Holomorphic volume forms on representation varieties of surfaces with boundary

TL;DR

This paper generalizes Witten's volume form formula to surfaces with boundary by introducing a holomorphic volume form on representation spaces, with explicit computations for simple cases and SL(N,C).

Contribution

It introduces a holomorphic volume form on representation varieties of surfaces with boundary, extending Witten's formula to include boundary contributions.

Findings

Explicit volume and symplectic forms computed for simple surfaces.

Holomorphic volume form introduced for boundary representation spaces.

Extension of Witten's formula to surfaces with boundary.

Abstract

For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the other one is the power of the Atiyah-Bott-Goldman symplectic form. We introduce an holomorphic volume form on the space of representations of the circle, so that, for surfaces with boundary, it appears as peripheral term in the generalization of Witten's formula. We compute explicit volume and symplectic forms for some simple surfaces and for the Lie group SL(N,C).