The prequantum line bundle on the moduli space of flat $SU(N)$ connections on a Riemann surface and the homotopy of the large $N$ limit

TL;DR

This paper proves the degree of the prequantum line bundle on the moduli space of flat $SU(2)$ connections is 1 and explores the homotopy equivalence of the large $N$ limit of the moduli space to $ ext{CP}^ Infty$, with applications to unitary connections.

Contribution

It establishes the degree of the prequantum line bundle on the moduli space and demonstrates the homotopy equivalence of the stable moduli space to $ ext{CP}^ Infty$ as $N$ approaches infinity.

Findings

Prequantum line bundle on the moduli space has degree 1.

Homotopy equivalence between the stable moduli space and $ ext{CP}^ Infty$.

Applications to the stable moduli space of flat unitary connections.

Abstract

We show that the prequantum line bundle on the moduli space of flat $SU(2)$ connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of flat $SU(N)$ connections, in the limit as $N$ tends to infinity, and $\mathbb{C}P^\infty$. Applications to the stable moduli space of flat unitary connections are also discussed.