arXiv subjects
Publications and source records attributed to A. Stern.
We find an infinite number of noncommutative geometries which posses a differential structure. They generalize the two dimensional noncommutative plane, and have infinite dimensional representations. Upon applying generalized coherent states we are able to take the continuum limit, where we recover the punctured plane with non constant Poisson structures.
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.
We construct a Dirac operator on the quantum sphere $S^2_q$ which is covariant under the action of $SU_q(2)$. It reduces to Watamuras' Dirac operator on the fuzzy sphere when $q\to 1$. We argue that our Dirac operator may be useful in constructing $SU_q(2)$ invariant field theories on $S^2_q$ following the Connes-Lott approach to noncommutative geometry.
We calculated the Coulomb drag contribution to the resistivity and the transresistivity for a double layer system, in which the passive layer is a Wigner crystal pinned by impurities, and the active one is a metal. We found that in quasi one dimensional systems both quantities are suppressed at low temperatures by the constraints of energy and momentum conservation in electron-phonon scattering. In two dimensions both quantities are ~T^4, which is consistent with Bloch law for the contribution of electron-phonon interaction to the resistivity. Strong impurities in the Wigner crystal decrease the transresistivity significantly when their number reaches the logarithm of the number of sites in the crystal. In contrast, the drag contribution to the resistivity is found to be independent of the impurities, at least as long as their density is kept zero in the thermodynamic limit.
We construct a star product associated with an arbitrary two dimensional Poisson structure using generalized coherent states on the complex plane. From our approach one easily recovers the star product for the fuzzy torus, and also one for the fuzzy sphere. For the latter we need to define the `fuzzy' stereographic projection to the plane and the fuzzy sphere integration measure, which in the commutative limit reduce to the usual formulae for the sphere.
We show how to map the Belavin-Polyakov instantons of the O(3)-nonlinear $σ-$model to a dual theory where they then appear as nontopological solitons. They are stationary points of the Euclidean action in the dual theory, and moreover, the dual action and the O(3)-nonlinear $σ-$model action agree on shell.
We construct a deformation of the quantum algebra Fun(T^*G) associated with Lie group G to the case where G is replaced by a quantum group G_q which has a bicovariant calculus. The deformation easily allows for the inclusion of the current algebra of left and right invariant one forms. We use it to examine a possible generalization of the Gauss law commutation relations for gauge theories based on G_q.
We construct dual Lagrangians for $G/H$ models in two space-time dimensions for arbitrary Lie groups $G$ and $H\subset G$. Our approach does not require choosing coordinates on $G/H$, and allows for a natural generalization to Lie-Poisson duality. For the case where the target metric on $G/H$ is induced from the invariant metric on $G$, the dual system is a gauged Higgs model, with a nonconstant metric and a coupling to an antisymmetric tensor. The dynamics for the gauge connection is governed by a $BF$-term. Lie-Poisson duality is relevant once we allow for a more general class of target metrics, as well as for couplings to an antisymmetric tensor, in the primary theory. Then the dual theory is written on a group $\tilde G$ dual to $G$, and the gauge group $H$ (which, in general, is not a subgroup of $\tilde G$) acts nonlinearly on $\tilde G$. The dual system therefore gives a nonlinear realization of a gauge theory. All dual descriptions are shown to be canonically equivalent to the corresponding primary descriptions, at least at the level of the current algebra.
In this paper we study the relation between the conventional Fermion-Chern-Simons (FCS) theory of the half-filled Landau level (nu=1/2), and alternate descriptions that are based on the notion of neutral quasi-particles that carry electric dipole moments. We have previously argued that these two approaches are equivalent, and that e.g., the finite compressibility obtained in the FCS approach is also obtained from the alternate approach, provided that one properly takes into account a peculiar symmetry of the dipolar quasiparticles --- the invariance of their energy to a shift of their center of mass momentum. Here, we demonstrate the equivalence of these two approaches in detail. We first study a model where the charge and flux of each fermion is smeared over a radius Q^{-1} where results can be calculated to leading order in the small parameter Q/k_f. We study two dipolar-quasiparticle descriptions of the nu=1/2 state in the small-Q model and confirm that they yield the same density response function as in the FCS approach. We also study the single-particle Green's function and the effective mass, for one form of dipolar quasiparticles, and find the effective mass to be infra-red divergent, exactly as in the FCS approach. Finally, we propose a form for a Fermi-liquid theory for the dipolar quasiparticles, which should be valid in the physical case where Q is infinite.
The Hamiltonian formalism offers a natural framework for discussing the notion of Poisson Lie T-duality. This is because the duality is inherent in the Poisson structures alone and exists regardless of the choice of Hamiltonian. Thus one can pose alternative dynamical systems possessing nonabelian T-duality. As an example, we find a dual Hamiltonian formulation of the O(3) nonlinear sigma-model. In addition, starting from a general quadratic Hamiltonian, we easily recover the known dynamical systems having Poisson Lie T-duality.
We find a one-parameter family of Lagrangian descriptions for classical general relativity in terms of tetrads which are not c-numbers. Rather, they obey exotic commutation relations. These noncommutative properties drop out in the metric sector of the theory, where the Christoffel symbols and the Riemann tensor are ordinary commuting objects and they are given by the usual expression in terms of the metric tensor. Although the metric tensor is not a c-number, we argue that all measurements one can make in this theory are associated with c-numbers, and thus that the common invariant sector of our one--parameter family of deformed gauge theories (for the case of zero torsion) is physically equivalent to Einstein's general relativity.
