- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Four-dimensional spectrum generating algebra for superstrings

Renann Lipinski Jusinskas1 and Aravindhan Srinivasan2,3

PDF

Phys. Rev. D 113, L021902 – Published 20 January, 2026

DOI: https://doi.org/10.1103/phw3-yrsv

Abstract

We present the spectrum generating algebra for the hybrid string with manifest N=1 super Poincaré symmetry in R3,1×R6. Our Del-Giudice–Di-Vecchia–Fubini operators establish a one-to-one correspondence with the conventional superstring spectrum, while making its four-dimensional structure manifest. We also discuss part of the antifield spectrum, and introduce a simple realization of the supersymmetry charges involving the R6 directions. As an application, we algebraically compute the helicity partition function. These results provide new tools for analyzing four-dimensional on-shell superspaces and may extend naturally to phenomenologically relevant compactifications.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (46)

  1. N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, The hierarchy problem and new dimensions at a millimeter, Phys. Lett. B 429, 263 (1998).
  2. L. Randall and R. Sundrum, A large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83, 3370 (1999).
  3. N. B. Agmon, A. Bedroya, M. J. Kang, and C. Vafa, Lectures on the string landscape and the Swampland, arXiv:2212.06187.
  4. R. L. Workman et al. (Particle Data Group), Review of particle physics, Chapter 85—extra dimensions, Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
  5. W. Z. Feng, D. Lust, and O. Schlotterer, Massive supermultiplets in four-dimensional superstring theory, Nucl. Phys. B861, 175 (2012).
  6. N. Berkovits, Covariant quantization of the Green-Schwarz superstring in a Calabi-Yau background, Nucl. Phys. B431, 258 (1994).
  7. N. Berkovits, A new description of the superstring, arXiv:hep-th/9604123.
  8. N. Berkovits and C. Vafa, N=4 topological strings, Nucl. Phys. B433, 123 (1995).
  9. N. Berkovits, C. Vafa, and E. Witten, Conformal field theory of AdS background with Ramond-Ramond flux, J. High Energy Phys. 03 (1999) 018.
  10. N. Berkovits, M. Bershadsky, T. Hauer, S. Zhukov, and B. Zwiebach, Superstring theory on AdS2×S2 as a coset supermanifold, Nucl. Phys. B567, 61 (2000).
  11. N. Berkovits, Quantization of the superstring in Ramond-Ramond backgrounds, Classical Quantum Gravity 17, 971 (2000).
  12. L. Eberhardt, M. R. Gaberdiel, and R. Gopakumar, The worldsheet dual of the symmetric product CFT, J. High Energy Phys. 04 (2019) 103.
  13. L. Eberhardt, M. R. Gaberdiel, and R. Gopakumar, Deriving the AdS3/CFT2 correspondence, J. High Energy Phys. 02 (2020) 136.
  14. U. M. Portugal, Super Yang-Mills action from hybrid superstring field theory, J. High Energy Phys. 01 (2025) 176.
  15. N. Berkovits and U. M. Portugal, Heterotic string field theory with manifest spacetime supersymmetry, J. High Energy Phys. 05 (2025) 050.
  16. N. Berkovits and M. M. Leite, First massive state of the superstring in superspace, Phys. Lett. B 415, 144 (1997).
  17. K. Benakli, N. Berkovits, C. A. Daniel, and M. Lize, Higher-spin states of the superstring in an electromagnetic background, J. High Energy Phys. 12 (2021) 112.
  18. L. Dolan and E. Witten, Vertex operators for AdS(3) background with Ramond-Ramond flux, J. High Energy Phys. 11 (1999) 003.
  19. M. R. Gaberdiel and K. Naderi, The physical states of the hybrid formalism, J. High Energy Phys. 10 (2021) 168.
