Instantons to the people: the power of one-form symmetries

TL;DR

This paper demonstrates that the non-perturbative dynamics of certain supersymmetric gauge theories can be fully understood through RG equations of surface operators linked to one-form symmetries, modeled by a non-autonomous Toda chain.

Contribution

It introduces a systematic method to compute multi-instanton corrections using RG equations of surface operators, connecting non-perturbative dynamics to integrable systems.

Findings

Derived recurrence relations for instanton corrections.

Applied method to compute up to two-instantons for E6 and G2 gauge groups.

Established a link between surface operator RGEs and non-autonomous Toda chains.

Abstract

We show that the non-perturbative dynamics of $\mathcal{N}=2$ super Yang-Mills theories in a self-dual $\Omega$-background and with an arbitrary simple gauge group is fully determined by studying renormalization group equations of vevs of surface operators generating one-form symmetries. The corresponding system of equations is a {\it non-autonomous} Toda chain, the time being the RG scale. We obtain new recurrence relations which provide a systematic algorithm computing multi-instanton corrections from the tree-level one-loop prepotential as the asymptotic boundary condition of the RGE. We exemplify by computing the $E_6$ and $G_2$ cases up to two-instantons.