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String Theory on AdS Orbifolds

Emil Martinec, Will McElgin

TL;DR

Martinec and McElgin develop and analyze $Z_N$ orbifolds of ${ m AdS}_3 imes{ m S}^3$ in both bosonic and supersymmetric string theories, using ${ m SL}(2, ext{R})$ and ${ m SU}(2)$ WZW and parafermion formalisms. They construct twist operators for the twisted sectors, derive conformal dimensions, and introduce fractional spectral flow that extends the spectrum to windings by $1/N$, embedding the orbifold in a twisted ${ m N}=4$ spacetime superVirasoro algebra with central charge $c=N ilde c$, consistent with AdS gravity. The analysis reveals $4(N-1)$ twisted-moduli from the orbifold and additional untwisted moduli from spectral flow, interpretable as blowup modes and internal fluxes, and discusses D-branes and RR charges at the singularity. The work connects conical defect geometries in AdS/CFT to a controlled string-theoretic framework, highlighting tachyon instabilities in non-supersymmetric cases, the role of fractional spectral flow in the spacetime CFT, and potential links to near-extremal BTZ physics via a large-$N$ limit.

Abstract

We consider worldsheet string theory on $Z_N$ orbifolds of $AdS_3$ associated with conical singularities. If the orbifold action includes a similar twist of $S^3$, supersymmetry is preserved, and there is a moduli space of vacua arising from blowup modes of the orbifold singularity. We exhibit the spectrum, including the properties of twisted sectors and states obtained by fractional spectral flow. A subalgebra of the spacetime superconformal symmetry remains intact after the $Z_N$ quotient, and serves as the spacetime symmetry algebra of the orbifold.

String Theory on AdS Orbifolds

TL;DR

Martinec and McElgin develop and analyze orbifolds of in both bosonic and supersymmetric string theories, using and WZW and parafermion formalisms. They construct twist operators for the twisted sectors, derive conformal dimensions, and introduce fractional spectral flow that extends the spectrum to windings by , embedding the orbifold in a twisted spacetime superVirasoro algebra with central charge , consistent with AdS gravity. The analysis reveals twisted-moduli from the orbifold and additional untwisted moduli from spectral flow, interpretable as blowup modes and internal fluxes, and discusses D-branes and RR charges at the singularity. The work connects conical defect geometries in AdS/CFT to a controlled string-theoretic framework, highlighting tachyon instabilities in non-supersymmetric cases, the role of fractional spectral flow in the spacetime CFT, and potential links to near-extremal BTZ physics via a large- limit.

Abstract

We consider worldsheet string theory on orbifolds of associated with conical singularities. If the orbifold action includes a similar twist of , supersymmetry is preserved, and there is a moduli space of vacua arising from blowup modes of the orbifold singularity. We exhibit the spectrum, including the properties of twisted sectors and states obtained by fractional spectral flow. A subalgebra of the spacetime superconformal symmetry remains intact after the quotient, and serves as the spacetime symmetry algebra of the orbifold.

Paper Structure

This paper contains 17 sections, 128 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Bosonic WZW model for $SL(2,\mathbb{R})$
  3. The Rotational Orbifold
  4. Twist Ground States
  5. $SL(2,\mathbb{R})$ Parafermions
  6. Fractional Spectral Flow
  7. The $\left(AdS_{3}\times S^{3}\right)\!/\mathbb{Z}_{N}$ Orbifold
  8. Bosonic Structure
  9. The Supersymmetric Orbifold
  10. The $AdS_3/{\mathbb Z}_N$ Orbifold of the Superstring
  11. D-branes
  12. Moduli Space
  13. The Spacetime CFT
  14. The Spacetime Superconformal Algebra
  15. The Spacetime Charges and Spectral Flow
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Fractional spectral flow of a timelike geodesic producing a string in the $q=2$ twisted sector for the $N=6$ orbifold.
  • Figure 2: Charges of the spectral flow of the chiral primary states (the red dots) associated with the orbifold spacetimes for the sector corresponding to the $N=2$ orbifold. The plot shows the unitarity bound (the circumscribed green parabola) on all of the states of the $\mathcal{N}=4$ superconformal algebra as well as the more restrictive bound (the blue polygon) that applies to the sector of the $N=2$ orbifold. The plot also shows the lower bound (the inscribed black parabola) for the emergence of BTZ black holes.