Mirror Mirror On The Wall: On 2d Black Holes and Liouville Theory

TL;DR

The work presents a domain-wall based derivation of the duality between the $2$-D Euclidean black hole and supersymmetric Liouville theory by embedding both theories on the worldvolume of domain walls in a 3D ${\cal N}=4$ abelian-Higgs model with three vacua. It shows two complementary descriptions: (i) a 1+1D ${\rm N}=(2,2)$ sigma-model with target the cigar $SL(2,\mathbb{R})/U(1)$ at level $k$, reproducing the 2D black hole CFT in the IR, and (ii) a 3D effective theory for small $k$ in which integrating out matter yields a Liouville-like theory for the relative wall motion, with complex coordinate $Y=\zeta R/4+i\chi$ and Liouville potential $|e^{-Y}|^2$ and scale $\mu=\zeta M/4$. The analysis uses Manton’s method to compute wall forces and shows how the RG flow, including a dilaton-like contribution, drives the system toward the conformal fixed point, linking classical domain-wall dynamics to quantum mirror dualities. The results further connect to brane constructions and suggest generalizations to toric sigma-models and their mirrors, offering a physical mechanism by which instanton effects in the 2D theory arise from domain-wall dynamics in 3D.

Abstract

We present a novel derivation of the duality between the two-dimensional Euclidean black hole and supersymmetric Liouville theory. We realise these (1+1)-dimensional conformal field theories on the worldvolume of domain walls in a (2+1)-dimensional gauge theory. We show that there exist two complementary descriptions of the domain wall dynamics, resulting in the two mirror conformal field theories. In particular, effects which are usually attributed to worldsheet instantons are captured by the classical scattering of domain walls.

Figures (4)

• Figure 1: The geometry of the two-dimensional Euclidean black hole
• Figure 2: Two domain walls. (No, really). The separation $R$ is a modulus.
• Figure 3: Two domain walls in the D2-D6 system. In contrast to other pictures, the sheets represent the D6-branes. The vacua of the field theory occur when the D2-brane lies vertically within a D6-brane. The domain walls correspond to places where the D2-brane interpolates horizontally between the D6-branes.
• Figure 4: Three domain walls. When the outer two become heavy (shown by the shading) the light, middle domain wall is restricted to bounce between them. This gives a realisation of the ${\bf CP}^1$ sigma-model.