Cosmic No Hair for Collapsing Universes

It is shown that all contracting, spatially homogeneous, orthogonal Bianchi cosmologies that are sourced by an ultra-stiff fluid with an arbitrary and, in general, varying equation of state asymptote to the spatially flat and isotropic universe in the neighbourhood of the big crunch singularity. This result is employed to investigate the asymptotic dynamics of a collapsing Bianchi type IX universe sourced by a scalar field rolling down a steep, negative exponential potential. A toroidally compactified version of M*-theory that leads to such a potential is discussed and it is shown that the isotropic attractor solution for a collapsing Bianchi type IX universe is supersymmetric when interpreted in an eleven-dimensional context.Comment: Extended discussion to include Kantowski-Sachs universe. In press, Classical and Quantum Gravit

The many faces of OSp(1|32)

We show that the complete superalgebra of symmetries, including central charges, that underlies F-theories, M-theories and type II string theories in dimensions 12, 11 and 10 of various signatures correspond to rewriting of the same OSp(1|32) algebra in different covariant ways. One only has to distinguish the complex and the unique real algebra. We develop a common framework to discuss all signatures theories by starting from the complex form of OSp(1|32). Theories are distinguished by the choice of basis for this algebra. We formulate dimensional reductions and dualities as changes of basis of the algebra. A second ingredient is the choice of a real form corresponding to a specific signature. The existence of the real form of the algebra selects preferred spacetime signatures. In particular, we show how the real d=10 IIA and IIB superalgebras for various signatures are related by generalized T-duality transformations that not only involve spacelike but also timelike directions. A third essential ingredient is that the translation generator in one theory plays the role of a central charge operator in the other theory. The identification of the translation generator in these algebras leads to the star algebras of Hull, which are characterized by the fact that the positive definite energy operator is not part of the translation generators. We apply our results to discuss different T-dual pictures of the D-instanton solution of Euclidean IIB supergravity.Comment: 30 pages, Latex, using lscape.st

U-duality as General Coordinate Transformations, and Spacetime Geometry

We show that the full global symmetry groups of all the D-dimensional maximal supergravities can be described in terms of the closure of the internal general coordinate transformations of the toroidal compactifications of D=11 supergravity and of type IIB supergravity, with type IIA/IIB T-duality providing an intertwining between the two pictures. At the quantum level, the part of the U-duality group that corresponds to the surviving discretised internal general coordinate transformations in a given picture leaves the internal torus invariant, while the part that is not described by internal general coordinate transformations can have the effect of altering the size or shape of the internal torus. For example, M-theory compactified on a large torus T^n can be related by duality to a compactification on a small torus, if and only if n\ge 3. We also discuss related issues in the toroidal compactification of the self-dual string to D=4. An appendix includes the complete results for the toroidal reduction of the bosonic sector of type IIB supergravity to arbitrary dimensions D\ge3.Comment: Latex, 28 page

Timelike Hopf Duality and Type IIA^* String Solutions

The usual T-duality that relates the type IIA and IIB theories compactified on circles of inversely-related radii does not operate if the dimensional reduction is performed on the time direction rather than a spatial one. This observation led to the recent proposal that there might exist two further ten-dimensional theories, namely type IIA^* and type IIB^*, related to type IIB and type IIA respectively by a timelike dimensional reduction. In this paper we explore such dimensional reductions in cases where time is the coordinate of a non-trivial U(1) fibre bundle. We focus in particular on situations where there is an odd-dimensional anti-de Sitter spacetime AdS_{2n+1}, which can be described as a U(1) bundle over \widetilde{CP}^n, a non-compact version of CP^n corresponding to the coset manifold SU(n,1)/U(n). In particular, we study the AdS_5\times S^5 and AdS_7\times S^4 solutions of type IIB supergravity and eleven-dimensional supergravity. Applying a timelike Hopf T-duality transformation to the former provides a new solution of the type IIA^* theory, of the form \widetilde{CP}^2\times S^1\times S^5. We show how the Hopf-reduced solutions provide further examples of ``supersymmetry without supersymmetry.'' We also present a detailed discussion of the geometrical structure of the Hopf-fibred metric on AdS_{2n+1}, and its relation to the horospherical metric that arises in the AdS/CFT correspondence.Comment: Latex, 26 page

