Many moduuli spaces in algebraic geometry can be expressed as quotients in the sense of Mumford's geoemtric invariant theory [9] of nonsingular complex projective varieties $X$ by actions of complex reductive groups $G$. Any such quotient can also be identified with a symplectic quotient or Marsden-Weinstein reduction of the variety $X$ by a maximal compact subgroup $K$ of the reductive group $G$ [8, 9, 10]. This symplectic quotient is $\mu^{-1}(O)/K$ where $\mu\colon X\to {\bf k}^*$ is a moment map for the action of $K$ on $X$ quiepped with a suitable symplectic form.
In [13] Witten describes a method of aclculating intersection pairings of cohomology classes on the quotient $\mu^{-1}(O)/K$ when $K$ acts freely on $\mu^{-1}(O)$ so that the quotient in nonsingular. The calculation is based on a localisation formula for equivariant cohomology which is analogous to the abelian localisation formula proved by Berline and Vergne [2] and independently by Atiyah and Bott [1], as a consequence of the Duistermaat-Heckman integration formula [4]. Wittetn's formula applies to nonabelian as well as abelian compact group actions. He employs it in an infinite-dimensional setting to give formulas for intersection pairings of cohomology classes in the moduli spaces of stable holomorphic bundles of coprime rank and degree over a fixed compact Riemann surface. These formulas agree with the calculations of Thaddeus [12] in the rank two case, and are related to other recent work on the Verlinde formulas such as [3, 11].
arA different, though closely related, approach [6] is based on the abelain localisation formula applied to a maximal torus $T$ of $K$. When $K$ acts freely on $\mu^{-1}(O)$ its inclusion in $X$ induces a surjection from the $K$-equivariant cohomology $K_K^*(X;{\bf Q})$ of $X$ to the cohomology $H^*(\mu^{-1}(O)/K;{\bf Q})$ of the symplectic quotient [8]. Witten's formula expresses the evaluation of the image $\sigma_o$ of $\sigma\in H^*_K(X;{\bf Q})$ against the fundamental class $[\mu^{-1}(O)/K]$ of the quotient in terms of the asymptotic behaviour of a certain integral. The method used to derive this in [6] gives as a byproduct a "residue formula" for $ sigma_o[ mu^{-1}(O)/K]$ as a sum over the connected components in $X$ of the fixed point set of the maximal torus $T$ of $K$. It can be generalised to allow for singularities in the quotient provided that interaction cohomology is used [7].
The moduli spaces of holomorphic bundles over a fixed compact Riemann surface can be regarded as symplectic quotients of finite-dimensional group actions on "extended moduli spaces" [5] and algebro-geometric versions of these, as well as the infinite-dimensional actions by sed by Witten. This leads to another derivation of Witten's formulas for pairings on the moduli spaces when the rank and degree of the bundles are corpime, and to formulas for intersection cohomology when the rank and degree are not coprime [7].
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