2016/07/31 by Samuel Monnier · 1 citation
Mathematics · Physics and Astronomy · #Abelian group #BRST quantization #Black Holes and Theoretical Physics #Cobordism #Cohomology #Gauge theory #Homotopy and Cohomology in Algebraic Topology #Introduction to gauge theory #Invariant (physics) #Manifold (fluid mechanics) #Quantum field theory #Topological Materials and Phenomena #Topological quantum field theory #hep-th #math-ph #math.AT #math.MP #msc:57R56 #msc:81T13 #msc:81T45 #msc:81T70
paper · pdf · doi:10.1142/s0129055x17500155
published as Rev. Math. Phys., 29, 1750015 (2017) · 114 pages. v2: Sign correction in (4.16) and equations using (4.16), correction in the definition of E-chain in Appendix D
openalex created_date 2016/07/22 · openalex publication_date 2017/04/12 · arxiv created 2018/08/09 · arxiv updated 2018/08/13 · openalex updated_date 2026/08/05
We construct invertible field theories generalizing abelian prequantum spin Chern–Simons theory to manifolds of dimension [Formula: see text] endowed with a Wu structure of degree [Formula: see text]. After analyzing 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 3-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 [Formula: see text] case, the gauged 7-dimensional topological field theories constructed here are essentially the anomaly field theories of the 6-dimensional conformal field theories with [Formula: see text] supersymmetry, as will be discussed elsewhere.