Topological field theories on manifolds with Wu structures
Samuel Monnier
TL;DR
The work develops a higher-dimensional generalization of spin Chern-Simons theory by incorporating Wu structures of degree $2\ell+2$ and encoding gauge data in a local system of lattices. A generalized cohomology framework (E-theory) is used to define a gauge-invariant action, while a discrete gauging procedure yields finite-sum partition functions with novel anomalies that are controlled by quadratic refinements and Arf invariants. The state space is constructed via a discrete geometric quantization of finite groups with a Heisenberg structure, including a metaplectic-like correction to handle orientation subtleties; Wilson operators realize a Heisenberg action, and the gluing law is established, including anomalous cases. The construction links to 6d (2,0) SCFT anomalies through a 7d anomaly theory and provides concrete tools (torus checks, tori boundary reductions, and explicit lattice cases) for understanding higher Wu Chern-Simons theories and their quantization, with potential applications to anomalies in high-dimensional (2,0) theories and beyond.
Abstract
We construct invertible field theories generalizing abelian prequantum spin Chern-Simons theory to manifolds of dimension 4k+3 endowed with a Wu structure of degree 2k+2. After analysing the anomalies of a certain discrete symmetry, we gauge it, producing topological field theories whose path integral reduces to a finite sum, akin to Dijkgraaf-Witten theories. We take a general point of view where the Chern-Simons gauge group and its couplings are encoded in a local system of integral lattices. The Lagrangian of these theories has to be interpreted as a class in a generalized cohomology theory in order to obtain a gauge invariant action. We develop a computationally friendly cochain model for this generalized cohomology and use it in a detailed study of the properties of the Wu Chern-Simons action. In the three-dimensional spin case, the latter provides a definition of the "fermionic correction" introduced recently in the literature on fermionic symmetry protected topological phases. In order to construct the state space of the gauged theories, we develop an analogue of geometric quantization for finite abelian groups endowed with a skew-symmetric pairing. The physical motivation for this work comes from the fact that in the k = 1 case, the gauged 7-dimensional topological field theories constructed here are essentially the anomaly field theories of the 6-dimensional conformal field theories with (2,0) supersymmetry, as will be discussed elsewhere.