Noncommutative Chern-Simons Theory Research Articles

Dualities in Quantum Hall System and Noncommutative Chern-Simons Theory

We discuss different dualities of QHE in the framework of the noncommutative Chern-Simons theory. First, we consider the Morita or T-duality transformation on the torus which maps the abelian noncommutative CS description of QHE on the torus into the nonabelian commutative description on the dual torus. It is argued that the Ruijsenaars integrable many-body system provides the description of the QHE with finite amount of electrons on the torus. The new IIB brane picture for the QHE is suggested and applied to Jain and generalized hierarchies. This picture naturally links 2d $\sigma$-model and 3d CS description of the QHE. All duality transformations are identified in the brane setup and can be related with the mirror symmetry and S duality. We suggest a brane interpretation of the plateu transition in IQHE in which a critical point is naturally described by $SL(2,R)$ WZW model.

Absence of the holographic principle in noncommutative Chern-Simons theory

We examine noncommutative Chern-Simons theory on a bounded spatial domain. We argue that upon `turning on' the noncommutativity, the edge observables, which characterized the commutative theory, move into the bulk. We show this to lowest order in the noncommutativity parameter appearing in the Moyal star product. If one includes all orders, the Hamiltonian formulation of the gauge theory ceases to exist, indicating that the Moyal star product must be modified in the presence of a boundary. Alternative descriptions are matrix models. We examine one such model, obtained by a simple truncation of Chern-Simons theory on the noncommutative plane, and express its observables in terms of Wilson lines.

The stringy quantum Hall fluid

Using branes in massive Type IIA string theory, and a novel decoupling limit, we provide an explicit correspondence between non-commutative Chern-Simons theory and the fractional quantum Hall fluid. The role of the electrons is played by D-particles, the background magnetic field corresponds to a RR 2-form flux, and the two-dimensional fluid is described by non-commutative D2-branes. The filling fraction is given by the ratio of the number of D2-branes and the number of D8-branes, and therefore by the ratio rank/level of the Chern-Simons gauge theory. Quasiparticles and quasiholes are realized as endpoints of fundamental strings on the D2-branes, and are found to possess fractional D-particle charges and fractional statistics.

String field theory, non-commutative Chern-Simons theory and Lie algebra cohomology

Motivated by noncommutative Chern-Simons theory, we construct an infinite class of field theories that satisfy the axioms of Witten's string field theory. These constructions have no propagating open string degrees of freedom. We demonstrate the existence of non-trivial classical solutions. We find Wilson loop-like observables in these examples.

Computing noncommutative Chern–Simons theory radiative corrections on the back of an envelope

We show that the renormalized U(N) noncommutative Chern-Simons theory can be defined in perturbation theory so that there are no loop corrections to the 1PI functional of the theory in an arbitrary homogeneous axial (time-like, light-like or space-like) gauge. We define the free propagators of the fields of the theory by using the Leibbrandt-Mandelstam prescription --which allows Wick rotation and is consistent with power-counting-- and regularize its Green functions with the help of a family of regulators which explicitly preserve the infinitesimal vector Grassmann symmetry of the theory. We also show that in perturbation theory the nonvanishing Green functions of the elementary fields of the theory are products of the free propagators.

Finite noncommutative Chern-Simons with a Wilson line and the quantum Hall effect

We present a finite dimensional matrix model associated to the noncommutative Chern-Simons theory, obtained by inserting a Wilson line. For a specific choice of the representation of the Wilson line the model is equivalent to the minimal modification of the matrix model which is compatible with finite dimensional matrices, and was introduced previously to study droplets of quantum Hall fluid. For other representations we obtain generalizations corresponding to regularized U(n) Chern-Simons theoris, representing multilayered quantum Hall fluids.

Matrix model and Chern–Simons theory

In this short note we would like to present a simple topological matrix model which has close relation with the noncommutative Chern–Simons theory.