Lisa C. Jeffrey

Localization for nonabelian group actions

Suppose X is a compact symplectic manifold acted on by a compact Lie group K (which may be nonabelian) in a Hamiltonian fashion, with moment map µ : X → Lie(K) ∗ and Marsden-Weinstein reduction MX = µ −1 (0)/K. There is then a natural surjective map �0 from the equivariant cohomology H ∗ K (X) of X to the cohomology H ∗ (MX). In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of MX of any �0 ∈ H ∗ (MX) whose degree is the dimension of MX, provided that 0 is a regular value of the moment map µ on X. This formula is given in terms of any class � ∈ H ∗ (X) for which �0(�) = �0, and involves ∗

Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation

We derive explicit formulas for the Chern-Simons-Witten invariants of lens spaces and torus bundles overS1, for arbitrary values of the levelk. Most of our results are for the groupG=SU(2), though some are for more general compact groups. We explicitly exhibit agreement of the limiting values of these formulas ask→∞ with the semiclassical approximation predicted by the Chern-Simons path integral.

Topological Lagrangians and cohomology

Witten [12] has interpreted the Donaldson invariants of four-manifolds by means of a topological Lagrangian. We show that this Lagrangian should be understood in terms of an infinite-dimensional analogue of the Gauss-Bonnet formula. Starting with a formula of Mathai and Quillen for the Thom class, we obtain a formula for the Euler class of a vector bundle, which formally yields the explicit form of Wittens Lagrangian. We use the same method to treat Lagrangians proposed for the Casson invariant.

Bohr-Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula

We show how the moduli space of flatSU(2) connections on a two-manifold can be quantized in the real polarization of [15], using the methods of [6]. The dimension of the quantization, given by the number of integral fibres of the polarization, matches the Verlinde formula, which is known to give the dimension of the quantization of this space in a Kähler polarization.

Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface

Let n and d be coprime positive integers, and define M(n,d) to be the moduli space of (semi)stable holomorphic vector bundles of rank n, degree d and fixed determinant on a compact Riemann surface E. This moduli space is a compact Kifhler manifold which has been studied from many different points of view for more than three decades (see for instance Narasimhan and Seshadri [41]). The subject of this article is the characterization of the intersection pairings in the cohomology ring1 H* (M (n, d)). A set of generators of this ring was described by Atiyah and Bott in their seminal 1982 paper [2] on the YangMills equations on Riemann surfaces (where in addition inductive formulas for the Betti numbers of M (n, d) obtained earlier using number-theoretic methods [13], [25] were rederived). By Poincare duality, knowledge of the intersection pairings between products of these generators (or equivalently knowledge of the evaluation on the fundamental class of products of the generators) completely determines the structure of the cohomology ring. In 1991 Donaldson [15] and Thaddeus [47] gave formulas for the intersection pairings between products of these generators in H* (M (2, 1)) (in terms of Bernoulli numbers). Then using physical methods, Witten [50] found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in H* (M (n, d)) for general rank n. These generalized his (rigorously proved) formulas [49] for the symplectic volume of M (n, d): For instance, the symplectic volume of M (2, 1) is given by

Group systems, groupoids, and moduli spaces of parabolic bundles

Moduli spaces of homomorphisms or, more generally, twisted homomorphisms from fundamental groups of surfaces to compact connected Lie groups, were connected with geometry through their identification with moduli spaces of holomorphic vector bundles (see [29]). Atiyah and Bott [2] initiated a new approach to the study of these moduli spaces by identifying them with moduli spaces of projectively fiat constant central curvature connections on principal bundles over Riemann surfaces, which they analyzed by methods of gauge theory. In particular, they showed that an invariant inner product on the Lie algebra of the Lie group in question induces a natural symplectic structure on a certain smooth open stratum. Although this moduli space is a finite-dimensional object, generally a stratified space which is locally semialgebraic [19] but sometimes a manifold, its symplectic structure (on the stratum just mentioned) was obtained by applying the method of symplectic reduction to the action of an infinite-dimensional group (the group of gauge transformations) on an infinite-dimensional symplectic manifold (the space of all connections on a principal bundle).

