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arXiv · 1812.11967

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

Zheyan Wan·Juven Wang
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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].

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Zheyan Wan, Juven Wang. 2018-12-31. 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. https://doi.org/10.4310/amsa.2019.v4.n2.a2

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