[Paper Review] (3+1)D Anomalous Twisted Gauge Theories with Global Symmetry

Peng Ye|arXiv (Cornell University)|Oct 27, 2016

Black Holes and Theoretical Physics3 citations

TL;DR

This paper demonstrates that in (3+1)D twisted gauge theories with discrete Abelian gauge group $G_g$, coupling a global symmetry $G_s$ to topological currents $\star\frac{1}{2\pi}db^I$ leads to an anomaly: the symmetry-enriched topological order (SET) is anomalous and cannot realize in (3+1)D alone, requiring a (4+1)D bulk to cancel the gauge anomaly upon gauging $G_s$. The anomaly is diagnosed via a concrete example, showing that $G_s$-gauging produces a gauge anomaly requiring a (4+1)D topological phase for cancellation.

ABSTRACT

In (3+1)D twisted gauge theories, global symmetry may be imposed on topological currents $\star\frac{1}{2\pi}db^I$ in a hydrodynamical way ($I=1,2,\cdots$, $\{b^I\}$ is a set of Kalb-Ramond gauge fields). This methodology has been applied before in the Chern-Simons theory of fractional quantum Hall liquids. We find that, in some twisted gauge theories (with discrete Abelian gauge group $G_g$), implementing a global symmetry (denoted by $G_s$) is always inconsistent. There are two consequences. First, the symmetry-enriched topological order (SET) of the ground state is anomalous, which cannot exist in (3+1)D system alone. It can exist as a boundary of 4+1D topological phases. Second, if $G_s$ is fully gauged, the resulting new gauge theory has gauge anomaly. A (4+1)D topological phase is required to cancel this anomaly. We elaborate this phenomenon via a concrete example.

Motivation & Objective

• To investigate the consistency of global symmetries in (3+1)D twisted gauge theories with discrete Abelian gauge groups.
• To determine whether global symmetries can be consistently coupled to topological currents $\star\frac{1}{2\pi}db^I$ in such theories.
• To identify conditions under which symmetry enrichment leads to anomalies that cannot be realized in (3+1)D alone.
• To demonstrate that gauging the global symmetry results in a gauge anomaly requiring a (4+1)D topological phase for cancellation.

Proposed method

• The authors analyze twisted gauge theories in (3+1)D with Kalb-Ramond fields $\{b^I\}$ and discrete Abelian gauge group $G_g$, focusing on topological currents $\star\frac{1}{2\pi}db^I$.
• They impose a global symmetry $G_s$ on these currents in a hydrodynamical manner, treating them as conserved currents coupled to background gauge fields.
• Using the framework of anomaly inflow, they show that the resulting symmetry-enriched topological order (SET) is anomalous and cannot exist as a standalone (3+1)D phase.
• They demonstrate that gauging $G_s$ leads to a gauge anomaly, which is canceled only by coupling to a (4+1)D topological phase.
• A concrete example is constructed to illustrate the anomaly mechanism, showing explicit field content and action structure.
• The analysis relies on the interplay between gauge fields, global symmetries, and topological invariants, with anomaly diagnosis via the descent of characteristic classes.

Experimental results

Research questions

• RQ1 Can a global symmetry $G_s$ be consistently implemented in a (3+1)D twisted gauge theory with discrete Abelian gauge group $G_g$?
• RQ2 Under what conditions does the coupling of $G_s$ to topological currents $\star\frac{1}{2\pi}db^I$ lead to an anomalous symmetry-enriched topological order (SET)?
• RQ3 Why is the SET phase not realizable in (3+1)D alone, and what is the role of a (4+1)D bulk in canceling the anomaly?
• RQ4 What happens when the global symmetry $G_s$ is gauged, and how does the resulting gauge anomaly relate to higher-dimensional topological phases?
• RQ5 How can a concrete example be constructed to realize and diagnose the anomaly in this framework?

Key findings

• The symmetry-enriched topological order (SET) arising from coupling $G_s$ to topological currents $\star\frac{1}{2\pi}db^I$ is anomalous and cannot exist as a standalone (3+1)D system.
• The anomaly arises because the global symmetry $G_s$ cannot be consistently gauged without introducing a gauge anomaly in (3+1)D.
• Upon gauging $G_s$, the resulting gauge theory exhibits a gauge anomaly that requires a (4+1)D topological phase to cancel.
• The (4+1)D topological phase is necessary to cancel the anomaly via anomaly inflow, ensuring consistency of the gauged theory.
• A concrete example is constructed where the anomaly is explicitly diagnosed, showing the necessity of the higher-dimensional bulk.
• The mechanism is rooted in the topological nature of the currents and the failure of global symmetry realization in the absence of a higher-dimensional anomaly-canceling bulk.