Integrable Chern-Simons Gauge Field Theory in 2+1 Dimensions

Abstract

The classical spin model in planar condensed media is represented as the U (1) Chern-Simons gauge field theory. When the vorticity of the continuous flow of the media coincides with the statistical magnetic field, which is necessary for the model's integrability, the theory admits zero curvature connection. This allows me to formulate the model in terms of gauge -invariant fields whose evolution is described by In the present paper I reduce the self-dual CS model from an integrable model in 2+1 dimensions. This model was first derived by Ishimori and has integrable dynamics of the magnetic vortices [7]. Its formulation in terms of the vorticity for continuous flow tends to clarify the physical meaning of the model and allows an extension for higher dimensions, admitting the bilinear Hirota representation [8,10]. The physical applications of the model are connected with the resolution of the anomalous behavior of the linear momentum in ferromagnets [9]. As it was shown in the latter paper by using the de-localized electron model of ferromagnets, the canonical momentum becomes well defined due to a fermionic background.The model considered in the present paper is a modification of the classical Heisenberg model for condensed media, having hydrodynamical flow with its vorticity related to a topological charge density. In this case a hydrodynamical vortex in the magnetic (anyon) fluid is also inducing a magnetic vorticity.The CS gauge field theory is then obtained by projecting spin variables on the tangent plane to the sphere (spin phase space). This idea has been recently applied to the classical Heisenberg model in [12]. Furthermore, the stationary magnetic vortices of the model have been related to the Chern-Simons solitons, while the topological charge of the former has been related to the electric charge of the latter. 2In the case of equality between hydrodynamical vorticity and the "statistical" magnetic field, the theory can be formulated for a zero curvature effective gauge field. This allows me to reformulate the model in terms of gauge invariant matter fields. As shown, their evolution is described by the DS equations and at the same time the fields satisfy the CS system. For the self-dual case, we reproduce CS solitons with an integrable evolution replacing the statistical gauge field with a velocity field.As a by-product we obtain: a) new reduction conditions for the DS-I model related to the Ishimori-I (IM-I) model, b) the 2+1 dimensional zero curvature formulation for the DS equation. Zero Curvature ConditionThe crucial moment of our construction is a zero curvature condition in 2+1 dimensional spacefor u(2)(u(1, 1)) Lie algebra valued connection J µ , (µ = 0, 1, 2). Decomposing the algebra for the diagonal and off-diagonal partsand parametrizingin terms of two U (1) gauge potentials W µ and V µ , and the q µ matter fields, we have the set of U (1) × U (1) gauge invariant equationsHere the covariant derivative is defined asFor the spatial part of the matter fields we introduceand denote D ± =...

On the Stability of Certain Spin Models in 2+1 Dimensions

“…6 we derive the connection between the Ishimori systems, which correspond to = −1, and the Davey-Stewartson equations, starting from our modified spin systems. This connection is well-known; see, for example, [6,9,11], or [16], at least in the focusing case μ = 1. The analysis in this paper can then be combined with the global analysis of the defocusing Davey-Stewartson II equation (see [13][14][15]), to give global solutions of the defocusing Ishimori system in the case of large classical data, constant outside a compact set (see Theorem 6.1).…”

Integrable vortex dynamics in anisotropic planar spin liquid model

“…The Ishimori model is the first example of integrable classical spin model in 2+1 dimensions [36]. It was shown to be gauge equivalent to the Davey-Stewartson equation, representing the 2+1 dimensional generalization of the Nonlinear Schrödinger equation [32], [33], [37]. Though it was solved in terms of the ∂ problem, for description of vortices and vortex lattices we propose more simple and elegant method.…”

The Cauchy problem for the planar spin-liquid model

“…The generalized Hasimoto transform can also be applied to the Ishimori model (1.5) to derive the Davey-Stewartson equations. We will not repeat the procedure here (see [27]). …”

“…In [5], global existence was only proved for equivariant solutions of the Heisenberg model (1.1). For the spin-liquid model (1.2), the velocity constraint (1.3) allows us to prove global existence of solutions without symmetry assumption (see [27]). …”