Nonabelian Superconductors: Vortices and Confinement in ${\cal N}=2$ SQCD

TL;DR

The paper demonstrates that in softly broken ${\cal N}=2$ ${\rm SU}(3)$ QCD with fundamental matter, nonabelian vortices exist in a vacuum where an $SU(2)\times U(1)$ subgroup remains unbroken. These vortices carry nonabelian flux with exact orientational zero modes forming a moduli space governed by $SU(2)_{C+F}$, and their low-energy dynamics on the worldsheet reduces to a supersymmetric $O(3)$ (and, more generally, ${\bf CP}^{N-2}$) sigma model in $(1+1)$ dimensions. Mirror symmetry reveals a mass gap and no spontaneous breaking of the dual nonabelian group, implying stable nonabelian flux tubes that confine nonabelian monopoles. The construction generalizes to $SU(N)$ with unbroken $SU(N-1)\times U(1)$, yielding CP^{N-2} moduli and affine Toda dual descriptions on the worldsheet, thereby offering a concrete, semi-classical realization of nonabelian confinement in four-dimensional gauge theories.

Abstract

We study nonabelian vortices (flux tubes) in SU(N) gauge theories, which are responsible for the confinement of (nonabelian) magnetic monopoles. In particular a detailed analysis is given of ${\cal N}=2$ SQCD with gauge group SU(3) deformed by a small adjoint chiral multiplet mass. Tuning the bare quark masses (which we take to be large) to a common value $m$, we consider a particular vacuum of this theory in which an SU(2) subgroup of the gauge group remains unbroken. We consider $5 \ge N_f \ge 4$ flavors so that the SU(2) sub-sector remains non asymptotically free: the vortices carrying nonabelian fluxes may be reliably studied in a semi-classical regime. We show that the vortices indeed acquire exact zero modes which generate global rotations of the flux in an $SU(2)_{C+F}$ group. We study an effective world-sheet theory of these orientational zero modes which reduces to an ${\cal N}=2$ O(3) sigma model in (1+1) dimensions. Mirror symmetry then teaches us that the dual SU(2) group is not dynamically broken.

Figures (6)

• Figure 1: Vortex profile functions $\phi_1(r)$ and $\phi_2(r)$ of the $(1,0)$-string. Note $\phi_1(0)=0$.
• Figure 2: The profile functions $f_3(r)$ (lower curve) and $f_8(r)$ (upper curve) for the $(1,0)$-string.
• Figure 3: Lattice of $(n,k)$ vortices.
• Figure 4: Reduced lattice of ${\mathbb Z}$ vortices.
• Figure 5: Interpolating between the $(1,0)$-string and $(0,1)$-string.