TATE COHOMOLOGY OF CIRCLE ACTIONS AS A HEISENBERG GROUP JACK MORAVA Abstract.We study the Madsen-Tillmann spectrum CP1-1as a quotient of the Mahowald pro-object CP1-1, which is closely related to the Tate coho- mology of circle actions. That theory has an associated symplectic struc* *ture, whose symmetries define the Virasoro operations on the cohomology of mod* *uli space constructed by Kontsevich and Witten. 1.Tate cohomology of circle actions 1.1 If E is a geometric bordism theory (such as integral homology), its Tate co- homology t*TE can be constructed by tom Dieck stabilization from the geometric theory of E-manifolds with T-action, the action required to be free on the boun* *dary [8]. If E is multiplicative, so is t*TE; there is a cofibration sequence . .!.E*BT+ ! t*TE ! E-*-2BT+ ! . . . in which the boundary map sends a T-manifold with boundary to the quotient of i* *ts boundary by the (free) T-action. When E is complex-oriented [eg MU or HZ] this sequence reduces to a short exact sequence which identifies t*TE with the Laure* *nt series ring E*((e)) obtained by inverting the Euler class in E*(BT) = E*[[e]], and the boundary map can be calculated as a formal residue; more precisely, the formal Laurent series f maps to the residue of fd logEat e = 0, where d logEis the invariant differential of the formal group law of E. When E is not complex- orientable, tTE can behave very differently [4], as the Segal conjecture shows:* * up to a profinite completion, Y tTS0 ~ S0 _ S1 BT=C where C runs through proper subgroups of T. There is a related but simpler theory ø*TE defined by manifolds with free T-act* *ion on the boundary alone, which fits in an exact sequence . .!.E* ! ø*TE ! E-*-2BT+ ! . .;. ignoring the interior T-action defines a truncation t*TE ! ø*TE . ____________ Date: 15 September 2001. 1991 Mathematics Subject Classification. 19Dxx, 57Rxx, 83Cxx. The author was supported in part by the NSF. 1 2 JACK MORAVA 1.2 It is natural to think of t*TE as the E-homology of a version [1] of Mahowa* *ld's pro-spectrum CP-11, constructed from the inverse system {CP-1k= T h(-kj)} of Thom spectra defined by the filtered vector bundle . . .kj (k + 1)j . . . over CP 1. To be more precise, tTE ~ E ^ S2CP-11 as spectra, and there is a similar equivalence øTE ~ E ^ S2CP-11. From this point of view, the morphism from t*TE to ø*TE is the E-homology of the collapse map CP-11 ! CP-11. The cobordism class defined by a family of complex-oriented surfaces with a free circle action on its boundary, parametrized by X, defines an element of ø-2TMU(X+ ) = [X+ , CP-11^ MU] ; the image of this class under the Thom homomorphism from MU to HZ is the homomorphism H*(CP-11, Z) ! H*(X+ , Z) defined by the classifying map of Madsen and Tillmann, sketched below. 1.3 When E = HZ, the symmetric bilinear form f, g 7! (f, g) = rese=0(fg de) on the Laurent series ring tTHZ is nondegenerate, and the involution I(z) = z-1 on T defines a symplectic form {f, g} = (I(f), g) which restricts to zero on the subspaces of elements of degree 0 and < 0. The Tate cohomology thus has an intrinsic inner product, with canonical polarization and involution. The functor ~Gwhich sends the commutative ring A to the set of formal Laurent series X p G~(A) = {g = gkxk+1 2 A((x)) | g0 2 Ax , gk 2 A if0 > k} k 0 (i.e. with g0 a unit and gk nilpotent when k is negative) in fact takes values* * in the category of groups, with formal composition of series as the operation. Th* *is group of invertible `nil-Laurent' series has a linear representation on the abe* *lian group valued functor A 7! A((x)), but it is a little too large to be convenient* *ly representable; in a certain sense it is an ind-pro-algebraic analog of the grou* *p of diffeomorphisms of the circle. TATE COHOMOLOGY OF CIRCLE ACTIONS AS A HEISENBERG GROUP 3 1.4 It is tempting to interpret ~Gas a group of automorphisms of the Tate coho- mology, but the most obvious action does not preserve the symplectic structure. Kontsevich-Witten theory suggests a better alternative: there is an embedding p ek 7! fl-k-1_(x) : t*THZ ! R(( x)) 2 of symplectic modules, defined using the fractional divided power s fls(x) = ___x___ (1,+ s) in which the symplectic structure on the target is defined by u, v 7! {u, v} = resx=0udv . [The reals are a notational convenience: some powers of ß have been ignored.] Over any field of characteristic zero, the square root ofpan invertible nil-Lau* *rent series in x is an invertible odd nil-Laurent series in x, and it makes better* * sense to think ofpthe group ~G-1=2of such series [8 x1.3] as symplectic automorphisms* * of t*THQ R(( x)). The half-integral shift comes ultimately from the fact that t* *THZ is not Spanier-Whitehead self-dual; rather, its dual is most naturally interpre* *ted as its own double suspension. 