THE VERLINDE FORMULA FOR PARABOLIC BUNDLES

Abstract

Let Σ^g be a compact Riemann surface of genus g, and G = SU(n). The central element c = diag(e^2πid/n, …, e^2πid/n) for d coprime to n is introduced. The Verlinde formula is proved for the Riemann–Roch number of a line bundle over the moduli space [Mscr ]_{g, 1}(c, Λ) of representations of the fundamental group of a Riemann surface of genus g with one boundary component, for which the loop around the boundary is constrained to lie in the conjugacy class of cexp(Λ) (for Λ ∈ t₊), and also for the moduli space [Mscr ]_{g, b}(c, Λ) of representations of the fundamental group of a Riemann surface of genus g with s + 1 boundary components for which the loop around the 0th boundary component is sent to the central element c and the loop around the jth boundary component is constrained to lie in the conjugacy class of exp(Λ^(j)) for Λ^(j) ∈ t₊. The proof is valid for Λ^(j) in suitable neighbourhoods of 0.