In this paper we show the renormalizability of the translation invariant noncommutative Chern-Simons theory, motivated by the work done on noncommutative scalar field theory [06]. We add a new term to the bilinear part of the action. In addition, we prove, the finiteness of the theory at one- and two-loop level despite this modification. Finally we perform the one-loop two point functions of the gluon contribution.
In this paper, we derive the energy momentum tensor for the translation invariant noncommutative Tanasa scalar field model. The Wilson regularization procedure is used to improve this tensor and the local conservation property is recovered. The same question is addressed in the case where the Moyal star product is deformed including the tetrad fields. It provides with an extension of the recent work [J. Phys. A: Math. Theor. 43 (2010) 155202], regarding the computation and properties of the Noether currents to the renormalizable models.
We show that the noncommutative Wess-Zumino model is renormalizable to all orders of perturbation theory. The noncommutative scalar potential by itself is non-renormalizable but the Yukawa terms demanded by supersymmetry improve the situation turning the theory into a renormalizable one. As in the commutative case, there are neither quadratic nor linear divergences. Hence, the IR/UV mixing does not give rise to quadratic infrared poles.
Noncommutative Maxwell-Chern-Simons theory in 3-dimensions is defined in terms of star product and noncommutative fields. 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.
We study the effect of noncommutative Chern-Simons term on fundamental fermions. In particular the one-loop contribution to the magnetic moment to order $þ$ is calculated.
We derive the noncommutative Chern-Simons action induced by Dirac fermions coupled to a background gauge field, for the fundamental, antifundamental, and the adjoint representation. We discuss properties of the noncommutative Chern-Simons action showing in particular that the Seiberg-Witten formula maps it into the standard commutative Chern-Simons action.
By analyzing the odd parity part of the gauge field two and three point vertex functions, the one-loop radiative correction to the Chern-Simons coefficient is computed in noncommutative Chern-Simons-Higgs model at zero and at high temperature. At high temperature, we show that the static limit of this correction is proportional to $T$ but the first noncommutative correction increases as $T\log T$. Our results are analytic functions of the noncommutative parameter.
We show that the coefficient of the three-dimensional Chern-Simons action on the noncommutative plane must be quantized. Similar considerations apply in other dimensions as well.
We study the possibility of establishing the dual equivalence between the noncommutative Maxwell-Chern-Simons theory and the noncommutative self-dual theory. It turns to be that whereas in the commutative case the Maxwell-Chern-Simons theory can be mapped into the sum of the self-dual theory and the Chern-Simons theory, in the noncommutative case such a mapping is possible only for the theory with the modified Maxwell term.
We examine noncommutative Chern Simons theory using operator regularization. Both the zeta-function and the eta-function are needed to determine one loop effects. The contributions to these functions coming from the two point function is evaluated. The U(N) noncommutative model smoothly reduces to the SU(N) commutative model as the noncommutative parameter theta_{mu nu} vanishes.
We construct exact soliton solutions to the Chern-Simons-Higgs system in noncommutative space, for non-relativistic and relativistic models. In both cases we find regular vortex-like solutions to the BPS equations which approach the ordinary selfdual non-topological and topological solitons when the noncommutative parameter $θ$ goes to zero.
We consider the Dunne-Jackiw-Pi-Trugenberger model of a U(N) Chern-Simons gauge theory coupled to a nonrelativistic complex adjoint matter on noncommutative space. Soliton configurations of this model are related the solutions of the chiral model on noncommutative plane. A generalized Uhlenbeck's uniton method for the chiral model on noncommutative space provides explicit Chern-Simons solitons. Fundamental solitons in the U(1) gauge theory are shaped as rings of charge `n' and spin `n' where the Chern-Simons level `n' should be an integer upon quantization. Toda and Liouville models are genera…
A novel approach to the analysis of a noncommutative Chern--Simons gauge theory with matter coupled in the adjoint representation has been discussed. The analysis is based on a recently proposed closed form Seiberg--Witten map which is exact in the noncommutative parameter.
We study a noncommutative nonrelativistic fermionic field theory in 2+1 dimensions coupled to the Chern-Simons field. We perform a perturbative analysis of model and show that up to one loop the ultraviolet divergences are canceled and the infrared divergences are eliminated by the noncommutative Pauli term.
