On the Geometry of the Quantum Poincare Group

We review the construction of the multiparametric inhomogeneous orthogonal quantum group ISO_qr(N) as a projection from SO_qr(N+2), and recall the conjugation that for N=4 leads to the quantum Poincare group. We study the properties of the universal enveloping algebra U_qr(iso(N)), and give an R-matrix formulation. A quantum Lie algebra and a bicovariant differential calculus on twisted ISO(N) are found.Comment: 12 pages, Latex. Contribution to the proceedings of the 30-th Arhenshoop Symposium on the Theory of Elementary Particles. August 1996. To appear in Nucl. Phys. B Proc. Sup

Deformation quantization of principal bundles

We outline how Drinfeld twist deformation techniques can be applied to the deformation quantization of principal bundles into noncommutative principal bundles, and more in general to the deformation of Hopf-Galois extensions. First we twist deform the structure group in a quantum group, and this leads to a deformation of the fibers of the principal bundle. Next we twist deform a subgroup of the group of authomorphisms of the principal bundle, and this leads to a noncommutative base space. Considering both deformations we obtain noncommutative principal bundles with noncommutative fiber and base space as well.Comment: 20 pages. Contribution to the volume in memory of Professor Mauro Francaviglia. Based on joint work with Pierre Bieliavsky, Chiara Pagani and Alexander Schenke

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 \cite{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 analized.Comment: Latex, 13 pages. Added ref. [4] and [7] (page 12

Twisting all the way: from algebras to morphisms and connections

Given a Hopf algebra H and an algebra A that is an H-module algebra we consider the category of left H-modules and A-bimodules, where morphisms are just right A-linear maps (not necessarily H-equivariant). Given a twist F of H we then quantize (deform) H to H^F, A to A_\star and correspondingly the category of left H-modules and A-bimodules to the category of left H^F-modules and A_\star-bimodules. If we consider a quasitriangular Hopf algebra H, a quasi-commutative algebra A and quasi-commutative A-bimodules, we can further construct and study tensor products over A of modules and of morphisms, and their twist quantization. This study leads to the definition of arbitrary (i.e., not necessarily H-equivariant) connections on quasi-commutative A-bimodules, to extend these connections to tensor product modules and to quantize them to A_\star-bimodule connections. Their curvatures and those on tensor product modules are also determined.Comment: 15 pages. Proceedings of the Julius Wess 2001 workshop of the Balkan Summer Institute 2011, 27-28.8.2011 Donji Milanovac, Serbi

Noncommutative gravity at second order via Seiberg-Witten map

We develop a general strategy to express noncommutative actions in terms of commutative ones by using a recently developed geometric generalization of the Seiberg-Witten map (SW map) between noncommutative and commutative fields. We apply this general scheme to the noncommutative vierbein gravity action and provide a SW differential equation for the action itself as well as a recursive solution at all orders in the noncommutativity parameter \theta. We thus express the action at order \theta^n+2 in terms of noncommutative fields of order at most \theta^n+1 and, iterating the procedure, in terms of noncommutative fields of order at most \theta^n. This in particular provides the explicit expression of the action at order \theta^2 in terms of the usual commutative spin connection and vierbein fields. The result is an extended gravity action on commutative spacetime that is manifestly invariant under local Lorentz rotations and general coordinate transformations.Comment: 14 page

Noncommutative Chern-Simons gauge and gravity theories and their geometric Seiberg-Witten map

We use a geometric generalization of the Seiberg-Witten map between noncommutative and commutative gauge theories to find the expansion of noncommutative Chern-Simons (CS) theory in any odd dimension $D$ and at first order in the noncommutativity parameter $\theta$. This expansion extends the classical CS theory with higher powers of the curvatures and their derivatives. A simple explanation of the equality between noncommutative and commutative CS actions in $D=1$ and $D=3$ is obtained. The $\theta$ dependent terms are present for $D\geq 5$ and give a higher derivative theory on commutative space reducing to classical CS theory for $\theta\to 0$. These terms depend on the field strength and not on the bare gauge potential. In particular, as for the Dirac-Born-Infeld action, these terms vanish in the slowly varying field strength approximation: in this case noncommutative and commutative CS actions coincide in any dimension. The Seiberg-Witten map on the $D=5$ noncommutative CS theory is explored in more detail, and we give its second order $\theta$-expansion for any gauge group. The example of extended $D=5$ CS gravity, where the gauge group is $SU(2,2)$, is treated explicitly.Comment: 18 pages, LaTeX. Added clarifications, added reference. Matches published version on JHE

Global Seiberg-Witten maps for U(n)-bundles on tori and T-duality

Seiberg-Witten maps are a well-established method to locally construct noncommutative gauge theories starting from commutative gauge theories. We revisit and classify the ambiguities and the freedom in the definition. Geometrically, Seiberg-Witten maps provide a quantization of bundles with connections. We study the case of U(n)-vector bundles on two-dimensional tori, prove the existence of globally defined Seiberg-Witten maps (induced from the plane to the torus) and show their compatibility with Morita equivalence.Comment: 28 pages. Revised version: sharpened in Sec. 4.3 the study of the Seiberg-Witten maps for sections in the adjoint, related to their ordering ambiguities; added sum of connections for tensor product bundles in Sec. 5; improved in Sec. 5.1 the compatibility between Seiberg-Witten map and T-duality transformation

Proof of a Symmetrized Trace Conjecture for the Abelian Born-Infeld Lagrangian

In this paper we prove a conjecture regarding the form of the Born-Infeld Lagrangian with a U(1)^2n gauge group after the elimination of the auxiliary fields. We show that the Lagrangian can be written as a symmetrized trace of Lorentz invariant bilinears in the field strength. More generally we prove a theorem regarding certain solutions of unilateral matrix equations of arbitrary order. For solutions which have perturbative expansions in the matrix coefficients, the solution and all its positive powers are sums of terms which are symmetrized in all the matrix coefficients and of terms which are commutators.Comment: 9 pages, LaTeX, no figures, theorem generalized and a new method of proof include

Extended gravity theories from dynamical noncommutativity

In this paper we couple noncommutative (NC) 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}.Comment: 12 pages, LaTeX. Added section 7 on NC field equations and perturbative solution