Heat Kernel and Moduli Spaces

In this paper we describe a proof of the formulas of Witten [W1], [W2] about the symplectic volumes and the intersection numbers of the moduli spaces of principal bundles on a compact Riemann surface. It is known that these formulas give all the information needed for the Verlinde formula. The main idea of the proof is to use the heat kernel on compact Lie groups, in a way very similar to the heat kernel proof of the Atiyah-Singer index formula and the Atiyah-Bott fixed point formula. The Reidemeister torsion comes into the picture, through a beautiful observation of Witten, as the symplectic volume of the moduli space. It plays the role similar to that played by the Ray-Singer torsion in the path-integral computations on the space of connections. The basic idea is as follows. Consider a smooth map between two compact smooth manifolds f : M → N . Let H(t, x, x0) be the heat kernel of the Laplace-Beltrami operator on N with x0 a fixed regular value of f . Because of the basic properties of the heat kernel, we know that for any continuous function a(y) on M , when t goes to zero, ∫