The stable moduli space of flat connections over a surface

TL;DR

This paper determines the homotopy type of the stabilized moduli space of flat unitary connections over surfaces, revealing a connection to symmetric products and algebraic K-theory, using gauge theory and deformation K-theory techniques.

Contribution

It provides explicit homotopy type calculations for the moduli space over various surfaces, extending known results and connecting to algebraic K-theory conjectures.

Findings

Homotopy type over orientable surfaces is the infinite symmetric product.

Homotopy over non-orientable surfaces is a union of two tori.

Results relate to the Quillen-Lichtenbaum conjectures in algebraic K-theory.

Abstract

We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the homotopy type of the infinite symmetric product of M^g, generalizing a well-known fact for the torus. Over a non-orientable surface, we show that this space is homotopy equivalent to a disjoint union of two tori, whose common dimension corresponds to the rank of the first (co)homology group of the surface. Similar calculations are provided for products of surfaces, and show a close analogy with the Quillen-Lichtenbaum conjectures in algebraic K-theory. The proofs utilize Tyler Lawson's work in deformation K-theory, and rely heavily on Yang-Mills theory and gauge theory.