5D AGT conjecture for circular quivers

TL;DR

This paper extends the AGT correspondence to 5D circular quiver gauge theories by relating q-Virasoro conformal blocks on elliptic surfaces to instanton partition functions and Shiraishi functions, confirming their equivalence.

Contribution

It introduces a novel integral representation for q-Virasoro conformal blocks on elliptic surfaces and verifies their correspondence with 5D gauge theory partition functions.

Findings

Confirmed equivalence between conformal blocks and gauge theory partition functions.

Extended AGT correspondence to 5D circular quiver theories.

Established connection with Shiraishi functions for degenerate cases.

Abstract

The best way to represent generic conformal blocks is provided by the free-field formalism, where they acquire a form of multiple Dotsenko-Fateev-like integrals of the screening operators. Degenerate conformal blocks can be described by the same integrals with special choice of parameters. Integrals satisfy various recurrent relations, which for the special choice of parameters reduce to closed equations. This setting is widely used in explaining the AGT relation, because similar integral representations exist also for Nekrasov functions. We extend this approach to the case of q-Virasoro conformal blocks on elliptic surface -- generic and degenerate. For the generic case, we check equivalence with instanton partition function of a 5d circular quiver gauge theory. For the degenerate case, we check equivalence with partition function of a defect in the same theory, also known as the Shiraishi function. We find agreement in both cases. This opens a way to re-derive the sophisticated equation for the Shiraishi function as the equation for the corresponding integral, what seems straightforward, but remains technically involved and is left for the future.