Gluing Formulas for Volume Forms on Representation Varieties of Surfaces

2022/02/22 by Erdal, Esma Dirican
#Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2202.11155

Abstract

Let Σ_g,n be a compact oriented surface with genus g≥ 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 π₁(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 \mathbbS¹. 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.

Related

• A note on Reidemeister Torsion of G-Anosov Representations
• Volume of representation varieties
• Torsion volume forms
• Components of spaces of surface group representations
• A new proof of gluing formula for analytic torsion forms