The Verlinde Formula for Parabolic Bundles

Let Σ g be a compact Riemann surface of genus g , and G = SU( n ). The central element c = diag( e 2π id / n , …, e 2π id / 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 c exp(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 j th 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.