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 h...

We compute the index of the Dirac operator on spin Riemannian manifolds with conical singularities, acting from L(Σ) to Lq(Σ−) with p, q > 1. When 1 + np − n q > 0 we obtain the usual Atiyah-Patodi-Singer formula, but with a spectral cut at n+1 2 − n q instead of 0 in the definition of the eta invariant. In particular we reprove Chou’s formula for the L index. For 1+ np − n q ≤ 0 the index form...

Let Γ be a discrete finitely generated group. Let M̂ → T be a Γ-equivariant fibration, with fibers diffeomorphic to a fixed even dimensional manifold with boundary Z. We assume that Γ → M̂ → M̂/Γ is a Galois covering of a compact manifold with boundary. Let (D(θ))θ∈T be a Γ-equivariant family of Dirac-type operators. Under the assumption that the boundary family is L-invertible , we define an inde...

Fermionic Brownian paths are defined as paths in a space para-metrised by anticommuting variables. Stochastic calculus for these paths, in conjunction with classical Brownian paths, is described; Brownian paths on supermanifolds are developed and applied to establish a Feynman-Kac formula for the twisted Laplace-Beltrami operator on differential forms taking values in a vector bundle. This form...

We provide the first construction of exact solitary waves of large amplitude with an arbitrary distribution of vorticity. We use continuation to construct a global connected set of symmetric solitary waves of elevation, whose profiles decrease monotonically on either side of a central crest. This generalizes the classical result of Amick and Toland.

We define and prove a noncommutative generalization of a formula relating the Maslov index of a triple of Lagrangian subspaces of a symplectic vector space to eta-invariants associated to a pair of Lagrangian subspaces. The noncommutative Maslov index, defined for modules over a C∗-algebra A, is an element in K0(A). The generalized formula calculates its Chern character in the de Rham homology ...