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Duality of the Fermionic 2d Black Hole and N=2 Liouville Theory as Mirror Symmetry

Kentaro Hori, Anton Kapustin

TL;DR

This paper establishes a precise supersymmetric FZZ-type duality by realizing the fermionic 2d Black Hole as the IR limit of a gauged linear sigma-model and showing its mirror description as ${\cal N}=2$ Liouville theory. The authors compute the exact IR central charge, demonstrate the emergence of a linear dilaton, and prove rigidity against marginal deformations, thereby pinning the IR fixed point to the ${\cal SL}(2,\mathbb{R})/U(1)$ Kazama–Suzuki coset for all $k>0$. They derive the dual theory via T-duality, integrate out heavy fields, and show that the low-energy limit reduces to ${\cal N}=2$ Liouville with central charge $c=3\left(1+\frac{2}{k}\right)$, establishing a mirror correspondence. The work further generalizes to orbifolds, multi-field models, and squashed toric sigma-models, revealing a broad class of dilatonic backgrounds with mirror LG descriptions and providing insights into the relationship between coset and LG/LG-like theories. These results have broad implications for understanding mirror symmetry in non-compact backgrounds and for constructing dilatonic string vacua with controlled IR behavior.

Abstract

We prove the equivalence of the SL(2,R)/U(1) Kazama-Suzuki model, which is a fermionic generalization of the 2d Black Hole, and N=2 Liouville theory. We show that this duality is an example of mirror symmetry. The essential part of the derivation is to realize the fermionic 2d Black Hole as the low energy limit of a gauged linear sigma-model. Liouville theory is obtained by dualizing the charged scalar fields and taking into account the vortex-instanton effects, as proposed recently in non-dilatonic models. The gauged linear sigma-model we study has many useful generalizations which we briefly discuss. In particular, we show how to construct a variety of dilatonic superstring backgrounds which generalize the fermionic 2d Black Hole and admit a mirror description in terms of Toda-like theories.

Duality of the Fermionic 2d Black Hole and N=2 Liouville Theory as Mirror Symmetry

TL;DR

This paper establishes a precise supersymmetric FZZ-type duality by realizing the fermionic 2d Black Hole as the IR limit of a gauged linear sigma-model and showing its mirror description as Liouville theory. The authors compute the exact IR central charge, demonstrate the emergence of a linear dilaton, and prove rigidity against marginal deformations, thereby pinning the IR fixed point to the Kazama–Suzuki coset for all . They derive the dual theory via T-duality, integrate out heavy fields, and show that the low-energy limit reduces to Liouville with central charge , establishing a mirror correspondence. The work further generalizes to orbifolds, multi-field models, and squashed toric sigma-models, revealing a broad class of dilatonic backgrounds with mirror LG descriptions and providing insights into the relationship between coset and LG/LG-like theories. These results have broad implications for understanding mirror symmetry in non-compact backgrounds and for constructing dilatonic string vacua with controlled IR behavior.

Abstract

We prove the equivalence of the SL(2,R)/U(1) Kazama-Suzuki model, which is a fermionic generalization of the 2d Black Hole, and N=2 Liouville theory. We show that this duality is an example of mirror symmetry. The essential part of the derivation is to realize the fermionic 2d Black Hole as the low energy limit of a gauged linear sigma-model. Liouville theory is obtained by dualizing the charged scalar fields and taking into account the vortex-instanton effects, as proposed recently in non-dilatonic models. The gauged linear sigma-model we study has many useful generalizations which we briefly discuss. In particular, we show how to construct a variety of dilatonic superstring backgrounds which generalize the fermionic 2d Black Hole and admit a mirror description in terms of Toda-like theories.

Paper Structure

This paper contains 20 sections, 152 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Gauged Linear Sigma-Model
  3. Flow to 2d Black Hole I: One-Loop Approximation
  4. An Exact Computation of the Central Charge
  5. $\overline{Q}_+$-cohomology
  6. The current and its anomaly
  7. The superconformal algebra
  8. Ambiguity and its resolution
  9. Flow to 2d Black Hole II: Exact Treatment
  10. General remarks
  11. Marginal deformations of the bosonic coset
  12. Marginal deformations of the supersymmetric coset
  13. Liouville Theory As The Mirror
  14. The Dual Theory
  15. Liouville theory as the IR limit of the dual theory
  16. ...and 5 more sections

Figures (1)

  • Figure 1: RG evolution of the cigar metric. We plotted $F(v,t)/\tanh v$ as a function of $v$ for several values of the rescaled RG time $\tau=t/(4\pi)$.