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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.01804 |
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| _version_ | 1866910097149526016 |
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| author | Poghosyan, Armen Poghosyan, Hasmik |
| author_facet | Poghosyan, Armen Poghosyan, Hasmik |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01804 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $W_n$ Light One-Point Torus Conformal Block Poghosyan, Armen Poghosyan, Hasmik High Energy Physics - Theory 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. |
| title | The $W_n$ Light One-Point Torus Conformal Block |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2604.01804 |