Showing papers in "International Journal of Geometric Methods in Modern Physics in 2014"
TL;DR: In this article, the authors consider modified gravity which may describe the early-time inflation and/or late-time cosmic acceleration of the universe and discuss the properties of F(R), F(G), string-inspired and scalar-Einstein-Gauss-Bonnet gravities.
Abstract: We consider modified gravity which may describe the early-time inflation and/or late-time cosmic acceleration of the universe. In particular, we discuss the properties of F(R), F(G), string-inspired and scalar-Einstein–Gauss–Bonnet gravities, including their FRW equations and fluid or scalar-tensor description. Simplest accelerating cosmologies are investigated and possibility of unified description of the inflation with dark energy is described. The cosmological reconstruction program which permits to get the requested universe evolution from modified gravity is developed. As some extension, massive F(R) bigravity which is ghost-free theory is presented. Its scalar-tensor form turns out to be the easiest formulation. The cosmological reconstruction method for such bigravity is presented. The unified description of inflation with dark energy in F(R) bigravity turns out to be possible.
118 citations
TL;DR: In this article, the authors studied the Kerr-Newman black hole solutions in the context of f(R) modified gravity theories and concluded that, in the absence of a cosmological constant, the existence of BHs is determined by the sign of a parameter h dependent of the mass, charge, the spin and the scalar curvature.
Abstract: In the context of f(R) modified gravity theories, we study the Kerr-Newman black hole solutions. We study nonzero constant scalar curvature solutions and discuss the metric tensor that satisfies the modified field equations. We conclude that, in the absence of a cosmological constant, the black holes (BHs) existence is determined by the sign of a parameter h dependent of the mass, the charge, the spin and the scalar curvature. Different values of this parameter lead to diverse astrophysical objects, such as extremal and marginal extremal BHs. Thermodynamics of BHs are then studied, as well as their local and global stability. We analyze these features in a large variety of f(R) models. We remark the main differences with respect to general relativity and show the rich thermodynamical phenomenology that characterizes this framework.
84 citations
TL;DR: In this paper, the weak field limit of f(R, 𝒢) gravity taking into account analytic functions of the Ricci scalar R and the Gauss-Bonnet invariant and the Lagrangian Lagrangians is discussed.
Abstract: We discuss in detail the weak field limit of f(R, 𝒢) gravity taking into account analytic functions of the Ricci scalar R and the Gauss–Bonnet invariant 𝒢. Specifically, we develop, in metric formalism, the Newtonian, Post-Newtonian (PN) and Parametrized Post-Newtonian (PPN) limits starting from general f(R, 𝒢) Lagrangian. The special cases of f(R) and f(𝒢) gravities are considered. In the case of the Newtonian limit of f(R, 𝒢) gravity, a general solution in terms of Green's functions is achieved.
73 citations
TL;DR: In this paper, a modified Mimetic gravity (MMG) is proposed as a generalization of general relativity, which contains a physical metric which is the function of an auxiliary (unphysical) metric and a Lyra's metric.
Abstract: A modified Mimetic gravity (MMG) is proposed as a generalization of general relativity. The model contains a physical metric which is the function of an auxiliary (unphysical) metric and a Lyra's metric. We construct different kinds of conformally invariant models in different levels of the expansion parameter λ. This model phenomenologically has been extended to higher-order forms. Cosmology of a certain class of such models has been investigated in detail. A cosmological solution has been proposed in inhomogeneous form of scalar field. For homogeneous case, energy conditions are widely investigated. We have shown that the system evaluated at intervals shorter than a certain time Tc meets all the energy conditions.
62 citations
TL;DR: In this paper, the Hartle criterion is used to select correlated regions in the configuration space of dynamical variables whose meaning is related to the emergence of classical observable universes, and the existence of conserved quantities gives selection rules that allow to recover classical behaviors in cosmic evolution.
Abstract: We summarize the use of Noether symmetries in Minisuperspace Quantum Cosmology. In particular, we consider minisuperspace models, showing that the existence of conserved quantities gives selection rules that allow to recover classical behaviors in cosmic evolution according to the so-called Hartle criterion. Such a criterion selects correlated regions in the configuration space of dynamical variables whose meaning is related to the emergence of classical observable universes. Some minisuperspace models are worked out starting from Extended Gravity, in particular coming from scalar-tensor, f(R) and f(T) theories. Exact cosmological solutions are derived.
