On the Noncommutative Geometry of Twisted Spheres

We describe noncommutative geometric aspects of twisted deformations, in particular of the spheres of Connes and Landi and of Connes and Dubois Violette, by using the differential and integral calculus on these spaces that is covariant under the action of their corresponding quantum symmetry groups. We start from multiparametric deformations of the orthogonal groups and related planes and spheres. We show that only in the twisted limit of these multiparametric deformations the covariant calculus on the plane gives, by a quotient procedure, a meaningful calculus on the sphere. In this calculus, the external algebra has the same dimension as the classical one. We develop the Haar functional on spheres and use it to define an integral of forms. In the twisted limit (differently from the general multiparametric case), the Haar functional is a trace and we thus obtain a cycle on the algebra. Moreover, we explicitly construct the *-Hodge operator on the space of forms on the plane and then by quotient on the sphere. We apply our results to even spheres and compute the Chern–Connes pairing between the character of this cycle, i.e. a cyclic 2n-cocycle, and the instanton projector defined in math.QA/0107070.     

Spatial harmonics modelling of planar periodic segmented waveguides

A theoretical approach based on Bloch theorem and spatial harmonics has been used to investigate the propagation characteristics in planar periodic segmented waveguides. This analytical method allows to evaluate the field distribution, the effective index and the propagation losses of these structures, taking into account both the propagating and the counterpropagating field components, thus evidencing all the phenomena which can take place in: Bragg reflections, leaky waves, etc. Results for TE fields are presented and compared with those obtained using a paraxial 2D FD-BPM method and a Padé based one, showing that no more than (1, 1) order approximants are needed to provide good estimates of the device characteristics.     

Real Forms of Quantum Orthogonal Groups,q-Lorentz Groups in any Dimension

We review known real forms of the quantum orthogonal groups SO _q (N). New *-conjugations are then introduced and we contruct all real forms of quantum orthogonal groups. We thus give an RTT formulation of the *-conjugations on SO _q (N) that is complementary to the U _q (g) *-structure classification of Twietmeyer. In particular, we easily find and describe the real forms SO _q (N-1,1) for any value of N. Quantum subspaces of the q-Minkowski space are analyzed.     

R-matrix formulation of the quantum inhomogeneous groups ISO q,r (N) and ISp q,r (N)

The quantum commutationsRTT=TTR and the orthogonal (symplectic) conditions for the inhomogeneous multiparametricq-groups of theB _n ,C _n ,D _n type are found in terms of theR-matrix ofB _n+1 ,C _n+1 ,D _n+1 .A consistent Hopf structure on these inhomogeneousq-groups is constructed by means of a projection fromB _n+1 ,C _n+1 ,D _n+1 .Real forms are discussed; in particular, we obtain theq-groups ISO _q,r (n+1,n–1), including the quantum Poincaré group. The inhomogeneusq-groups do not contain dilatations when the parameters satisfy certain conditions. For example, we find a dilatation-freeq-Poincaré group depending on one real parameterq.     

Twisted Gauge Theories

Gauge theories on a space-time that is deformed by the Moyal–Weyl product are constructed by twisting the coproduct for gauge transformations. This way a deformed Leibniz rule is obtained, which is used to construct gauge invariant quantities. The connection will be enveloping algebra valued in a particular representation of the Lie algebra. This gives rise to additional fields, which couple only weakly via the deformation parameter θ and reduce in the commutative limit to free fields. Consistent field equations that lead to conservation laws are derived and some properties of such theories are discussed.     

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 $\star $ -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 $D_{IJK}$ .     

Dynamical Noncommutativity and Noether Theorem in Twisted $${\phi^{\star 4}}$$ Theory

A -product is defined via a set of commuting vector fields , and used in a theory coupled to the fields. The -product is dynamical, and the vacuum solution , reproduces the usual Moyal product. The action is invariant under rigid translations and Lorentz rotations, and the conserved energy–momentum and angular momentum tensors are explicitly derived.      

Universal enveloping algebra and differential calculi on inhomogeneous orthogonal q-groups

We review the construction of the multiparametric quantum group ISO_q,r(N) as a projection from SO_q,r (N + 2) and show that it is a bicovariant bimodule over SO_q,r(N). The universal enveloping algebra U_q,r(iso(N)), characterized as the Hopf algebra of regular functionals on ISO_q,r(N), is found as a Hopf subalgebra of U_q,r(so(N + 2)) and is shown to be a bicovariant bimodule over U_q,r(so(N)).

An R-matrix formulation of U_q,r(iso(N)) is given and we prove the pairing U_q,r(iso(N)) — ISO_q,r(N)). We analyze the subspaces of U_q,r(iso(N)) that define bicovariant differential calculi on ISO_q,r(N).     

Extended gravity theories from dynamical noncommutativity

In this paper we couple noncommutative vielbein gravity to scalar fields. Noncommutativity is encoded in a $\star $ -product between forms, given by an abelian twist (a twist with commuting vector fields). A geometric generalization of the Seiberg–Witten map for abelian twists yields an extended theory of gravity coupled to scalars, where all fields are ordinary (commutative) fields. The vectors defining the twist can be related to the scalar fields and their derivatives, and hence acquire dynamics. Higher derivative corrections to the classical Einstein–Hilbert and Klein–Gordon actions are organized in successive powers of the noncommutativity parameter $\theta ^{AB}$ .