Localization and the quantization conjecture

Let (M, ω) be a compact symplectic manifold with a Hamiltonian action of a compact Lie group K. Suppose that 0 is a regular value of the moment map μ: M → Lie(K)∗, so that the Marsden-Weinstein reduction Mred = μ−(0)K is a symplectic orbifold. In our earlier paper (Quart. J. Math., 47, 1996) we proved a formula (the residue formula) for η0eω0[Mred] for any η0 ϵ H∗(Mred), where ω0 is the induced symplectic form on Mred. This formula is given in terms of the restrictions of classes in the equivariant cohomology H∗T(M) of M to the components of the fixed point set of a maximal torus T in M. In this paper, we consider a line bundle L on Mfor which c1(L) = ω. If M is given a K-invariant complex structure compatible with ω we may apply the residue formula when η0 is the Todd class of Mred to obtain a formula for the Riemann-Roch number RR(Lred) of the induced line bundle Lred on Mred when K acts freely on μ−1(0). More generally when 0 is a regular value of μ, so that Mred is an orbifold and Lred is an orbifold bundle, Kawasakis Riemann-Roch theorem for orbifolds can be applied, in combination with the residue formula. Using the holomorphic Lefschetz formula we similarly obtain a formula for the K-invariant Riemann-Roch number RRK(L) of L. We show that the formulae obtained for RR(Lred) and RRK(L) are almost identical and in many circumstances (including when K is a torus) are the same. Thus in these circumstances a special case of the residue formula is equivalent to the conjecture of Guillemin and Sternberg (Invent. Math. 67 (1982), 515–538) (proved in various degrees of generality by Guillemin and Sternberg themselves and others including Sjamaar, Guillemin, Vergne and Meinrenken) that RR(Lred) = RRK(L).

Group cohomology construction of the cohomology of moduli spaces of flat connections on 2-manifolds

We use group cohomology and the de Rham complex on simplicial manifolds to give explicit differential forms representing generators of the cohomology rings of moduli spaces of representations of fundamental groups of 2-manifolds. These generators are constructed using the de Rham representatives for the cohomology of classifying spaces BK where K is a compact Lie group; such representatives (universal characteristic classes) were found by Bott and Shulman. Thus our representatives for the generators of the cohomology of moduli spaces are given explicitly in terms of the Maurer-Cartan form. This work solves a problem posed by Weinstein, who gave a corresponding construction (following Karshon and Goldman) of the symplectic forms on these moduli spaces. We also give a corresponding construction of equivariant differential forms on the extended moduli space X, which is a finite dimensional symplectic space equipped with a Hamiltonian action of K for which the symplectic reduced space is the moduli space of representations of the 2-manifold fundamental group in K.

Half density quantization of the moduli space of flat connections and witten's semiclassical manifold invariants☆

LET PpB denote the moduli space of flat connections on (the trivial SU(2) bundle on) a compact, oriented two-manifold P. The space s”, is a stratified space containing an open dense set Yg which is a symplectic manifold of (real) dimension 6g - 6. There exists a natural line bundle 9 -+ ,Fb, with hermitian metric ( , ), whose restriction to Y9 has a connection V with curvature given by the symplectic form o on Y9. If we equip the two-manifold Xg with a metric, the resulting Hodge star operator turns ,Yg into a Klhler manifold, with w as the Kahler form. Setting aside for the moment concern with the singularities of Pg,, we arrive at the usual setting for geometric quantization: we are given a symplectic manifold (qg, o), a line bundle 3’ + 9, with connection of curvature CO, and a polarization of the space of sections of 3. The quantization CYY~(C~) = H”(Pg, Yk) of this system has been of importance in attempts to construct a topological field theory. A topological field theory would assign to every two-manifold Cg the quantization XDk(Cg) of the system (Pg, CO, 9, V), which must be proved independent of the choice of Kahler structure on -Fg, coming from a choice of metric on X9. The space Xk(Xg) can be given the structure of a Hilbert space with the metric given by ((s1,s~)) = j(si>s~)~~~-~. A topological field theory would also assign an element sN of the quantization Xk(Xg) to every three-manifold N3 whose boundary is given by 8N3 = Cg. If this assignment satisfied the axioms of topological field theory, the space Zk(Eg) would be equipped with a representation of the group Diff(Xg) of diffeomorphisms of Eg. Furthermore, we would obtain invariants of closed three-manifolds as follows. Let N be a closed three-manifold, and let N = H be a

Intersection pairings in moduli spaces of holomorphic bundles on a Riemann surface

are coprime. This space had long been studied by algebraicgeometers (see for instance Narasimhan and Seshadri 1965 [22]), but a new view-point on it was revealed by the seminal 1982 paper [1] of Atiyah and Bott on theYang-Mills equations on Riemann surfaces. In this paper a set of generators forthe (rational) cohomology ring of