Abstract
If Σ is a Riemann surface, let M(Σ, G) be the moduli space of conjugacy classes of representations of the fundamental group of Σ in a compact Lie group G. In his paper “Quantum gauge theories in two dimensions” (Commun. Math. Phys. 141 (1991) 153-209) Witten defined a volume on this space using Reidemeister-Ray-Singer torsion, and proved this volume is equal to the symplectic volume. Witten also defined a volume on the corresponding moduli space M(Σ0 , G) when Σ0 is a non-orientable 2-manifold. The latter space does not admit a symplectic structure. We show that in this case Witten’s volume can be obtained from the Riemannian volume associated to a choice of Riemannian metric on Σ0 .