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.