Flowing between string vacua for the critical non-Abelian vortex with a deformation of N=2 Liouville theory

This is the authors' abstract. We don't add key points for this paper.

It has been shown that non-Abelian solitonic vortex string supported in four-dimensional (4D) <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="script">N</a:mi><a:mo>=</a:mo><a:mn>2</a:mn></a:math> supersymmetric QCD (SQCD) with the U(2) gauge group and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msub><d:mi>N</d:mi><d:mi>f</d:mi></d:msub><d:mo>=</d:mo><d:mn>4</d:mn></d:math> quark flavors becomes a critical superstring. This string propagates in the ten-dimensional space formed by a product of the flat 4D space and an internal space given by a Calabi-Yau noncompact threefold, namely, the conifold. The spectrum of low-lying closed string states was found and interpreted as a spectrum of hadrons in 4D <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi mathvariant="script">N</f:mi><f:mo>=</f:mo><f:mn>2</f:mn></f:math> SQCD. In particular, the lowest string state appears to be a massless Bogomol’nyi-Prasad-Sommerfield (BPS) baryon associated with the deformation of the complex structure modulus <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>b</i:mi></i:math> of the conifold. It was recently shown that the Coulomb branch of the associated string sigma model which opens up at strong coupling can be described by <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi mathvariant="script">N</k:mi><k:mo>=</k:mo><k:mn>2</k:mn></k:math> Liouville theory. Building on these results we switch on quark masses in 4D <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi mathvariant="script">N</n:mi><n:mo>=</n:mo><n:mn>2</n:mn></n:math> SQCD and study the interpolation of the initial U(2) SQCD with <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>N</q:mi><q:mi>f</q:mi></q:msub><q:mo>=</q:mo><q:mn>4</q:mn></q:math> quarks to the final SQCD with the U(4) gauge group and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>N</s:mi><s:mi>f</s:mi></s:msub><s:mo>=</s:mo><s:mn>8</s:mn></s:math> quarks. To find the true string vacuum which arises due to the mass deformation we solve the effective supergravity equations of motion associated with the deformed world sheet Liouville theory. We show that the massless BPS baryon <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>b</u:mi></u:math> survives the deformation and that finding of the spectrum of low-lying massive hadrons in the final SQCD is linked to the Calogero problem.