The $W_n$ Light One-Point Torus Conformal Block

TL;DR

This paper derives an explicit representation for the light one-point torus $W_n$ conformal block in Toda field theory, simplifying the instanton sum via arm-length contributions and confirming results for specific cases.

Contribution

It provides a new explicit formula for the light one-point torus $W_n$ conformal block for any $n\,geq 2$, connecting AGT correspondence with simplified instanton sums.

Findings

Derived an explicit representation for the light torus $W_n$ conformal block.

Showed that only specific arm-length boxes contribute in the light limit.

Confirmed the formula matches known results for $n=2$ and discussed the $W_3$ case.

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.