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Superstrings on PP-Wave Backgrounds and Symmetric Orbifolds

Yasuaki Hikida, Yuji Sugawara

TL;DR

This work analyzes superstrings on pp-wave backgrounds with NSNS-flux obtained via the Penrose limit of $AdS_3\times S^3\times M^4$, quantizing covariantly to construct the space-time SUSY algebra and DDF operators. The spectrum splits into short-string (discrete, non-propagating) and long-string (continuous, propagating) sectors, with short strings embedding into the single-particle Hilbert space of the symmetric orbifold dual $Sym^M(M^4)$ (and its $\mathbb{Z}_2$-orbifold for K3). By contracting the AdS$_3\times$S$^3$ currents to the $H_6$ WZW model, the authors obtain explicit vertex operators and a complete DDF framework to analyze BPS and almost-BPS states, drawing a precise correspondence with the symmetric orbifold CFT in the large-$k$ (BMN-like) limit while highlighting a set of missing, likely non-perturbative, excitations. The extension to $H_6\times T^4/\mathbb{Z}_2$ confirms the matching with $Sym^M(T^4/\mathbb{Z}_2)$ (K3) structures and emphasizes the role of twisted sectors in the duality. Overall, the paper lays groundwork for a Pontryagin-like NSNS pp-wave/ symmetric orbifold duality and points to non-perturbative phenomena as key future directions.

Abstract

We study the superstring theory on pp-wave background with NSNS-flux that is realized as the Penrose limit of AdS_3 x S^3 x M^4, where M^4 is T^4 or T^4/Z_2(~ K3). Quantizing this system in the covariant gauge, we explicitly construct the space-time supersymmetry algebra and the complete set of DDF operators. We analyse the spectrum of physical states by using the spectrally flowed representations of current algebra. This spectrum is classified by the ``short string sectors'' and the ``long string sectors'' as in AdS_3 string theory. The states of the latter propagate freely along the transverse plane of pp-wave background, but the states of the former do not. We compare the short string spectrum with the BPS and almost BPS states which have large R-charges in the symmetric orbifold conformal theory, which is known as the candidate of dual theory of superstrings on AdS_3 x S^3 x M^4. We show that every short string states can be embedded successfully in the single particle Hilbert space of symmetric orbifold conformal theory.

Superstrings on PP-Wave Backgrounds and Symmetric Orbifolds

TL;DR

This work analyzes superstrings on pp-wave backgrounds with NSNS-flux obtained via the Penrose limit of , quantizing covariantly to construct the space-time SUSY algebra and DDF operators. The spectrum splits into short-string (discrete, non-propagating) and long-string (continuous, propagating) sectors, with short strings embedding into the single-particle Hilbert space of the symmetric orbifold dual (and its -orbifold for K3). By contracting the AdSS currents to the WZW model, the authors obtain explicit vertex operators and a complete DDF framework to analyze BPS and almost-BPS states, drawing a precise correspondence with the symmetric orbifold CFT in the large- (BMN-like) limit while highlighting a set of missing, likely non-perturbative, excitations. The extension to confirms the matching with (K3) structures and emphasizes the role of twisted sectors in the duality. Overall, the paper lays groundwork for a Pontryagin-like NSNS pp-wave/ symmetric orbifold duality and points to non-perturbative phenomena as key future directions.

Abstract

We study the superstring theory on pp-wave background with NSNS-flux that is realized as the Penrose limit of AdS_3 x S^3 x M^4, where M^4 is T^4 or T^4/Z_2(~ K3). Quantizing this system in the covariant gauge, we explicitly construct the space-time supersymmetry algebra and the complete set of DDF operators. We analyse the spectrum of physical states by using the spectrally flowed representations of current algebra. This spectrum is classified by the ``short string sectors'' and the ``long string sectors'' as in AdS_3 string theory. The states of the latter propagate freely along the transverse plane of pp-wave background, but the states of the former do not. We compare the short string spectrum with the BPS and almost BPS states which have large R-charges in the symmetric orbifold conformal theory, which is known as the candidate of dual theory of superstrings on AdS_3 x S^3 x M^4. We show that every short string states can be embedded successfully in the single particle Hilbert space of symmetric orbifold conformal theory.

Paper Structure

This paper contains 14 sections, 117 equations.

Table of Contents

  1. Introduction
  2. Space-time Superalgebra of PP-Wave Background
  3. Superstring Theory on PP-Wave Background with NSNS-Flux
  4. $H_6$ Super WZW Model as a Penrose Limit
  5. Hilbert Space of $H_6$ Super WZW Model
  6. Physical Vertices of Superstring on PP-Wave Background
  7. Spectrum of Physical States
  8. Comparison with $Sym^{M}(T^4)$ $\sigma$-Model
  9. Short Review of $Sym^{M}(T^4)$ $\sigma$-Model
  10. "PP Wave Limit" of $Sym^{M}(T^4)$ $\sigma$-Model and Comparison with the Superstring Spectrum
  11. Extension to the Case of $H_6 \times T^4/\hbox{\bf Z}_2$
  12. Spectrum of Superstring on $H_6 \times T^4/\hbox{\bf Z}_2$
  13. Comparison with $Sym^M(T^4/\hbox{\bf Z}_2)$ $\sigma$-Model
  14. Conclusion