Non-Abelian Chern–Simons vortices with generic gauge groups

Abstract

We study non-Abelian Chern-Simon BPS-saturated vortices enjoying N = 2 supersymmetry in d = 2 + 1 dimensions, with generic gauge groups of the form U (1) × G ′ , with G ′ being a simple group, allowing for orientational modes in the solutions. We will keep the group as general as possible and utilizing the powerful moduli matrix formalism to provide the moduli spaces of vortices and derive the corresponding master equations. Furthermore, we study numerically the vortices applying a radial Ansatz to solve the obtained master equations and we find especially a splitting of the magnetic fields, when the coupling constants for the trace-part and the traceless part of the Chern-Simons term are varied, such that the Abelian magnetic field density can become negative near the origin of the vortex while the non-Abelian part stays positive, and vice versa.

On Non-Topological Solutions for Planar Liouville Systems of Toda-Type

“…Motivated by the delicate issue of "'quark confinement"', Gudnason in [33,34] introduced a non-abelian Chern-Simons model formulated within a N = 2 Supersymmetric (SUSY) Field Theory, with a general gauge group of the type: G = U (1) × G ′ allowing solutions with orientational modes. For the model in [33,34], the author identifies the BPS-sector of the theory and the corresponding self-dual equations. In particular, when G ′ = SO (2) or G ′ = U S p (2), Gudnason in [33,34] introduced some meaningful physical ansatz on the structure of the vortex solutions, by which (as in [40]) the corresponding self-dual equations reduced to the following set of Master's equations:…”

“…In this context, a successful way to detect vortices is to identify the BPS-sector of the theory, since in such regime vortex configurations simply correspond to (static) solutions of the so called self-dual equations of Bogomolnyi type, and saturate the minimal energy allowed by the system. For this reason, it has been useful to invoke "duality" and formulate the theory within the general framework of N = 2 Supersymmetric (SUSY) Field Theory, in this direction see for example: [27,28,31,33,34,55,60] for more details . We observe that, when the Chern-Simons Lagrangian is taken into account then the theory can attain self-duality only with the help of a six-order scalar potential field, see [26,67,71], in place of the more familiar quadratic (double-well) potential of the Maxwell-Higgs model, see [40].…”

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Motivated by the study of non-abelian Chern Simons vortices of non-topological type in Gauge Field Theory, see e.g. [33,34], [26], we analyse the solvability of the following (normalised) Liouville-type system in presence of singular sources:2π R 2 e u1 and β 2 = 1 2π R 2 e u2 , with τ > 0 and N > 0. We identify necessary and sufficient conditions on the parameter τ and the "flux" pair: (β 1 , β 2 ), which ensure the radial solvability of (1) τ .

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Small Data Global Existence and Decay for Relativistic Chern–Simons Equations

“…In fact, even when we assume the sharp decay rate 17) where N = 1 √ t 2 −r 2 S is the normalization of S, the gauge covariant vector field method discussed so far seems to only lead to a weak decay rate…”

“…The supersymmetric Chern-Simons model was discussed in [16,17,37]. Topological solutions were constructed by Yang [51].…”