We characterize = 1 vacua of type-II theories in terms of generalized complex structure on the internal manifold M. The structure group of T(M)⊕T*(M) being SU(3) × SU(3) implies the existence of two pure spinors Φ1 and Φ2. The conditions for preserving = 1 supersymmetry turn out to be simple generalizations of equations that have appeared in the context of = 2 and topological strings. They are (d+H∧)Φ1 = 0 and (d+H∧)Φ2 = FRR. The equation for the first pure spinor implies that the internal space is a twisted generalized Calabi-Yau manifold of a hybrid complex-symplectic type, while the RR-fields serve as an integrability defect for the second.

http://ipht.cea.fr/Docspht/articles/t05/079/public/1126-6708_2005_11_020.pdf

Generalized structures of N=1 vacua

We consider the possibility that the dark matter which is required to explain the dynamics of the neutral hydrogen clouds at large distances from the galactic centre could be in the form of a Bose–Einstein condensate. To study the condensate we use the non-relativistic Gross–Pitaevskii equation. By introducing the Madelung representation of the wavefunction, we formulate the dynamics of the system in terms of the continuity equation and of the hydrodynamic Euler equations. Hence dark matter can be described as a non-relativistic, Newtonian Bose–Einstein gravitational condensate gas, whose density and pressure are related by a barotropic equation of state. In the case of a condensate with quartic non-linearity, the equation of state is polytropic with index n = 1. In the framework of the Thomas–Fermi approximation the structure of the Newtonian gravitational condensate is described by the Lane–Emden equation, which can be exactly solved. General relativistic configurations with quartic non-linearity are studied, by numerically integrating the structure equations. The basic parameters (mass and radius) of the Bose–Einstein condensate dark matter halos sensitively depend on the mass of the condensed particle and of the scattering length. To test the validity of the model we fit the Newtonian tangential velocity equation of the model with a sample of rotation curves of low surface brightness and dwarf galaxies, respectively. We find a very good agreement between the theoretical rotation curves and the observational data for the low surface brightness galaxies. The deflection of photons passing through the dark matter halos is also analysed, and the bending angle of light is computed. The bending angle obtained for the Bose–Einstein condensate is larger than that predicted by standard general relativistic and dark matter models. The angular radii of the Einstein rings are obtained in the small angle approximation. Therefore the study of the light deflection by galaxies and the gravitational lensing could discriminate between the Bose–Einstein condensate dark matter model and other dark matter models.

https://iopscience.iop.org/article/10.1088/1475-7516/2007/06/025/pdf

Can dark matter be a Bose?Einstein condensate?

We compute leading-order corrections to the entropy of any thermodynamic system due to small statistical fluctuations around equilibrium. When applied to black holes, these corrections are shown to be of the form −k ln(Area). For BTZ black holes, k = 3/2, as found earlier. We extend the result to anti-de Sitter Schwarzschild and Reissner–Nordstrom black holes in arbitrary dimensions. Finally we examine the role of conformal field theory in black-hole entropy and its corrections.

https://iopscience.iop.org/article/10.1088/0264-9381/19/9/302/pdf

General logarithmic corrections to black-hole entropy

We show that the supersymmetry transformations for type II string theories on six-manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kahler form eiJ and the holomorphic form Ω. The equations are explicitly symmetric under exchange of the two pure spinors and a choice of even or odd-rank RR field. This is mirror symmetry for manifolds with torsion. Moreover, RR fluxes affect only one of the two equations: eiJ is closed under the action of the twisted exterior derivative in IIA theory, and similarly Ω is closed in IIB. Modulo a different action of the B–field, this means that supersymmetric SU(3)-structure manifolds are all generalized Calabi-Yau manifolds, as defined by Hitchin. An equivalent, and somewhat more conventional, description is given as a set of relations between the components of intrinsic torsions modified by the NS flux and the Clifford products of RR fluxes with pure spinors, allowing for a classification of type II supersymmetric vacua on six-manifolds. We find in particular that supersymmetric six-manifolds are always complex for IIB backgrounds while they are twisted symplectic for IIA.

