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On AGT conjecture

V. A. Fateev, A. V. Litvinov

TL;DR

The notes verify the AGT correspondence for the $\mathcal{N}=2^{*}$ $U(2)$ theory by connecting the Nekrasov instanton partition function to the torus one-point Liouville conformal block. A Zamolodchikov-style recursive structure is derived for the torus block, and the Nekrasov integral is shown to satisfy the same recursion, with a precise Young-diagram representation and a clear mapping of parameters. The Seiberg-Witten prepotential emerges from the $\varepsilon_1,\varepsilon_2\to0$ limit via a degenerate-field WKB analysis on the torus, yielding a parametric energy equation and a closed form for the instanton part involving the elliptic Weierstrass function. Together, these results provide a concrete realization of AGT for this theory and outline extensions to other flavors and higher-rank/quiver theories.

Abstract

In these notes we consider relation between conformal blocks and the Nekrasov partition function of certain $\mathcal{N}=2$ SYM theories proposed recently by Alday, Gaiotto and Tachikawa. We concentrate on $\mathcal{N}=2^{*}$ theory, which is the simplest example of AGT relation.

On AGT conjecture

TL;DR

The notes verify the AGT correspondence for the theory by connecting the Nekrasov instanton partition function to the torus one-point Liouville conformal block. A Zamolodchikov-style recursive structure is derived for the torus block, and the Nekrasov integral is shown to satisfy the same recursion, with a precise Young-diagram representation and a clear mapping of parameters. The Seiberg-Witten prepotential emerges from the limit via a degenerate-field WKB analysis on the torus, yielding a parametric energy equation and a closed form for the instanton part involving the elliptic Weierstrass function. Together, these results provide a concrete realization of AGT for this theory and outline extensions to other flavors and higher-rank/quiver theories.

Abstract

In these notes we consider relation between conformal blocks and the Nekrasov partition function of certain SYM theories proposed recently by Alday, Gaiotto and Tachikawa. We concentrate on theory, which is the simplest example of AGT relation.

Paper Structure

This paper contains 5 sections, 101 equations, 2 figures.

Table of Contents

  1. Recursive formula for the one-point toric conformal block
  2. $\mathcal{N}=2^{*}$ $U(2)$ Nekrasov instanton partition function
  3. Seiberg-Witten prepotential
  4. Structure of singularities of the integral \ref{['Zn-integral']}
  5. Useful formulae

Figures (2)

  • Figure 1: Integration contour in \ref{['Zn-integral']} surrounds poles $P_1$ and $P_2$ while poles $P_1-Q$ and $P_2-Q$ lie outside. When two of them collide (for example $P_1$ and $P_2-Q$) the integral \ref{['Zn-integral']} occur a pole.
  • Figure 2: Integration contour $\mathcal{C}_i$ in \ref{['general-Young-integral-proof']}.