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

2019/12/31 by Zheyan Wan, Juven Wang, Yunqin Zheng
Physics and Astronomy · Mathematics · #hep-th #cond-mat.str-el #hep-ph #math.AT

paper · pdf · doi:10.4310/amsa.2020.v5.n2.a2

published as Annals of Mathematical Sciences and Applications, Vol. 5, No. 2, 171-257 (2020) · Sequel to: arXiv:1711.11587, arXiv:1812.11967, arXiv:1904.00994 arXiv:1910.14668, arXiv:1912.13514. Accepted by Annals of Mathematical Sciences and Applications, Vol. 5, No. 2 (2020)

arxiv created 2020/09/03 · arxiv updated 2020/10/14

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 → G → G' → G_L → 1. By enumerating all possible G_L 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 G_L contains a graded fermion parity ℤ₂^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]× 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)×SU(2)×U(1))/ℤ_q. We comment on SPTs protected by Lorentz symmetry, and the symmetry-extended trivialization for their boundary states.

Citations

• Time Reversal, SU(N) Yang-Mills and Cobordisms: Interacting Topological Superconductors/Insulators and Quantum Spin Liquids in 3+1D
• Higher Anomalies, Higher Symmetries, and Cobordisms I: Classification of Higher-Symmetry-Protected Topological States and Their Boundary Fermionic/Bosonic Anomalies via a Generalized Cobordism Theory
• Higher Anomalies, Higher Symmetries, and Cobordisms III: QCD Matter Phases Anew