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

A. Morozov, Sh. Shakirov

TL;DR

The authors study the matrix-model realization of the AGT correspondence by comparing two equivalent descriptions of the non-Gaussian β-ensemble partition function $Z_{DF}$ in the DV phase: (i) a contour-based, Dotsenko-Fateev (DF) approach yielding a $q$-expansion in the multi-Penner potential and Virasoro conformal blocks, and (ii) a quasiclassical, CIV-DV expansion in filling numbers $N_a$ and parameter $q$, valid at fixed $N_a$ and exact in $q$. They focus on the simplest nontrivial case of a 3-Penner potential, deriving a double expansion in $( ext{ħ},q)$ from the DV method and a corresponding $( ext{ħ},q)$-series from the contour/CFT approach, and show complete agreement of coefficients in the overlapping regime. The paper also develops the generalized CIV-DV prepotential for polynomial and logarithmic potentials, extending to arbitrary β and higher genera, with explicit formulas up to genus 3/2 and systematicWard-identity-based calculations of Gaussian correlators. A major result is the demonstrated equivalence $Z^{(CFT)} = Z^{(DV)}$ for the 3-Penner case, reinforcing the matrix-model perspective on AGT and providing a bridge to Seiberg-Witten-type structures via the generalized CIV-DV prepotential. The appendices elaborate the polynomial-potential case, providing comprehensive β-deformed, higher-genus CIV-DV prepotentials and illustrating homogeneity properties at genus zero and beyond.

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.

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

TL;DR

The authors study the matrix-model realization of the AGT correspondence by comparing two equivalent descriptions of the non-Gaussian β-ensemble partition function in the DV phase: (i) a contour-based, Dotsenko-Fateev (DF) approach yielding a -expansion in the multi-Penner potential and Virasoro conformal blocks, and (ii) a quasiclassical, CIV-DV expansion in filling numbers and parameter , valid at fixed and exact in . They focus on the simplest nontrivial case of a 3-Penner potential, deriving a double expansion in from the DV method and a corresponding -series from the contour/CFT approach, and show complete agreement of coefficients in the overlapping regime. The paper also develops the generalized CIV-DV prepotential for polynomial and logarithmic potentials, extending to arbitrary β and higher genera, with explicit formulas up to genus 3/2 and systematicWard-identity-based calculations of Gaussian correlators. A major result is the demonstrated equivalence for the 3-Penner case, reinforcing the matrix-model perspective on AGT and providing a bridge to Seiberg-Witten-type structures via the generalized CIV-DV prepotential. The appendices elaborate the polynomial-potential case, providing comprehensive β-deformed, higher-genus CIV-DV prepotentials and illustrating homogeneity properties at genus zero and beyond.

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.

Paper Structure

This paper contains 19 sections, 154 equations, 1 figure.

Table of Contents

  1. Introduction
  2. 1)
  3. 2)
  4. 1) vs. 2)
  5. A toy model: B-function
  6. $\hbar$-expansion for the generic case
  7. $\hbar$-expansion for the 3-Penner case
  8. $q$-expansion for the 3-Penner case. Comparison
  9. Conclusion
  10. $\bullet$
  11. $\bullet$
  12. Appendix. $\hbar$-expansion for the polynomial case. Generalized CIV-DV prepotential
  13. The case of $\beta = 1$ and genus zero: ordinary CIV-DV prepotential
  14. Generalisation to $\beta = 1$ and higher genera
  15. Generalisation to arbitrary $\beta$ and genus zero
  16. ...and 4 more sections

Figures (1)

  • Figure 1: Filling numbers $N_i$ describe distribution of eigenvalues between critical points $\mu_i$ of $V(z)$.