Noncommutative Chern-Simons Theory Research Articles

Lowest Landau level theory of the bosonic Jain states

Quantum Hall systems offer the most familiar setting where strong inter-particle interactions combine with the topology of single particle states to yield novel phenomena. Despite our mature understanding of these systems, an open challenge has been to to develop a microscopic theory capturing both their universal and non-universal properties, when the Hamiltonian is restricted to the non-commutative space of the lowest Landau level. Here we develop such a theory for the Jain sequence of bosonic fractional quantum Hall states at fillings $\nu={p\over p+1}$. Building on a lowest Landau level description of a parent composite fermi liquid at $\nu = 1$, we describe how to dope the system to reach the Jain states. Upon doping, the composite fermions fill non-commutative generalizations of Landau levels, and the Jain states correspond to integer composite fermion filling. Using this approach, we obtain an approximate expression for the bosonic Jain sequence gaps with no reference to any long-wavelength approximation. Furthermore, we show that the universal properties, such as Hall conductivity, are encoded in an effective non-commutative Chern-Simons theory, which is obtained on integrating out the composite fermions. This theory has the same topological content as the familiar Abelian Chern-Simons theory on commutative space.

Hall viscosity and geometric response in the Chern-Simons matrix model of the Laughlin states

We study geometric aspects of the Laughlin fractional quantum Hall (FQH) states using a description of these states in terms of a matrix quantum mechanics model known as the Chern-Simons matrix model (CSMM). This model was proposed by Polychronakos as a regularization of the noncommutative Chern-Simons theory description of the Laughlin states proposed earlier by Susskind. Both models can be understood as describing the electrons in a FQH state as forming a noncommutative fluid, i.e., a fluid occupying a noncommutative space. Here we revisit the CSMM in light of recent work on geometric response in the FQH effect, with the goal of determining whether the CSMM captures this aspect of the physics of the Laughlin states. For this model we compute the Hall viscosity, Hall conductance in a non-uniform electric field, and the Hall viscosity in the presence of anisotropy (or intrinsic geometry). Our calculations show that the CSMM captures the guiding center contribution to the known values of these quantities in the Laughlin states, but lacks the Landau orbit contribution. The interesting correlations in a Laughlin state are contained entirely in the guiding center part of the state/wave function, and so we conclude that the CSMM accurately describes the most important aspects of the physics of the Laughlin FQH states, including the Hall viscosity and other geometric properties of these states which are of current interest.

Noncommutative Chern-Simons theory and exotic geometry emerging from the lowest Landau level

We relate the collective dynamic internal geometric degrees of freedom to the gauge fluctuations in $\ensuremath{\nu}=1/m(\mathrm{m}\text{ }\text{ }\mathrm{odd})$ fractional quantum Hall effects. In this way, in the lowest Landau level, a highly nontrivial quantum geometry in two-dimensional guiding center space emerges from these internal geometric modes. Using the Dirac bracket method, we find that this quantum geometric field theory is a topological noncommutative Chern-Simons theory. Topological indices, such as the guiding center angular momentum (also called the shift) and the guiding center spin, which characterize the fractional quantum Hall (FQH) states besides the filling factor, are naturally defined. A noncommutative K-matrix Chern-Simons theory is proposed as a generalization to a large class of Abelian FQH topological orders.

Noncommutative Chern-Simons gauge and gravity theories and their geometric Seiberg-Witten map

We use a geometric generalization of the Seiberg-Witten map between noncommutative and commutative gauge theories to find the expansion of noncommutative Chern-Simons (CS) theory in any odd dimension $D$ and at first order in the noncommutativity parameter $\theta$. This expansion extends the classical CS theory with higher powers of the curvatures and their derivatives. A simple explanation of the equality between noncommutative and commutative CS actions in $D=1$ and $D=3$ is obtained. The $\theta$ dependent terms are present for $D\geq 5$ and give a higher derivative theory on commutative space reducing to classical CS theory for $\theta\to 0$. These terms depend on the field strength and not on the bare gauge potential. In particular, as for the Dirac-Born-Infeld action, these terms vanish in the slowly varying field strength approximation: in this case noncommutative and commutative CS actions coincide in any dimension. The Seiberg-Witten map on the $D=5$ noncommutative CS theory is explored in more detail, and we give its second order $\theta$-expansion for any gauge group. The example of extended $D=5$ CS gravity, where the gauge group is $SU(2,2)$, is treated explicitly.

