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THE VERLINDE FORMULA FOR PARABOLIC BUNDLES

Published online by Cambridge University Press:  05 July 2001

LISA C. JEFFREY
Affiliation:
Mathematics Department, University of Toronto, Toronto, Ontario M5S 3G3, Canada
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Abstract

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

Type
Research Article
Copyright
The London Mathematical Society 2001

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Footnotes

This material is based on work supported by grants from NSERC and the Alfred P. Sloan Foundation.