2.Madsen-Tillmann and Kontsevich-Witten Madsen and Tillmann construct a map a BDiff(Fg) ! 1 CP-11 g 0 which is compatible with gluing of surfaces; in particular, it defines a lax fu* *nctor from the two-dimensional topological gravity category [10] to a topological cat* *egory with one object and an H-space of morphisms. [Reversing the orientation of a surface corresponds to the involution I.] The point of this note is to identif* *y a suitable subgroup of ~G-1=2as the motivic automorphisms of this functor. 2.1 Here is a quick account of one component of [7]: if F Rn is a closed two- manifold embedded smoothly in a high-dimensional Euclidean space, its Pontrjagi* *n- Thom construction Rn+! F maps compactified Euclidean space to the Thom space of the normal bundle of the embedding. The tangent plane to F is classifi* *ed by a map ø : F ! Grass2,nto the Grassmannian of oriented two-planes in Rn, and the canonical two-plane bundle j over this space has a complementary (n-2)-plane bundle, which I will call (n - j). The normal bundle is the pullback along ø * *of (n - j); composing the map induced on Thom spaces with the collapse defines Rn+! F ! Grass(n-j)2,n. The space Emb (F ) of embeddings of F in Rn becomes highly connected as n in- creases, and the group Diff(F ) of orientation-preserving diffeomorphisms of F * *acts freely on it, defining a compatible family Rn+^DiffEmb(F ) ! Grass(n-j)2,n which can be interpreted as a morphism BDiff(F ) ! lim nGrass(n-j)2,n:= 1 CP-11. 4 JACK MORAVA 2.2 Madsen and Tillmann show their construction factors through an infinite loo* *pspace map Z x B +1! 1 CP-11! Q(CP+1) , in which the last arrow is defined by collapsing the bottom two-cell in a cofib* *ration S-2 ! CP-11! CP+1 . The fiber 2QS0 of the induced map of loop spaces is torsion, so the rational cohomology of 1 CP-11is isomorphic to the algebra of symmetric functions on the subspace of non-negative powers in t*THQ. This algebra is thus canonically isomorphic to the Fock representation [10 x2.2] of the Heisenberg algebra of th* *at symplectic module; but this representation possesses a canonical Virasoro actio* *n, defining a homomorphism H*(Z x B +1, Q) ! Symm (H*(CP+1)) 2 (G~-1=2- representations) . In Kontsevich-Witten theory the usual generators bk 2 H*(CP+1), k 0, map to symmetric functions Tracefl-k-1_( 2) ~ -(2k - 1)!! Trace -2k-1 = tk( ) 2 of a positive-definite Hermitian matrix ; this leads to a construction of the * *appro- priate twisted Virasoro representation in terms of Schur Q-functions [2,6]. 2.3 The homomorphism limMU*+n-2(T h(n - j)) ! MU*-2(BDiff(F )) defined on cobordism by the Madsen-Tillmann construction sends the Thom class to a kind of Euler class: according to Quillen's conventions, the Thom class is* * the zero-section of (n - j), regarded as a cobordism class of maps between manifold* *s. Its image is the class defined by the fiber product Zn _____________//Grass2,n | | | | fflffl| fflffl| Rn+^DiffEmb(F )_____//Grass(n-j)2,n; it is the space of equivalence classes, under the action of Diff(F ), of pairs * *(x, OE), with x 2 OE(F ) Rn a point of the surface (ie, in the zero-section of ), and* * OE an embedding. The image is thus the element [Zn ! Rn+^DiffEmb] 7! MUn-2(SnBDiff(F )) defined by the tautological family F xDiffEDiff(F ) of surfaces over the classi* *fying space of the diffeomorphism group. This class is primitive in the Hopf-like str* *ucture defined by gluing [9 x2.2], so the class = exp(th(-j)v) 2 MU0Q( 1 CP-11)[[v]] of finite unordered configurations of points on the universal surface (with v a* * book- keeping indeterminate) defines a multiplicative transformation ~ *: H*(Q(CP+1), Q) ! H*(MU, Q[[v]]) TATE COHOMOLOGY OF CIRCLE ACTIONS AS A HEISENBERG GROUP 5 with properties analogous to the Chern character of a vector bundle. It sends t* *he Fock representation described above to an algebra of cohomological characterist* *ic numbers. 