manifolds, together with a choice of equivalence class of Riemannian metric on the cobordism. Riemannian geometry thus provides the gravity category with a smooth structure. A (projective) Hilbert-space representation of a topological gravity category, along the lines considered by Segal in his definition of a conformal field theory, is thus very close to a quantum theory of gravity. When d = 1 we can see this more explicitly: the Riemann moduli space is the quotient of the space of conformal structures on a closed connected surface by the group of its orientation-preserving diffeomorphisms, which acts with finite isotropy when the genus exceeds one. This defines a monoidal functor from the two-dimensional gravity category to Segal's, which (away from closed surfaces of low genus) is a rational homology isomorphism on morphism spaces. Consequently, any conformal field theory in Segal's sense defines a quantum theory of two-dimensional gravity. 1.3 Examples: i) From this point of view, there is no a priori reason to limit ourselves to smooth manifolds. We could begin with a two-category of topological manifolds, and replace its morphism categories by their classifying spaces, as before: there are plenty of non-smoothable four-manifolds! ii) In higher dimensions, the category of manifolds and equivalence classes of h-cobordisms is a groupoid, with the Whitehead group of an object as its automorphisms. In low dimensions these categories are still quite mysteri- ous. iii) We can consider classes of manifolds with extra structure: for example, by requiring that the Stiefel-Whitney class w2 vanish, we can define a gravity category of four-dimensional Spin-manifolds. [The set of Spin-structures on such a manifold is a principal homogeneous space over its first mod two cohomology group, but is not naturally isomorphic to that group.] iv) Similarly, the four-dimensional gravity category of SpinC-manifolds is defined by cobordisms endowed with a complex line bundle with Chern class lifting w2. Ex. iii) can be regarded as the subcategory of Ex. iv) defined by objects with trivial Chern class. It is natural to think of the morphism categories in 4 PRETTY GOOD GRAVITY Ex. iii) as graded by elements of the middle homology lattice; for example, algebraic surfaces lie on the quadric c21= 2Ø + 3oe. [Note that reversing orientation changes the signature, but not the Euler characteristic.] When d is odd, the morphisms of a d + 1-dimensional gravity category are naturally graded by Euler characteristic: the correction term in the formula Ø(W O W 0) = Ø(W ) + Ø(W 0) - Ø(W \ W 0) is zero. When d is one, the Euler characteristic counts the number of handles or loops in the usual quantum or genus expansion; it defines a zeroth Mum- ford class ~0. If we exclude closed manifolds from our morphism spaces, and thus do not admit the empty set as a plausible object, this grading is bounded below. The signature defines a similar grading, when d = 3. Many interesting decorations of gravity categories are possible: Lorentz cobordism [28, 33], defined by a nowhere-vanishing vector field oriented suitably at the boundary, is one example. Restricting the objects (e.g. to be unions of (standard, or homology) spheres, or contact manifolds [19]) is another alternative. Witten's original two-dimensional theory [34] admits singular (stable) algebraic curves as morphisms; this compactifies its mor- phism spaces, and Kontsevich has shown (as Witten conjectured) that the resulting theory has a well-behaved vacuum state. 2 Pretty good theories of topological gravity A Riemannian metric g on an oriented closed connected two-manifold defines a Hodge operator *g on its harmonic forms. This operator squares to -1 on one-forms, and so defines a complex structure on the de Rham co- homology H1dR( ). The space of isomorphism classes of complex structures on a real Euclidean space of dimension 2g is the quotient SO (2g)=U(g), so we get a map ø : BDiff+( ) ! (Metrics)=(Diff+( )) ! SO=U in the large genus limit. This can be constructed more generally by working with differential forms which vanish on the boundary. Orthogonal sum of vector spaces makes an H-space of the target of ø, and it is not hard to see that if and 0 are surfaces with geodesic boundaries, then gluing Morava 5 them c times along some sets of compatible boundary components defines a homotopy-commutative diagram BDiff+( ) x BDiff+( 0)____//_BDiff+( O 0) |fixfi| |fi| fflffl| fflffl| SO=U x SO=U ____________//SO=U. [The intersection form on the middle homology of O 0is the direct sum of the intersection forms of and 0, together with a split hyperbolic intersection form of rank c - 1, which has a canonical complex structure [32 IV x4].] This is perhaps the simplest example of a theory of two-dimensional topo- logical gravity: it is a monoidal homotopy-functor to a topological category with one object and the H-space SO =U of morphisms [25]. The functor is a version of the Jacobian, which refines the infinite symmetric product