Consistent Orientation of Moduli Spaces
Daniel S. Freed, Michael J. Hopkins, Constantin Teleman
TL;DR
The paper constructs a 2D TQFT as the a priori dimensional reduction of 3D Chern-Simons theory, realized via push-pull maps on twisted $K$-theory of moduli spaces of flat $G$-connections and yielding the Frobenius/Verlinde ring $K^{\tau+\dim G}_G(G)$. Central to the construction are consistent $K$-orientations (twistings) encoded by universal orientations, whose existence is established using Madsen–Tillmann spectra and the universal symbol, yielding a level and coherent pushforwards along bordisms. The theory is defined over $\mathbb{Z}$ and extends to families, where anomalies appear as extra twists $\tau_{\mathscr{X}/S}$ tracked by twisted $K$-theory; this anomaly is interpreted as an invertible TQFT. The framework clarifies how universal orientations determine the theory beyond the level data and connects to the Verlinde/fusion structure via the Pontrjagin product on twisted equivariant $K$-theory, with a canonical universal orientation $\mu$ providing a natural baseline. Overall, the work provides an algebraic-topological, a priori construction of a 1–2 dimensional TQFT rooted in twisted $K$-theory and universal orientation theory, with implications for representations of loop groups and generalized orientation data.
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.