- 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
  • Open Access

Critical non-Abelian vortex string and 2D N=2 black hole

Evgenii Ievlev1, Andrei Marshakov2,3, Gleb Sumbatian4, and Alexei Yung4,5

PDF

Phys. Rev. D 112, 105010 – Published 10 November, 2025

DOI: https://doi.org/10.1103/rz16-99g6

Abstract

It has been shown that the non-Abelian vortex string in 4D N=2 supersymmetric QCD (SQCD) with the U(2) gauge group and Nf=4 flavors becomes a critical superstring. Its 10D target space is a product of the flat 4D space and an internal noncompact Calabi-Yau threefold, namely, the conifold. It was also shown that the Coulomb branch of the associated string sigma model, which opens up at strong coupling, can be described by N=2 Liouville theory. We continue here the study of the recently proposed mass deformation of the U(2) theory with Nf=4, interpolating to SQCD with the U(4) gauge group and Nf=8 quarks, by analyzing the mass-deformed N=2 Liouville theory on the string world sheet, and show that it is always described by the trumpet geometry of the target space, which is T-dual to the 2D N=2 supersymmetric black hole. We use this correspondence to find the low-lying hadron spectrum in the deformed SQCD, and explain the expected increase in the number of hadronic states in the theory with more gauge fields and quarks by considering the near-Hagedorn behavior of the 2D black hole.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (65)

