Noncommutative Gauge Field Research Articles

Lifting Bratteli diagrams between Krajewski diagrams: Spectral triples, spectral actions, and AF algebras

In this paper, we present a framework to construct sequences of spectral triples on top of an inductive sequence defining an $AF$-algebra. One aim of this paper is to lift arrows of a Bratteli diagram to arrows between Krajewski diagrams. The spectral actions defining Non-commutative Gauge Field Theories associated to two spectral triples related by these arrows are compared (tensored by a commutative spectral triple to put us in the context of Almost Commutative manifolds). This paper is a follow up of a previous one in which this program was defined and physically illustrated in the framework of the derivation-based differential calculus, but the present paper focuses more on the mathematical structure without trying to study the physical implications.

Implications of Seiberg–Witten map on Type-I superconductors

Guided by Pippard superconductivity, we incorporate the Seiberg–Witten map in the classical London theory of Type-I superconductors when an external magnetic field is applied. After defining the noncommutative Maxwell potentials, we derive the London equation for the supercurrent as a function of the noncommutative parameter, up to second order in gauge fields expansions. Keeping track of the effects of noncommutative gauge fields, we argue that noncommutative magnetic field effects can be cast in the London penetration length similar to nonlocal Pippard superconductivity. Also, we show that the flux quantization remains consistent relative to the commutative case. Our effective London penetration length reduces to the standard one in the commutative limit. These results allow us to argue that the framework of noncommutative electrodynamics can give some insights into the anisotropic and nonlocal structure of superconductivity and Condensed Matter Systems, perhaps out of the ultra-microscopic scale regime.

Noncommutative D=5 Chern–Simons gravity: Kaluza–Klein reduction and chiral gravitational anomaly

Actions for noncommutative (NC) gauge field theories can be expanded perturbatively in powers of the noncommutativity parameter theta using the Seiberg–Witten map between ordinary classical fields and their NC counterparts. The leading order term represents classical (theta =0) action while higher-order terms give us theta -dependent NC corrections that ought to capture some aspects of quantum gravity. Building on previous work of Aschieri and Castellani on NC Chern–Simons (CS) gauge and gravity theories, showing that non-trivial theta -dependence exists only for spacetime dimensions Dge 5, we investigate a correlated effect of these extra spatial dimensions and noncommutativity on four-dimensional physics, up to first-order in theta . Assuming that one spatial dimension is compactified into a circle, we apply the Kaluza–Klein reduction procedure on the NC D=5 CS theory for the conformal gauge group SO(4, 2), to obtain an effective, theta -dependent four-dimensional theory of gravity that has Einstein–Hilbert gravity with negative cosmological constant as its commutative limit. We derive field equations for this modified theory of gravity and study the effect of NC interactions on some classical geometries, such as the AdS-Schwarzschild black hole. We find that this NC background spacetime gives rise to chiral gravitational anomaly due to the nonvanishing theta -dependent Pontryagin density.

Derivation-based noncommutative field theories on AF algebras

In this paper, we start the investigation of a new natural approach to “unifying” noncommutative gauge field theories (NCGFT) based on approximately finite-dimensional ([Formula: see text]) [Formula: see text]-algebras. The defining inductive sequence of an [Formula: see text] [Formula: see text]-algebra is lifted to enable the construction of a sequence of NCGFT of Yang–Mills–Higgs types. This paper focuses on derivation-based noncommutative field theories. A mathematical study of the ingredients involved in the construction of a NCGFT is given in the framework of [Formula: see text] [Formula: see text]-algebras: derivation-based differential calculus, modules, connections, metrics and Hodge *-operators, and Lagrangians. Some physical applications concerning mass spectra generated by Spontaneous Symmetry Breaking Mechanisms (SSBM) are proposed using numerical computations for specific situations.

