Bootstrapping the $S^5$ partition function

TL;DR

This paper proposes a new Higgs branch-like expression for the $S^5$ partition function of $U(N)$ SQCD, using defect partition functions and non-perturbative equations, connecting to $q$-Virasoro CFT structures.

Contribution

It introduces a novel Higgs branch-like formula for the $S^5$ partition function based on defect functions and their relation to $q$-Virasoro conformal blocks.

Findings

The defect partition functions satisfy non-perturbative Schwinger-Dyson equations.

The $S^5$ partition function can be expanded in terms of $q$-Virasoro conformal blocks.

The approach aligns with the BPS/CFT correspondence.

Abstract

We consider $U(N)$ SQCD on $S^5$ and propose a Higgs branch-like expression for its partition function. We support the result by arguing that the knowledge of certain BPS codimension 2 and 4 defects arising from Higgsing is enough to reconstruct the bulk partition function, and that the defect partition functions satisfy a set of non-perturbative Schwinger-Dyson equations. We show that the result is consistent with, and naturally come from, the BPS/CFT perspective. In this language, the defect partition functions are identified with free boson correlators of the $q$-Virasoro modular triple, and the constraint equations with Ward identities satisfied by the corresponding Dotsenko-Fateev $q$-conformal blocks, providing a natural basis to expand the $S^5$ partition function.