Topological field theories on manifolds with Wu structures

TL;DR

This paper constructs and analyzes invertible topological field theories on manifolds with Wu structures, generalizing abelian Chern-Simons theories to higher dimensions and exploring their anomalies, gauge invariance, and state spaces.

Contribution

It introduces a new class of invertible topological field theories on Wu-structured manifolds, with a detailed cohomological framework and applications to fermionic phases and higher-dimensional anomalies.

Findings

Developed a cochain model for generalized cohomology used in the theories.

Provided a definition of the fermionic correction in spin Chern-Simons theory.

Connected the theories to anomaly field theories of 6D (2,0) supersymmetric conformal field theories.

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.