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

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