Noncommutative vortex solitons

Abstract

We consider the noncommutative Abelian-Higgs theory and investigate general static vortex configurations including recently found exact multi-vortex solutions. In particular, we prove that the self-dual BPS solutions cease to exist once the noncommutativity scale exceeds a critical value. We then study the fluctuation spectra about the static configuration and show that the exact non BPS solutions are unstable below the critical value. We have identified the tachyonic degrees as well as massless moduli degrees. We then discuss the physical meaning of the moduli degrees and construct exact time-dependent vortex configurations where each vortex moves independently. We finally give the moduli description of the vortices and show that the matrix nature of moduli coordinates naturally emerges.

Transmogrifying fuzzy vortices

“…Interestingly enough, choosing q (2) = 0 does not affect this conclusion. Again, this is not altogether unexpected since q (2) is just the angular momentum quantum number of the vortex [16]. Convergence of the coefficient sequence for φ 2 bounds the angular momentum quantum number to the range 0 ≤ q (2) < k. However, since none of the arguments presented here depends essentially on q (2) we can, without any loss of generality, set q (2) = 0.…”

“…This is not unlike the situation with the conventional Nielsen-Olesen vortex. However this particular hurdle was recently surmounted in [1,2,23,37] where a noncommutative deformation of the two-dimensional configuration space of the Abelian-Higgs model allows for the construction of exact vortex solutions. The fact that the noncommutative version of the theory seems so much richer than its commutative counterpart is by now not surprising [7,8].…”

“…where {q (1) , q (2) } is a set of integers related to the topological charge and, respectively, angular momentum quantum number of the vortex as we show below. For the U(1) gauge field we take the cylindrically symmetric ansatz…”

“…Solutions of this system will be the noncommutative generalizations of the semilocal vortex of [17]. In the spirit of [2,23], we begin with an ansatz for the Higgs doublet and the gauge field. To this end the most general vortex-like solution of the BPS equations which maintain the cylindrical symmetry is of the form 3…”

“…Fuzzy deformations of the background space have, in only a few years, not only yielded a wealth of new solitonic solutions but also several new insights into old solutions to a host of field theories (see [4,12,33] for excellent reviews). The noncommutative Abelian-Higgs model, for example, exhibits exact vortex solutions [1,2,23,37] whose moduli space metric can be computed explicitly in the large noncommutativity limit [34].…”