Divergences on projective modules and non-commutative integrals

A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first order differential calculus on such a module which admits a hom-connection or a divergence. Properties of integrals associated to this divergence are studied, in particular the formula of integration by parts is derived. Specific examples include inner calculi on a noncommutative algebra, the Berezin integral on the supercircle and integrals on Hopf algebras.Comment: 13 pages; v2 construction of projective modules has been generalise

Constraints on dark energy and cosmic topology

A non-trivial spatial topology of the Universe is a potentially observable attribute, which can be probed through the circles-in-the-sky for all locally homogeneous and isotropic universes with no assumptions on the cosmological parameters. We show how one can use a possible circles-in-the-sky detection of the spatial topology of globally homogeneous universes to set constraints on the dark energy equation of state parameters.Comment: 6 pages, 1 figure. To appear in Int. J. Mod. Phys. A (2009). From a talk presented at the Seventh Alexander Friedmann International Seminar on Gravitation and Cosmolog

DRA method: Powerful tool for the calculation of the loop integrals

We review the method of the calculation of multiloop integrals suggested in Ref.\cite{Lee2010}.Comment: 6 pages, contribution to ACAT2011 proceedings, Uxbridge, London, September 5-9, 2011, typos are correcte

Observational constraints on modified gravity models and the Poincar\'e dodecahedral topology

We study the constraints that spatial topology may place on the parameters of models that account for the accelerated expansion of the universe via infrared modifications to general relativity, namely the Dvali-Gabadadze-Porrati braneworld model as well as the Dvali-Turner and Cardassian models. By considering the Poincar\'e dodecahedral space as the circles-in-the-sky observable spatial topology, we examine the constraints that can be placed on the parameters of each model using type Ia supernovae data together with the baryon acoustic peak in the large scale correlation function of the Sloan Digital Sky Survey of luminous red galaxies and the Cosmic Microwave Background Radiation shift parameter data. We show that knowledge of spatial topology does provide relevant constraints, particularly on the curvature parameter, for all models.Comment: Revtex4, 10 pages, 1 table, 12 figures; version to match the one to be published in Physical Review

Constraints on the cosmological density parameters and cosmic topology

A nontrivial topology of the spatial section of the universe is an observable, which can be probed for all locally homogeneous and isotropic universes, without any assumption on the cosmological density parameters. We discuss how one can use this observable to set constraints on the density parameters of the Universe by using a specific spatial topology along with type Ia supenovae and X-ray gas mass fraction data sets.Comment: 11 pages, 4 figures. To appear in Int. J. Mod. Phys. D (2006). Invited talk delivered at the 2nd International Workshop on Astronomy and Relativistic Astrophysic

Minkowski superspaces and superstrings as almost real-complex supermanifolds

In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that mathematicians study are real or (almost) complex ones, while Minkowski superspaces are completely different objects. They are what we call almost real-complex supermanifolds, i.e., real supermanifolds with a non-integrable distribution, the collection of subspaces of the tangent space, and in every subspace a complex structure is given. An almost complex structure on a real supermanifold can be given by an even or odd operator; it is complex (without "always") if the suitable superization of the Nijenhuis tensor vanishes. On almost real-complex supermanifolds, we define the circumcised analog of the Nijenhuis tensor. We compute it for the Minkowski superspaces and superstrings. The space of values of the circumcised Nijenhuis tensor splits into (indecomposable, generally) components whose irreducible constituents are similar to those of Riemann or Penrose tensors. The Nijenhuis tensor vanishes identically only on superstrings of superdimension 1|1 and, besides, the superstring is endowed with a contact structure. We also prove that all real forms of complex Grassmann algebras are isomorphic although singled out by manifestly different anti-involutions.Comment: Exposition of the same results as in v.1 is more lucid. Reference to related recent work by Witten is adde

Property (T) and rigidity for actions on Banach spaces

We study property (T) and the fixed point property for actions on $L^p$ and other Banach spaces. We show that property (T) holds when $L^2$ is replaced by $L^p$ (and even a subspace/quotient of $L^p$), and that in fact it is independent of $1\leq p<\infty$. We show that the fixed point property for $L^p$ follows from property (T) when 1. For simple Lie groups and their lattices, we prove that the fixed point property for $L^p$ holds for any $1< p<\infty$ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.Comment: Many minor improvement

The Poisson sigma model on closed surfaces

Using methods of formal geometry, the Poisson sigma model on a closed surface is studied in perturbation theory. The effective action, as a function on vacua, is shown to have no quantum corrections if the surface is a torus or if the Poisson structure is regular and unimodular (e.g., symplectic). In the case of a Kahler structure or of a trivial Poisson structure, the partition function on the torus is shown to be the Euler characteristic of the target; some evidence is given for this to happen more generally. The methods of formal geometry introduced in this paper might be applicable to other sigma models, at least of the AKSZ type.Comment: 32 pages; references adde

Topology Change in Canonical Quantum Cosmology

We develop the canonical quantization of a midisuperspace model which contains, as a subspace, a minisuperspace constituted of a Friedman-Lema\^{\i}tre-Robertson-Walker Universe filled with homogeneous scalar and dust fields, where the sign of the intrinsic curvature of the spacelike hypersurfaces of homogeneity is not specified, allowing the study of topology change in these hypersurfaces. We solve the Wheeler-DeWitt equation of the midisuperspace model restricted to this minisuperspace subspace in the semi-classical approximation. Adopting the conditional probability interpretation, we find that some of the solutions present change of topology of the homogeneous hypersurfaces. However, this result depends crucially on the interpretation we adopt: using the usual probabilistic interpretation, we find selection rules which forbid some of these topology changes.Comment: 23 pages, LaTex file. We added in the conclusion some comments about path integral formalism and corrected litle misprinting

Notes on Stein-Sahi representations and some problems of non L2L^2 harmonic analysis

We discuss one natural class of kernels on pseudo-Riemannian symmetric spaces.Comment: 40p