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

A

bstract

Five-dimensional

= 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 Ω-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 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 SU(4) and USp(4) with antisymmetric hypermultiplets and Spin(7) and 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

$$ {B}_N^{\left(1,2,3\right)} $$ B N 1 2 3

families, the cubic 1-form anomalies of the rank-two theories

$$ {\mathbb{P}}^2\cup {\mathbbm{F}}_3 $$ ℙ 2 ∪ F 3

and

$$ {\mathbb{P}}^2\cup {\mathbbm{F}}_6 $$ ℙ 2 ∪ F 6

, and several mixed 1-form–flavour and 1-form–SU(2) _R anomalies.