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

TL;DR

The paper develops a generalized cobordism framework that extends the Freed–Hopkins approach to include higher groups and higher classifying spaces, enabling the systematic classification of higher-Symmetry-Protected Topological states and their boundary anomalies. By introducing the generalized Thom–Madsen–Tillmann spectrum formalism and leveraging Adams spectral sequence techniques, it ties deformations of symmetric invertible QFTs to torsion cobordism groups $ ext{TP}_d(H imesG)=[MT(H imesG),\Sigma^{d+1}I\Z]_{ ext{tors}}$, and computes explicit low-dimensional (co)bordism groups for a wide range of $H$ and higher-group structures. Warm-up examples connect these cobordism calculations to perturbative chiral anomalies, U(1) SPTs, Witten’s SU(2) anomaly, and a new SU(2) anomaly, clarifying how bulk higher-SPTs encode boundary anomalies. The extensive calculations in sections 5 and 4 provide detailed invariants for ${ ext{O, SO, Spin, Pin}^{ abla}}$ bordisms with classifying spaces and higher fibrations such as $ ext{B}^2 ext{Z}_n$ and $ ext{B PSU}(n)$, offering a practical catalog for identifying and classifying higher-SPTs and their anomalous boundaries. Overall, the work delivers a concrete mathematical framework to classify higher-dimensional anomalies and protected boundary theories in condensed matter and QFT, with broad implications for understanding symmetry, topology, and quantum phases.

Abstract

By developing a generalized cobordism theory, we explore the higher global symmetries and higher anomalies of quantum field theories and interacting fermionic/bosonic systems in condensed matter. Our essential math input is a generalization of Thom-Madsen-Tillmann spectra, Adams spectral sequence, and Freed-Hopkins's theorem, to incorporate higher-groups and higher classifying spaces. We provide many examples of bordism groups with a generic $H$-structure manifold with a higher-group $\mathbb{G}$, and their bordism invariants --- e.g. perturbative anomalies of chiral fermions [originated from Adler-Bell-Jackiw] or bosons with U(1) symmetry in any even spacetime dimensions; non-perturbative global anomalies such as Witten anomaly and the new SU(2) anomaly in 4d and 5d. Suitable $H$ such as SO/Spin/O/Pin$^\pm$ enables the study of quantum vacua of general bosonic or fermionic systems with time-reversal or reflection symmetry on (un)orientable spacetime. Higher 't Hooft anomalies of $d$d live on the boundary of $(d+1)$d higher-Symmetry-Protected Topological states (SPTs) or symmetric invertible topological orders (i.e., invertible topological quantum field theories at low energy); thus our cobordism theory also classifies and characterizes higher-SPTs. Examples of higher-SPT's anomalous boundary theories include strongly coupled non-Abelian Yang-Mills gauge theories and sigma models, complementary to physics obtained in [arXiv:1810.00844, 1812.11955, 1812.11968, 1904.00994].

Figures (79)

• Figure 1: $\mathcal{A}_2(1)$
• Figure 2: Adams chart of $\operatorname{Ext}_{\mathcal{A}_2}^{s,t}(\operatorname{H}^*(M{\rm SO},\mathbb{Z}_2),\mathbb{Z}_2)$
• Figure 3: Adams chart of $\operatorname{Ext}_{\mathcal{A}_3}^{s,t}(\operatorname{H}^*(M{\rm SO},\mathbb{Z}_3),\mathbb{Z}_3)$
• Figure 4: Adams chart of $\operatorname{Ext}_{\mathcal{A}_2(1)}^{s,t}(\mathbb{Z}_2,\mathbb{Z}_2)$. The dashed arrows indicate the possible differentials.
• Figure 5: Adams chart of $\operatorname{Ext}_{\mathcal{A}_2(1)}^{s,t}(L_2,\mathbb{Z}_2)$. The arrows indicate the differential $d_1$, the dashed line indicates the extension.

Theorems & Definitions (132)

• Definition 1
• Definition 2
• Definition 3
• Example 4
• Example 5
• Example 6
• Definition 7
• Example 8
• Definition 9: Axioms
• Theorem 10: Change of rings/Frobenius reciprocity