Superintegrability as the hidden origin of Nekrasov calculus

TL;DR

This paper proposes that superintegrability in matrix models explains the emergence of Nekrasov functions and the AGT relations, providing a conceptual foundation for these connections beyond technical tricks.

Contribution

It introduces the idea that superintegrability underpins Nekrasov functions in matrix models, extending factorization properties to multi-logarithmic models and generalizing to multi-matrix and non-Gaussian models.

Findings

Superintegrability explains Nekrasov functions in matrix models.

Factorization extends from single characters to bilinear and multi-character averages.

The approach generalizes to multi-matrix models and other phases, supporting AGT relations.

Abstract

Once famous and a little mysterious, AGT relations between Nekrasov functions and conformal blocks are now understood as the Hubbard-Stratanovich duality in the Dijkgraaf-Vafa (DV) phase of a peculiar Dotsenko-Fateev multi-logarithmic matrix model. However, it largely remains a collection of somewhat technical tricks, lacking a clear and generalizable conceptual interpretation. Our new claim is that the Nekrasov functions emerge in matrix models as a straightforward implication of superintegrability, factorization of peculiar matrix model averages. Recently, we demonstrated that, in the Gaussian Hermitian model, the factorization property can be extended from averages of single characters to their bilinear combinations. In this paper, we claim that this is true also for multi-logarithmic matrix models, where factorized are just the point-split products of two characters. It is this enhanced superintegrability that is responsible for existence of the Nekrasov functions and the AGT relations. This property can be generalized both to multi-matrix models, thus leading to AGT relations for multi-point conformal blocks, and to DV phases of other non-Gaussian models.