  1. A. Hanany and D. Tong, Vortices, instantons and branes, J. High Energy Phys. 07 (2003) 037.
  2. R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi, and A. Yung, Non-Abelian superconductors: Vortices and confinement in N=2 SQCD, Nucl. Phys. B673, 187 (2003).
  3. M. Shifman and A. Yung, Non-Abelian string junctions as confined monopoles, Phys. Rev. D 70, 045004 (2004).
  4. A. Hanany and D. Tong, Vortex strings and four-dimensional gauge dynamics, J. High Energy Phys. 04 (2004) 066.
  5. D. Tong, TASI lectures on solitons, arXiv:hep-th/0509216.
  6. M. Eto, Y. Isozumi, M. Nitta, K. Ohashi, and N. Sakai, Solitons in the Higgs phase: The moduli matrix approach, J. Phys. A 39, R315 (2006).
  7. M. Shifman and A. Yung, Supersymmetric solitons and how they help us understand non-Abelian gauge theories, Rev. Mod. Phys. 79, 1139 (2007). for an expanded version see Supersymmetric Solitons, (Cambridge University Press, Cambridge, England, 2009).
  8. D. Tong, Quantum vortex strings: A review, Ann. Phys. (Amsterdam) 324, 30 (2009).
  9. M. Shifman and A. Yung, Critical string from non-Abelian vortex in four dimensions, Phys. Lett. B 750, 416 (2015).
  10. P. Koroteev, M. Shifman, and A. Yung, Non-Abelian vortex in four dimensions as a critical string on a conifold, Phys. Rev. D 94, 065002 (2016).
  11. P. Candelas and X. C. de la Ossa, Comments on conifolds, Nucl. Phys. B342, 246 (1990).
  12. A. Neitzke and C. Vafa, Topological strings and their physical applications, arXiv:hep-th/0410178.
  13. M. Shifman and A. Yung, Critical non-Abelian vortex in four dimensions and little string theory, Phys. Rev. D 96, 046009 (2017).
  14. D. Kutasov, Introduction to Little String Theory, in Superstrings and Related Matters 2001, Proceedings of the ICTP Spring School of Physics, Edited by C. Bachas, K. S. Narain, and S. Randjbar-Daemi (2002), pp. 165–209.
  15. D. Ghoshal and C. Vafa, c=1 string as the topological theory of the conifold, Nucl. Phys. B453, 121 (1995).
  16. E. Ivanov and S. Krivonos, U(1) supersymmetric extension of the Liouville equation, Lett. Math. Phys. 7, 523 (1983).
  17. D. Kutasov and N. Seiberg, Noncritical superstrings, Phys. Lett. B 251, 67 (1990).
  18. A. Giveon and D. Kutasov, Little string theory in a double scaling limit, J. High Energy Phys. 10 (1999) 034.
  19. A. Giveon, D. Kutasov, and O. Pelc, Holography for noncritical superstrings, J. High Energy Phys. 10 (1999) 035.
  20. M. Shifman and A. Yung, Hadrons of N=2 supersymmetric QCD in four dimensions from little string theory, Phys. Rev. D 98, 085013 (2018).
  21. P. Gavrylenko, E. Ievlev, A. Marshakov, I. Monastyrskii, and A. Yung, 2D sigma models on non-compact Calabi-Yau and N=2 Liouville theory, Phys. Rev. D 111, 106003 (2025).
  22. A. Yung, Flowing between string vacua for the critical non-Abelian vortex with a deformation of N=2 Liouville theory, Phys. Rev. D 110, 025004 (2024).
  23. E. Witten, String theory and black holes, Phys. Rev. D 44, 314 (1991).
  24. S. Mukhi and C. Vafa, Two-dimensional black hole as a topological coset model of c=1 string theory, Nucl. Phys. B407, 667 (1993).
  25. H. Ooguri and C. Vafa, Two-dimensional black hole and singularities of CY manifolds, Nucl. Phys. B463, 55 (1996).
  26. A. Giveon, Target space duality and stringy black holes, Mod. Phys. Lett. A 06, 2843 (1991).
  27. R. Dijkgraaf, H. Verlinde, and E. Verlinde, String propagation in a black hole geometry, Nucl. Phys. B371, 269 (1992).
  28. K. Hori and A. Kapustin, Duality of the fermionic 2-D black hole and N=2 Liouville theory as mirror symmetry, J. High Energy Phys. 08 (2001) 045.
  29. P. Fayet and J. Iliopoulos, Spontaneously broken supergauge symmetries and goldstone spinors, Phys. Lett. 51B, 461 (1974).
  30. M. Shifman and A. Yung, Lessons from supersymmetry: “Instead-of-Confinement” mechanism, Int. J. Mod. Phys. A 29, 1430064 (2014).
  31. E. Ievlev, M. Shifman, and A. Yung, String baryon in four-dimensional N=2 supersymmetric QCD from the 2D-4D correspondence, Phys. Rev. D 102, 054026 (2020).
  32. For a review see e.g. A. Achucarro and T. Vachaspati, Semilocal and electroweak strings, Phys. Rep. 327, 347 (2000).
  33. M. Shifman and A. Yung, Non-Abelian semilocal strings in N=2 supersymmetric QCD, Phys. Rev. D 73, 125012 (2006).
  34. M. Eto, J. Evslin, K. Konishi, G. Marmorini, M. Nitta, K. Ohashi, W. Vinci, and N. Yokoi, On the moduli space of semilocal strings and lumps, Phys. Rev. D 76, 105002 (2007).
  35. M. Shifman, W. Vinci, and A. Yung, Effective world-sheet theory for non-Abelian semilocal strings in N=2 supersymmetric QCD, Phys. Rev. D 83, 125017 (2011).
  36. E. Witten, Phases of N=2 theories in two dimensions, Nucl. Phys. B403, 159 (1993).
  37. N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. Phys. B431, 484 (1994).
  38. N. Dorey, The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms, J. High Energy Phys. 11 (1998) 005.
  39. N. Dorey, T. J. Hollowood, and D. Tong, The BPS spectra of gauge theories in two and four dimensions, J. High Energy Phys. 05 (1999) 006.
  40. E. Ievlev and A. Yung, Critical non-Abelian vortex and holography for little string theory, Phys. Rev. D 104, 114033 (2021).
  41. Y. Nakayama, Liouville field theory: A decade after the revolution, Int. J. Mod. Phys. A 19, 2771 (2004).
  42. J. Teschner, Operator product expansion and factorization in the H+(3) WZNW model, Nucl. Phys. B571, 555 (2000).
  43. O. Aharony, A. Giveon, and D. Kutasov, LSZ in LST, Nucl. Phys. B691, 3 (2004).
  44. P. Baseilhac and V. Fateev, Expectation values of local fields for a two-parameter family of integrable models and related perturbed conformal field theories, Nucl. Phys. B532, 567 (1998).
  45. L. J. Dixon, M. E. Peskin, and J. D. Lykken, N=2 superconformal symmetry and SO(2,1) current algebra, Nucl. Phys. B325, 329 (1989).
  46. P. M. S. Petropoulos, Comments on SU(1,1) string theory, Phys. Lett. B 236, 151 (1990).
  47. S. Hwang, Cosets as gauge slices in SU(1,1) strings, Phys. Lett. B 276, 451 (1992).
  48. J. M. Evans, M. R. Gaberdiel, and M. J. Perry, The no ghost theorem for AdS(3) and the stringy exclusion principle, Nucl. Phys. B535, 152 (1998).
  49. J. M. Evans, M. R. Gaberdiel, and M. J Perry, The no-ghost theorem and strings on AdS3, in Proceedings of the 1998 ICTP Spring School of Physics, Nonperturbative Aspects of Strings, Branes and Supersymmetry, Edited by M. J. Duff et al. (1998), pp. 435–444.
  50. A. Strominger, Massless black holes and conifolds in string theory, Nucl. Phys. B451, 96 (1995).
  51. T. Eguchi and Y. Sugawara, D-branes in singular Calabi-Yau n fold and N=2 Liouville theory, Nucl. Phys. B598, 467 (2001).
  52. E. Alvarez, L. Alvarez-Gaume, and Y. Lozano, An introduction to T duality in string theory, Nucl. Phys. B, Proc. Suppl. 41, 1 (1995).
  53. T. G. Mertens, Hagedorn string thermodynamics in curved spacetimes and near black hole horizons, Ph.D. thesis, Gent University, 2015.
  54. J. Polchinski, Critical behavior of random surfaces in one dimension, Nucl. Phys. B346, 253 (1990).
  55. I. S. Grandshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, translated from Russian (Elsevier/Academic Press, Amsterdam, 2007), 7th ed.
  56. R. Hagedorn, Statistical thermodynamics of strong interactions at high-energies, Nuovo Cimento Suppl. 3, 147 (1965).
  57. J. J. Atick and E. Witten, The Hagedorn transition and the number of degrees of freedom of string theory, Nucl. Phys. B310, 291 (1988).
  58. V. Fainberg and A. Marshakov, A propagator for the fermionic string, Phys. Lett. B 211, 81 (1988).
  59. L. Susskind, Some speculations about black hole entropy in string theory, arXiv:hep-th/9309145.
  60. G. T. Horowitz and J. Polchinski, A correspondence principle for black holes and strings, Phys. Rev. D 55, 6189 (1997).
  61. A. Giveon, D. Kutasov, E. Rabinovici, and A. Sever, Phases of quantum gravity in AdS3 and linear dilaton backgrounds, Nucl. Phys. B719, 3 (2005).
  62. D. Kutasov, Accelerating branes and the string/black hole transition, arXiv:hep-th/0509170.
  63. V. Kazakov, I. K. Kostov, and D. Kutasov, A matrix model for the two-dimensional black hole, Nucl. Phys. B622, 141 (2002).
  64. Y. Chen and J. Maldacena, String scale black holes at large D, J. High Energy Phys. 01 (2022) 095.
  65. E. Witten, Introduction to black hole thermodynamics, arXiv:2412.16725.

Outline

Information

Phys. Rev. D 112, 105010– Published 10 November, 2025
  • Received 31 August 2025
  • Accepted 17 October 2025
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