We write down couplings of the fields on a single BPS Dp-brane with noncommutative world-volume coordinates to the RR-forms in type II theories, in a manifestly background independent way. This generalises the usual Chern-Simons action for a commutative Dp-brane. We show that the noncommutative Chern-Simons terms can be mapped to Myers terms on a collection of infinitely many D-instantons. We also propose Chern-Simons couplings for unstable non-BPS branes, and show that condensation of noncommutative tachyons on these branes leads to the correct Myers terms on the decay products.
Applying a master action technique we obtain the dual of the noncommutative Maxwell-Chern-Simons theory. The equivalence between the Maxwell-Chern-Simons theory and the self-dual model in commutative space-time does not survive in the non-commutative setting. We also point out an ambiguity in the Seiberg-Witten map.
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.
We investigate BPS soliton solutions of U(N) Chern-Simons gauge theory coupled to a scalar field in noncommutative plane. With a scalar field in the fundamental representation, we show that the BPS equation becomes that of abelian Chern-Simons theory in the unitary gauge. We also find a class of particular solutions for the BPS equation with scalar field in the adjoint representation.
In this work we propose the Z -> gamma gamma decay as a process strictly forbidden in the standard model, suitable for the search of noncommutativity of coordinates at very short distances. We computed the Z -> gamma gamma partial width in the framework of the recently proposed one-loop renormalizable gauge sector of the noncommutative standard model. New experimental possibilities at LHC are analyzed and a firm bound to the scale of noncommutativity parameter is set around 1 TeV.
Within the superfield approach, we prove the absence of UV/IR mixing in the three-dimensional noncommutative supersymmetric Maxwell-Chern-Simons theory at any loop order and demonstrate its finiteness in one, three and higher loop orders.
We compute a Chern-Simons term induced by the fermions on noncommutative torus interacting with two U(1) gauge fields. For rational noncommutativity θ\propto P/Q we find a new mixed term in the action which involves only those fields which are (2π)/Q periodic, like the fields in a crystal with Q^2 nodes.
In this paper the hamiltonian analysis of the pure Chern-Simons theory on the noncommutative plane is performed. We use the techniques of geometric quantization to show that the classical reduced phase space of the theory has nontrivial topology and that quantization of the symplectic structure on this space is possible only if the Chern-Simons coefficient is quantized. Also we show that the physical Hilbert space of the theory is one dimensional and give an explicit expression for the vacuum wavefunction. This vacuum state is found to be related to the noncommutative Wess-Zumino-Witten action…
We analyze the structure of noncommutative pure Chern-Simons theory systematically in the axial gauge. We show that there is no IR/UV mixing in this theory in this gauge. In fact, we show, using the usual BRST identities as well as the identities following from vector supersymmetry, that this is a free theory. As a result, the tree level Chern-Simons coefficient is not renormalized. It also holds that the Chern-Simons coefficient is not modified at finite temperature. As a byproduct of our analysis, we prove that the ghosts completely decouple in the axial gauge in a noncommutative gauge theor…
In this paper, we study the three-dimensional noncommutative Maxwell-Chern-Simons theory. In the present analysis, a complete account for the gauge field two-point function renormalizability is presented and physical significant quantities are carefully established. The respective form factor expressions from the gauge field self-energy are computed at one-loop order. More importantly, an analysis of the gauge field dispersion relation, in search of possible noncommutative anomalies and infrared finiteness, is performed for three special cases, with particular interest in the highly noncommuta…
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…
In this paper we study the one-loop shift in the coupling constant in a noncommutative pure U(N) Chern-Simons gauge theory in three dimensions. The one-loop shift is shown to be a constant proportional to $N$, independent of noncommutativity parameters, and non-vanishing for U(1) theory. Possible physical and mathematical implications of this result are discussed.
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.