58 citations
TL;DR: In this article, the links between Finsler Geometry and the geometry of spacetimes are briefly revisited, and prospective ideas and results are explained, with special attention paid to geometric problems with a direct motivation in Relativity and other parts of Physics.
Abstract: Recent links between Finsler Geometry and the geometry of spacetimes are briefly revisited, and prospective ideas and results are explained. Special attention is paid to geometric problems with a direct motivation in Relativity and other parts of Physics.
50 citations
TL;DR: In this paper, the Lie point symmetries of the Schrodinger and the Klein-Gordon equations in a general Riemannian space were determined and related with the homothetic and conformal algebra of the metric of the space.
Abstract: We determine the Lie point symmetries of the Schrodinger and the Klein–Gordon equations in a general Riemannian space. It is shown that these symmetries are related with the homothetic and the conformal algebra of the metric of the space, respectively. We consider the kinematic metric defined by the classical Lagrangian and show how the Lie point symmetries of the Schrodinger equation and the Klein–Gordon equation are related with the Noether point symmetries of this Lagrangian. The general results are applied to two practical problems: (a) The classification of all two- and three-dimensional potentials in a Euclidean space for which the Schrodinger equation and the Klein–Gordon equation admit Lie point symmetries; and (b) The application of Lie point symmetries of the Klein–Gordon equation in the exterior Schwarzschild spacetime and the determination of the metric by means of conformally related Lagrangians.
50 citations
TL;DR: In this paper, the authors consider a two-component universe filled with barotropic fluid and van der Waals gas, and they study two different models of the two components of total fluid, assuming the linear equation of state (EoS) as the first component.
Abstract: In this paper, we consider Universe filled with two-component fluid. We study two different models. In the first model we assume barotropic fluid with the linear equation of state (EoS) as the first component of total fluid. In the second model we assume van der Waals gas as the first component of total fluid. In both models, the second component assumed generalized ghost dark energy. We consider also interaction between components and discuss, numerically, cosmological quantities for two different parametrizations of EoS which varies with time. We consider this as a toy model of our Universe. We fix parameters of the model by using generalized second law of thermodynamics. Comparing our results with some observational data suggests interacting barotropic fluid with EoS parameter and generalized ghost dark energy as an appropriate model to describe our Universe.
47 citations
TL;DR: Weyl compatibility as discussed by the authors is an algebraic property for symmetric tensors and vectors that is strictly related to Riemann compatibility, which generalizes the Codazzi condition while preserving much of its geometric implications.
Abstract: We introduce the new algebraic property of Weyl compatibility for symmetric tensors and vectors. It is strictly related to Riemann compatibility, which generalizes the Codazzi condition while preserving much of its geometric implications. In particular, it is shown that the existence of a Weyl compatible vector implies that the Weyl tensor is algebraically special, and it is a necessary and sufficient condition for the magnetic part to vanish. Some theorems (Derdzinski and Shen [11], Hall [15]) are extended to the broader hypothesis of Weyl or Riemann compatibility. Weyl compatibility includes conditions that were investigated in the literature of general relativity (as in McIntosh et al. [16, 17]). A simple example of Weyl compatible tensor is the Ricci tensor of an hypersurface in a manifold with constant curvature.
47 citations
TL;DR: In this article, an invariant and canonical contraction between covariant indices was introduced for singular semi-Riemannian manifolds, which is applicable even for degenerate metrics.
Abstract: On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices. In this paper, we develop the geometry of singular semi-Riemannian manifolds. First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth. We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate.
45 citations
TL;DR: In this article, the main identities for the linear Finsler connection are presented in the general case, and then specialized to some notable cases like Berwald's, Cartan's or Chern-Rund's.
Abstract: We give an introduction to (pseudo-)Finsler geometry and its connections. For most results we provide short and self-contained proofs. Our study of the Berwald nonlinear connection is framed into the theory of connections over general fibered spaces pioneered by Mangiarotti, Modugno and other scholars. The main identities for the linear Finsler connection are presented in the general case, and then specialized to some notable cases like Berwald's, Cartan's or Chern–Rund's. In this way it becomes easy to compare them and see the advantages of one connection over the other. Since we introduce two soldering forms we are able to characterize the notable Finsler connections in terms of their torsion properties. As an application, the curvature symmetries implied by the compatibility with a metric suggest that in Finslerian generalizations of general relativity the mean Cartan torsion vanishes. This observation allows us to obtain dynamical equations which imply a satisfactory conservation law. The work ends with a discussion of yet another Finsler connection which has some advantages over Cartan's and Chern–Rund's.
