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By developing a generalized cobordism theory, we explore the higher globalsymmetries and higher anomalies of quantum field theories and interactingfermionic/bosonic systems in condensed matter. Our essential math input is ageneralization of Thom-Madsen-Tillmann spectra, Adams spectral sequence, andFreed-Hopkins's theorem, to incorporate higher-groups and higher classifyingspaces. We provide many examples of bordism groups with a generic $H$-structuremanifold with a higher-group $\mathbb{G}$, and their bordism invariants ---e.g. perturbative anomalies of chiral fermions [originated fromAdler-Bell-Jackiw] or bosons with U(1) symmetry in any even spacetimedimensions; non-perturbative global anomalies such as Witten anomaly and thenew 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 withtime-reversal or reflection symmetry on (un)orientable spacetime. Higher 'tHooft anomalies of $d$d live on the boundary of $(d+1)$dhigher-Symmetry-Protected Topological states (SPTs) or symmetric invertibletopological orders (i.e., invertible topological quantum field theories at lowenergy); thus our cobordism theory also classifies and characterizeshigher-SPTs. Examples of higher-SPT's anomalous boundary theories includestrongly coupled non-Abelian Yang-Mills gauge theories and sigma models,complementary to physics obtained in [arXiv:1810.00844, 1812.11955, 1812.11968,1904.00994].

annals of mathematical sciences and applications

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