Higher Anomalies, Higher Symmetries, and Cobordisms I
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 Wan1;2 and Juven Wang3;4 1 1Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China 2School of Mathematical Sciences, USTC, Hefei 230026, China 3School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA 4Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA 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 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± enables arXiv:1812.11967v2 [hep-th] 11 Jul 2019 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 dd 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.
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