T. R. Ramadas
Chennai Mathematical Institute
October 4, 2023
Integrals over the SU$(2)$ character variety and lattice gauge theory.: Given a genus-$g$ Riemann surface $\Sigma$, the moduli space of rank two vector bundles with trivial determinant is, by the Narasimhan-Seshadri Theorem, in bijection with the space of (equivalence classes of) representations in SU$(2)$ of the fundamental group of the surface . In the latter avatar, this space has a symplectic structure and a corresponding finite measure, the Liouville measure. Normalising to total mass one gives a probability measure. There is a natural class of real-valued functions parameterised by isotopy classes of loops on the surface. These are called Wilson loop functions by physicists and Goldman functions by mathematicians. I present a simple scheme to compute joint distributions of these functions for families of loops. This is possible because of the miracle of symplectic geometry called the Duistermaat-Heckman formalism (whose applicability in this context is due to L. Jeffrey and J. Weitsman) and a continuous analogue of the Verlinde algebra.
Chennai Mathematical Institute
October 4, 2023
Integrals over the SU$(2)$ character variety and lattice gauge theory.: Given a genus-$g$ Riemann surface $\Sigma$, the moduli space of rank two vector bundles with trivial determinant is, by the Narasimhan-Seshadri Theorem, in bijection with the space of (equivalence classes of) representations in SU$(2)$ of the fundamental group of the surface . In the latter avatar, this space has a symplectic structure and a corresponding finite measure, the Liouville measure. Normalising to total mass one gives a probability measure. There is a natural class of real-valued functions parameterised by isotopy classes of loops on the surface. These are called Wilson loop functions by physicists and Goldman functions by mathematicians. I present a simple scheme to compute joint distributions of these functions for families of loops. This is possible because of the miracle of symplectic geometry called the Duistermaat-Heckman formalism (whose applicability in this context is due to L. Jeffrey and J. Weitsman) and a continuous analogue of the Verlinde algebra.
The large $g$ asymptotics can be easily read off.
This leads to the second (speculative) part of the talk, which suggests an approach to rigorous analysis of (lattice) gauge theories, as a preliminary step to quantum-field-theoretic constructions.
This is based on the preprint: https://arxiv.org/abs/2206.07455