The Power of Nekrasov Functions

TL;DR

This paper explores the use of Nekrasov functions as a basis for decomposing conformal blocks, providing insights into the AGT conjecture and its relation to hypergeometric integrals and free field formalism.

Contribution

It demonstrates the effectiveness of Nekrasov functions in decomposing conformal blocks across various chiral algebras and offers a proof of the AGT relation for specific external states.

Findings

Nekrasov functions serve as a natural basis for conformal block decomposition.

The AGT conjecture relates to hypergeometric integrals and free field formalism.

Proof of AGT relation for certain external states using hypergeometric conformal blocks.

Abstract

The recent AGT suggestion to use the set of Nekrasov functions as a basis for a linear decomposition of generic conformal blocks works very well not only in the case of Virasoro symmetry, but also for conformal theories with extended chiral algebra. This is rather natural, because Nekrasov functions are introduced as expansion basis for generalized hypergeometric integrals, very similar to those which arise in expansion of Dotsenko-Fateev integrals in powers of alpha-parameters. Thus, the AGT conjecture is closely related to the old belief that conformal theory can be effectively described in the free field formalism, and it can actually be a key to clear formulating and proof this long-standing hypothesis. As an application of this kind of reasoning we use knowledge of the exact hypergeometric conformal block for complete proof of the AGT relation for a restricted class of external states.