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D-branes and extended characters in SL(2,R)/U(1)

Angelos Fotopoulos, Vasilis Niarchos, Nikolaos Prezas

TL;DR

The paper develops a modular bootstrap framework for D-branes in the SL(2,ℝ)/U(1) coset at integer level $k$, utilizing extended coset characters and an ${ m N}=2$ embedding to organize boundary data. By constructing extended Ishibashi states and applying a Cardy-like ansatz, it identifies four Cardy-consistent brane classes: A-type class-2, A-type class-2′, B-type class-1, and B-type class-2, with explicit wavefunctions and open-string spectra. The D0-, D1-, and novel D2-branes emerge from this analysis, with D2-branes partially covering the cigar and carrying a ${ m Z}_2$ Wilson line, while some candidate class-3 constructions fail Cardy consistency. Comparisons with Ribault–Schomerus results from Euclidean AdS3 descent highlight both agreements and discrepancies, informing the understanding of non-rational boundary CFTs and suggesting directions for further factorization constraints and holographic connections.

Abstract

We present a detailed study of D-branes in the axially gauged SL(2,R)/U(1) coset conformal field theory for integer level k. Our analysis is based on the modular bootstrap approach and utilizes the extended SL(2,R)/U(1) characters and the embedding of the parafermionic coset algebra in the N=2 superconformal algebra. We propose three basic classes of boundary states corresponding to D0-, D1- and D2-branes. We verify that these boundary states satisfy the Cardy consistency conditions and discuss their physical properties. The D0- and D1-branes agree with those found in earlier work by Ribault and Schomerus using different methods (descent from the Euclidean AdS3 model). The D2-branes are new. They are not, in general, space-filling but extend from the asymptotic circle at infinity up to a circular boundary at some distance from the tip of the cigar.

D-branes and extended characters in SL(2,R)/U(1)

TL;DR

The paper develops a modular bootstrap framework for D-branes in the SL(2,ℝ)/U(1) coset at integer level , utilizing extended coset characters and an embedding to organize boundary data. By constructing extended Ishibashi states and applying a Cardy-like ansatz, it identifies four Cardy-consistent brane classes: A-type class-2, A-type class-2′, B-type class-1, and B-type class-2, with explicit wavefunctions and open-string spectra. The D0-, D1-, and novel D2-branes emerge from this analysis, with D2-branes partially covering the cigar and carrying a Wilson line, while some candidate class-3 constructions fail Cardy consistency. Comparisons with Ribault–Schomerus results from Euclidean AdS3 descent highlight both agreements and discrepancies, informing the understanding of non-rational boundary CFTs and suggesting directions for further factorization constraints and holographic connections.

Abstract

We present a detailed study of D-branes in the axially gauged SL(2,R)/U(1) coset conformal field theory for integer level k. Our analysis is based on the modular bootstrap approach and utilizes the extended SL(2,R)/U(1) characters and the embedding of the parafermionic coset algebra in the N=2 superconformal algebra. We propose three basic classes of boundary states corresponding to D0-, D1- and D2-branes. We verify that these boundary states satisfy the Cardy consistency conditions and discuss their physical properties. The D0- and D1-branes agree with those found in earlier work by Ribault and Schomerus using different methods (descent from the Euclidean AdS3 model). The D2-branes are new. They are not, in general, space-filling but extend from the asymptotic circle at infinity up to a circular boundary at some distance from the tip of the cigar.

Paper Structure

This paper contains 29 sections, 204 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $SL(2,\mathbb{R})/U(1)$ characters and the torus partition sum
  3. Representations and standard characters
  4. Embedding the $SL(2,\mathbb{R})/U(1)$ parafermion theory in ${\cal N}=2$
  5. Extended characters
  6. Character decomposition of the torus partition function
  7. Modular transformations of the extended coset characters
  8. Modular bootstrap for $SL(2,\mathbb{R})/U(1)$
  9. Generalities on the modular bootstrap
  10. Extended coset Ishibashi states from ${\cal N}=2$ decomposition
  11. Ishibashi state decompositions: ${\cal N}=2$ A-type $\rightarrow$ coset B-type
  12. Ishibashi state decompositions: ${\cal N}=2$ B-type $\rightarrow$ coset A-type
  13. Modular bootstrap and the Cardy ansatz for coset boundary states
  14. A-type boundary states
  15. B-type boundary states
  16. ...and 14 more sections

Figures (1)

  • Figure 1: D-branes on the cigar and corresponding $AdS_3$ branes