Scale-Dependent Price Fluctuations for the Indian Stock Market

Classic studies of the probability density of price fluctuations $g$ for stocks and foreign exchanges of several highly developed economies have been interpreted using a {\it power-law} probability density function $P(g) \sim g^{-(\alpha+1)}$ with exponent values $\alpha > 2$, which are outside the L\'evy-stable regime $0 < \alpha < 2$. To test the universality of this relationship for less highly developed economies, we analyze daily returns for the period Nov. 1994--June 2002 for the 49 largest stocks of the National Stock Exchange which has the highest volume of trade in India. We find that $P(g)$ decays as an {\it exponential} function $P(g) \sim \exp(-\beta g)$ with a characteristic decay scales $\beta = 1.51 \pm 0.05$ for the negative tail and $\beta = 1.34 \pm 0.04$ for the positive tail, which is significantly different from that observed for developed economies. Thus we conclude that the Indian stock market may belong to a universality class that differs from those of developed countries analyzed previously.Comment: 7 pages, 8 figure

Complexified sigma model and duality

We show that the equations of motion associated with a complexified sigma-model action do not admit manifest dual SO(n,n) symmetry. In the process we discover new type of numbers which we called `complexoids' in order to emphasize their close relation with both complex numbers and matroids. It turns out that the complexoids allow to consider the analogue of the complexified sigma-model action but with (1+1)-worldsheet metric, instead of Euclidean-worldsheet metric. Our observations can be useful for further developments of complexified quantum mechanics.Comment: 15 pages, Latex, improved versio

From p-branes to Cosmology

We study the relationship between static p-brane solitons and cosmological solutions of string theory or M-theory. We discuss two different ways in which extremal p-branes can be generalised to non-extremal ones, and show how wide classes of recently discussed cosmological models can be mapped into non-extremal p-brane solutions of one of these two kinds. We also extend previous discussions of cosmological solutions to include some that make use of cosmological-type terms in the effective action that can arise from the generalised dimensional reduction of string theory or M-theory.Comment: Latex, 24 pages, no figur

Time-like T-duality algebra

When compactifying M- or type II string-theories on tori of indefinite space-time signature, their low energy theories involve sigma models on E_{n(n)}/H_n, where H_n is a not necessarily compact subgroup of E_{n(n)} whose complexification is identical to the complexification of the maximal compact subgroup of E_{n(n)}. We discuss how to compute the group H_n. For finite dimensional E_{n(n)}, a formula derived from the theory of real forms of E_n algebra's gives the possible groups immediately. A few groups that have not appeared in the literature are found. For n=9,10,11 we compute and describe the relevant real forms of E_n and H_n. A given H_n can correspond to multiple signatures for the compact torus. We compute the groups H_n for all compactifications of M-, M*-, and M'-theories, and type II-, II*- and II'-theories on tori of arbitrary signature, and collect them in tables that outline the dualities between them. In an appendix we list cosets G/H, with G split and H a subgroup of G, that are relevant to timelike toroidal compactifications and oxidation of theories with enhanced symmetries.Comment: LaTeX, 37 pages, 1 eps-figure, uses JHEP.cls; v2. corrected typo's in tables 16 and 17, minor changes to tex

ALMA Science Verification Data: Millimeter Continuum Polarimetry of the Bright Radio Quasar 3C 286

We present full-polarization observations of the compact, steep-spectrum radio quasar 3C~286 made with the ALMA at 1.3~mm. These are the first full-polarization ALMA observations, which were obtained in the framework of Science Verification. A bright core and a south-west component are detected in the total intensity image, similar to previous centimeter images. Polarized emission is also detected toward both components. The fractional polarization of the core is about 17\%, this is higher than the fractional polarization at centimeter wavelengths, suggesting that the magnetic field is even more ordered in the millimeter radio core than it is further downstream in the jet. The observed polarization position angle (or EVPA) in the core is $\sim$\,$39^{\circ}$, which confirms the trend that the EVPA slowly increases from centimeter to millimeter wavelengths. With the aid of multi-frequency VLBI observations, we argue that this EVPA change is associated with the frequency-dependent core position. We also report a serendipitous detection of a sub-mJy source in the field of view, which is likely to be a submillimeter galaxy.Comment: 10 pages, 9 figures, Accepted for publication in the Ap