Toric 2-group anomalies via cobordism

Abstract

2-group symmetries arise in physics when a 0-form symmetry $G^{[0]}$ and a 1-form symmetry $H^{[1]}$ intertwine, forming a generalised group-like structure. Specialising to the case where both $G^{[0]}$ and $H^{[1]}$ are compact, connected, abelian groups (i.e. tori), we analyse anomalies in such `toric 2-group symmetries' using the cobordism classification. As a warm up example, we use cobordism to study various 't Hooft anomalies (and the phases to which they are dual) in Maxwell theory defined on non-spin manifolds. For our main example, we compute the 5th spin bordism group of $B|\mathbb{G}|$ where $\mathbb{G}$ is any 2-group whose 0-form and 1-form symmetry parts are both $\mathrm{U}(1)$, and $|\mathbb{G}|$ is the geometric realisation of the nerve of the 2-group $\mathbb{G}$. By leveraging a variety of algebraic methods, we show that ${\Omega }_5^{\mathrm{Spin}}(B|\mathbb{G}|)\cong \mathbb{Z}/m$ where $m$ is the modulus of the Postnikov class for $\mathbb{G}$, and we reproduce the expected physics result for anomalies in 2-group symmetries that appear in 4d QED. Moving down two dimensions, we recap that any (anomalous) $\mathrm{U}(1)$ global symmetry in 2d can be enhanced to a toric 2-group symmetry, before showing that its associated local anomaly reduces to at most an order 2 anomaly, when the theory is defined with a spin structure.

arXiv comment65 pages, 11 figures. Includes a mathematical appendix written by Arun Debray. Matches version to be published

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