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On AGT relation in the case of U(3)

A. Mironov, A. Morozov

TL;DR

This work tests the AGT correspondence in the $W_3$ sector by restricting to a very simple but nontrivial case: a 4-point block with two free fields yielding $c=2$, and imposing $W$-null-vector constraints on external states. By constructing the $W_3$ block from level-one and level-two Verma-module data and matching it to Nekrasov functions, the authors derive explicit linear relations between the Virasoro/$W$ primaries and Nekrasov moduli ($\boldsymbol{a}$, $\mu_f$), including a necessary $U(1)$ factor $\nu$. They demonstrate level-by-level consistency, deriving the denominator $v^2=\Delta_{\boldsymbol{\alpha}}^3-w_{\boldsymbol{\alpha}}^2$ and explicit numerators, with special attention to cases where external states are zero or nonzero and where a degenerate momentum imposes additional constraints. A complete proof is provided for a very special hypergeometric block, illustrating how Nekrasov sums generalize hypergeometric series in the space of $W_N$ blocks under special-state restrictions. Overall, the results bolster the view that Nekrasov functions encode the full conformal-block structure under appropriate $W$-null-vector constraints and shed light on the $W_N$ extension of AGT, including the role of the $U(1)$ factor and projective transformations.

Abstract

We consider the AGT relation, expressing conformal blocks for the Virasoro and W-algebras in terms of Nekrasov's special functions, in the simplest case of the 4-point functions for the first non-trivial W_3 algebra. The standard set of Nekrasov functions is sufficient only if additional null-vector restriction is imposed on a half of the external $W$-primaries and this is just the case when the conformal blocks are fully dictated by W-symmetry and do not depend on a particular model. Explicit checks confirm that the AGT relation survives in this restricted case, as expected.

On AGT relation in the case of U(3)

TL;DR

This work tests the AGT correspondence in the sector by restricting to a very simple but nontrivial case: a 4-point block with two free fields yielding , and imposing -null-vector constraints on external states. By constructing the block from level-one and level-two Verma-module data and matching it to Nekrasov functions, the authors derive explicit linear relations between the Virasoro/ primaries and Nekrasov moduli (, ), including a necessary factor . They demonstrate level-by-level consistency, deriving the denominator and explicit numerators, with special attention to cases where external states are zero or nonzero and where a degenerate momentum imposes additional constraints. A complete proof is provided for a very special hypergeometric block, illustrating how Nekrasov sums generalize hypergeometric series in the space of blocks under special-state restrictions. Overall, the results bolster the view that Nekrasov functions encode the full conformal-block structure under appropriate -null-vector constraints and shed light on the extension of AGT, including the role of the factor and projective transformations.

Abstract

We consider the AGT relation, expressing conformal blocks for the Virasoro and W-algebras in terms of Nekrasov's special functions, in the simplest case of the 4-point functions for the first non-trivial W_3 algebra. The standard set of Nekrasov functions is sufficient only if additional null-vector restriction is imposed on a half of the external -primaries and this is just the case when the conformal blocks are fully dictated by W-symmetry and do not depend on a particular model. Explicit checks confirm that the AGT relation survives in this restricted case, as expected.

Paper Structure

This paper contains 27 sections, 161 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The simplest example: two free fields with $c=2$
  3. Structure of the $W_3$ algebra
  4. $4$-point conformal block for the $W_3$ algebra
  5. Nekrasov's formulas
  6. AGT relation
  7. Denominator
  8. Numerator. The case of $\ \vec{\alpha}_1=\ldots=\vec{\alpha}_4=0$
  9. Numerator. The case of arbitrary $\vec{\alpha}_2$ and $\vec{\alpha}_4$ with no ambiguity in the $W$-conformal block
  10. The case of $\xi=2$
  11. The $U(1)$ factor $\nu$ and projective transformations
  12. General case, all $\vec{\alpha}\neq 0$, $\vec{\alpha}_1$ and $\vec{\alpha}_3$ special
  13. Intermediate summary
  14. Some CFT considerations
  15. $W$-algebra and Shapovalov form
  16. ...and 12 more sections

Figures (1)

  • Figure 1: Roots and minimal weights of $sl(3)$. The roots have the length $\sqrt{2}$, the minimal weights have $\sqrt{2/3}$. All angles are integer multiples of $30^\circ$. The two simple roots and the corresponding two fundamental weights are marked by black circles. The third positive root, marked with a white circle is also $\vec{\rho}$, the half-sum of all (three) positive roots or the sum of (two) fundamental weights. The vectors $\vec{\lambda}_{1,2,3}$ are shown by thick lines, they form a Weyl-invariant triple of minimal weights.