Consistent Orientation of Moduli Spaces

TL;DR

This paper constructs a two-dimensional topological quantum field theory from three-dimensional Chern-Simons theory, using moduli space orientations and complex K-theory to define a Frobenius ring structure.

Contribution

It introduces a method to consistently orient moduli spaces in the reduction of Chern-Simons theory, linking to twisted equivariant K-theory and Madsen-Tillmann spectra.

Findings

Constructs the 2D TQFT via moduli space correspondences.

Shows how orientations induce twistings in K-theory.

Connects the theory to twisted equivariant K-theory of Lie groups.

Abstract

We give an a priori construction of the two-dimensional reduction of three-dimensional quantum Chern-Simons theory. This reduction is a two-dimensional topological quantum field theory and so determines to a Frobenius ring, which here is the twisted equivariant K-theory of a compact Lie group. We construct the theory via correspondence diagrams of moduli spaces, which we "linearize" using complex K-theory. A key point in the construction is to consistently orient these moduli spaces to define pushforwards; the consistent orientation induces twistings of complex K-theory. The Madsen-Tillmann spectra play a crucial role.