Nonrelativistic Chern-Simons Theory Research Articles

Three-dimensional (higher-spin) gravities with extended Schr\xf6dinger and l-conformal Galilean symmetries

We show that an extended 3D Schrödinger algebra introduced in [1] can be reformulated as a 3D Poincaré algebra extended with an SO(2) R-symmetry generator and an SO(2) doublet of bosonic spin-1/2 generators whose commutator closes on 3D translations and a central element. As such, a non-relativistic Chern-Simons theory based on the extended Schrödinger algebra studied in [1] can be reinterpreted as a relativistic Chern-Simons theory. The latter can be obtained by a contraction of the SU(1, 2) × SU(1, 2) Chern-Simons theory with a non principal embedding of SL(2, ℝ) into SU(1, 2). The non-relativisic Schrödinger gravity of [1] and its extended Poincaré gravity counterpart are obtained by choosing different asymptotic (boundary) conditions in the Chern-Simons theory. We also consider extensions of a class of so-called l-conformal Galilean algebras, which includes the Schrödinger algebra as its member with l = 1/2, and construct ChernSimons higher-spin gravities based on these algebras.

Interacting SUSY-singlet matter in non-relativistic Chern–Simons theory

We construct an example of supersymmetric Chern–Simons-matter theory with a matter field transforming as a singlet representation of the supersymmetry algebra, where the bosonic and fermionic degrees of freedom do not match. This is obtained as a non-relativistic limit of the Chern–Simons-matter theory in 1+2 dimensions, where the particle and anti-particle coexist. We also study the index to investigate the mismatch of bosonic and fermionic degrees of freedom.

A quantum Hall fluid of vortices

In this note we demonstrate that vortices in a non-relativistic Chern-Simons theory form a quantum Hall fluid. We show that the vortex dynamics is controlled by the matrix mechanics previously proposed by Polychronakos as a description of the quantum Hall droplet. As the number of vortices becomes large, they fill the plane and a hydrodynamic treatment becomes possible, resulting in the non-commutative theory of Susskind. Key to the story is the recent D-brane realisation of vortices and their moduli spaces.

Conformal symmetry and the nonlinear Schrödinger equation

We show that the width of the wave-packet of a class of generalized nonlinear Schrodinger equations (NLSE) trapped in an arbitrary time-dependent harmonic well in any dimensions is universally determined by the same Hill's equation. This class of generalized NLSE is characterized by a dynamical O(2,1) symmetry in absence of the trap. As an application, we study the dynamical instabilities of the rotating as well as non-rotating Bose-Einstein condensates in one and two dimensions. We also show exact extended parametric resonance in a non-relativistic Chern-Simons theory producing a gauged NLSE.

Noncommutative Chern-Simons solitons

The Chern-Simons theories on a noncommutative plane, which are shown to describe the quantum Hall liquid, are 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 multivortices 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 vortices, 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 a certain noncommutativity scale.

The Topology of Nontopological Chern–Simons Vortices

The quantized magnetic flux Φ=−4π N(sign κ), N=0, ±1, ...of nontopological vortices in nonrelativistic Chern–Simons theory is related to the topological degree of the S2 → S2 mapping defined by lifting the problem to Riemann spheres. Regular solutions with finite degree only arise for rational functions whose topological degree, N, is the common number of zeros and poles on the Riemann sphere, also called their algebraic order.

Vortex Solutions of the Liouville Equation

The most general vortex solution of the Liouville equation (which arises in nonrelativistic Chern–Simons theory) is associated with rational functions, f(z)=P(z) ∖Q∥z), where P(z) and Q(z) are both polynomials, deg P<deg Q≡ N. This allows us to prove that the solution depends on 4N parameters without the use of an index theorem, as well as the flux quantization Φ=-Nπ(sign k).

Semilocal self-dual Chern-Simons solitons.

