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$A_n$-Triality

Mina Aganagic, Nathan Haouzi, Shamil Shakirov

TL;DR

The paper proves an A_n triality linking 4d ${\cal N}=2$ gauge theories, 3d vortex theories on vortices, and ${A}_n$ Toda CFT blocks, established via gauge/vortex duality and Coulomb-gas representations. The core insight is that the 3d vortex partition function in an ${\Omega}$-background equals the ${q}$-deformed ${A}_n$ Toda conformal block, while both match the Nekrasov partition function of a 5d ${\cal N}=1$ ${A}_n$ quiver gauge theory ${\cal T}_{5d}$; in genus-zero, full-puncture cases, this yields a concrete triality. Lifting to 5d and using a $q$-deformation are essential to control cases where the 4d theory is non-Lagrangian, with spectral duality and contour choices clarifying the Toda side. The results connect to topological-string large-$N$ duality and enrich the web of BPS/CFT correspondences by embedding the triality into a broader geometric and string-theoretic framework.

Abstract

A_n-type AGT correspondence anticipates that conformal blocks of A_n Toda CFT are related to partition functions of a family of 4d N=2 SCFTs. We use gauge/vortex duality to both give a precise form of the correspondence, and to prove it. Gauge/vortex duality relates the 4d theories and the 2d theories living on its vortices. Partition functions of the 2d theories on vortices provide Coulomb-gas representation of A_n Toda conformal blocks with discrete internal momenta. This gives a triality of relations between the gauge theory, its vortices and the Toda CFT. We prove that A_n triality holds for conformal blocks of A_n Toda on a sphere with all full punctures. The lift to one higher dimensional theories, compactified on a circle of arbitrary radius, and q-deformation of the Toda CFT, play a key role.

$A_n$-Triality

TL;DR

The paper proves an A_n triality linking 4d gauge theories, 3d vortex theories on vortices, and Toda CFT blocks, established via gauge/vortex duality and Coulomb-gas representations. The core insight is that the 3d vortex partition function in an -background equals the -deformed Toda conformal block, while both match the Nekrasov partition function of a 5d quiver gauge theory ; in genus-zero, full-puncture cases, this yields a concrete triality. Lifting to 5d and using a -deformation are essential to control cases where the 4d theory is non-Lagrangian, with spectral duality and contour choices clarifying the Toda side. The results connect to topological-string large- duality and enrich the web of BPS/CFT correspondences by embedding the triality into a broader geometric and string-theoretic framework.

Abstract

A_n-type AGT correspondence anticipates that conformal blocks of A_n Toda CFT are related to partition functions of a family of 4d N=2 SCFTs. We use gauge/vortex duality to both give a precise form of the correspondence, and to prove it. Gauge/vortex duality relates the 4d theories and the 2d theories living on its vortices. Partition functions of the 2d theories on vortices provide Coulomb-gas representation of A_n Toda conformal blocks with discrete internal momenta. This gives a triality of relations between the gauge theory, its vortices and the Toda CFT. We prove that A_n triality holds for conformal blocks of A_n Toda on a sphere with all full punctures. The lift to one higher dimensional theories, compactified on a circle of arbitrary radius, and q-deformation of the Toda CFT, play a key role.

Paper Structure

This paper contains 35 sections, 117 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The $A_n$ Triality
  3. Some Finer Points
  4. ${\cal T}_{5d}$ and M5 Branes
  5. M5 branes and ${\cal T}_{4d}$
  6. M5 branes and ${\cal T}_{5d}$
  7. Classification of Punctures in 5d
  8. ${\cal T}_{5d}$ is a 5d Quiver Gauge Theory
  9. A Class of 5d $A_n$-Type Quiver Gauge Theories
  10. From $S$-curve To $A_n$-Quiver Gauge Theory
  11. 4d vs. 5d
  12. Partition Function of ${\cal T}_{5d}$
  13. Other Realizations of ${\cal T}_{5d}$ in String Theory
  14. Examples of 5d Quivers, $S$-curves and Web Diagrams
  15. ${\cal V}_{3d}$ and Vortices in ${\cal T}_{5d}$
  16. ...and 20 more sections

Figures (10)

  • Figure 1: Quiver for gauge theories of type $A_n$.
  • Figure 2: A web diagram for 5d $A_n$ quiver theories, see section 2.5.
  • Figure 3: The $A_4$ quiver gauge theory with ${\vec{m}} = (4,0,0)$, the corresponding web diagram, and the $C$ curve, which is a sphere with three full punctures.
  • Figure 4: The $A_4$ quiver gauge theory with ${\vec{m}} = (8,0,0)$, the corresponding web diagram, and the $C$ curve, which is a sphere with four full punctures.
  • Figure 5: The $A_4$ quiver gauge theory with ${\vec{m}} = (2,1,0)$, the corresponding web diagram, and the $C$ curve, which is a sphere with two full punctures and one puncture of type $[1^2 2^1]$.
  • ...and 5 more figures