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Matrix model version of AGT conjecture and generalized Selberg integrals

A. Mironov, Al. Morozov, And. Morozov

TL;DR

This work presents a conceptual derivation of a matrix-model (Dotsenko-Fateev) representation for conformal blocks in 2d CFT that yields a matrix-model version of the AGT conjecture. By combining free-field correlators with Dotsenko-Fateev screenings, the authors construct universal Virasoro structure constants $C_{ abla_1,Y_1; abla_2,Y_2}^{ abla,Y}$ via generalized Selberg integrals $I_Y$, and demonstrate their exact match with the standard CFT $3j$-symbols after an analytic continuation in the screening number. The approach unifies Virasoro intertwiner theory with Selberg/Jackson polynomial frameworks, yielding a constructive path to connect conformal blocks with Nekrasov functions in the DV-matrix-model phase. While the full AGT conjecture is not fully proven in this paper, the matrix-model (DF) route provides a conceptually transparent and computationally workable bridge between CFT data and gauge-theory data, with clear extensions to higher-level blocks and to $W$-algebras. The results suggest deep dualities between DF-type integrals and LNS-type Nekrasov integrals, and offer a concrete program toward a complete proof of AGT.

Abstract

Operator product expansion (OPE) of two operators in two-dimensional conformal field theory includes a sum over Virasoro descendants of other operator with universal coefficients, dictated exclusively by properties of the Virasoro algebra and independent of choice of the particular conformal model. In the free field model, these coefficients arise only with a special "conservation" relation imposed on the three dimensions of the operators involved in OPE. We demonstrate that the coefficients for the three unconstrained dimensions arise in the free field formalism when additional Dotsenko-Fateev integrals are inserted between the positions of the two original operators in the product. If such coefficients are combined to form an $n$-point conformal block on Riemann sphere, one reproduces the earlier conjectured $β$-ensemble representation of conformal blocks, thus proving this (matrix model) version of the celebrated AGT relation. The statement can also be regarded as a relation between the $3j$-symbols of the Virasoro algebra and the slightly generalized Selberg integrals $I_Y$, associated with arbitrary Young diagrams. The conformal blocks are multilinear combinations of such integrals and the remaining part of the original AGT conjecture relates them to the Nekrasov functions which have exactly the same structure.

Matrix model version of AGT conjecture and generalized Selberg integrals

TL;DR

This work presents a conceptual derivation of a matrix-model (Dotsenko-Fateev) representation for conformal blocks in 2d CFT that yields a matrix-model version of the AGT conjecture. By combining free-field correlators with Dotsenko-Fateev screenings, the authors construct universal Virasoro structure constants via generalized Selberg integrals , and demonstrate their exact match with the standard CFT -symbols after an analytic continuation in the screening number. The approach unifies Virasoro intertwiner theory with Selberg/Jackson polynomial frameworks, yielding a constructive path to connect conformal blocks with Nekrasov functions in the DV-matrix-model phase. While the full AGT conjecture is not fully proven in this paper, the matrix-model (DF) route provides a conceptually transparent and computationally workable bridge between CFT data and gauge-theory data, with clear extensions to higher-level blocks and to -algebras. The results suggest deep dualities between DF-type integrals and LNS-type Nekrasov integrals, and offer a concrete program toward a complete proof of AGT.

Abstract

Operator product expansion (OPE) of two operators in two-dimensional conformal field theory includes a sum over Virasoro descendants of other operator with universal coefficients, dictated exclusively by properties of the Virasoro algebra and independent of choice of the particular conformal model. In the free field model, these coefficients arise only with a special "conservation" relation imposed on the three dimensions of the operators involved in OPE. We demonstrate that the coefficients for the three unconstrained dimensions arise in the free field formalism when additional Dotsenko-Fateev integrals are inserted between the positions of the two original operators in the product. If such coefficients are combined to form an -point conformal block on Riemann sphere, one reproduces the earlier conjectured -ensemble representation of conformal blocks, thus proving this (matrix model) version of the celebrated AGT relation. The statement can also be regarded as a relation between the -symbols of the Virasoro algebra and the slightly generalized Selberg integrals , associated with arbitrary Young diagrams. The conformal blocks are multilinear combinations of such integrals and the remaining part of the original AGT conjecture relates them to the Nekrasov functions which have exactly the same structure.

Paper Structure

This paper contains 14 sections, 92 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The free field formulas for $C_{\alpha_1\alpha_2}^{\alpha_1+\alpha_2}$
  3. The CFT formulas for $C_{\alpha_1\alpha_2}^{\,\alpha}$
  4. Operator product with screening insertions
  5. Structure constants $C_{\alpha_1\alpha_2}^{\,\alpha_1+\alpha_2+bN}$ from Selberg integrals. Level one
  6. Level two
  7. Higher levels
  8. Virasoro intertwiners
  9. From OPE to conformal blocks
  10. Towards a proof of the AGT conjecture
  11. Conclusion
  12. The proof of the matrix-model version of AGT conjecture
  13. Towards a proof of the remaining part of the AGT conjecture
  14. Extension from Virasoro to $W$

Figures (2)

  • Figure 1: A comb-like conformal block from AGTAlMo2, for which the AGT relation is known in the case of $0=x_1\ll x_2\ll x_3\ll x_4\ll \ldots$ Shown are the $\alpha$-parameters, the dimensions are equal to $\Delta = \alpha(\alpha-b+1/b)$. The intermediate dimensions are parameterized by the $N$-variables, after analytical continuation in $N$ they take arbitrary (continuum) values.
  • Figure 2: A star-like conformal block, for which the AGT relation (the corresponding set of Nekrasov function) is yet unknown. The "matrix-model" or Dotsenko-Fateev representation, a weak form of the AGT relation, is readily available and provided by eqs.(\ref{['CB2']}) and (\ref{['OPEfs']}) of the present paper. The brackets in subscripts of the $N$-variables define the integration segments of the screening insertions. From the point of view of the conformal blocks, they refer to a particular corner of the $x$-moduli space, see AlMo2.