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

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.