construction (which takes disjoint union to Cartesian product). The Siegel moduli space for abelian varieties has the rational cohomology of an integral symplectic group, and a version of Hirzebruch's proportionality principle im- plies that the stable rational cohomology of this moduli space agrees with the cohomology of SO =U. 2.1.1 In general, a topological quantum field theory HF (with values in some category of modules over a ringspectrum k) assigns to a suitable d-manifold V , a module-spectrum HF (V ), such that i) the construction is exponential, in the sense that HF (V t V 0) ~=HF (V ) ^ HF (V 0) ; ii) there is a pairing Trace : HF (Vop) ^ HF (V ) ! k which is nondegenerate, in the sense that the induced map from HF (Vop) to the functional k-dual of HF (V ) is an isomorphism; iii) there is a natural transformation øW : BDiff+(W ) ! HF (@W ) 6 PRETTY GOOD GRAVITY subject to a monoidal axiom: if @W = Vopt V 0, etc., then the diagram BDiff+(W ) x BDiff+(W 0)_____________//BDiff+(W O W 0) fixfi|| fi|| fflffl| fflffl| HF (Vop) ^ (HF (V 0) ^ HF (Vo0p)) ^ HF_(V_00)//_HF(Vop) ^ HF.(V 00) commutes up to homotopy. The smash product of two such functors yields another. 2.1.2 Objects in the two-dimensional gravity category are just collections of circles, which can be indexed by nonnegative integers. In this case, a theory is defined by a dualizable k-module spectrum M, together with a system øpq2 (M^(p+q))*(BDiff+( )) = [BDiff+( ), M ^k . .^.kM]* of characteristic classes for bundles of connected surfaces with p incom- ing and q outgoing boundary components, which behave compatibly under gluing. The example above is deceptive, for in that case M agrees with the group ring k = S[SO =U], so the multiple smash product simplifies. The topological category with one object, and Tillmann's group-completion a BDiff+( g) ! Z x B +1 g 0 as its space of morphisms, defines the universal example of a theory of this type; the cohomology homomorphism defined by the induced map Z x B +1! SO=U factors through the classical map which kills the Mumford classes in degree divisible by four. In more general cases related to quantum cohomology [20, 24], M will be a Frobenius object in the category of spectra, and the theory can be reformulated in terms of a family of natural transformations p+qH*(M) ! H*(BDiff+( )) . 2.2 The Hodge-theoretic construction described above has a close analogue for four-manifolds, which is also classical in a way: the wall-crossing for- mulas [17] of Donaldson theory are its descendants. As in dimension two, Morava 7 its construction is based on properties of the intersection form on middle cohomology: If W is an compact connected oriented four-manifold with @W a union of homology spheres then the intersection form x, y 7! = (x [ y)[W, @W ] on the integral lattice B = H2(W, @W, Z) is unimodular. In dimension four, Wu's formula implies that q(x) = modulo two, so the form q is even if the manifold admits a Spin-structure [16 x5.7.6]. On a SpinC-manifold the intersection form is even or odd depending on the parity of the Chern class of its associated complex line bundle. By a fundamental theorem of Freedman [13] any unimodular quadratic form can arise as the intersection form of a closed topological four-manifold; but by similarly fundamental results of Donaldson [6, 9] the intersection form of a closed smooth four-manifold is either indefinite, or diagonalizable over the integers. As in two dimensions, the action of a diffeomorphism on homology defines a monodromy representation Diff+(W ) ! Aut+(B, q) = SO(B) which factors through ß0(Diff+(W )); it is convenient to think of its kernel [18] as an analogue, for four-manifolds, of the Torelli group of surface theory. 2.3 Let b = b+ + b- be the rank, and oe = b+ - b- the signature, of the inner product space defined by q on B R. For our purposes the indefinite lattices are the most interesting: these are classified by their rank, signatur* *e, and type (even if q(x) 0 mod two, otherwise odd). In the indefinite case, the manifold Grass-(B) of maximal negative-definite subspaces of B R is a noncompact (contractible) symmetric space defined by a cell of dimension b+ b- in the usual Grassmannian of b- -planes in b-space. The orthogonal group of the lattice acts on this cell with finite isotropy, so the canonical homotopy-to-geometric quotient map BSO (B) ! Grass-(B)=SO (B) is a rational homology isomorphism. If B and B0 are indefinite lattices, then the construction which sends a pair of negative definite subspaces in 8 PRETTY GOOD GRAVITY the real span of each, to their orthogonal sum in the real span of the direct sum lattice, defines a map Grass-(B) x Grass-(B0) ! Grass-(B B0) which is equivariant with respect to the Whitney sum homomorphism SO (B) x SO(B0) ! SO(B B0) The Grothendieck group of the category of even indefinite unimodular lat- tices is free abelian on two generators, corresponding to the hyperbolic plane and the E8 lattice [30 V x2]. The `Hasse-Minkowski' spectrum KEIU defined by the algebraic K-theory of the category of such lattices is the group com- pletion of the monoid constructed from the disjoint union of the classifying spaces of their orthogonal groups; the tensor product of two such lattices defines another, making this a commutative ring-spectrum. 