Canonical deformation of N=2 AdS4 supergravity

It is known that one can define a consistent theory of extended, $N=2$ anti-de Sitter (AdS) Supergravity (SUGRA) in $D=4$. Besides the standard gravitational part, this theory involves a single $U(1)$ gauge field and a pair of Majorana vector-spinors that can be mixed into a pair of charged spin-$3/2$ gravitini. The action for $N=2$ $AdS_{4}$ SUGRA is invariant under $SO(1,3)\times U(1)$ gauge transformations, and under local SUSY. We present a geometric action that involves two "inhomogeneous" parts: an orthosymplectic $OSp(4\vert 2)$ gauge-invariant action of the Yang-Mills type, and a supplementary action invariant under purely bosonic $SO(2,3)\times U(1)\sim Sp(4)\times SO(2)$ sector of $OSp(4\vert 2)$, that needs to be added for consistency. This action reduces to $N=2$ $AdS_{4}$ SUGRA after gauge fixing, for which we use a constrained auxiliary field in the manner of Stelle and West. Canonical deformation is performed by using the Seiberg-Witten approach to noncommutative (NC) gauge field theory with the Moyal product. The NC-deformed action is expanded in powers of the deformation parameter $\theta^{\mu\nu}$ up to the first order. We show that $N=2$ $AdS_{4}$ SUGRA has non-vanishing linear NC correction in the physical gauge, originating from the additional, purely bosonic action. For comparison, simple $N=1$ Poinacar\'{e} SUGRA can be obtained in the same manner, directly from an $OSp(4\vert 1)$ gauge-invariant action. The first non-vanishing NC correction is quadratic in $\theta^{\mu\nu}$ and therefore exceedingly difficult to calculate. Under Wigner-In\"{o}n\"{u} (WI) contraction, $N=2$ AdS superalgebra reduces to $N=2$ Poincar\'{e} superalgebra, and it is not clear whether this relation holds after canonical deformation. We present the linear NC correction to $N=2$ $AdS_{4}$ SUGRA explicitly, discuss its low-energy limit, and what remains of it after WI contraction.

Schwarzschild instanton in emergent gravity

In the bottom-up approach of emergent gravity, we attempt to find symplectic gauge fields emerging from Euclidean Schwarzschild instanton, which is studied as electromagnetism defined on the symplectic space [Formula: see text]. Geometrical engineering with the emergent metric sets up the Seiberg–Witten map between commutative and non-commutative gauge fields, preparing the ground for the evaluation of topological invariants in terms of the underlying gauge theory quantities.

Highly effective action from largeNgauge fields

Recently John H. Schwarz put forward a conjecture that the world-volume action of a probe D3-brane in an AdS5 x S5 background of type IIB superstring theory can be reinterpreted as the highly effective action (HEA) of four-dimensional N=4 superconformal field theory on the Coulomb branch. We argue that the HEA can be derived from the noncommutative (NC) field theory representation of the AdS/CFT correspondence and the Seiberg-Witten (SW) map defining a spacetime field redefinition between ordinary and NC gauge fields. It is based only on the well-known facts that the master fields of large N matrices are higher-dimensional NC U(1) gauge fields and the SW map is a local coordinate transformation eliminating U(1) gauge fields known as the Darboux theorem in symplectic geometry.

Forbidden and invisible Z boson decays in a covariant θ-exact noncommutative standard model

The triple neutral gauge boson and direct U(1)Y-neutrino interactions, being absent in ordinary field theory, can arise quite naturally in noncommutative gauge field theories. Using non-perturbative methods and a Seiberg–Witten map-based covariant approach to noncommutative gauge theory, we have found θ-exact expressions for both interactions, thereby eliminating previous restrictions to low-energy phenomena. In particular, we obtain for the first time the covariant, θ-exact, triple neutral gauge boson interactions within the noncommutative standard model gauge sector including an additional gauge-field deformation freedom. Finally, we discuss implications for Z → γγ and decays, and show that our results behave quite reasonably throughout all interaction energy scales.

Noncommutative gauge fields coupled to noncommutative gravity

We present a noncommutative (NC) version of the action for vielbein gravity coupled to gauge fields. Noncommutativity is encoded in a twisted star product between forms, with a set of commuting background vector fields defining the (abelian) twist. A first order action for the gauge fields avoids the use of the Hodge dual. The NC action is invariant under diffeomorphisms and twisted gauge transformations. The Seiberg-Witten map, adapted to our geometric setting and generalized for an arbitrary abelian twist, allows to re-express the NC action in terms of classical fields: the result is a deformed action, invariant under diffeomorphisms and usual gauge transformations. This deformed action is a particular higher derivative extension of the Einstein-Hilbert action coupled to Yang-Mills fields, and to the background vector fields defining the twist. Here noncommutativity of the original NC action dictates the precise form of this extension. We explicitly compute the first order correction in the NC parameter of the deformed action, and find that it is proportional to cubic products of the gauge field strength and to the symmetric anomaly tensor D_{IJK}.