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…
We study issues of duality in 3D field theory models over a canonical noncommutative spacetime and obtain the noncommutative extension of the Self-Dual model induced by the Seiberg-Witten map. We apply the dual projection technique to uncover some properties of the noncommutative Maxwell-Chern-Simons theory up to first-order in the noncommutative parameter. A duality between this theory and a model similar to the ordinary self-dual model is estabilished. The correspondence of the basic fields is obtained and the equivalence of algebras and equations of motion are directly verified. We also com…
Using the superfield formalism and implementing the canonical noncommutativity, the Kahlerian effective superpotential is evaluated in the three-dimensional noncommutative supersymmetric Chern--Simons-matter model at the two-loop order. The computation of the Kahlerian effective superpotential is enough to determine whether the model can exhibit spontaneous (super) symmetry breaking. It is shown that the model possesses a spontaneous gauge symmetry broken phase, generating masses for the scalar and gauge superfields at the two-loop order. Just as for the commutative version, in the noncommutat…
The noncommutative (NC) massive quantum electrodynamics in $2+1$ dimensions is considered. We show explicitly that the one-loop effective action arising from the integrating out the fermionic fields leads to the ordinary NC Chern-Simons and NC Maxwell action at the long wavelength limit (large fermion mass). In the next to leading order, the higher-derivative contributions to NC Chern-Simons are obtained. Moreover, the gauge invariance of the outcome action is carefully discussed. We then consider the higher-derivative modification into the pure NC Chern-Simons Lagrangian density and evaluate …
We have conclusively established the duality between noncommutative Maxwell-Chern-Simons theory and Self-Dual model, the latter in ordinary spacetime, to the first non-trivial order in the noncommutativity parameter $θ^{μν}$, with $θ^{0i}=0$. This shows that the former theory is free for marginally noncommutative spacetimes. A $θ$-generalized covariant mapping between the variables of the two models in question has been derived explicitly, that converts one model to the other, including the symplectic structure and action.
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 the solutions of two spatially separated vortices and determine the moduli dependence of the size and the angular momentum. …
In this work we study the three-dimensional ${\cal N}=2$ supersymmetric Chern-Simons-matter model in a noncommutative space-time. We construct the action of the noncommutative $U(N)$ non-Abelian model in terms of explicit ${\cal N}=2$ supervariables by dimensionally reducing a four-dimensional ${\cal N}=1$ supermultiplet. We also obtain the on-shell ${\cal N}=2$ supersymmetric model writing it in terms of ${\cal N}=1$ superfields. In the noncommutative Abelian case, we show that linear UV divergences are cancelled in Feynman diagrams and logarithmic divergences are absent up to one-loop order,…
We consider the Maxwell-Chern-Simons theory in noncommutative three dimensional space-time. We show that the Seiberg-Witten map is ambiguous due to the dimensional coupling constant. To get the dual theory we start from a master action obtained by promoting the global shift invariance to a local one. We also obtain the mapping between the observables of the two equivalent theories. We show that the equivalence between the Maxwell-Chern-Simons theory and the self-dual model in commutative space-time does not survive in the non-commutative setting.
We investigate U(N) Chern-Simons theories on noncommutative plane. We show that for the theories to be consistent quantum mechanically, the coefficient of the Chern-Simons term should be quantized $κ= n/2π$ with an integer $n$. This is a surprise for the U(1) gauge theory. When uniform background charge density $ρ_e$ is present, the quantization rule changes to $κ+ρ_eθ= n/2π$ with noncommutative parameter $θ$. With the exact expression for the angular momentum, we argue in the U(1) theory that charged particles in the symmetric phase carry fractional spin $1/2n$ and vortices in the broken phas…
We investigate the spectrum of the gauge theory with Chern-Simons term on the noncommutative plane, a modification of the description of the Quantum Hall fluid recently proposed by Susskind. We find a series of the noncommutative massive ``plane wave'' solutions with polarization dependent on the magnitude of the wave-vector. The mass of each branch is fixed by the quantization condition imposed on the coefficient of the noncommutative Chern-Simons term. For the radially symmetric ansatz a vortex-like solution is found and investigated. We derive a nonlinear difference equation describing thes…
A U(N) Chern-Simons theory on noncommutative $\mathbb{R}^{3}$ is constructed as a $\q$-deformed field theory. The model is characterized by two symmetries: the BRST-symmetry and the topological linear vector supersymmetry. It is shown that the theory is finite and $\q_{\m\n}$-independent at the one loop level and that the calculations respect the restriction of the topological supersymmetry. Thus the topological $\q$-deformed Chern-Simons theory is an example of a model which is non-singular in the limit $\q \to 0$.
