The matrix model version of AGT conjecture and CIV-DV prepotential

TL;DR

This paper compares two different expansion methods for conformal blocks and prepotentials in beta-ensembles, demonstrating their agreement in overlapping regimes and analyzing their distinct integral representations.

Contribution

It establishes the equivalence of two expansion approaches for conformal blocks and prepotentials, extending the CIV-DV prepotential to arbitrary beta and non-polynomial potentials.

Findings

The two expansions coincide in the overlapping region.

Different integral contours are used in each method, yet yield consistent results.

The formulas describe the lowest terms of the q_a-expansion and N_a-expansion.

Abstract

Recently exact formulas were provided for partition function of conformal (multi-Penner) beta-ensemble in the Dijkgraaf-Vafa phase, which, if interpreted as Dotsenko-Fateev correlator of screenings and analytically continued in the number of screening insertions, represents generic Virasoro conformal blocks. Actually these formulas describe the lowest terms of the q_a-expansion, where q_a parameterize the shape of the Penner potential, and are exact in the filling numbers N_a. At the same time, the older theory of CIV-DV prepotential, straightforwardly extended to arbitrary beta and to non-polynomial potentials, provides an alternative expansion: in powers of N_a and exact in q_a. We check that the two expansions coincide in the overlapping region, i.e. for the lowest terms of expansions in both q_a and N_a. This coincidence is somewhat non-trivial, since the two methods use different integration contours: integrals in one case are of the B-function (Euler-Selberg) type, while in the other case they are Gaussian integrals.