Gluing formulas for volume forms on representation varieties of surfaces

Abstract

Let $Σ_{g,n}$ 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 $π_1(Σ_{g,0})$ in any semi-simple Lie group. In the present paper, we first obtain a multiplicative gluing formula for the twisted Reidemeister torsion of $Σ_{g,0}$ in terms of torsions of $Σ_{2,2},$ $Σ_{2,1},$ and boundary circles $\mathbb{S}^1.$ Then, by using Heusener and Porti's results on $Σ_{g,n},$ we show that the symplectic volume form on the representation variety of $Σ_{g,0}$ can be expressed as a product of the holomorphic symplectic volume forms on the relative representation varieties of surfaces $Σ_{2,1}$ and $Σ_{2,2}.$

16 pages, redundancies removed in C1,C2 and Thm 5.5 and Cor 5.8 added. Comments welcome!