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# Annals of Mathematical Sciences and Applications

## Volume 5 (2020)

### Number 2

### Higher anomalies, higher symmetries, and cobordisms II: Lorentz symmetry extension and enriched bosonic / fermionic quantum gauge theory

Pages: 171 – 257

DOI: https://dx.doi.org/10.4310/AMSA.2020.v5.n2.a2

#### Authors

#### Abstract

We systematically study Lorentz symmetry extensions in quantum field theories (QFTs) and their ’t Hooft anomalies via cobordism. The total symmetry $G^\prime$ can be expressed in terms of the extension of Lorentz symmetry $G_\textrm{Lorentz}$ by an internal global symmetry $G$ as $1 \to G \to G^\prime \to G_\textrm{Lorentz} \to 1$. By enumerating all possible $G_\textrm{Lorentz}$ and symmetry extensions, other than the familiar $\mathrm{SO / Spin / O / Pin^\pm}$ groups, we introduce a new $\mathrm{EPin}$ group (in contrast to $\mathrm{DPin}$), and provide natural physical interpretations to exotic groups $\mathrm{E(d)}, \mathrm{EPin(d)}, \mathrm{ (SU(2) \times E(d)) / \mathbb{Z}_2}$, $\mathrm{(SU(2) \times EPin(d)) / \mathbb{Z}^\pm_2}$, etc. By Adams spectral sequence, we systematically classify all possible $d\mathrm{d}$ Symmetry Protected Topological states (SPTs as invertible TQFTs) and $(d-1)\mathrm{d}$ ’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 $G_\textrm{Lorentz}$ contains a graded fermion parity $\mathbb{Z}^F_2$), 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 $I_{[1]} \times G_\textrm{Lorentz}$ by higher cobordism invariants as well as QFT analysis. We focus on the simply connected $G = \mathrm{SU}(2)$ and briefly comment on non-simply connected $\mathrm{G = SO(3)}$, $\mathrm{U(1)}$, other simple Lie groups, and Standard Model gauge groups $(\mathrm{SU}(3) \times SU(2) \times U(1)) / \mathbb{Z}_q$. We comment on SPTs protected by Lorentz symmetry, and the symmetry-extended trivialization for their boundary states.

#### Keywords

algebraic topology, quantum field theory, gauge theory, quantum anomaly, ’t Hooft anomaly, cohomology theory, cobordism theory, topological insulators/superconductors, symmetry protected topological states, invertible topological orders, invertible topological quantum field theory, spectral sequences

Received 7 April 2020

Accepted 3 September 2020

Published 13 October 2020