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

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 GL by an internal global symmetry G as 1 → G → G' → GL → 1. By enumerating all possible GL and symmetry extensions, other than the familiar SO/Spin/O/Pin± 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)×E(d))/Z₂, (SU(2)×EPin(d))/Z₂±, etc. By Adams spectral sequence, we systematically classify all possible dd 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≤ 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 GL contains a graded fermion parity Z₂F), 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]× GL 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)×SU(2)×U(1))/Zq. We comment on SPTs protected by Lorentz symmetry, and the symmetry-extended trivialization for their boundary states.