Noncommutative Chern–Simons Description of the Fractional Quantum Hall Edge

Noncommutative Chern–Simons (NCCS) theory is a workable description for the fractional quantum Hall fluid. We apply and generalize the NCCS theory to the physically important case with an edge. From relabeling symmetry of electrons and incompressibility of the fluid, we obtain a constraint and reduce the two-dimensional NCCS theory to a one-dimensional chiral Tomonaga–Luttinger liquid theory, which contains additional interaction terms. Further, we calculate one-loop corrections to the boson and electron propagators and obtain a new tunneling exponent, which agrees with experiments.

The noncommutative BTZ black hole in polar coordinates

Based on the equivalence between the three-dimensional gravity and the Chern–Simons theory, we obtain a noncommutative BTZ black hole solution as a solution of U(1, 1) × U(1, 1) noncommutative Chern–Simons theory using the Seiberg–Witten map. The Seiberg–Witten map is carried out in noncommutative polar coordinates whose commutation relation is equivalent to the usual canonical commutation relation in the rectangular coordinates up to first order in the noncommutativity parameter θ. The solution exhibits a characteristic of noncommutative polar coordinates in such a way that the apparent horizon and the Killing horizon coincide only in the nonrotating limit showing the effect of noncommutativity between the radial and angular coordinates.

On consistency of noncommutative Chern-Simons theory

We consider the noncommutative extension of Chern-Simons theory. We show the the theory can be fully expanded in power series of the noncommutative parameter theta and that no non-analytical sector exists. The theory appears to be unstable under radiative corrections, but we show that the infinite set of instabilities, to all orders in \hbar and in theta, is confined to a BRS exact cocycle. We show also that the theory is anomaly free. The quantum theory cannot be written in terms of the Groenewald-Moyal star product, and hence doubts arise on the interpretation of the noncommutative nature of the underlying spacetime. Nonetheless, the deformed theory is well defined as a quantum field theory, and the beta function of the Chern-Simons coupling constant vanishes, as in the ordinary Chern-Simons theory.

Supersymmetric non-Abelian non-commutative Chern–Simons theory

Abstract In this work, we study the three-dimensional non-Abelian non-commutative supersymmetric Chern–Simons model with the U ( N ) gauge group. Using a superfield formulation, we prove that, for the pure gauge theory, the Green functions are one-loop finite in any gauge, if the gauge superpotential belongs to the fundamental representation of u ( N ) ; this result also holds when matter in the fundamental representation is included. However, the cancellation of both ultraviolet and ultraviolet/infrared divergences only happens in a special gauge if the coupling of the matter is in the adjoint representation. We also look into the finite one-loop quantum corrections to the effective action: in the pure gauge sector the Maxwell term receives a quantum correction while, in the matter sector, the Chern–Simons term is generated, inducing a shift in the classical Chern–Simons coefficient.

Noncommutative AdS3 with quantized cosmological constant**Dedicated to Rafael Sorkin in his 60th year.

We examine a recent deformation of three-dimensional anti-de Sitter gravity based on noncommutative Chern–Simons theory with gauge group U(1, 1) × U(1, 1). In addition to a noncommutative analogue of 3D gravity, the theory contains two gauge fields which decouple in the commutative limit. It is well known that the level is quantized in noncommutative Chern–Simons theory. Here we show that it implies that the cosmological constant goes like −1 over an integer squared. We also construct the noncommutative AdS3 vacuum by applying a Seiberg–Witten map from the commutative case. The procedure is repeated for the case of a conical space resulting from a massive spinning particle.