We consider a nonrelativistic Chern-Simons theory of planar matter fields interacting with the Chern-Simons gauge field in a SU(N${)}_{\mathrm{global}}$\ifmmode\times\else\texttimes\fi{}U(1${)}_{\mathrm{local}}$ invariant fashion. We find that this model admits static zero-energy self-dual soliton solutions. We also present a set of exact soliton solutions. The exact time-dependent solutions are also obtained, when this model is considered in the background of an external uniform magnetic field. \textcopyright{} 1996 The American Physical Society.

Operator ordering problem of the nonrelativistic Chern-Simons theory.

The operator ordering problem due to the quantization or regularization ambiguity in the Chern-Simons theory exists. However, we show that this can be avoided if we require Galilei covariance of the nonrelativistic Abelian Chern-Simons theory even at the quantum level for the extended sources. The covariance can be recovered only by choosing some particular operator orderings for the generators of the Galilei group depending on the quantization ambiguities of the gauge-matter commutation relation. We show that the desired ordering for the unusual prescription is not the same as the well-known normal ordering but still satisfies all the necessary conditions. Furthermore we show that the equations of motion can be expressed in a similar form regardles of the regularization ambiguity. This suggests that the different regularization prescriptions do not change the physics. On the other hand, for the case of point sources the regularization prescription is uniquely determined, and only the orderings, which are equivalent to the usual one, are allowed.

AN EXAMPLE OF q-DEFORMED FIELD THEORY

The nonrelativistic Chern–Simons theory with the single-valued anyonic field is proposed as an example of q-deformed field theory. The corresponding q-deformed algebra interpolating between bosons and fermions, both in position and momentum space, is analyzed. A possible generalization to a space with an arbitrary dimension is suggested.

On Schrödinger superalgebras

Using the supersymplectic framework of Berezin, Kostant, and others, two types of supersymmetric extensions of the Schrödinger algebra (itself a conformal extension of the Galilei algebra) were constructed. An ‘I-type’ extension exists in any space dimension, and for any pair of integers N+ and N−. It yields an N=N++N− superalgebra, which generalizes the N=1 supersymmetry Gauntlett et al. found for a free spin-1/2 particle, as well as the N=2 supersymmetry of the fermionic oscillator found by Beckers et al. In two space dimensions, new, ‘exotic’ or ‘IJ-type’ extensions arise for each pair of integers ν+ and ν−, yielding an N=2(ν++ν−) superalgebra of the type discovered recently by Leblanc et al. in nonrelativistic Chern–Simons theory. For the magnetic monopole the symmetry reduces to o(3)×osp(1/1), and for the magnetic vortex it reduces to o(2)×osp(1/2).

Nonrelativistic Cherns-Simons theory for the repulsive Bose gas.

We propose a new nonrelativistic Chern-Simons theory based on a simple modification of the standard Lagrangian. This admits asymptotically nonvanishing field configurations and is applicable to the description of systems of repulsive bosons. The new model supports topological vortices and has a self-dual limit, both in the pure Chern-Simons and in the mixed Chern-Simons-Maxwell cases. The analysis is based on a new formulation of the Chern-Simons theories as constrained Hamiltonian systems.

Finite and infinite symmetries in (2+1)-Dimensional field theory

We describe the role of SO(2,1) conformal symmetry in non-relativistic Chern-Simons theory and in effective field theories for the eikonal regime: how the symmetry acts, how it controls the nature of solutions, how it expands to an infinite group on the manifold of static solutions thereby rendering the static problem completly integrable.

Classical and quantal nonrelativistic Chern-Simons theory.

We construct a nonrelativistic field theory for the second-quantized N-body system of point particles with Chern-Simons interactions. Various properties of this model are discussed: its obvious and hidden symmetries, its relation to a relativistic field theory, and its supersymmetric formulation. We present classical, static solutions--solitons--that satisfy a self-dual equation, which is equivalent to the Liouville equation; hence, it is completely solvable. The dynamical role of the Chern-Simons interaction is demonstrated: the interaction does not merely change statistics but also provides the forces that bind the classical solitons.