Abstract
Let Σg be a compact Riemann surface of genus g, and
G = SU(n). The central element c =
diag(e2πid/n, …, e2πid/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.
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