Gluing Formulas for Volume Forms on Representation Varieties of Surfaces

TL;DR

This paper establishes a multiplicative gluing formula for volume forms on representation varieties of surfaces, linking Reidemeister torsion and symplectic forms through surface decompositions.

Contribution

It introduces a novel multiplicative gluing formula for Reidemeister torsion and symplectic volume forms on surface representation varieties.

Findings

Gluing formula for Reidemeister torsion in terms of simpler surfaces.

Expression of symplectic volume form as a product of volume forms on smaller surfaces.

Connection between torsion, symplectic forms, and surface decompositions.

Abstract

Let $\Sigma_{g,n}$ be a compact oriented surface with genus $g\geq 2$ bordered by $n$ circles. Due to Witten, the twisted Reidemeister torsion coincides with a power of the Atiyah-Bott-Goldman-Narasimhan symplectic form on the space of representations of $\pi_1(\Sigma_{g,0})$ in any semi-simple Lie group. In the present paper, we first obtain a multiplicative gluing formula for the twisted Reidemeister torsion of $\Sigma_{g,0}$ in terms of torsions of $\Sigma_{2,2},$ $\Sigma_{2,1},$ and boundary circles $\mathbb{S}^1.$ Then, by using Heusener and Porti's results on $\Sigma_{g,n},$ we show that the symplectic volume form on the representation variety of $\Sigma_{g,0}$ can be expressed as a product of the holomorphic symplectic volume forms on the relative representation varieties of surfaces $\Sigma_{2,1}$ and $\Sigma_{2,2}.$