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

TL;DR

This work extends the framework of Lorentz symmetry extensions in quantum field theories by employing cobordism and Adams spectral sequence methods to classify all consistent $d$-dimensional G'-SPTs and $(d-1)$-dimensional ’t Hooft anomalies for $d\le 5$, with a focus on SU(2) internal symmetry and various Lorentz groups. It introduces new symmetry structures ${\rm E}(d)$ and ${\rm EPin}(d)$, analyzes two principal extension classes for each Lorentz type, and computes the corresponding bordism and cobordism invariants that encode anomalies. By gauging the internal SU(2) symmetry, the authors derive enriched bosonic/fermionic Yang–Mills theories and identify emergent one-form symmetries, showing how SPT data influence gauge dynamics and boundary states. The results illuminate how symmetry-extension can trivialize or realize bulk-boundary interplay, with explicit examples across ${\rm SO}(d)$, ${\rm Spin}(d)$, ${\rm O}(d)$, and Pin$^{\pm}(d)$ contexts, and they provide a roadmap for constructing Lorentz-symmetry-enriched gauge theories with discrete theta terms in various dimensions. The work has implications for topological phases, anomaly inflow, and the classification of both bosonic and fermionic gauge theories in condensed matter and high-energy contexts.

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.

Figures (21)

• Figure 1: $E_2$ page of the Adams spectral sequence with symmetry ${\rm SO}(d)\times {\rm SU}(2)$. The Bordism group $\Omega_*^{{\rm SO} \times {\rm SU}(2)}$ and the invariants can be read off from the this chart.
• Figure 2: The $\mathcal{A}_2(1)$-module structure of $\operatorname{H}^{*+3}(M{\rm SO}(3),\mathbb{Z}_2)$ below degree 5.
• Figure 3: $E_2$ page of the Adams spectral sequence with symmetry $({\rm Spin}(d) \times {\rm SU}(2))/\mathbb{Z}_2$. The Bordism group $\Omega_*^{({{\rm Spin} \times {\rm SU}(2)})/{\mathbb{Z}_2}}$ and the invariants can be read off from the this chart.
• Figure 4: The $\mathcal{A}_2(1)$-module structure of $\operatorname{H}^*(\mathrm{B}{\rm SU}(2),\mathbb{Z}_2)$ below degree 5.
• Figure 5: $E_2$ page of the Adams spectral sequence with symmetry ${\rm Spin}(d) \times {\rm SU}(2)$. The Bordism group $\Omega_*^{{{\rm Spin} \times {\rm SU}(2)}}$ and the invariants can be read off from the this chart.