Toric 2-group anomalies via cobordism

Abstract

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|𝔾| where 𝔾 is any 2-group whose 0-form and 1-form symmetry parts are both U(1), and |𝔾| is the geometric realisation of the nerve of the 2-group 𝔾. By leveraging a variety of algebraic methods, we show that $$ {\varOmega}_5^{\textrm{Spin}}\left(B\left|\mathbbm{G}\right|\right)\cong \mathbb{Z}/m $$ Ω 5 Spin B G ≅ ℤ / m where m is the modulus of the Postnikov class for 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) 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.