2.4 A Riemannian metric g on W defines a Hodge operator *g on harmonic forms, but now this operator squares to +1 on the middle cohomology. The function which assigns to g, the *g = -1-eigenspace of harmonic two-forms vanishing on @W , maps the space of Riemannian metrics to the negative- definite Grassmannian Grass-(B) equivariantly with respect to the action of Diff+(W ). If W and W 0are four-manifolds bounded (as above) by homology spheres, and if W O W 0results from gluing these manifolds along a collection of compatible boundary components, then the quadratic module of W O W 0is canonically isomorphic to B B0; hence the cohomology representation of the diffeomorphism group defines a monoidal functor from the gravity cate- gory of Spin four-manifolds bounded by standard spheres, to the topological category with one object, and the Hasse-Minkowski spectrum as morphisms. There is a similar functor defined on the category with homology spheres as objects, but the resulting lattice is no longer necessarily indefinite [9 x1.2.* *3]. The higher algebraic K-theory of such lattices has apparently not received much attention. It is remarkable that the relatively naive constructions sketched above already define pretty good theories of topological gravity. The j-invariant of Atiyah-Patodi-Singer [3] is much more sophisticated; to find an interpretation in these terms for it, analogous to the way Floer homology globalizes the Casson invariant, would be extremely interesting. Morava 9 3 Toward a parametrized Donaldson theory A good theory of gravity shouldn't exist in a vacuum: it deserves to be coupled to some nontrivial matter. Donaldson [8] and Moore and Witten [23] have suggested the study of equivariant supersymmetric Yang-Mills theory parameterized by classifying spaces of diffeomorphism groups. A fragment of such a theory is sketched below. 3.1 Suppose for simplicity that W is closed. The graded space Bun *(W ) of gauge equivalence classes of connections on SU (2)-bundles over W has components indexed by the second Chern class of the bundle. Let D* be the subspace of Metricsx Bun*(W ) consisting of pairs (g, A), where A is a connection on an SU (2)-bundle over W with curvature two-form *g(FA) = -FA antiselfdual with respect to the metric g. The standard transversality argu- ments of Donaldson theory [9 x4.3] imply that this space is a manifold, with fiber of dimension 8c2- 3_2(oe + Ø) above the metric g; at least, provided this metric admits no reducible antiselfdual connections. These reducible con- nections define an interesting kind of distinguished boundary for the space of antiselfdual connections. 3.2 Reducible connections on W are parametrized by the wall arrangement Wall(B) = {H 2 Grass-(B) | H \ B 6= {0} } of the lattice B: it is the set of maximal negative-definite subspaces of B R containing a lattice point. This is a union of smooth submanifolds of codimension b- , filtered by the increasing family Walld(B) of subspaces consisting of maximal negative-definite H containing a lattice point x with 0 > q(x) -d (which is a locally finite union of manifolds [14]). The orthogonal group of B acts naturally on these arrangements, as well as on the quotient spaces WallSd(B) = Grass-(B)=Walld(B) (which are roughly the S-duals of the wall arrangements). If B and B0 are two indefinite lattices, then the orthogonal direct sum map defines a commutative diagram Grass-(B) x Grass-(B0)_____//Grass-(B B0) | | | | fflffl| fflffl| WallSd(B) ^ WallSd0(B0)___//WallSd+d0(B B0) 10 PRETTY GOOD GRAVITY which is equivariant, with respect to the Whitney sum on orthogonal groups. The equivariant cohomology H*SO(WallS*) defines yet another variant of a topological gravity theory, but there seems to be little known about such essentially arithmetic invariants. 3.3 If g is in the complement of the preimage Metrics0dof Walldin the space Metricsof metrics on W , then no SU (2)-bundle with Chern class less than -d admits a connection with *g-antiselfdual curvature. Thus if D0ddenotes the space of pairs (g, A) such that A is gauge equivalent to a connection induced from a line bundle with curvature antiselfdual with respect to g, then (Dd, D0d) ! (Metrics, Metrics0d) x Bund(W ) is a kind of Diff+(W )-equivariant cycle, of relative finite dimension above the space of metrics. It cannot be expected to be proper, but Donaldson theory has developed sophisticated methods to deal with such issues [7]: let SP1d(W+ ) be the space of finitely supported functions f from W to the integers, such that X f(x) = d , x2W and let __ a D d= Dix SP1d-i(W+ ) 0 i d be the analogue of the Uhlenbeck-Donaldson compactification of Dd in the stratified space a ____ Metricsx ( Bun i(W ) x SP1d-i(W+ )) = Metricsx Bund(W ) . 