Flux Compactification for the Critical Non-Abelian Vortex and Quark Masses

It has been shown that non-Abelian solitonic vortex strings supported in four-dimensional (4D) ${\mathcal N}=2\;$ supersymmetric QCD (SQCD) with the U($N=2$) gauge group and $N_f=4$ quark flavors behave as critical superstrings. In addition to four translational moduli non-Abelian strings under consideration carry six orientational and size moduli. Together they form a ten-dimensional space required for a superstring to be critical. The target space of the string sigma model is a product of the flat four-dimensional space $\mathbb{R}^4$ and a Calabi-Yau non-compact threefold $Y_6$, namely, the conifold. The spectrum of low lying closed string states in the associated type IIA string theory was found and interpreted as a spectrum of hadrons in 4D ${\mathcal N}=2\;$ SQCD. In particular, the lowest string state appears to be a massless BPS baryon associated with the deformation of the complex structure modulus $b$ of the conifold. Here we address a problem of switching on quark masses in 4D SQCD, which classically breaks the world sheet conformal invariance in the string sigma model. To avoid this problem we follow a standard string theory approach and use a flux "compactification" to lift the complex structure modulus of the conifold. Namely, we find a solution of supergravity equations of motion with non-zero NS 3-form flux. It produces a potential for the baryon $b$, which leads to the run-away vacuum. Using field theory arguments we interpret 3-form flux in terms of a particular choice of quark masses in 4D SQCD. At the run-away vacuum the conifold degenerates to lower dimensions. We interpret this as a flow from a non-Abelian string to an Abelian one.