The Wn light one-point torus conformal block

Abstract

A bstract 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 ∗ U( n ) supersymmetric Yang-Mills theory, which we then examine in the limit b → 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 ≥ 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.