Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders

TL;DR

This paper establishes a systematic framework linking gauge anomalies in various dimensions to symmetry-protected topological orders and topological orders, providing a unified classification scheme for both gauge and gravitational anomalies.

Contribution

It introduces a cohomology-based classification of gauge anomalies, including nonABJ anomalies, and connects gravitational anomalies to topological orders in higher dimensions.

Findings

Classifies gauge anomalies using cohomology groups and ohomology groups.

Relates gauge anomalies to SPT orders in one-higher dimension.

Connects gravitational anomalies to topological orders.

Abstract

In this paper, we systematically study gauge anomalies in bosonic and fermionic weak-coupling gauge theories with gauge group G (which can be continuous or discrete). We show a very close relation between gauge anomalies and symmetry-protected trivial (SPT) orders [also known as symmetry-protected topological (SPT) orders] in one-higher dimensions. Using such an idea, we argue that, in d space-time dimensions, the gauge anomalies are described by the elements in Free[H^{d+1}(G,R/Z)]\oplus H_\pi^{d+1}(BG,R/Z). The well known Adler-Bell-Jackiw anomalies are classified by the free part of the group cohomology class H^{d+1}(G,R/Z) of the gauge group G (denoted as Free[H^{d+1}(G,\R/\Z)]). We refer other kinds of gauge anomalies beyond Adler-Bell-Jackiw anomalies as nonABJ gauge anomalies, which include Witten SU(2) global gauge anomaly. We introduce a notion of \pi-cohomology group, H_\pi^{d+1}(BG,R/Z), for the classifying space BG, which is an Abelian group and include Tor[H^{d+1}(G,R/Z)] and topological cohomology group H^{d+1}(BG,R/Z) as subgroups. We argue that H_\pi^{d+1}(BG,R/Z) classifies the bosonic nonABJ gauge anomalies, and partially classifies fermionic nonABJ anomalies. Using the same approach that shows gauge anomalies to be connected to SPT phases, we can also show that gravitational anomalies are connected to topological orders (ie patterns of long-range entanglement) in one-higher dimension.