Yukawa couplings and seesaw neutrino masses in noncommutative gauge theory

We consider Yukawa couplings in a θ-exact approach to noncommutative gauge field theory and show that both Dirac and singlet Majorana neutrino mass terms can be consistently accommodated. This shows that in fact the whole neutrino-mass extended standard model on noncommutative spacetime can be formulated in the new nonperturbative (in θ) approach which eliminates the previous restriction of Seiberg–Witten map based theories to low-energy phenomena. Spacetime noncommutativity induced couplings between neutrinos and photons as well as Z-bosons appear quite naturally in the model. We derive relevant Feynman rules for the type I seesaw mechanism.

Noncommutative $\varepsilon$-graded connections

We introduce the new notion of \varepsilon -graded associative algebras which takes its roots from the notion of commutation factors introduced in the context of Lie algebras ([Scheunert, 1979]). We define and study the associated notion of \varepsilon -derivation-based differential calculus, which generalizes the derivation-based calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with various examples of \varepsilon -graded commutative algebras, in particular some graded matrix algebras and the Moyal algebra. This last example also permits us to interpret mathematically a noncommutative gauge field theory.

On UV/IR mixing in noncommutative gauge field theories

In formulating gauge field theories on noncommutative (NC) spaces it is suggested that particles carrying gauge invariant quantities should not be viewed as pointlike, but rather as extended objects whose sizes grow linearly with their momenta. This and other generic properties deriving from the nonlocal character of interactions (showing thus unambiguously their quantum-gravity origin) lead to a specific form of UV/IR mixing as well as to a pathological behavior at the quantum level when the noncommutativity parameter theta is set to be arbitrarily small. In spite of previous suggestions that in a NC gauge theory based on the theta-expanded Seiberg-Witten (SW) maps UV/IR mixing effects may be under control, a fairly recent study of photon self-energy within a SW theta-exact approach has shown that UV/IR mixing is still present. We study the self-energy contribution for neutral fermions in the theta-exact approach of NC QED, and show by explicit calculation that all but one divergence can be eliminated for a generic choice of the noncommutativity parameter theta. The remaining divergence is linked to the pointlike limit of an extended object.

Noncommutative two-dimensional gauge theories

We elaborate on the dynamics of noncommutative two-dimensional gauge field theories. We consider U(N) gauge theories with fermions in either the fundamental or the adjoint representation. Noncommutativity leads to a rather non-trivial dependence on theta (the noncommutativity parameter) and to a rich dynamics. In particular the mass spectrum of the noncommutative U(1) theory with adjoint matter is similar to that of ordinary (commutative) two-dimensional large-NSU(N) gauge theory with adjoint matter. The noncommutative version of the ʼt Hooft model receives a non-trivial contribution to the vacuum polarization starting from three-loops order. As a result the mass spectrum of the noncommutative theory is expected to be different from that of the commutative theory.

Classical noncommutative electrodynamics with external source

In a $U(1)_{\star}$-noncommutative (NC) gauge field theory we extend the Seiberg-Witten (SW) map to include the (gauge-invariance-violating) external current and formulate - to the first order in the NC parameter - gauge-covariant classical field equations. We find solutions to these equations in the vacuum and in an external magnetic field, when the 4-current is a static electric charge of a finite size $a$, restricted from below by the elementary length. We impose extra boundary conditions, which we use to rule out all singularities, $1/r$ included, from the solutions. The static charge proves to be a magnetic dipole, with its magnetic moment being inversely proportional to its size $a$. The external magnetic field modifies the long-range Coulomb field and some electromagnetic form-factors. We also analyze the ambiguity in the SW map and show that at least to the order studied here it is equivalent to the ambiguity of adding a homogeneous solution to the current-conservation equation.

