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The $W_n$ Light One-Point Torus Conformal Block

Armen Poghosyan, Hasmik Poghosyan

Abstract

We study the light asymptotic limit of the one-point torus conformal block in $A_{n-1}$ Toda field theory. Through the AGT correspondence, this problem can be translated into the computation of the instanton partition function of four-dimensional ${\cal N}=2^{\ast}$ $U(n)$ supersymmetric Yang--Mills theory, which we then examine in the limit $b\to 0$ at fixed conformal dimensions. We show that, in this regime, the instanton sum simplifies drastically: for each Young diagram, only boxes with specific arm lengths contribute to the bifundamental factors. Exploiting this property, we derive an explicit representation for the light one-point torus $W_n$ conformal block valid for arbitrary $n\ge 2$. As a consistency check, we specialize our construction to the Liouville case $n=2$ and compare it with the previously known hypergeometric representation of the torus block in the light limit. We also discuss the $W_3$ case and its relation to a known alternative representation obtained by the shadow formalism.

The $W_n$ Light One-Point Torus Conformal Block

Abstract

We study the light asymptotic limit of the one-point torus conformal block in Toda field theory. Through the AGT correspondence, this problem can be translated into the computation of the instanton partition function of four-dimensional supersymmetric Yang--Mills theory, which we then examine in the limit at fixed conformal dimensions. We show that, in this regime, the instanton sum simplifies drastically: for each Young diagram, only boxes with specific arm lengths contribute to the bifundamental factors. Exploiting this property, we derive an explicit representation for the light one-point torus conformal block valid for arbitrary . As a consistency check, we specialize our construction to the Liouville case and compare it with the previously known hypergeometric representation of the torus block in the light limit. We also discuss the case and its relation to a known alternative representation obtained by the shadow formalism.

Paper Structure

This paper contains 10 sections, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on $A_{n-1}$ Toda Theory, ${\cal N}=2^{\ast}$ SYM Instanton Counting and AGT Duality
  3. $W_n$ conformal field theories
  4. Instanton counting for ${\cal N}=2^{\ast}$ SYM theory
  5. AGT duality
  6. The light asymptotic limit of the instanton partition function
  7. The Liouville one-point torus conformal block
  8. $W_n$ one-point light conformal block
  9. Summary
  10. Another representation for the $W_3$ light conformal block

Figures (2)

  • Figure 1: The arm and leg lengths with respect to the Young diagram (outlined by the thick black line) are $A(s_1)=-2$, $L(s_1)=-2$, $A(s_2)=2$, $L(s_2)=3$, $A(s_3)=-3$, and $L(s_3)=-4$. This Young diagram is uniquely characterized by the infinite sequence of nonnegative integers integers $\{3,0,2,1,0,0,0,\ldots\}$.
  • Figure 2: A Young diagram $Y_u$ is uniquely specified by the infinite sequence of nonnegative integers $\{\ell_{u,0}, \ell_{u,1}, \ell_{u,2}, \ldots\}$. For the diagram shown here, this sequence is $\{2,1,3,0,1,0,0,0,\ldots\}$, where the dots indicate that all subsequent entries are zero. The purple boxes have arm length zero and therefore form the set $Y_{u,0}$, while the gray boxes have arm length one and form the set denoted by $Y_{u,1}$.