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



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.)


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.


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.

We thank Daniel Freed, Meng Guo, Michael Hopkins, Anton Kapustin, Pavel Putrov, and Edward Witten for conversations. JW thanks the collaborators for a previous collaboration on Ref. [35]. JW thanks the participants of Developments in Quantum Field Theory and Condensed Matter Physics (November 5–7, 2018) at Simons Center for Geometry and Physics at SUNY Stony Brook University for giving valuable feedback where this work is reported [71]. JW thanks the feedback from the attendants of IAS seminar [72]. ZW gratefully acknowledges the support from NSFC grants 11431010, 11571329. JW gratefully acknowledges the Corning Glass Works Foundation Fellowship and NSF Grant PHY-1606531. This work is also supported by NSF Grant DMS-1607871 “Analysis, Geometry and Mathematical Physics” and Center for Mathematical Sciences and Applications at Harvard University.

Received 1 July 2019

Accepted 19 July 2019

Published 2 October 2019