Chennai Mathematical Institute

Seminars




Mathematics Colloquium
Date: Wednesday, February 8, 2023
Time: 3:30 pm to 4:30 pm
Venue: Seminar Hall
Symplectic integrals on the moduli space of rank two vector bundles

T. R. Ramadas
Chennai Mathematical Institute.
08-02-23


Abstract

Dear all,

There will be two series of mathematics talks this semester at CMI.

The Fields Medal Lecture series: 4 talks on the work and impact of the recent Fields medallists, accessible to undergraduate students. These will be scheduled on 4 Fridays in the semester, from 2 pm to 3 pm in the Seminar Hall. We will try to organize these in hybrid mode.

The Mathematics Colloquium talks: research talks accessible to a wide audience. These will be in-person, scheduled on Wednesdays, from 3:30 pm to 4:30 pm in the Seminar Hall.

More details of upcoming colloquium talks can be found here - https://adityakarnataki.github.io/CMI-Math-Coll.html

The details of the first Mathematics Colloquium talk of the semester are below:

Mathematics Colloquium
Speaker: T. R. Ramadas, CMI
Date: Wednesday, February 8, 2023
Time: 3:30 pm to 4:30 pm
Venue: Seminar Hall

Title: Symplectic integrals on the moduli space of rank two vector bundles

Abstract: 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, the space $ \cM_g $ has a symplectic structure and a corresponding finite measure, the Liouville measure. Normalised to total mass one this gives a probability measure. There is a natural class of real-valued functions $ W_C: \cM_g \to [0,1] $, parameterised by isotopy classes of loops, $ C \subset \Sigma $. These are called Wilson loop functions by physicists and Goldman functions by mathematicians. With applications in mind, I present a simple scheme to compute joint distributions of these functions for families of loops. This is possible because of a 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 Verlinde algebra is well-known to physicists, representation theorists and algebraic geometers in the context of "generalised theta functions".