2.4 Kontsevich-Witten theory uses`a more_sophisticated configuration space, whi* *ch maps the rational homology of Q( g 0M g) (suitably interpreted, for small g) t* *o a similar ring of characteristic numbers; this homology contains a fundamental cl* *ass a __ X __ [Q( M g)] = exp( [M g]v3(g-1)) g 0 for the moduli space of not necessarily connected curves. Witten's tau-function* * is the image of this `highest-weight' vector under the analog of ~ *; it is invari* *ant under the subalgebra of Virasoro generated by operators Lk of cohomological degree k, with k -1. 3.Afterthoughts, and possible generalizations 3.1 Witten has proposed a generalization of 2D topological gravity which encom- passes surfaces with higher spin structures: for a closed smooth surface F an r* *-spin structure is roughly a complex line bundle L together with a fixed isomorphism L r ~=TF of two-plane bundles, but for surfaces with nodes or marked points the necessary technicalities are formidable [5]. The group of automorphisms of such* * a structure is an extension of its group of diffeomorphisms by the group of rth r* *oots of unity, and there is a natural analog of the group completion of the category defined by such surfaces. The generalized Madsen-Tillmann construction maps this loopspace to the Thom spectrum T h(-jr), and it is reasonable to expect that th* *is map is equivariant with respect to automorphisms of the group of roots of unity. This fits with some classical homotopy theory: if (for simplicity) r = p is pri* *me, multiplication by an integer u relatively prime to p in the H-space structure of CP 1 defines a morphism T h(-jp) ! T h(-jup) of spectra, and the classification of fiber-homotopy equivalences of vector bun* *dles yields an equivalence of T h(-jup) with T h(-jp) after p-completion. There is an analogous decomposition of tTHZp and a corresponding decomposition of the associated Fock representations [10 x2.4]. 3.2 In an extension of her work Tillmann also considers categories of surfaces mapped to some parameter space X, which has interesting connections with both Tate and quantum cohomology. When X is a smooth compact almost-complex manifold, its Hodge-deRham cohomology admits a natural action of the Lie algebra sl2(R), generated [12 IV x4] by the Hodge dimension operator H, multiplication * *by the first Chern class c1(X) = E, and its adjoint F = *E*. Recently Givental [3 x8.1] has shown that earlier work of (the schools of) Eguchi, Dubrovin, and oth* *ers on the Virasoro structure of quantum cohomology can be formulated in terms of (what I like to think of as) t*TH(X, R), polarized by the twisted involution 1_H -1_H IGiv= e2 e-E IeE e 2 . It would be very interesting if this polarization could be understood in terms * *of the equivariant geometry of the free loopspace of X. 6 JACK MORAVA 3.3 I owe thanks to R. Cohen, E. Getzler, A. Givental, J. Greenlees, I. Madsen, N. Strickland, and U. Tillmann (at least), for their forebearance in the face o* *f my continued misunderstanding of things they have tried patiently to tell me. I ho* *pe I'm finally starting to get it right. References 1. R.L. Cohen, J.D.S. Jones, G.B. Segal, Floer's infinite dimensional Morse the* *ory and homotopy theory, in the Floer Memorial Volume, Birkhäuser, Progress in Mathematics 13* *3 (1995) 297-326 2. P. DiFrancesco, C. Itzykson, J.-B. Zuber, Polynomial averages in the Kontsev* *ich model, CMP 151 (1993) 193-219 3. A. Givental, Gromov - Witten invariants and quantization of quadratic hamilt* *onians, available at math.AG/0108100 4. J. Greenlees, A rational splitting theorem for the universal space for almos* *t free actions, Bull. London Math. Soc. 28 (1996) 183-189 5. T. Jarvis, T. Kimura, A. Vaintrob, Moduli spaces of higher spin curves and i* *ntegrable hierar- chies, available at math.AG/9905034 6. T. J'ozefiak, Symmetric functions in the Kontsevich-Witten intersection theo* *ry of the moduli space of curves, Lett. Math. Phys. 33 (1995) 347-351 7. I. Madsen, U. Tillmann, The stable mapping-class group and Q(CP1+), Aarhus p* *reprint 14 (1999); and further work in in progress 8. J. Morava, Cobordism of involutions revisited, revisited, in the Boardman Fe* *stschrift, Con- temporary Math. 239 (1999) 9. __-, Topological gravity in dimensions two and four, available at math.QA/99* *08006 10.__-, An algebraic analog of the Virasoro group, Czech. J. Phys. 51 (2001) 11.__-, A rudimentary theory of topological 4D gravity, Adv. Th. and Math. Phys* *. (to appear); available at math.DG/0007018 12.J.P. Serre, Algebres de Lie semisimples complexes, Benjamin (1966) Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218 E-mail address: jack@math.jhu.edu