Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders (1303.1803v4)

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.

Understanding Gauge Anomalies Through Symmetry-Protected Topological Orders

The paper by Xiao-Gang Wen addresses the classification of gauge anomalies within bosonic and fermionic gauge theories using symmetry-protected topological (SPT) orders. The exploration of the intricate relationships between gauge anomalies, traditionally understood as obstructions to well-defined gauge theories, and SPT phases offers a novel approach to classify and understand these phenomena.

Key Concepts and Structure

Wen introduces a systematic approach to studying gauge anomalies by leveraging SPT phases in one higher dimension. Gauge anomalies, which manifest as a failure of gauge invariance, are closely related to the non-on-site symmetries of SPT states. These symmetries, when gauged, can lead to anomalous gauge theories. The classification relies on group cohomology classes $\cH^{d+1}(G,\R/\Z)$. The paper distinguishes the well-established Adler-Bell-Jackiw (ABJ) anomalies from the lesser-known nonABJ anomalies, the latter encompassing Witten's global $SU(2)$ anomaly and other anomalies arising from discrete gauge groups.

Numerical Results and Claims

• Classification of Anomalies: Wen proposes that bosonic gauge anomalies for gauge group $G$ in $d$ space-time dimensions are classified by Free$[\cH^{d+1}(G,)]\oplus H_\pi^{d+1}(BG,\R/\Z)$. Fermionic gauge anomalies are partially described by the same framework.
• Group Cohomology and $\pi$-Cohomology: Through careful analysis, it is shown that $\pi$-cohomology offers a broader classification than group cohomology, potentially capturing aspects beyond traditional approaches. However, it remains speculative whether it perfectly encapsulates all anomalies, particularly fermionic ones.

Implications and Future Directions

The implications of this research are profound. By connecting gauge anomalies to SPT orders, the paper provides a framework where each gauge anomaly corresponds to a distinct SPT phase in one additional dimension. This linkage offers theoretical insights into the nature of anomalies within quantum field theories and suggests potential directions for exploring non-perturbative definitions of chiral gauge theories. The work encourages further investigation into the role of intrinsic topological order in understanding gravitational anomalies.

Concluding Thoughts

Wen's exploration into gauge anomalies using topological approaches opens new avenues for comprehending complex theoretical physics concepts. The integration of SPT orders into anomaly classification not only enriches the theoretical landscape but also posits fresh questions regarding the nature of entanglement, symmetry, and topology in higher-dimensional quantum field theories. This paper is a catalyst for future work aimed at unraveling the subtleties of gauge theories, symmetry-breaking phenomena, and their implications across various dimensions in physics.