Higher Anomalies, Higher Symmetries, and Cobordisms II: Lorentz Symmetry Extension and Enriched Bosonic/Fermionic Quantum Gauge Theory

TL;DR

This paper classifies all possible Lorentz symmetry extensions and anomalies in quantum field theories using cobordism, introduces new symmetry groups, and explores their implications for bosonic and fermionic gauge theories in various dimensions.

Contribution

It systematically enumerates symmetry extensions beyond familiar groups, introduces the EPin group, and classifies all possible SPTs and anomalies up to 5 dimensions using cobordism.

Findings

Introduction of the EPin group and exotic symmetry groups.

Complete classification of SPTs and 't Hooft anomalies in dimensions up to 5.

Discovery of new fermionic Yang-Mills theories with applications to condensed matter systems.

Abstract

We systematically study Lorentz symmetry extensions in quantum field theories (QFTs) and their 't Hooft anomalies via cobordism. The total symmetry $G'$ can be expressed in terms of the extension of Lorentz symmetry $G_L$ by an internal global symmetry $G$ as $1 \to G \to G' \to G_L \to 1$. By enumerating all possible $G_L$ and symmetry extensions, other than the familiar SO/Spin/O/Pin$^{\pm}$ groups, we introduce a new EPin group (in contrast to DPin), and provide natural physical interpretations to exotic groups E($d$), EPin($d$), (SU(2)$\times$E(d))/$\mathbb{Z}_2$, (SU(2)$\times$EPin(d))/$\mathbb{Z}_2^{\pm}$, etc. By Adams spectral sequence, we systematically classify all possible $d$d Symmetry Protected Topological states (SPTs as invertible TQFTs) and $(d-1)$d 't Hooft anomalies of QFTs by co/bordism groups and invariants in $d\leq 5$. We further gauge the internal $G$, and study Lorentz symmetry-enriched Yang-Mills theory with discrete theta terms given by gauged SPTs. We not only enlist familiar bosonic Yang-Mills but also discover new fermionic Yang-Mills theories (when $G_L$ contains a graded fermion parity $\mathbb{Z}_2^F$), applicable to bosonic (e.g., Quantum Spin Liquids) or fermionic (e.g., electrons) condensed matter systems. For a pure gauge theory, there is a one form symmetry $I_{[1]}$ associated with the center of the gauge group $G$. We further study the anomalies of the emergent symmetry $I_{[1]}\times G_L$ by higher cobordism invariants as well as QFT analysis. We focus on the simply connected $G=$SU(2) and briefly comment on non-simply connected $G=$SO(3), U(1), other simple Lie groups, and Standard Model gauge groups (SU(3)$\times$SU(2)$\times$U(1))/$\mathbb{Z}_q$. We comment on SPTs protected by Lorentz symmetry, and the symmetry-extended trivialization for their boundary states.