Non-Abelian monopoles in the Higgs phase

Abstract

We use the moduli matrix approach to study the moduli space of 1/4 BPS kinks supported by vortices in the Higgs phase of N = 2 supersymmetric U(N) gauge theories when non-zero masses for the matter hypermultiplets are introduced. We focus on the case of degenerate masses. In these special cases vortices acquire new orientational degrees of freedom, and become "non-Abelian". Kinks acquire new degrees of freedom too, and we will refer to them as "non-Abelian". As already noticed for the Abelian case, non-Abelian kinks must correspond to non-Abelian monopoles of the unbroken phase of SU(N) Yang-Mills. We show, in some special cases, that the moduli spaces of the two objects are in one-to-one correspondence. We argue that the correspondence holds in the most general case. The consequence of our result is two-fold. First, it gives an alternative way to construct non-Abelian monopoles, in addition to other wellknown techniques (Nahm transform, spectral curves, rational maps). Second, it opens the way to the study of the quantum physics of non-Abelian monopoles, by considering the simpler non-Abelian kinks.

Quantum energies and tensorial central charges of confined monopoles

“…If the first N masses form q groups of n r degenerate masses the surviving symmetry group is given by [23] 12) with r n r = N . It supports monopoles with typical size 1/∆m, the inverse mass difference, and are confined by flux tubes of width ∼ 1/g √ ξ, see figure 1.…”

“…8 To make contact with our four-dimensional N = 2 SQCD Lagrangian (2.14) we choose a chiral representation for the four-dimensional gamma matrices γ µ and decompose the four-component spinors into Weyl spinors as follows: 23) and the charge conjugation matrix is given by C 4D = i γ 2 γ 0 . Inserting these decompositions into the six-dimensional Lagrangian (2.20) and taking all fields independent of the extra coordinates x 5,6 one finds the original four-dimensional Lagrangian (2.14) upon the following identifications:…”

“…In the case of static fields and using temporal gauge one has, in analogy to before, e i ϑ = ±1 and the equations reduce to 23) where the sign of the φ terms is uncorrelated with the sign of the D term. As shown above they determine the signs of the charges Z and T v .…”

“…The field configuration that we have in mind in the following is that of (multiple) confined monopoles, as depicted in figure 1. 23 The axial orientation of the field configurations implies that the asymptotic boundary has the form of an infinite cylinder, see figure 1, and accordingly we have to specify the asymptotic behavior: i.) At x 3 → ±∞ the boundary is given by the infinite discs D ± at which the fields behave like (multiple) vortices, though in general different vortices at D + and D − .…”

“…It is determined by the difference in the flux through the discs D ± . Following [23] one can express this flux in terms of the individual contributions to the vortex number according…”