The Power of Nekrasov Functions

TL;DR

The paper investigates extending the AGT correspondence from Virasoro to $W^{(N)}$ algebras by showing that Nekrasov functions can serve as a natural, analytic-continuation basis for conformal blocks, analogous to generalized hypergeometric structures arising from Dotsenko-Fateev integrals. It develops the Nekrasov construction for $SU(N)$ quivers, including the treatment of chiral/anti-chiral contributions, and demonstrates that Fateev-Litvinov blocks arise when external states are chosen to isolate a single diagram, thereby proving AGT in this restricted setting. Explicit checks are performed for $SU(2)$/Virasoro and $SU(3)$/W^{(3)} cases at low levels, confirming that the corresponding conformal blocks are reproduced by the Nekrasov expansions and that non-chiral corrections account for Virasoro deviations. The work also discusses parameter-count mismatches and the need for potential extensions of the Nekrasov basis to fully resolve non-Virasoro AGT relations, outlining directions for future proofs and applications in conformal field theory and related string-theoretic frameworks.

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.

Figures (1)

• Figure 1: Three examples of Young diagrams: a generic type diagram $Y = [64332221]$, a column $Y=[6]$ and a chain $Y=[11111111] = [1^8]$. With the chains and columns are associated the chiral and anti-chiral Nekrasov functions respectively. Horizontal and vertical directions are called $+$ and $-$, because the "natural" position of the Young diagram is rotated by $45^\circ$ counterclockwise.