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ADE Little String Theory on a Riemann Surface (and Triality)

Mina Aganagic, Nathan Haouzi

TL;DR

This work develops a comprehensive ADE generalization of the six-dimensional (2,0) theory in its little-string regime by realizing it as IIB string theory on ADE singularities and compactifying on a Riemann surface with codimension-two defects. The authors derive a low-energy 5d ADE quiver gauge theory describing the little string on a sphere with three full punctures, and show that its partition function matches the $q$-deformed ADE Toda CFT three-point block, thereby proving a broad triality between 5d gauge theories, 3d vortex theories from D3 branes, and $q$-deformed Toda CFT. Central to the construction are the geometric data of ADE singularities, the weight system ${\cal W}_{\cal S}$ encoding defects, and the interplay between monopole/Hitchin integrable systems and Seiberg–Witten geometry, connected through a gauge/vortex duality. The results extend the A_n AHS correspondence to all ADE groups, offering a robust string-theoretic framework to compute and relate partition functions across dimensions, and providing a rich structure for future explorations in gauge theory, integrable systems, and mathematical physics.

Abstract

We initiate the study of (2,0) little string theory of ADE type using its definition in terms of IIB string compactified on an ADE singularity. As one application, we derive a 5d ADE quiver gauge theory that describes the little string compactified on a sphere with three full punctures, at low energies. As a second application, we show the partition function of this theory equals the 3-point conformal block of ADE Toda CFT, q-deformed. To establish this, we generalize the A_n triality of \cite{AHS} to all ADE Lie algebras; IIB string perspective is crucial for this as well.

ADE Little String Theory on a Riemann Surface (and Triality)

TL;DR

This work develops a comprehensive ADE generalization of the six-dimensional (2,0) theory in its little-string regime by realizing it as IIB string theory on ADE singularities and compactifying on a Riemann surface with codimension-two defects. The authors derive a low-energy 5d ADE quiver gauge theory describing the little string on a sphere with three full punctures, and show that its partition function matches the -deformed ADE Toda CFT three-point block, thereby proving a broad triality between 5d gauge theories, 3d vortex theories from D3 branes, and -deformed Toda CFT. Central to the construction are the geometric data of ADE singularities, the weight system encoding defects, and the interplay between monopole/Hitchin integrable systems and Seiberg–Witten geometry, connected through a gauge/vortex duality. The results extend the A_n AHS correspondence to all ADE groups, offering a robust string-theoretic framework to compute and relate partition functions across dimensions, and providing a rich structure for future explorations in gauge theory, integrable systems, and mathematical physics.

Abstract

We initiate the study of (2,0) little string theory of ADE type using its definition in terms of IIB string compactified on an ADE singularity. As one application, we derive a 5d ADE quiver gauge theory that describes the little string compactified on a sphere with three full punctures, at low energies. As a second application, we show the partition function of this theory equals the 3-point conformal block of ADE Toda CFT, q-deformed. To establish this, we generalize the A_n triality of \cite{AHS} to all ADE Lie algebras; IIB string perspective is crucial for this as well.

Paper Structure

This paper contains 46 sections, 128 equations, 12 figures.

Table of Contents

  1. Introduction
  2. $(2,0)$ Little String Theory on ${\cal C}$
  3. Little String on ${\cal C}$ with Codimension Two Defects
  4. Elements of Holomogy
  5. Homology Classes of Defects
  6. Gauge Theory Description of The System
  7. $(2,0)$ CFT From The Little String
  8. Partition Function of Little String Theory on ${\cal C}$
  9. Integrable Systems, Bogomolny and Hitchin Equations
  10. Monopoles on ${\mathbb R}\times T^2$
  11. Hitchin-Type System on ${\cal C}$
  12. The Weight System ${\cal W}_{\cal S}$
  13. Defects in $(2,0)$ CFT and Little String
  14. Coulomb Branch for Finite and Infinite $m_s$
  15. ADE Little String and D3 Branes
  16. ...and 31 more sections

Figures (12)

  • Figure 1: The vanishing cycles of $A_n$ singularity $S_a$ (in black) and the dual non-compact cycles $S_a^*$ (in blue). For any ADE singularity, $S_a^*$ is constructed as the fiber of the cotangent bundle $T^*S_a$ over a generic point on $S_a$.
  • Figure 2: 5d gauge theory describing $A_n$ little string with 3 full punctures.
  • Figure 3: 5d gauge theory describing $D_n$ little string with 3 full punctures.
  • Figure 4: 5d gauge theory describing $E_6$ little string with 3 full punctures.
  • Figure 5: 5d gauge theory describing $E_7$ little string with 3 full punctures.
  • ...and 7 more figures