Noncommutative Chern-Simons solitons

Abstract

The Chern-Simons theories on a noncommutative plane, which is shown to be describing the quantum Hall liquid, is considered. We introduce matter fields fundamentally coupled to the noncommutative Chern-Simons field. Exploiting BPS equations for the nonrelativistic Chern-Simons theory, we find the exact solutions of multi vortices that are closely packed and exponentially localized. We determine the position, the size and the angular momentum explicitly. We then construct solutions of two spatially separated vortex, and determine the moduli dependence of the size and the angular momentum. We also consider the relativistic Chern-Simons theory and find nontopological solutions whose properties are similar to the nonrelativistic counterpart. However, unlike the nonrelativistic case, there are two branches of solutions for a given magnetic field and they cease to exist below certain noncommutativity scale. * Electronic Mail: dsbak@mach.uos.ac.kr † The Seiberg-Witten map is a way to find the local gauge invariant quantities [32].

Galilean noncommutative gauge theory: symmetries and vortices

“…Let us now turn to the noncommutative version of the above theory [8,9]. The Lagrangian L l = L l matter + L * f ield is formally the same as in the commutative case, Eq.…”

“…In order to have a reference for the fermionic case (Section 7), we review briefly the main results of Lozano, Moreno, and Schaposnik [8], and of Bak, Kim, Soh, and Yee [9], respectively. These authors consider the previous NC-CS gauge theory to which they add a quartic "left-potential" V = (λ/4)ψ ⋆ψ = (λ/4)ρ l .…”

“…In the NC case, vortices are in turn constructed by solving these equations [8,9] using a rather involved technique we do not reproduce here. We note for further reference that their SD (D + ψ = 0) solution is regular for κ < 0, and their ASD (D − ψ = 0) solution is regular for κ > 0, respectively.…”

LLL vs. LLM: Half BPS sector of SYM ≡ quantum Hall system

“…(2.26)). Quasiparticle states, however, were obtained within the model introduced in [24] in the context of commutative Chern-Simons model and then in [25,26] extended to the noncommutative Chern-Simons.…”

“…with [y i , y j ] = iθǫ ij , (4.1) then reduces to noncommutative Chern-Simons (NCCS) action in 2+1 dimensions [12,25]. In this notation…”