Induced magnetic moment in noncommutative Chern–Simons scalar QED

We compute the one loop, $O(\th)$ correction to the vertex in the noncommutative Chern-Simons theory with scalar fields in the fundamental representation. Emphasis is placed on the parity odd part of the vertex, since the same leads to the magnetic moment structure. We find that, apart from the commutative term, a $\th$-dependent magnetic moment type structure is induced. In addition to the usual commutative graph, cubic photon vertices also give a finite $\th$ dependent contribution. Furthermore, the two two-photon vertex diagrams, that give zero in the commutative case yield finite $\th$ dependent terms to the vertex function.

Time dependent solitons of noncommutative Chern-Simons theory coupled to scalar fields

We study one- and two-soliton solutions of noncommutative Chern-Simons theory coupled to a nonrelativistic or a relativistic scalar field. In the nonrelativistic case, we find a tower of new stationary time-dependent solutions, all with the same charge density, but with increasing energies. The dynamics of these solitons cannot be studied using traditional moduli space techniques, but we do find a nontrivial symplectic form on the phase space indicating that the moduli space is not flat. In the relativistic case we find the metric on the two soliton moduli space.

Aspects of noncommutative descriptions of planar systems in high magnetic fields

We study some aspects of recent proposals to use the noncommutative Chern–Simons theory as an effective description of some planar condensed matter models in strong magnetic fields, such as the quantum Hall effect. We present an alternative justification for such a description, which may be extended to other planar systems where a uniform magnetic field is present.

Edge currents in non-commutative Chern-Simons theory from a new matrix model

This paper discusses the formulation of the non-commutative Chern-Simons (CS) theory where the spatial slice, an infinite strip, is a manifold with boundaries. As standard star products are not correct for such manifolds, the standard non-commutative CS theory is not also appropriate here. Instead we formulate a new finite-dimensional matrix CS model as an approximation to the CS theory on the strip. A work which has points of contact with ours is due to Lizzi, Vitale and Zampini where the authors obtain a description for the fuzzy disc. The gauge fields in our approach are operators supported on a subspace of finite dimension N+\eta of the Hilbert space of eigenstates of a simple harmonic oscillator with N, \eta \in Z^+ and N \neq 0. This oscillator is associated with the underlying Moyal plane. The resultant matrix CS theory has a fuzzy edge. It becomes the required sharp edge when N and \eta goes to infinity in a suitable sense. The non-commutative CS theory on the strip is defined by this limiting procedure. After performing the canonical constraint analysis of the matrix theory, we find that there are edge observables in the theory generating a Lie algebra with properties similar to that of a non-abelian Kac-Moody algebra. Our study shows that there are (\eta+1)^2 abelian charges (observables) given by the matrix elements (\cal A_i)_{N-1 N-1} and (\cal A_i)_{nm} (where n or m \geq N) of the gauge fields, that obey certain standard canonical commutation relations. In addition, the theory contains three unique non-abelian charges, localized near the N^th level. We show that all non-abelian edge observables except these three can be constructed from the abelian charges above. Using the results of this analysis we discuss the large N and \eta limit.

SOLITONS ON NONCOMMUTATIVE TORUS AS ELLIPTIC CALOGERO–GAUDIN MODELS, BRANES AND LAUGHLIN WAVE FUNCTIONS

For the noncommutative torus [Formula: see text], in the case of the noncommutative parameter [Formula: see text], we construct the basis of Hilbert space ℋn in terms of θ functions of the positions zi of n solitons. The wrapping around the torus generates the algebra [Formula: see text], which is the Zn × Zn Heisenberg group on θ functions. We find the generators g of a local elliptic su (n), which transform covariantly by the global gauge transformation of [Formula: see text]. By acting on ℋn we establish the isomorphism of [Formula: see text] and g. We embed this g into the L-matrix of the elliptic Gaudin and Calogero–Moser models to give the dynamics. The moment map of this twisted cotangent [Formula: see text] bundle is matched to the D-equation with the Fayet–Illiopoulos source term, so the dynamics of the noncommutative solitons become that of the brane. The geometric configuration (k, u) of the spectral curve det |L(u) - k| = 0 describes the brane configuration, with the dynamical variables zi of the noncommutative solitons as the moduli T⊗ n/Sn. Furthermore, in the noncommutative Chern–Simons theory for the quantum Hall effect, the constrain equation with quasiparticle source is identified also with the moment map equation of the noncommutative [Formula: see text] cotangent bundle with marked points. The eigenfunction of the Gaudin differential L-operators as the Laughlin wave function is solved by Bethe ansatz.

