Let Σ~g be a compact Riemann surface of genus g, and G = SU(n). The central element c = diag(e~(2πid), …, e~(2πid)) 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 u_(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 c exp(Λ) (for Λ ∈ t_+), and also for the moduli space u_(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.
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