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Modular Invariant Critical Superstrings on Four-dimensional Minkowski Space $\times$ Two-dimensional Black Hole

Shun'ya Mizoguchi

TL;DR

The paper develops a modular invariant, tachyon-free critical superstring model on four-dimensional Minkowski space × a two-dimensional black hole using the SL(2,\hbox{R})/U(1) Kazama-Suzuki coset with N=2, c=9 characters. By integrating over a Liouville momentum along the noncompact direction and employing specific theta identities, the authors achieve modular invariance and construct the partition function Z(\tau) that respects a particular GSO projection. The resulting four-dimensional massless spectrum coincides with tensionless string content, hinting at a dual description of Type II strings on a conifold in terms of two intersecting NS5-branes and relating to D=6, N=4 gauged supergravity. The work provides a noncompact Gepner-like framework for singular Calabi-Yau backgrounds and suggests avenues for further generalizations to other Calabi-Yau singularities.

Abstract

Extending the seminal work of Bilal and Gervais, we construct a tachyon-free, modular invariant partition function for critical superstrings on four-dimensional Minkowski x two-dimensional black hole. This model may be thought of as an SL(2,R)/U(1) version of Gepner models and corresponds to a conifold point on the moduli space of Calabi-Yau compactifications. We directly deal with N=2, c=9 unitary superconformal characters. Modular invariance is achieved by requiring the string to have a momentum along an extra noncompact direction, in agreement with the picture of singular CFTs advocated by Witten. The four-dimensional massless spectrum coincides with that of the tensionless strings, suggesting a possible dual description of type II strings on a conifold in terms of two intersecting NS5-branes. An interesting relation to D=6, N=4 gauged supergravity is also discussed.

Modular Invariant Critical Superstrings on Four-dimensional Minkowski Space $\times$ Two-dimensional Black Hole

TL;DR

The paper develops a modular invariant, tachyon-free critical superstring model on four-dimensional Minkowski space × a two-dimensional black hole using the SL(2,\hbox{R})/U(1) Kazama-Suzuki coset with N=2, c=9 characters. By integrating over a Liouville momentum along the noncompact direction and employing specific theta identities, the authors achieve modular invariance and construct the partition function Z(\tau) that respects a particular GSO projection. The resulting four-dimensional massless spectrum coincides with tensionless string content, hinting at a dual description of Type II strings on a conifold in terms of two intersecting NS5-branes and relating to D=6, N=4 gauged supergravity. The work provides a noncompact Gepner-like framework for singular Calabi-Yau backgrounds and suggests avenues for further generalizations to other Calabi-Yau singularities.

Abstract

Extending the seminal work of Bilal and Gervais, we construct a tachyon-free, modular invariant partition function for critical superstrings on four-dimensional Minkowski x two-dimensional black hole. This model may be thought of as an SL(2,R)/U(1) version of Gepner models and corresponds to a conifold point on the moduli space of Calabi-Yau compactifications. We directly deal with N=2, c=9 unitary superconformal characters. Modular invariance is achieved by requiring the string to have a momentum along an extra noncompact direction, in agreement with the picture of singular CFTs advocated by Witten. The four-dimensional massless spectrum coincides with that of the tensionless strings, suggesting a possible dual description of type II strings on a conifold in terms of two intersecting NS5-branes. An interesting relation to D=6, N=4 gauged supergravity is also discussed.

Paper Structure

This paper contains 8 sections, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $SL(2,\hbox{R})/U(1)$ Kazama-Suzuki model
  3. Free-field realizations
  4. Representations of classical SL(2,R)
  5. Interpretation --- the $SL(2,\hbox{R})$ WZW model
  6. Modular Invariant Partition Function
  7. Mass Spectra
  8. Conclusion

Figures (2)

  • Figure 1: The NS sector. The thick lines and dots show the $N=2$ unitary representations used as the internal CFT. The former correspond to the propagating modes along the cigar, while the latter are the bound states.
  • Figure 2: The R sector. $Q$ represents the mean value of $U(1)$ charges of two lowest $L_0$ states if $h\neq 3/8$. The $U(1)$ charge of the lowest weight state is $Q\pm 1/2$ depending on the convention (which of ${\rm P}^\pm$ representations is considered). If $h=3/8$, the lowest $L_0$ state is unique because the other becomes null. The (superficial) asymmetry at the bottom is due to the convention used here; we consider ${\rm P}^+$ representations so that the $U(1)$ charge of the lowest weight state is $Q-1/2$.