Following M. F. Atiyah and R. Bott [AB] and E. Witten [W], we consider the space of flat connections on the trivial SU(2) bundle over a surface M, modulo the space of gauge transformations. We describe on this quotient space a natural hermitian line-bundle with connection and prove that if the surface M is now endowed with a complex structure, this line bundle is isomorphic to the determinant bundle. We show heuristically how path-integral quantisation of the Chern-Simons action yields holomorphic sections of this bundle.

We show how the moduli space of flat SU(2) connections on a twomanifold 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 Kahler polarization.

Let M be a symplectic manifold, equipped with a Hamiltonian action of a torus T . We give an explicit formula for the rational cohomology ring of the symplectic quotient M//T in terms of the cohomology ring of M and fixed point data. Under some restrictions, our formulas apply to integral cohomology. In certain cases these methods enable us to show that the cohomology of the reduced space is torsion-free. SUSAN TOLMAN AND JONATHAN WEITSMAN Section 8 of [JK]) which computes M red κ(α) in terms of fixed point data on M (in other words the components of the fixed point set of the maximal torus, the characteristic classes of their normal bundles, the weights of the action of the maximal torus on the normal bundles, and the values of the moment map on the components of the fixed point set). Since, by Poincaré duality, β ∈ H * T (M; Q) is in the kernel of κ exactly if the integral of κ(αβ) over M red is zero for all α ∈ H * T (M, Q), the kernel of κ can, in principle, be computed from their results, in the case of rational cohomology. However, we do not know of any direct method of relating their formulas to ours. See also [GK, P, V].In this paper we give a description of the kernel of κ in terms of fixed point data. For according to a theorem of F. Kirwan, the natural restriction map from the equivariant cohomology of M to the equivariant cohomology of the fixed point set is an injection. We compute the kernel of κ in terms of the image of this map. This image is well understood in many examples. Moreover, it is determined by the subset of one and zero dimensional orbits (see [GKM, TW] and references therein). Our methods also give a basis for the kernel of κ, and, under some restrictions, allow us to compute the integral cohomology rings of symplectic quotients.Our main result is the following. Theorem 2. Let a torus T act on a compact symplectic manifold M with a moment map φ : M −→ t * . Assume that 0 is a regular value of φ. Let F denote the set of fixed points. For all ξ ∈ t, define M ξ := {m ∈ M | φ(m), ξ ≤ 0}, K ξ := {α ∈ H * (M; Q) | α| F ∩M ξ = 0 }, and K := ξ∈t

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