Gluing Formulas for Volume Forms on Representation Varieties of Surfaces

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 $\Sigma_{2,2}.$