Using the tools of q--differential calculus and quantum Lie algebras associated to quantum groups, we find a one--parameter family of q-gauge theories associated to the quantum group $ISO_q(3,1)$. Although the gauge fields, that is the spin--connection and the vierbeins are non--commuting objects depending on a deformation parameter, $q$, it is possible to construct out of them a metric theory which is insensitive to the deformation. The Christoffel symbols and the Riemann tensor are ordinary commuting objects. Hence it is argued that torsionless Einstein's General Relativity is the common invariant sector of a one--parameter family of deformed gauge theories.
It is well known that the Fractional Quantum Hall Effect (FQHE) may be effectively represented by a Chern-Simons theory. In order to incorporate QH Skyrmions, we couple this theory to the topological spin current, and include the Hopf term. The cancellation of anomalies for chiral edge states, and the proviso that Skyrmions may be created and destroyed at the edge, fixes the coefficients of these new terms. Consequently, the charge and the spin of the Skyrmion are uniquely determined. For those two quantities we find the values $eνN_{Sky}$ and $νN_{Sky}/2$, respectively, where $e$ is electron charge, $ν$ is the filling fraction and $N_{Sky}$ is the Skyrmion winding number. We also add terms to the action so that the classical spin fluctuations in the bulk satisfy the standard equations of a ferromagnet, with spin waves that propagate with the classical drift velocity of the electron.
A new formal scheme is presented in which Einstein's classical theory of General Relativity appears as the common, invariant sector of a one-parameter family of different theories. This is achieved by replacing the Poincare` group of the ordinary tetrad formalism with a q-deformed Poincare` group, the usual theory being recovered at q=1. Although written in terms of noncommuting vierbein and spin-connection fields, each theory has the same metric sector leading to the ordinary Einstein-Hilbert action and to the corresponding equations of motion. The Christoffel symbols and the components of the Riemann tensor are ordinary commuting numbers and have the usual form in terms of a metric tensor built as an appropriate bilinear in the vierbeins. Furthermore we exhibit a one-parameter family of Hamiltonian formalisms for general relativity, by showing that a canonical formalism a` la Ashtekar can be built for any value of q. The constraints are still polynomial, but the Poisson brackets are not skewsymmetric for q different from 1.
The usual description of 2+1 dimensional Einstein gravity as a Chern-Simons (CS) theory is extended to a one parameter family of descriptions of 2+1 Einstein gravity. This is done by replacing the Poincare' gauge group symmetry by a q-deformed Poincare' gauge group symmetry, with the former recovered when q-> 1. As a result, we obtain a one parameter family of Hamiltonian formulations for 2+1 gravity. Although formulated in terms of noncommuting dreibeins and spin-connection fields, our expression for the action and our field equations, appropriately ordered, are identical in form to the ordinary ones. Moreover, starting with a properly defined metric tensor, the usual metric theory can be built; the Christoffel symbols and space-time curvature having the usual expressions in terms of the metric tensor, and being represented by c-numbers. In this article, we also couple the theory to particle sources, and find that these sources carry exotic angular momentum. Finally, problems related to the introduction of a cosmological constant are discussed.
We construct a Chern-Simons action for q-deformed gauge theory which is a simple and straightforward generalization of the usual one. Space-time continues to be an ordinary (commuting) manifold, while the gauge potentials and the field strengths become q-commuting fields. Our approach, which is explicitly carried out for the case of `minimal' deformations, has the advantage of leading naturally to a consistent Hamiltonian structure that has essentially all of the features of the undeformed case. For example, using the new Poisson brackets, the constraints form a closed algebra and generate q-deformed gauge transformations.
We apply the SL(2,C) lattice Kac-Moody algebra of Alekseev, Faddeev and Semenov-Tian-Shansky to obtain a new lattice description of the SU(2) chiral model in two dimensions. The system has a global quantum group symmetry and it can be regarded as a deformation of two different theories. One is the nonabelian Toda lattice which is obtained in the limit of infinite central charge, while the other is a nonstandard Hamiltonian description of the chiral model obtained in the continuum limit.
We write down a general geometric action principle for spinning strings in $d$-dimensional Minkowski space, which is formulated without the use of Grassmann coordinates. Instead, it is constructed in terms of the pull-back of a left invariant Maurer-Cartan form on the $d$-dimensional Poincaré group to the world sheet. The system contains some interesting special cases. Among them are the Nambu string (as well as, null and tachyionic strings) where the spin vanishes, and also the case of a string with a spin current - but no momentum current. We find the general form for the Virasoro generators, and show that they are first class constraints in the Hamiltonian formulation of the theory. The current algebra associated with the momentum and angular momentum densities are shown, in general, to contain rather complicated anomaly terms which obstruct quantization. As expected, the anomalies vanish when one specializes to the case of the Nambu string, and there one simply recovers the algebra associated with the Poincaré loop group. We speculate that there exist other cases where the anomalies vanish, and that these cases give the bosonization of the known pseudoclassical formulations of spinning strings.