On AGT relation in the case of U(3)

TL;DR

This paper investigates the AGT relation for 4-point conformal blocks in the W_3 algebra, confirming its validity under specific null-vector restrictions using Nekrasov functions.

Contribution

It demonstrates that the AGT relation holds for W_3 algebra conformal blocks when null-vector restrictions are applied, extending the understanding of AGT correspondence.

Findings

AGT relation confirmed for W_3 algebra with null-vector restrictions

Explicit checks support the validity of the relation in this case

Nekrasov functions are sufficient under the imposed restrictions

Abstract

We consider the AGT relation, expressing conformal blocks for the Virasoro and W-algebras in terms of Nekrasov's special functions, in the simplest case of the 4-point functions for the first non-trivial W_3 algebra. The standard set of Nekrasov functions is sufficient only if additional null-vector restriction is imposed on a half of the external $W$-primaries and this is just the case when the conformal blocks are fully dictated by W-symmetry and do not depend on a particular model. Explicit checks confirm that the AGT relation survives in this restricted case, as expected.