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

Understand

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.