ArXiv · 2026
We study the light asymptotic limit of the one-point torus conformal block in Aₙ₋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^∗ 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 Wₙ 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 W₃ case and its relation to a known alternative representation obtained by the shadow formalism.
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