Bootstrapping 4d $\mathcal{N}=2$ gauge theories: the case of SQCD

TL;DR

This paper establishes exact relations in 4d $ =2$ gauge theories linking flavor current integrals to derivatives of sphere free energy, and uses these to bound CFT data in $SU(2)$ SQCD with $SO(8)$ flavor symmetry.

Contribution

It introduces a novel method combining supersymmetric localization and conformal bootstrap to constrain 4d $ =2$ gauge theories, specifically applying it to $SU(2)$ SQCD.

Findings

Bounds on unprotected scaling dimensions as a function of $ au$

Results match free theory limits

Demonstrates $SL(2,b Z)$ duality and $SO(8)$ triality effects

Abstract

We derive exact relations between certain integrals of the conserved flavor current four point function in 4d $\mathcal{N}=2$ conformal field theories (CFTs) and derivatives of the mass deformed sphere free energy, which can be computed exactly for gauge theories using supersymmetric localization. For conformal gauge theories with flavor groups of rank greater than one, there are at least two such integrated constraints, which can then be combined with the numerical conformal bootstrap to bound CFT data as a function of the complexified gauge coupling $\tau$. We apply this strategy to the case of $SU(2)$ conformal SQCD with flavor group $SO(8)$, where we compute bounds on unprotected scaling dimensions as a function of $\tau$ that match the free theory limit, and exhibit the expected mixing between the action of the $SL(2,\mathbb{Z})$ duality group and $SO(8)$ triality.