Topological field theories on manifolds with Wu structures

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.Comment: 113 page

Remarks on the Green–Schwarz Terms of Six-Dimensional Supergravity Theories

“…It is known how to make sense of half-integer level Chern-Simons theories on spin 3manifolds: those are the so-called spin Chern-Simons theories [11,12,13,14] that play a central role in the quantum Hall effect. Their higher-dimensional generalization have recently been studied in [15] and the results of that paper will play a central role in the construction of the Green-Schwarz terms in the present paper.…”

IIB flux non-commutativity and the global structure of field theories

“…) We emphasize that our viewpoint here, focusing purely on a careful analysis of the original construction of (2, 0) theories in ten dimensional type IIB string theory, is complementary to existing viewpoints on the partition function of (2, 0) theories. One such viewpoint is that of relative QFTs articulated by Freed and Teleman in [12], where one views the (2, 0) theories as furnishing the boundary degrees of freedom for certain non-invertible seven dimensional TQFTs [13][14][15]. For the A N cases one can also study the question using holography [16].…”

A Modern Point of View on Anomalies

“…The net anomaly due to the coupling the self‐dual field to the background gauge field is therefore At the level of anomaly field theories, the quarter Dai–Freed theory whose partition function is gets tensored by a certain Wu Chern–Simons theory, whose partition function on a manifold U such that is . Wu Chern–Simons theories can be seen as quadratic Chern–Simons theories of degree Abelian gauge field at ‘half‐integer level’, thereby generalizing spin Chern–Simons theories to higher dimension. The definition of the anomaly field theory of charged self‐dual fields is implicit in [] and will be discussed in more detail elsewhere.…”

“…The gravitational part of the anomaly simply gets multiplied by the signature of Λ, while the gauge part becomes where G is now a ‐form valued in and is the signature of the lattice . The flux quantization of G is now the following . Let Γ (2) be the quotient of by the radical of the induced pairing.…”

“…The flux quantization of G is now the following. [22] Let (2) be the quotient of /2 by the radical of the induced pairing. The reduction modulo 2 of the periods of 2G in (2) should coincide with those of ν ⊗ γ , where ν is the degree 2 + 2 Wu class of W and γ is the unique characteristic element in (2) , i.e.…”