Annals of Mathematical Sciences and Applications

Volume 4 (2019)

Number 2

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

Pages: 107 – 311

DOI: https://dx.doi.org/10.4310/AMSA.2019.v4.n2.a2

Authors

Zheyan Wan (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China; and School of Mathematical Sciences, University of Science and Technology of China, Hefei, China)

Juven Wang (School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, U.S.A.; and Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.)

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 theorem, to incorporate higher-groups and higher classifying spaces. We provide many examples of bordism groups with a generic $H$-structure manifold with a $(d + 1)\mathrm{-th}$ higher-group $\mathbb{G}$, and their $(d + 1)\mathrm{d}$ bordism invariants, which systematically classify anomalies of $d\mathrm{d}$ spacetime dimensions — perturbative (e.g. chiral fermions [originated from Adler–Bell–Jackiw] or bosons with $\mathrm{U}(1)$ symmetry in any even d) and non-perturbative global anomalies (e.g. Witten anomaly and the new $\mathrm{SU}(2)$ anomaly in $4\mathrm{d}$ and $5\mathrm{d}$). Suitable $H$ such as $\mathrm{SO / Spin / O / Pin^{\pm}}$ 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 $d\mathrm{d}$ live on the boundary of $(d + 1)\mathrm{d}$ 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.

Keywords

quantum field theory, gauge theory, quantum anomaly, ’t Hooft anomaly, cohomology theory, cobordism theory, topological insulators/superconductors, symmetry protected topological states, invertible topological orders, invertible topological quantum field theory, spectral sequences

2010 Mathematics Subject Classification

Primary 18G40, 55N22. Secondary 57R56, 81T13.

Received 1 July 2019

Accepted 19 July 2019

Published 2 October 2019