Einstein-Riemann Gravity on Deformed Spaces

A differential calculus, differential geometry and the E-R Gravity theory are studied on noncommutative spaces. Noncommutativity is formulated in the star product formalism. The basis for the gravity theory is the infinitesimal algebra of diffeomorphisms. Considering the corresponding Hopf algebra we find that the deformed gravity is based on a deformation of the Hopf algebra.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 2006, Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Quantum groups and q-lattices in phase space

Quantum groups lead to an algebraic structure that can be realized on quantum spaces. These are noncommutative spaces that inherit a well defined mathematical structure from the quantum group symmetry. In turn such quantum spaces can be interpreted as noncommutative configuration spaces for physical systems which carry a symmetry like structure. These configuration spaces will be generalized to noncommutative phase space. The definition of the noncommutative phase space will be based on a differential calculus on the configuration space which is compatible with the symmetry. In addition a conjugation operation will be defined which will allow us to define the phase space variables in terms of algebraically selfadjoint operators. An interesting property of the phase space observables will be that they will have a discrete spectrum. These noncommutative phase space puts physics on a lattice structure.Comment: 6 pages, Postscrip

Noncommutative Gravity and the *-Lie algebra of diffeomorphisms

We construct functions and tensors on noncommutative spacetime by systematically twisting the corresponding commutative structures. The study of the deformed diffeomorphisms (and Poincare) Lie algebra allows to construct a noncomutative theory of gravity.Comment: 12 pages. Presented at the Erice International School of Subnuclear Physics, 44th course, Erice, Sicily, 29.8- 7.9 2006, and at the Second workshop and midterm meeting of the MCRTN ``Constituents, Fundamental Forces and Symmetries of the Universe" Napoli, 9-13.10 200

q-deformed Hermite Polynomials in q-Quantum Mechanics

The q-special functions appear naturally in q-deformed quantum mechanics and both sides profit from this fact. Here we study the relation between the q-deformed harmonic oscillator and the q-Hermite polynomials. We discuss: recursion formula, generating function, Christoffel-Darboux identity, orthogonality relations and the moment functional.Comment: latex, 8 pages, no figures. accepted for publication in European Journal of Physics

A Note on Superfields and Noncommutative Geometry

We consider the supersymmetric field theories on the noncommutative $R^4$ using the superspace formalism on the commutative space. The terms depending on the parameter of the noncommutativity $\Theta$ are regarded as the interactions. In this way we construct the N=1 supersymmetric action for the U(N) vector multiplets and chiral multiplets of the fundamental, anti-fundamental and adjoint representations of the gauge group. The action for vector multiplets of the products gauge group and its bi-fundamental matters is also obtained. We discuss the problem of the derivative terms of the auxiliary fields.Comment: 13 pages, LaTeX, no figures, Note added is changed, one reference adde

Ingredients of supergravity

These notes give a summary of lectures given in Corfu in 2010 on basic ingredients in the study of supergravity. It also summarizes initial chapters of a forthcoming book `Supergravity' by the same authors.Comment: 8 pages, to be published in Fortsch. Phys. as proceedings of the 10th Hellenic School on Elementary Particle Physics and Gravity, Corfu 2010; v2: reference adde

QuasiSupersymmetric Solitons of Coupled Scalar Fields in Two Dimensions

We consider solitonic solutions of coupled scalar systems, whose Lagrangian has a potential term (quasi-supersymmetric potential) consisting of the square of derivative of a superpotential. The most important feature of such a theory is that among soliton masses there holds a Ritz-like combination rule (e.g. $M_{12}+M_{23}=M_{13}$), instead of the inequality ($M_{12}+M_{23}<M_{13}$) which is a stability relation generally seen in N=2 supersymmetric theory. The promotion from N=1 to N=2 theory is considered.Comment: 18 pages, 5 figures, uses epsbox.st