Quantum aspects of the Seiberg–Witten map in noncommutative Chern–Simons theory

Noncommutative Chern–Simons theory can be classically mapped to commutative Chern–Simons theory by the Seiberg–Witten map. We provide evidence that the equivalence persists at the quantum level by computing two and three-point functions of field strengths on the commutative side and their Seiberg–Witten transforms on the noncommutative side to the first nontrivial order in perturbation theory.

Noncommutative Maxwell–Chern–Simons theory, duality and a new noncommutative Chern–Simons theory in d=3

Noncommutative Maxwell–Chern–Simons theory in 3 dimensions is defined in terms of star product and noncommutative fields. The Seiberg–Witten map is employed to write it in terms of ordinary fields. A parent action is introduced and the dual action is derived. For spatial noncommutativity it is studied up to second order in the noncommutativity parameter θ. A new noncommutative Chern–Simons action is defined in terms of ordinary fields, inspired by the dual action. Moreover, a transformation between noncommuting and ordinary fields is proposed.

Gravitational fields on a noncommutative space

Noncommutative three-dimensional gravity can be described in terms of a noncommutative Chern–Simons theory. We extend this structure and also propose an action for gravitational fields on an even-dimensional noncommutative space. The action is worked out in some detail for fields on a noncommutative CP 2 and on S 4.

GEOMETRIC TRANSFORMATIONS AND NCCS THEORY IN THE LOWEST LANDAU LEVEL

Chern-Simons type gauge field is generated by the means of singular area preserving transformations in the lowest Landau level of electrons forming fractional quantum Hall state. Dynamics is governed by the system of constraints which correspond to the Gauss law in the non-commutative Chern-Simons gauge theory and to the lowest Landau level condition in the picture of composite fermions. Physically reasonable solution to this constraints corresponds to the Laughlin state. It is argued that the model leads to the non-commutative Chern-Simons theory of the QHE and composite fermions.

On gauge invariance of noncommutative Chern–Simons theories

Motivated by possible applications to condensed matter systems, in this paper we construct U( N) noncommutative Chern–Simons (NCCS) action for a disc and for a double-layer geometry, respectively. In both cases, gauge invariance severely constrains the form of the NCCS action. In the first case, it is necessary to introduce a group-valued boson field with a nonlocal chiral boundary action, whose gauge variation cancels that of the bulk action. In the second case, the coefficient matrix K in the double U( N) NCCS action is restricted to be of the form K=k 1 1 1 1 with integer k. We suggest that this double NCCS theory with U(1) gauge group describes the so-called Halperin ( kkk) state in a double-layer quantum Hall system. Possible physical consequences are addressed.

A NOTE ON NONCOMMUTATIVE CHERN–SIMONS MODEL ON MANIFOLDS WITH BOUNDARY

We study field theories defined in regions of the spatial noncommutative (NC) plane with a boundary present delimiting them, concentrating in particular on the U(1) NC Chern–Simons theory on the upper half-plane. We find that classical consistency and gauge invariance lead necessary to the introduction of K0-space of square integrable functions null together with all their derivatives at the origin. Furthermore the requirement of closure of K0 under the *-product leads to the introduction of a novel notion of the *-product itself in regions where a boundary is present, that in turn yields the complexification of the gauge group and to consider chiral waves in one sense or other. The canonical quantization of the theory is sketched identifying the physical states and the physical operators. These last ones include ordinary NC Wilson lines starting and ending on the boundary that yield correlation functions depending on points on the one-dimensional boundary. We finally extend the definition of the *-product to a strip and comment on possible relevance of these results to finite quantum Hall systems.