Spacetime noncommutativity and ultrahigh energy cosmic ray experiments

If new physics were capable of pushing the neutrino-nucleon inelastic cross section 3 orders of magnitude beyond the standard model prediction, then ultrahigh energy (UHE) neutrinos would have already been observed at neutrino observatories. We use such a constraint to reveal information on the scale of noncommutativity (NC) ${\ensuremath{\Lambda}}_{\mathrm{NC}}$ in noncommutative gauge field theories where neutrinos possess a tree-level coupling to photons in a generation-independent manner. In the energy range of interest (${10}^{10}$ to ${10}^{11}\text{ }\text{ }\mathrm{GeV}$), the $\ensuremath{\theta}$ expansion ($|\ensuremath{\theta}|\ensuremath{\sim}1/{\ensuremath{\Lambda}}_{\mathrm{NC}}^{2}$) and, therefore, the perturbative expansion, in terms of ${\ensuremath{\Lambda}}_{\mathrm{NC}}$, retains no longer its meaningful character, forcing us to resort to those NC field theoretical frameworks involving the full $\ensuremath{\theta}$ resummation. Our numerical analysis of the contribution to the process coming from the photon exchange impeccably pins down a lower bound on ${\ensuremath{\Lambda}}_{\mathrm{NC}}$ to be as high as 900 (450) TeV, depending on the estimates for the cosmogenic neutrino flux. If, on the other hand, one considers a surprising recent result that occurred in Pierre Auger Observatory data, that UHE cosmic rays are mainly composed of highly ionized Fe nuclei, then our bounds get weaker, due to the diminished cosmic neutrino flux. Nevertheless, we show that, even for the very high fraction of heavy nuclei in primary UHE cosmic rays, our method may still yield remarkable bounds on ${\ensuremath{\Lambda}}_{\mathrm{NC}}$, typically always above 200 TeV. Albeit, in this case, one encounters a maximal value for the Fe fraction, from which any useful information on ${\ensuremath{\Lambda}}_{\mathrm{NC}}$ can be drawn, delimiting thus the applicability of our method.

Deformation quantization of surjective submersions and principal fibre bundles

In this paper we establish a notion of deformation quantization of a surjective submersion which is specialized further to the case of a principal fibre bundle: the functions on the total space are deformed into a right module for the star product algebra of the functions on the base manifold. In case of a principal fibre bundle we require in addition invariance under the principal action. We prove existence and uniqueness of such deformations. The commutant within all differential operators on the total space is computed and gives a deformation of the algebra of vertical differential operators. Applications to noncommutative gauge field theories and phase space reduction of star products are discussed.

One-loop calculations for a translation invariant non-commutative gauge model

In this paper we discuss one-loop results for the translation invariant non-commutative gauge field model we introduced recently. This model relies on the addition of some carefully chosen extra terms in the action which mix long and short scales in order to circumvent the infamous UV/IR mixing, and which were motivated by the renormalizable non-commutative scalar model of Gurau et al. [arXiv:0802.0791].

Constraining spacetime noncommutativity with primordial nucleosynthesis

We discuss a constraint on the scale $\Lambda_{\rm NC}$ of noncommutative (NC) gauge field theory arising from consideration of the big bang nucleosynthesis (BBN) of light elements. The propagation of neutrinos in the NC background described by an antisymmetric tensor $\theta^{\mu\nu}$ does result in a tree-level vector-like coupling to photons in a generation-independent manner, raising thus a possibility to have an appreciable contribution of three light right-handed (RH) fields to the energy density of the universe at nucleosynthesis time. Considering elastic scattering processes of the RH neutrinos off charged plasma constituents at a given cosmological epoch, we obtain for a conservative limit on an effective number of additional doublet neutrinos, $\Delta N_\nu =1$, a bound $\Lambda_{\rm NC} \stackrel{>}{\sim}$ 3 TeV. With a more stringent requirement, $\Delta N_\nu \lesssim 0.2$, the bound is considerably improved, $\Lambda_{\rm NC} \stackrel{>}{\sim} 10^3$ TeV. For our bounds the $\theta$-expansion of the NC action stays always meaningful, since the decoupling temperature of the RH species is perseveringly much less than the inferred bound for the scale of noncommutativity.

Aspects of a noncommutative scalar/tensor duality

We study the noncommutative massless Kalb-Ramond gauge field coupled to a dynamical $U(1)$ gauge field in the adjoint representation together with a compensating vector field. We derive the Seiberg-Witten map and obtain the corresponding mapped action to first order in $\ensuremath{\theta}$. The (emergent) gravity structure found in other situations is not present here. The off-shell dual scalar theory is derived and it does not coincide with the Seiberg-Witten mapped scalar theory. Dispersion relations are also discussed. The $p$-form generalization of the Seiberg-Witten map to order $\ensuremath{\theta}$ is also derived.

Renormalizability and phenomenology of θ‐expanded noncommutative gauge field theory

AbstractIn this article we consider θ‐expanded noncommutative gauge field theory, constructed at the first order in noncommutative parameter θ, as an effective, anomaly free theory, with one‐loop renormalizable gauge sector. Related phenomenology with emphasis on the standard model forbidden decays, is discussed. Experimental possibilities of Z → γγ decay are analyzed and a firm bound to the scale of the noncommutativity parameter is set around few TeV's.