  20. C. A. Daniel, Vertex operators for the superstring with manifest d=6N=1 supersymmetry, J. High Energy Phys. 09 (2025) 017.
  21. A. Giveon, D. Kutasov, and N. Seiberg, Comments on string theory on AdS(3), Adv. Theor. Math. Phys. 2, 733 (1998).
  22. L. Eberhardt and M. R. Gaberdiel, String theory on AdS3 and the symmetric orbifold of Liouville theory, Nucl. Phys. B948, 114774 (2019).
  23. K. Naderi, DDF operators in the hybrid formalism, J. High Energy Phys. 12 (2022) 043.
  24. E. Del Giudice, P. Di Vecchia, and S. Fubini, General properties of the dual resonance model, Ann. Phys. (N.Y.) 70, 378 (1972).
  25. D. J. Gross and V. Rosenhaus, Chaotic scattering of highly excited strings, J. High Energy Phys. 05 (2021) 048.
  26. V. Rosenhaus, Chaos in a many-string scattering amplitude, Phys. Rev. Lett. 129, 031601 (2022).
  27. D. Biswas, R. Marotta, and I. Pesando, The Reggeon vertex for DDF states, J. High Energy Phys. 04 (2025) 073.
  28. M. Bianchi, M. Firrotta, J. Sonnenschein, and D. Weissman, Measure for chaotic scattering amplitudes, Phys. Rev. Lett. 129, 261601 (2022).
  29. M. Bianchi, M. Firrotta, J. Sonnenschein, and D. Weissman, Measuring chaos in string scattering processes, Phys. Rev. D 108, 066006 (2023).
  30. M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory Vol. 1: Introduction (Cambridge University Press, Cambridge, England, 2012), ISBN [Amazon][WorldCat], [Amazon][WorldCat].
  31. A. Aldi and M. Firrotta, String coherent vertex operators of Neveu-Schwarz and Ramond states, Nucl. Phys. B955, 115050 (2020).
  32. P. Mukhopadhyay, DDF construction and D-brane boundary states in pure spinor formalism, J. High Energy Phys. 05 (2006) 055.
  33. R. L. Jusinskas, Spectrum generating algebra for the pure spinor superstring, J. High Energy Phys. 10 (2014) 022.
  34. N. Berkovits and R. Lipinski Jusinskas, Light-cone analysis of the pure spinor formalism for the superstring, J. High Energy Phys. 08 (2014) 102.
  35. L. J. Dixon, J. A. Harvey, C. Vafa, and E. Witten, Strings on orbifolds, Nucl. Phys. B261, 678 (1985).
  36. L. J. Dixon, J. A. Harvey, C. Vafa, and E. Witten, Strings on orbifolds. 2., Nucl. Phys. B274, 285 (1986).
  37. N. Berkovits, Off-shell supersymmetry versus hermiticity in the superstring, Phys. Rev. Lett. 77, 2891 (1996).
  38. N. Berkovits and B. C. Vallilo, One loop N point superstring amplitudes with manifest d=4 supersymmetry, Nucl. Phys. B624, 45 (2002).
  39. D. Lüst, N. Mekareeya, O. Schlotterer, and A. Thomson, Refined partition functions for open superstrings with 4, 8 and 16 supercharges, Nucl. Phys. B876, 55 (2013).
  40. N. Arkani-Hamed, T. C. Huang, and Y. t. Huang, Scattering amplitudes for all masses and spins, J. High Energy Phys. 11 (2021) 070.
  41. O. Hohm, W. Siegel, and B. Zwiebach, Doubled α′-geometry, J. High Energy Phys. 02 (2014) 065.
  42. W. Siegel, Amplitudes for left-handed strings, arXiv:1512.02569.
  43. L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, J. High Energy Phys. 07 (2014) 048.
  44. E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252, 189 (2004).
  45. N. Berkovits, An alternative string theory in twistor space for N=4 superYang-Mills, Phys. Rev. Lett. 93, 011601 (2004).
  46. F. Cachazo, S. He, and E. Y. Yuan, Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett. 113, 171601 (2014).

Outline

Information

Phys. Rev. D 113, L021902– Published 20 January, 2026
  • Received 5 August 2025
  • Accepted 5 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation