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Critical Non-Abelian Vortex String and 2D N=2 Black Hole

E. Ievlev, A. Marshakov, G. Sumbatian, A. Yung

TL;DR

This work analyzes a mass deformation of the critical non-Abelian vortex string in 4D ${\mathcal{N}}=2$ SQCD, interpolating from U$(2)$ with $N_f=4$ to U$(4)$ with $N_f=8$. The authors show the mass-deformed worldsheet theory remains dual to a trumpet geometry, T-dual to a 2D ${\mathcal{N}}=2$ black hole with cigar geometry, enabling a holographic-like mapping to the 4D hadron spectrum. The low-lying spectrum is unchanged by the deformation, while the number of hadronic states grows as more quark flavors are included, explained via near-Hagedorn behavior of the 2D black hole. The work combines Liouville theory, T- and S-dualities, and effective gravity/Virasoro techniques to connect 4D SQCD hadrons to 2D string dynamics and black hole thermodynamics, offering a quantitative picture of state counting in strongly coupled regimes.

Abstract

It has been shown that the non-Abelian vortex string in 4D $\mathcal{N}=2$ SQCD with the U(2) gauge group and $N_f=4$ 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 $\mathcal{N}=2$ Liouville theory. We continue here the study of the recently proposed mass deformation of the U(2) theory with $N_f=4$, interpolating to SQCD with the U(4) gauge group and $N_f=8$ quarks, by analyzing the mass-deformed $\mathcal{N}=2$ 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 $\mathcal{N}=2$ 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.

Critical Non-Abelian Vortex String and 2D N=2 Black Hole

TL;DR

This work analyzes a mass deformation of the critical non-Abelian vortex string in 4D SQCD, interpolating from U with to U with . The authors show the mass-deformed worldsheet theory remains dual to a trumpet geometry, T-dual to a 2D black hole with cigar geometry, enabling a holographic-like mapping to the 4D hadron spectrum. The low-lying spectrum is unchanged by the deformation, while the number of hadronic states grows as more quark flavors are included, explained via near-Hagedorn behavior of the 2D black hole. The work combines Liouville theory, T- and S-dualities, and effective gravity/Virasoro techniques to connect 4D SQCD hadrons to 2D string dynamics and black hole thermodynamics, offering a quantitative picture of state counting in strongly coupled regimes.

Abstract

It has been shown that the non-Abelian vortex string in 4D SQCD with the U(2) gauge group and 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 Liouville theory. We continue here the study of the recently proposed mass deformation of the U(2) theory with , interpolating to SQCD with the U(4) gauge group and quarks, by analyzing the mass-deformed Liouville theory on the string world sheet, and show that it is always described by the trumpet geometry of the target space, which is -dual to the 2D 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.

Paper Structure

This paper contains 27 sections, 163 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Non-Abelian vortex string and $\mathcal{N}=2$ Liouville theory
  3. Four-dimensional ${\mathcal{N}}=2\;$ SQCD
  4. World-sheet theory of the non-Abelian string
  5. $\mathbb{WCP}(N,N)\;$ model
  6. ${\mathcal{N}}=2\;$ Liouville theory
  7. Primary operators on the world sheet
  8. 4D SQCD low-lying hadron spectrum
  9. Mass deformation
  10. Metric after mass deformation
  11. $T$-duality
  12. Trumpet geometry and fluctuations
  13. Switching on both superpotential and mass deformations
  14. String spectrum after mass deformation
  15. Tachyon equations
  16. ...and 12 more sections

Figures (3)

  • Figure 1: Cigar geometry of 2D black hole.
  • Figure 2: Trumpet geometry. Asymptotically, at $\rho\to\infty$, it turns into a cylinder of radius $Q$.
  • Figure 3: States with different baryonic charges. The blue dashed line corresponds to $l=\frac{B}{2},k=0,$ the green one to $l=0,k=\frac{B}{2}$. Red crosses are new states that emerge as we reduce $\mathcal{M}$.