Consistent Orientation of Moduli Spaces

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.

💡 Research Summary

The paper presents a purely topological construction of the two‑dimensional reduction of three‑dimensional quantum Chern‑Simons theory, using complex K‑theory as the linearization tool for moduli spaces of flat connections. The authors replace the infinite‑dimensional space of all gauge fields with the finite‑dimensional moduli stack (M_X) of flat (G)‑connections on a compact oriented surface (X). For a bordism (X\colon Y_0\to Y_1) they obtain restriction maps (s\colon M_X\to M_{Y_0}) and (t\colon M_X\to M_{Y_1}). The desired TQFT assigns to each closed one‑manifold (Y) the abelian group (A_Y=K^\bullet(M_Y)) and to each bordism the push‑pull homomorphism \