0 i d Completing the subspace D0dof reducible connections analogously defines a candidate __ __ ____ (D d, D0d) ! (Metrics, Metrics0d) x Bund(W ) for a Diff+(W )-equivariant Donaldson cycle. To extract homological information from this construction, note that a class z of dimension * in the rational homology of BDiff+(W ) maps to a sum, with rational coefficients, of homology classes defined by maps Z ! MetricsxDiff+pt of smooth manifolds Z. The fiber product of such a map with the projection __ ____ D d ! MetricsxDiff+Bund(W ) ! MetricsxDiff+pt Morava 11 defines a class of dimension * + 8d - 3_2(oe + Ø) in the rational homology of ____ (Metrics, Metrics0d) xDiff+Bund(W ) ; note that this admits a canonical map to the space WallSd^SO(B)SP 1d(W+ ) , which depends only on the lattice B. 3.4 The homotopy-to-geometric quotient map for the space of connections is a rational homology equivalence of Bun*(W ) with the space of based smooth maps from W+ to BSU (2) [9 x5.1.15], and the Pontrjagin class defines a rational homology isomorphism of the space of maps with the Eilenberg- MacLane space H(Z, 4). By the Dold-Thom theorem, ßiMaps (W+ , H(Z, 4)) ~=H4-i(W, Z) ~=Hi(W, Z) ~=ßi(SP 1(W+ )) so_as_far as rational (co)homology is concerned, we can replace the space Bun*(W ) with the free topological abelian group on W . [This identification uses Poincar'e duality, and hence requires a choice of orientation: the space of bundles is a contravariant functor, but the infinite symmetric product is covariant.] Combined with the constructions outlined above, this defines a generalized Donaldson invariant as a homomorphism Dd : H*(BDiff+, Q)) ! H*+8d-3_ (WallSd^SO SP1d, Q) 2(ff+ffl) with values in a group which depends only on the cohomology lattice B; indeed the rational homology of SP1 (W+ ) is the symmetric algebra on the homology of W , and the automorphic cohomology H*SO(B)(SP 1(W+ ), Q) = H*(SO (B), Sym(H*(W ))) contains the classical ring of automorphic forms for the orthogonal group [5] as the invariant elements of the symmetric algebra on B. This invariant generalizes the usual one, in the sense that Dd on a gener- ator of the zero-dimensional homology of BDiff+ is the classical invariant. [The usual convention is to interpret the antiselfdual cycle as a function on the cohomology of W , by taking its Kronecker product with exp(x), x 2 H*(W ).] A four-manifold is said to be of simple type, if the behavior of its classical invariant as a function of charge is not too complicated: in the present formalism, the condition is that Dd+1(1) 7! w0w24Dd(1) 12 PRETTY GOOD GRAVITY under the homomorphism induced by the restriction map from WallSd+1to WallSd(where w0 and w4 generate the homology in degrees zero and four of W ). This suggests ~Dd= (w0w24)-dDd 2 Hom -3_2(ff+ffl)(H*(BDiff+), H*(WallSd^SO SP10)) as the natural normalization for the generalized invariant. 4 On the inadequacy of the foregoing The preceding sketch defines at best a piece of a topological gravity functor. It is defined only for manifolds without boundary, but it behaves correctly under disjoint union: if W0 and W1 are two closed four-manifolds, then X Dd0(W0) Dd1(W1) 7! Dd(W0 [ W1) d=d0+d1 under the maps of x3.2; this is nothing but a definition of the generalized invariant for non-connected manifolds. In fact there is reason to think that these constructions may have wider validity. Some years ago, Atiyah [2] proposed a unification of the invariants of Donaldson and Floer, based on a theory of semi-infinite cycles in the polarized manifold of connections on a three-manifold. A theory of such cycles which behaves naturally under variation of the metric on a bounding four-manifold would yield a topological gravity theory for four-manifolds, taking values in generalized automorphic forms with coefficients in Floer homology. Many results which follow from Atiyah's program are known now to be true; but (mostly because of difficulty with compactifications), work on these questions has advanced without using his cycle calculus. I am told, how- ever, that recently there has been progress along the lines he suggested [26], though in Seiberg-Witten rather than Floer-Donaldson theory. Meanwhile, Bauer [4] and Furuta [15] have studied generalized Seiberg-Witten invari- ants from a homotopy-theoretic point of view, and Bauer has shown that his invariant behaves nicely under connected sum. The hope that these new developments can be extended to the context proposed in this paper has encouraged me to write this incomplete and probably naive account. Morava 13 Acknowledgements This research was supported by the NSF. It is a plea- sure to thank Stefan Bauer, Paul Feehan, Kenji Fukaya, Mikio Furuta, Peter Oszvath, Andrei Tjurin, and Richard Wentworth for helpful conversations about the material in this paper. It is, however, very speculative, and they deserve no blame for my excesses. References [1] M.F. Atiyah, Topological quantum field theories, IHES Publ. 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