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

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.