Abstract
Geometric and dynamical aspects of a coupled 4D-2D interacting quantum field theory - the gauged nonAbelian vortex - are investigated. The fluctuations of the internal 2D nonAbelian vortex zeromodes excite the massless 4D Yang-Mills modes and in general give rise to divergent energies. This means that the well-known 2D CP(N-1) zeromodes associated with a nonAbelian vortex become nonnormalizable. Moreover, all sorts of global, topological 4D effects such as the nonAbelian Aharonov-Bohm effect come into play. These topological global features and the dynamical properties associated with the fluctuation of the 2D vortex moduli modes are intimately correlated, as shown concretely here in a U(1) x SU(N) x SU(N) model with scalar fields in a bifundamental representation of the two SU(N) factor gauge groups.
Highlights
All sorts of global effects, such as nonAbelian Aharonov-Bohm phases and scattering, an obstruction to part of the “unbroken” gauge symmetry, nonAbelian statistics under the exchange of parallel vortices, Cheshire charges, etc. make their appearance, depending on the vortex orientations, {Bi}. These phenomena have been investigated in various general contexts [9]–[13] with gauge symmetry breaking, G → H, with π1(G/H) = 1, and more recently, in the context of concrete model, e.g., in a U0(1) × SUl(N ) × SUr(N ) gauge theory, with scalar fields in the bifundamental representation of the two SU(N ) gauge groups [14]–[16]
Let us remind ourselves that a characteristic feature of a color-flavor locked vacuum is the fact that all massless Nambu-Goldstone particles are eaten by the broken gauge fields, all of which become massive, maintaining mass degeneracy among them
As soon as U is taken to fluctuate along the string, nontrivial Aα fields are induced, whose precise form is dictated by the equations of motion. Their behavior at spatial infinity, is fixed by the requirement |DαQ − (DαQ)| → 0, which implies that Aα approaches the gauge fields belonging to H ⊂ G left unbroken by the vacuum configuration u(θ)Q0(∞)
Summary
The vortex solutions can be found by the BPS completion of the expression for the tension (for configurations depending only on the transverse coordinates x and y): T=. The BPS equations are : D1Q + iD2Q = 0 , F1(2r) − gr ta Tr(Q†Qta) = 0 , f12 + g0(Tr Q†Q − N ξ) = 0 , F1(2l) + gl ta Tr(Q† taQ) = 0. The BPS equations (2.8)–(2.9) show that the profile functions satisfy rQ1 − Q1. Which can be solved by numerical methods. These equations are identical to those found earlier for the global nonAbelian vortex, i.e., for gr = 0 or gl = 0, except for the fact that the gauge fields compensating the scalar winding energy ∂Q/∂θ are shared between the left and right SU(N ) fields.
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