Representations with Weighted Frames and Framed Parabolic Bundles

Abstract

There is a well-known correspondence between the symplectic variety of representations of the fundamental group of a punctured Riemann surface into a compact Lie group G, with fixed conjugacy classes at the punctures, and a complex variety of holomorphic bundles on the unpunctured surface with a parabolic structure at the puncture points. For G=SU(2), we build a symplectic variety of pairs (representations of the fundamental group into G, `weighted frame' at the puncture points), and a corresponding complex variety of moduli of `framed parabolic bundles', which encompass respectively all of the above spaces, in the sense that one can obtain the former from the latter by taking a symplectic quotient or a geometric invariant theory quotient. This allows us to explain certain features of the toric geometry of the SU(2) moduli spaces discussed by Jeffrey and Weitsman, by giving the actual toric variety associated with their integrable system.