The FZZ duality with boundary

TL;DR

This paper extends the FZZ duality to include boundary conditions, relating D-branes in the cigar model to those in sine-Liouville theory, and provides an alternative path integral proof including boundary effects.

Contribution

It introduces the boundary extension of the FZZ duality, constructs the boundary action for D2-branes, and offers a new path integral proof for the fermionic version of the duality.

Findings

Boundary FZZ duality relates D1-branes to D2-branes.

Constructed boundary action for D2-branes in sine-Liouville theory.

Path integral proof confirms duality with boundary conditions.

Abstract

The Fateev-Zamolodchikov-Zamolodchikov (FZZ) duality relates Witten's cigar model to sine-Liouville theory. This duality was proven in the path integral formulation and extended to the case of higher genus closed Riemann surfaces by Schomerus and one of the authors. In this note we further extend the duality to the case with boundary. Specifically, we relate D1-branes in the cigar model to D2-branes in the sine-Liouville theory. In particular, the boundary action for D2-branes in the sine-Liouville theory is constructed. We also consider the fermionic version of the FZZ duality. This duality was proven as a mirror symmetry by Hori and Kapustin, but we give an alternative proof in the path integral formulation which directly relates correlation functions. Also here the case with boundary is investigated and the results are consistent with those for branes in N=2 super Liouville field theory obtained by Hosomichi.