On AGT relations with surface operator insertion and a stationary limit of beta-ensembles

We present a summary of what is currently known about of the AGT relations for conformal blocks with the additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, in which the conformal blocks satisfy hypergeometric-type differential equations in the position of the degenerate operator. Special attention is devoted to the representation of the conformal block through using 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 Schrödinger/Baxter equations, considered earlier in this context, directly follow from the differential equations for the degenerate conformal block. This approach provides a powerful method for the evaluation of the single-deformed prepotentials, and even for the Seiberg–Witten prepotentials themselves. We primarily concentrate on the representative case of the insertion into the four-point block on a sphere and the one-point block on a torus.