2018/12/31 by Zheyan Wan, Juven Wang · 5 citations
Physics and Astronomy · Mathematics · #hep-th #cond-mat.str-el #hep-ph #math.AT #math.GT
paper · pdf · doi:10.4310/amsa.2019.v4.n2.a2
published as Annals of Mathematical Sciences and Applications, Vol. 4, No. 2, 107-311 (2019) · 174 pages, 79 figures, 20 tables, 132 theorems. A companion article with further detailed calculations supporting: arXiv:1810.00844, arXiv:1812.11955, arXiv:1812.11968, arXiv:1904.00994. A Primer introduction: arXiv:1711.11587
arxiv created 2019/07/11 · arxiv updated 2019/10/04
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 \mathbbG, 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. 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. Examples of higher-SPT's anomalous boundary theories include strongly coupled non-Abelian Yang-Mills gauge theories and sigma models, complementary to physics obtained in [arXiv:1810.00844, 1812.11955, 1812.11968, 1904.00994].