Quantization of some moduli spaces of parabolic vector bundles on CP^1

TL;DR

This paper explores the quantization of the symplectic structure on a specific moduli space of parabolic vector bundles over CP^1, linking it to non-abelian theta functions expressed via distributions on SU(2).

Contribution

It provides explicit formulas for non-abelian theta functions associated with these moduli spaces, connecting parabolic bundles on CP^1 to vector bundles on elliptic curves.

Findings

Explicit expressions for non-abelian theta functions.

Connection between parabolic bundles and elliptic curves.

Representation of theta functions via distributions on SU(2).

Abstract

We address quantization of the natural symplectic structure on a moduli space of parabolic vector bundles of parabolic degree zero over $\mathbf{CP}^1$ with four parabolic points and parabolic weights in {0,1/2}. Identifying such parabolic bundles as vector bundles on an elliptic curve, we obtain explicit expressions for the corresponding non-abelian theta functions. These non-abelian theta functions are described in terms of certain naturally defined distributions on the compact group SU(2).