Generalised global symmetries in 5d $\mathcal{N}=1$ theories from the blow-up equations

Abstract

Five-dimensional $\mathcal{N}=1$ superconformal field theories admit a rich variety of generalised global symmetries, including higher-form and 2-group symmetries and their 't Hooft anomalies. We show that this data can be extracted directly from the blow-up equations governing the instanton partition functions of such theories on the $\Omega$-background. The central object is the classical prefactor $\exp (-V_n)$ weighting each magnetic flux on the blown-up geometry: evaluated on a background for the electric 1-form symmetry, the fractional parts of its exponents encode the cubic self-anomaly of the 1-form symmetry and its mixed anomalies with the instanton, flavour, gravitational, and $\mathrm{SU}(2{)}_R$ symmetries. Combined with the faithful continuous global symmetry of the ultraviolet fixed point, determined from the superconformal index, the same data decides whether the theory possesses a 2-group symmetry or a mixed 't Hooft anomaly. We illustrate the method in gauge theories, including $\mathrm{SU}(4)$ and $\mathrm{USp}(4)$ with antisymmetric hypermultiplets and $\mathrm{Spin}(7)$ and $\mathrm{Spin}(8)$ with vector hypermultiplets, as well as in several families of non-Lagrangian theories. New results include the effective prepotentials of the $B_N$ and $B_N^{(1,2,3)}$ families, the cubic 1-form anomalies of the rank-two theories ${\mathbb{P}}^2\cup {\mathbb{F}}_3$ and ${\mathbb{P}}^2\cup {\mathbb{F}}_6$, and several mixed 1-form--flavour and 1-form--$\mathrm{SU}(2{)}_R$ anomalies.

63 pages, plus appendices. V2: new results and references added. V3: Published in JHEP