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

Xiao-Gang Wen

TL;DR

This work develops a π‑cohomology–based framework linking gauge anomalies in $d$‑dimensional weakly coupled theories to symmetry‑protected topological (SPT) orders in $(d+1)$ dimensions, with ABJ anomalies arising from the free part of $H^{d+1}(G,\mathbb{R}/\mathbb{Z})$ and nonABJ anomalies arising from the torsion and the new π‑cohomology class $\mathscr{H}_\pi^{d+1}(BG,\mathbb{R}/\mathbb{Z})$; it unifies descriptions for bosonic and fermionic cases and extends to gravitational anomalies via boundary topological orders. The framework gives concrete, computable classifications for a wide range of examples (e.g., bosonic/fermionic $Z_n$, $U(1)$, and mixed groups) and shows how anomaly inflow from $(d+1)$‑D bulk terms accounts for boundary gauge non‑invariance. It also provides a nonperturbative lattice construction to define anomaly‑free chiral gauge theories by stacking SPT states and gauging on‑site symmetries, and discusses the limitations of the π‑cohomology approach for some fermionic anomalies. Overall, the work reframes anomalies as obstructions that are captured by higher‑dimensional topological data, with implications for both the classification of anomalies and the nonperturbative realization of chiral gauge theories.

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_π^{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 π-cohomology group, H_π^{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_π^{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.

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

TL;DR

This work develops a π‑cohomology–based framework linking gauge anomalies in ‑dimensional weakly coupled theories to symmetry‑protected topological (SPT) orders in dimensions, with ABJ anomalies arising from the free part of and nonABJ anomalies arising from the torsion and the new π‑cohomology class ; it unifies descriptions for bosonic and fermionic cases and extends to gravitational anomalies via boundary topological orders. The framework gives concrete, computable classifications for a wide range of examples (e.g., bosonic/fermionic , , and mixed groups) and shows how anomaly inflow from ‑D bulk terms accounts for boundary gauge non‑invariance. It also provides a nonperturbative lattice construction to define anomaly‑free chiral gauge theories by stacking SPT states and gauging on‑site symmetries, and discusses the limitations of the π‑cohomology approach for some fermionic anomalies. Overall, the work reframes anomalies as obstructions that are captured by higher‑dimensional topological data, with implications for both the classification of anomalies and the nonperturbative realization of chiral gauge theories.

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_π^{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 π-cohomology group, H_π^{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_π^{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.

Paper Structure

This paper contains 39 sections, 96 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. A general discussion of gauge anomalies
  3. Study gauge anomalies in one-higher dimension and in zero-coupling limit
  4. Gauge anomalies and SPT states
  5. Gauge anomalies and gauge topological term in $(d+1)$ dimensions
  6. Simple examples of nonABJ gauge anomalies
  7. Bosonic $Z_2$ gauge anomaly in 1+1D
  8. Bosonic $Z_n$ gauge anomalies in 1+1D
  9. Bosonic $Z_2\times Z_2\times Z_2$ gauge anomalies in 1+1D
  10. Fermionic $Z_2\times Z_2$ gauge anomalies in 1+1D
  11. Bosonic $U(1)$ gauge anomalies in 2+1D
  12. Fermionic $U(1)$ gauge anomalies in 2+1D
  13. $U(1)\times [U(1)\rtimes Z_2]$ gauge anomalies in 2+1D
  14. Cohomology description
  15. Continuous gauge anomalies
  16. ...and 24 more sections

Figures (3)

  • Figure 1: (Color online) A point defect in 2D looks like a boundary of an effective 1D system, if we wrap the 2D space into a cylinder.
  • Figure 2: (Color online) A $Z_2$ gauge configuration with two identical holes on a torus that contains a unit of $Z_2$ flux in each hole. The $Z_2$ link variables are equal to $-1$ on the crossed links and $1$ on other links. If the 1+1D bosonic $Z_2$ gauge theory on the edge of one hole is anomalous, then such a $Z_2$ gauge configuration induces half unit of total $Z_2$ charge on the edge (representing a $Z_2$ gauge anomaly). Braiding those holes around each other reveals the fractional statistics of the holes. The edge states for one hole are degenerate with $\pm 1/2$$Z_2$ charge if there is a time-reversal symmetry.
  • Figure 3: (Color online) (a) A SPT state described by a cocycle $\nu \in \cH^{d+1}(G,\R/\Z)$ in $(d+1)$-dimensional space-time. After "gauging" the on-site symmetry $G$, we get a bosonic chiral gauge theory on one boundary and the "mirror" of the bosonic chiral gauge theory on the other boundary. (b) A stacking of a few SPT states in $(d+1)$-dimensional space-time described by cocycles $\nu_i$. If $\sum_i \nu_i=0$, then after "gauging" the on-site symmetry $G$, we get a anomaly-free chiral gauge theory on one boundary. We also get the "mirror" of the anomaly-free chiral gauge theory on the other boundary, which can be gapped without breaking the "gauge symmetry".