On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles

TL;DR

This paper explores the AGT relations involving surface operators, analyzing conformal blocks with degenerate operators, their representation via beta-ensembles, and the asymptotic behavior leading to deformed Seiberg-Witten prepotentials and Nekrasov functions.

Contribution

It provides a novel connection between conformal blocks with degenerate insertions and beta-ensemble resolvents, deriving asymptotics and differential equations for evaluating deformed prepotentials.

Findings

Conformal blocks with degenerate operators satisfy hypergeometric-type equations.

The asymptotics of conformal blocks relate to the generating differential of deformed Seiberg-Witten theory.

Differential equations for conformal blocks lead to evaluation methods for prepotentials.

Abstract

We present a summary of current knowledge about the AGT relations for conformal blocks with additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, when the conformal blocks satisfy hypergeometric-type differential equations in position of the degenerate operator. A special attention is devoted to representation of conformal block through the beta-ensemble resolvents and to its asymptotics in the limit of large dimensions (both external and intermediate) taken asymmetrically in terms of the deformation epsilon-parameters. The next-to-leading term in the asymptotics defines the generating differential in the Bohr-Sommerfeld representation of the one-parameter deformed Seiberg-Witten prepotentials (whose full two-parameter deformation leads to Nekrasov functions). This generating differential is also shown to be the one-parameter version of the single-point resolvent for the corresponding beta-ensemble, and its periods in the perturbative limit of the gauge theory are expressed through the ratios of the Harish-Chandra function. The Shr\"odinger/Baxter equations, considered earlier in this context, directly follow from the differential equations for the degenerate conformal block. This provides a powerful method for evaluation of the single-deformed prepotentials, and even for the Seiberg-Witten prepotentials themselves. We mostly concentrate on the representative case of the insertion into the four-point block on sphere and one-point block on torus.