/pdf/supersymmetric-backgrounds-from-generalized-calabi-yau-198yk8t1gw.pdf

Supersymmetric backgrounds from generalized Calabi-Yau manifolds

We perform a systematic search for = 1 Minkowski vacua of type II string theories on compact six-dimensional parallelizable nil- and solvmanifolds (quotients of six-dimensional nilpotent and solvable groups, respectively). Some of these manifolds have appeared in the construction of string backgrounds and are typically called twisted tori. We look for vacua directly in ten dimensions, using the a reformulation of the supersymmetry condition in the framework of generalized complex geometry. Certain algebraic criteria to establish compactness of the manifolds involved are also needed. Although the conditions for preserved = 1 supersymmetry fit nicely in the framework of generalized complex geometry, they are notoriously hard to solve when coupled to the Bianchi identities. We find solutions in a large-volume, constant-dilaton limit. Among these, we identify those that are T-dual to backgrounds of IIB on a conformal T6 with self-dual three-form flux, and hence conceptually not new. For all backgrounds of this type fully localized solutions can be obtained. The other new solutions need multiple intersecting sources (either orientifold planes or combinations of O-planes and D-branes) to satisfy the Bianchi identities; the full list of such new solution is given. These are so far only smeared solutions, and their localization is yet unknown. Although valid in a large-volume limit, they are the first examples of Minkowski vacua in supergravity which are not connected by any duality to a Calabi-Yau. Finally, we discuss a class of flat solvmanifolds that may lead to AdS4 vacua of type IIA strings.

https://iopscience.iop.org/article/10.1088/1126-6708/2007/05/031/pdf

A scan for new N = 1 vacua on twisted tori

The purpose of this paper is to further investigate the solution space of self-similar spherically symmetric perfect-fluid models and gain a deeper understanding of the physical aspects of these solutions We achieve this by combining the state space description of the homothetic approach with the use of the physically interesting quantities arising in the comoving approach We focus on three types of models First, we consider models that are natural inhomogeneous generalizations of the Friedmann universe; such models are asymptotically Friedmann in their past and evolve fluctuations in the energy density at later times Secondly, we consider so-called quasi-static models This class includes models that undergo self-similar gravitational collapse and is important for studying the formation of naked singularities If naked singularities do form, they have profound implications for the predictability of general relativity as a theory Thirdly, we consider a new class of asymptotically Minkowski self-similar spacetimes, emphasizing that some of them are associated with the self-similar solutions associated with the critical behaviour observed in recent gravitational collapse calculations

The state space and physical interpretation of self-similar spherically symmetric perfect-fluid models

Spatially homogeneous cosmological models with a positive cosmological constant are investigated, using dynamical systems methods. We focus on the future evolution of these models. In particular, we address the question whether there are models within this class that are de Sitter-like in the future, but are tilted.

Homogeneous cosmologies with a cosmological constant

Einstein's field equations for timelike self-similar spherically symmetric perfect-fluid models are investigated. The field equations are rewritten as a first-order system of autonomous differential equations. Dimensionless variables are chosen in such a way that the number of equations in the coupled system is reduced as far as possible and so that the reduced phase space becomes compact and regular. The system is subsequently analysed qualitatively using the theory of dynamical systems. Using this approach, we obtain a clear picture of the full phase space and the full space of solutions. Solutions of physical interest, e.g. the solution associated with criticality in black hole formation, are easily singled out. We also discuss the various `band structures' that are associated with certain one-parameter sets of solutions.

Timelike self-similar spherically symmetric perfect-fluid models

Einstein's field equations for spatially self-similar spherically symmetric perfect-fluid models are investigated. The field equations are rewritten as a first-order system of autonomous differential equations. Dimensionless variables are chosen in such a way that the number of equations in the coupled system is reduced as far as possible and so that the reduced phase space becomes compact and regular. The system is subsequently analysed qualitatively with the theory of dynamical systems.

Spatially self-similar spherically symmetric perfect-fluid models

Closed, spatially homogeneous cosmological models with a perfect fluid and a scalar field with exponential potential are investigated, using dynamical systems methods. First, we consider the closed Friedmann-Robertson-Walker models, discussing the global dynamics in detail. Next, we investigate Kantowski-Sachs models, for which the future and past attractors are determined. The global asymptotic behaviour of both the Friedmann-Robertson-Walker and the Kantowski-Sachs models is that they either expand from an initial singularity, reach a maximum expansion and thereafter recollapse to a final singularity (for all values of the potential parameter kappa), or else they expand forever towards a flat power-law inflationary solution (when kappa^2<2). As an illustration of the intermediate dynamical behaviour of the Kantowski-Sachs models, we examine the cases of no barotropic fluid, and of a massless scalar field in detail. We also briefly discuss Bianchi type IX models.

Martin Goliath

Papers

The state space and physical interpretation of self-similar spherically symmetric perfect-fluid models

Homogeneous cosmologies with a cosmological constant

Timelike self-similar spherically symmetric perfect-fluid models

Spatially self-similar spherically symmetric perfect-fluid models

Closed cosmologies with a perfect fluid and a scalar field