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The $W_n$ Light One-Point Torus Conformal Block
High Energy Physics - Theory · Submitted 2 Apr 2026
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Exploiting this property, we derive an explicit representation for the light one-point torus $W_n$ conformal block valid for arbitrary $n\ge 2$.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the light asymptotic limit of the one-point torus conformal block in $A_{n-1}$ Toda field theory. | contribution 0.91 | 0.92 | 0.06 | asserted 0.83 |
| s1 | 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. | method 0.95 | 0.62 | 0.06 | asserted 0.83 |
| s2 | 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. | result 0.92 | 0.92 | 0.15 | asserted 1.00 |
| s3 | Exploiting this property, we derive an explicit representation for the light one-point torus $W_n$ conformal block valid for arbitrary $n\ge 2$. | result 0.80 | 0.97 | 0.86 | asserted 1.00 |
| s4 | 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. | method 0.44 | 0.90 | 0.05 | asserted 0.93 |
| s5 | We also discuss the $W_3$ case and its relation to a known alternative representation obtained by the shadow formalism. | contribution 0.83 | 0.71 | 0.04 | asserted 0.81 |
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The paper
Original 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.
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What it cites · 26 sentences without a confident label
Uses the method or tool of 4 papers
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To derive the one-point $W_n$ torus conformal block in the light limit, we use the well-known AGT duality [ref] , which relates Liouville conformal blocks to the instanton partition functions [ref] of four-dimensional $N=2$ supersymmetric gauge theories.
Liouville Correlation Functions from Four-dimensional Gauge Theories · Introduction · 0.91
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To derive the one-point $W_n$ torus conformal block in the light limit, we use the well-known AGT duality [ref] , which relates Liouville conformal blocks to the instanton partition functions [ref] of four-dimensional $N=2$ supersymmetric gauge theories.
On combinatorial expansion of the conformal blocks arising from AGT conjecture · Introduction · 0.91
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To derive the one-point $W_n$ torus conformal block in the light limit, we use the well-known AGT duality [ref] , which relates Liouville conformal blocks to the instanton partition functions [ref] of four-dimensional $N=2$ supersymmetric gauge theories.
Seiberg-Witten Prepotential From Instanton Counting · Introduction · 0.91
Compares with 1 paper
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In Section [ref], we present a detailed derivation of the Liouville conformal block using our approach and compare our result with the expression previously obtained in the literature [ref] .
Various semiclassical limits of torus conformal blocks · Introduction · 1.00
Cites as background 19 papers
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This limit is of particular interest in the context of $AdS_3 / CFT_2$ holography [ref] .
Nonlinear W(infinity) Algebra as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity · Introduction · 0.85
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This limit is of particular interest in the context of $AdS_3 / CFT_2$ holography [ref] .
Quantum W-symmetry in AdS_3 · Introduction · 0.85
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This limit is of particular interest in the context of $AdS_3 / CFT_2$ holography [ref] .
Interface Minimal Model Holography and Topological String Theory · Introduction · 0.85
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