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

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 theorem, to incorporate higher-groups and higher classifying spaces. We provide many examples of bordism groups with a generic H-structure manifold with a (d + 1)-th higher-group C, and their (d+ 1)d bordism invariants, which systematically classify anomalies of dd spacetime dimensions perturbative (e.g. chiral fermions [originated from Adler-Bell-Jackiw] or bosons with U(1) symmetry in any even d) and non-perturbative global anomalies (e.g. Witten anomaly and the new SU(2) anomaly in 4d and 5d). Suitable H such as SO/Spin/O/Pin(+/-) 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 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, which include higher symmetric generalization of time-reversal invariant topological insulators/superconductors. Examples of higher-SPT's anomalous boundary theories include strongly coupled non-Abelian Yang-Mills (YM) gauge theories and sigma models, complementary to physics obtained in [arXiv:1810. 00844, 1812.11955, 1812.11968, 1904.00994]. This article is a companion with further detailed calculations supporting other shorter articles.