The relation of the noncommutative self-dual Chern-Simons (NCSDCS) system to the noncommutative generalizations of Toda and of affine Toda field theories is investigated more deeply. This paper continues the programme initiated in $JHEP {\bf 10} (2005) 071$, where it was presented how it is possible to define Toda field theories through second order differential equation systems starting from the NCSDCS system. Here we show that using the connection of the NCSDCS to the noncommutative chiral model, exact solutions of the Toda field theories can be also constructed by means of the noncommutativ…
Renormalizable $ϕ^{\star 4}_4$ models on Moyal space have been obtained by modifying the commutative propagator. But these models have a divergent "naive" commutative limit. We explain here how to obtain a coherent such commutative limit for a recently proposed translation-invariant model. The mechanism relies on the analysis of the uv/ir mixing in general Feynman graphs.
By introducing an appropriate parent action and considering a perturbative approach, we establish, up to fourth order terms in the field and for the full range of the coupling constant, the equivalence between the noncommutative Yang-Mills-Chern-Simons theory and the noncommutative, non-Abelian Self-Dual model. In doing this, we consider two different approaches by using both the Moyal star-product and the Seiberg-Witten map.
The three dimensional Chern-Simons theory on $\rr^2_θ\times \rr$ is studied. Considering the gauge transformations under the group elements which are going to one at infinity, we show that under arbitrary (finite) gauge transformations action changes with an integer multiple of $2π$ {\it if}, the level of noncommutaitive Chern-Simons is {\it quantized}. We also briefly discuss the case of the noncommutaitve torus and some other possible extensions.
We investigate the coordinate dependence of noncommutative theory by studying the solutions of noncommutative $U(1,1)\times U(1,1)$ Chern-Simons theory on $AdS_3$ in the polar and rectangular coordinates. We assume that only the space coordinates are noncommuting. The two coordinate systems are equivalent only up to first order in the noncommutativity parameter $θ$. We investigate the effect of this non-exact equivalence between the two coordinate systems in two cases, a conical solution and a BTZ black hole solution, using the Seiberg-Witten map. In each case, the noncommutative solutions in …
We continue our investigation of the quantum equivalence between commutative and noncommutative Chern-Simons theories by computing the complete set of two-loop quantum corrections to the correlation function of a pure open Wilson line and an open Wilson line with a field strength insertion, on the noncommutative side in a covariant gauge. The conjectured perturbative equivalence between the free commutative theory and the apparently interacting noncommutative one requires that the sum of these corrections vanish, and herein we exhibit the remarkable cancellations that enforce this. From this c…
We find soliton solutions of the noncommutative Maxwell-Chern-Simons theory confined to a finite quantum Hall droplet. The solitons are exactly as hypothesized in \cite{Manu}. We also find new variations on these solitons. We compute their flux and their energies. The model we consider is directly related to the model proposed by Polychronakos\cite{Poly} and studied by Hellerman and Van Raamsdonk\cite{HvR} where it was shown that it is equivalent to the quantum Hall effect.
We construct a deformed Wess-Zumino model on the noncommutative superspace where the Bosonic and Fermionic coordinates are no longer commutative with each other. Using the background field method, we calculate the primary one-loop effective action based on the deformed action. By comparing the two actions, we find that the deformed Wess-Zumino model is not renormalizable. To obtain a renormalizable model, we combine the primary one-loop effective action with the deformed action, and then calculate the secondary one-loop effective action based on the combined action. After repeating this proces…
We construct here the parametric representation of a translation-invariant renormalizable scalar model on the noncommutative Moyal space of even dimension $D$. This representation of the Feynman amplitudes is based on some integral form of the noncommutative propagator. All types of graphs (planar and non-planar) are analyzed. The rôle played by noncommutativity is explicitly shown. This parametric representation established allows to calculate the power counting of the model. Furthermore, the space dimension $D$ is just a parameter in the formulas obtained. This paves the road for the dimensi…
We apply the noncommutative fields method for gauge theory in three dimensions where the Chern-Simons term is generated in the three-dimensional electrodynamics. Under the same procedure, the Chern-Simons term is shown to be cancelled in the Maxwell-Chern-Simons theory for the appropriate value of the noncommutativity parameter. Hence the mutual interchange between Maxwell-Chern-Simons theory and pure Maxwell theory turns out to be generated within this method.
