Dilaton Cosmology, Noncommutativity and Generalized Uncertainty Principle

The effects of noncommutativity and of the existence of a minimal length on the phase space of a dilatonic cosmological model are investigated. The existence of a minimum length, results in the Generalized Uncertainty Principle (GUP), which is a deformed Heisenberg algebra between the minisuperspace variables and their momenta operators. We extend these deformed commutating relations to the corresponding deformed Poisson algebra. For an exponential dilaton potential, the exact classical and quantum solutions in the commutative and noncommutative cases, and some approximate analytical solutions in the case of GUP, are presented and compared.Comment: 16 pages, 3 figures, typos correcte

Sharp version of the Goldberg-Sachs theorem

We reexamine from first principles the classical Goldberg-Sachs theorem from General Relativity. We cast it into the form valid for complex metrics, as well as real metrics of any signature. We obtain the sharpest conditions on the derivatives of the curvature that are sufficient for the implication (integrability of a field of alpha planes)$\Rightarrow$(algebraic degeneracy of the Weyl tensor). With every integrable field of alpha planes we associate a natural connection, in terms of which these conditions have a very simple form.Comment: In this version we made a minor change in Remark 5.5 and simplified Section 6, starting at Theorem 6.

Self-dual Einstein Spaces, Heavenly Metrics and Twistors

Four-dimensional quaternion-Kahler metrics, or equivalently self-dual Einstein spaces M, are known to be encoded locally into one real function h subject to Przanowski's Heavenly equation. We elucidate the relation between this description and the usual twistor description for quaternion-Kahler spaces. In particular, we show that the same space M can be described by infinitely many different solutions h, associated to different complex (local) submanifolds on the twistor space, and therefore to different (local) integrable complex structures on M. We also study quaternion-Kahler deformations of M and, in the special case where M has a Killing vector field, show that the corresponding variations of h are related to eigenmodes of the conformal Laplacian on M. We exemplify our findings on the four-sphere S^4, the hyperbolic plane H^4 and on the "universal hypermultiplet", i.e. the hypermultiplet moduli space in type IIA string compactified on a rigid Calabi-Yau threefold.Comment: 44 pages, 1 figure; misprints correcte

∗\ast-SDYM fields and heavenly spaces: II. Reductions of the ∗\ast-SDYM system

Reductions of self-dual Yang-Mills (SDYM) system for $\ast$-bracket Lie algebra to the Husain-Park (HP) heavenly equation and to sl(N,{\boldmath{C}) SDYM equation are given. An example of a sequence of $su(N)$ chiral fields ($N\geq 2$) tending for $N\to\infty$ to a curved heavenly space is found.Comment: 18 page

Stabilization of internal space in noncommutative multidimensional cosmology

We study the cosmological aspects of a noncommutative, multidimensional universe where the matter source is assumed to be a scalar field which does not commute with the internal scale factor. We show that such noncommutativity results in the internal dimensions being stabilizedComment: 8 pages, 1 figure, to appear in IJMP