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 symmetries, 2-group symmetries, and their 't$~$Hooft anomalies. We show that this data can be extracted directly from the blow-up equations that govern the instanton partition functions of such theories on the $Ω$-background. The central object is the classical prefactor $\exp(-V_n)$ that weights 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, as well as 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, which we determine 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, for which we obtain new results: notably, 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 anomalies involving the 1-form and flavour symmetries.