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



Zheya Wan (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Juven Wang (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.; and School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, U.S.A.)

Yunqin Zheng (Department of Physics, Princeton University, Princeton, New Jersey, U.S.A.; Kavli Institute, University of Tokyo, Kashiwa, Chiba, Japan; and Institute of Solid State Physics, University of Tokyo, Kashiwa, Chiba, Japan)


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.


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