The Maxwell--Chern--Simons model with scaler matter in the adjoint representation is analyzed from an alternative approach which is regular in the $θ\to 0$ limit. This method is complementary to the usual operator formalism applied to explore the nonperturbative solutions which gives singular results in the $θ\to 0$ limit. The absence of any regular non-trivial lumpy solutions satisfying B--P--S bound has been conclusively demonstrated.
We discuss the renormalizability of the noncommutative U(1)Higgs-Kibble model formulated within the enveloping-algebra approach. We consider both the phase of the model with unbroken gauge symmetry and the phase with spontaneously broken gauge symmetry. We show that against all odds the gauge sector of the model is always one-loop renormalizable at first order in theta^{mu nu}, perhaps, hinting at the existence of a new symmetry of the gauge sector of the model. However, we also show that the matter sector of the model is non-renormalizable whatever the phase.
We study Maxwell-Chern-Simons theory in 2 noncommutative spatial dimensions and 1 temporal dimension. We consider a finite matrix model obtained by adding a linear boundary field which takes into account boundary fluctuations. The pure Chern-Simons has been previously shown to be equivalent to the Laughlin description of the quantum Hall effect. With the addition of the Maxwell term, we find that there exists a rich spectrum of excitations including solitons with nontrivial "magnetic flux" and quasi-holes with nontrivial "charges", which we describe in this article. The magnetic flux correspon…
We consider 2+1 dimensional noncommutative models of scalar and fermionic fields coupled to the Chern-Simons field. We show that, at least up to one loop, the model containing only a fermionic field in the fundamental representation minimally coupled to the Chern-Simons field is consistent in the sense that there are no nonintegrable infrared divergences. By contrast, dangerous infrared divergences occur if the fermion field belongs to the adjoint representation or if the coupling of scalar matter is considered instead. The superfield formulation of the supersymmetric Chern-Simons model is als…
Recently, a new type of renormalizable $ϕ^{\star 4}_{4}$ scalar model on the Moyal space was proved to be perturbatively renormalizable. It is translation-invariant and introduces in the action a $a/(θ^2p^2)$ term. We calculate here the $β$ and $γ$ functions at one-loop level for this model. The coupling constant $β_λ$ function is proved to have the same behaviour as the one of the $ϕ^4$ model on the commutative $\mathbb{R}^4$. The $β_a$ function of the new parameter $a$ is also calculated. Some interpretation of these results are done.
In this paper, the connection between the path integral representation of propagators in the coherent state basis with additional degrees of freedom \cite{rohwer} and the one without any such degrees of freedom \cite{sgfgs} is established. We further demonstrate that the path integral formalism developed in the noncommutative plane using the coherent state basis leads to a quantum mechanics involving a Chern-Simons term in momentum which is of noncommutative origin. The origin of this term from the Bopp-shift point of view is also investigated. A relativistic generalization of the action deriv…
We compute the one loop, $O(þ)$ 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 $þ$-dependent magnetic moment type structure is induced. In addition to the usual commutative graph, cubic photon vertices also give a finite $þ$ dependent contribution. Furthermore, the two two-photon vertex diagrams, that give zero in the commutative case yield finite $þ$ dependent t…
The noncommutative extension of two dimensional BF model is considered. It is shown that the realization of the noncommutative map via the Groenewold-Moyal star product leads to instabilities of the action, hence to a non renormalizable theory.
The one-loop contribution to vacuum polarization is calculated for the adjoint fermions in three dimensional noncommutative spaces, both at zero and finite temperature. At zero temperature, we confirm a previously found result for the parity odd part and subsequently analyze the even parity part, which exhibits UV/IR mixing. We discuss in detail, two regimes of the high temperature behavior of the parity odd part. When the thermal wavelength is much smaller, as compared to the noncommutativity scale, we find an interesting Fermi-Bose transmutation in the nonplanar part.
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 …
We analyze the renormalizability properties of pure gauge noncommutative SU(N) theory in the $θ$-expanded approach. We find that the theory is one-loop renormalizable to first order in $θ$.