TL;DR: In this article, the authors analyzed the Hawking radiation (HR) of a non-asymptotically flat (NAF) dyonic black hole (dBH) in four-dimensional (4D) EMD gravity by using one of the semiclassical approaches which is the so-called Hamilton-Jacobi (HJ) method.
Abstract: In this paper, we analyze the Hawking radiation (HR) of a non-asymptotically flat (NAF) dyonic black hole (dBH) in four-dimensional (4D) Einstein–Maxwell–Dilaton (EMD) gravity by using one of the semiclassical approaches which is the so-called Hamilton–Jacobi (HJ) method. We particularly motivate on the isotropic coordinate system (ICS) of the dBH in order to highlight the ambiguity to be appeared in the derivation of the Hawking temperature (TH) via the HJ method. Besides, it will be shown that the ICS allows us to write the metric of the dBH in form of the Fermat metric, which renders possible of identification of the refractive index (n) of the dBH. It is unraveled that the value of n and therefore the gravitational lensing effect is decisive on the tunneling rate of the HR. We also uncloak how one can resolve the discrepancy about the TH of the dBH in spite of that lensing effect.
TL;DR: In this paper, the notion of recurrent conformal 2-forms on a pseudo-Riemannian manifold of arbitrary signature was introduced, and it was shown that the Ricci tensor is Riemann compatible or equivalently, Weyl compatible.
Abstract: In this paper, we introduce the notion of recurrent conformal 2-forms on a pseudo-Riemannian manifold of arbitrary signature. Some theorems already proved for the same differential structure on a Riemannian manifold are proven to hold in this more general contest. Moreover other interesting results are pointed out; it is proven that if the associated covector is closed, then the Ricci tensor is Riemann compatible or equivalently, Weyl compatible: these notions were recently introduced and investigated by one of the present authors. Further some new results about the vanishing of some Weyl scalars on a pseudo-Riemannian manifold are given: it turns out that they are consequence of the generalized Derdzinski–Shen theorem. Topological properties involving the vanishing of Pontryagin forms and recurrent conformal 2-forms are then stated. Finally, we study the properties of recurrent conformal 2-forms on Lorentzian manifolds (space-times). Previous theorems stated on a pseudo-Riemannian manifold of arbitrary signature are then interpreted in the light of the classification of space-times in four or in higher dimensions.
TL;DR: In this article, the authors constructed a group field cosmology with third quantized gauge symmetry, and argued that the process that changes the topology of spacetime is unitarity process.
Abstract: In this paper we will analyze the black hole information paradox in group field cosmology. We will first construct a group field cosmology with third quantized gauge symmetry. Then we will argue that in this group field cosmology the process that changes the topology of spacetime is unitarity process. Thus, the information paradox from this perspective appears only because we are using a second quantized formalism to explain a third quantized process. A similar paradox would also occur if we analyze a second quantized process in first quantized formalism. Hence, we will demonstrate that in reality there is no information paradox but only a breakdown of the second quantized formalism.
TL;DR: In this paper, a modified gravity model, f(T, 𝒯) is able to reproduce different epochs of the cosmological history.
Abstract: Motivated by the newly proposal for gravity as the effect of the torsion scalar T and trace of the energy momentum tensor 𝒯, we investigate the cosmological reconstruction of different models of the Universe Our aim here is to show that how this modified gravity model, f(T, 𝒯) is able to reproduce different epochs of the cosmological history We explicitly show that f(T, 𝒯) can be reconstructed for ΛCDM as the most popular and consistent model Also we study the mathematical reconstruction of f(T, 𝒯) for a flat cosmological background filled by two fluids mixture Such model describes phantom–non-phantom era as well as the purely phantom cosmology We extend our investigation to more cosmological models like perfect fluid, Chaplygin gas and massless scalar field In each case we obtain some specific forms of f(T, 𝒯) These families of f(T, 𝒯) contain arbitrary function of torsion and trace of the energy momentum
TL;DR: In this article, the authors considered interacting closed string tachyon with generalized cosmic Chaplygin gas as a cosmological model of the universe and obtained cosmology parameters and discussed about fixed point for stability analysis.
Abstract: In this paper, we consider interacting closed string tachyon with generalized cosmic Chaplygin gas as a cosmological model of Universe. We obtained cosmological parameters and discuss about fixed point for stability analysis. We find appropriate conditions where the theory is stable.
TL;DR: The main aim of as mentioned in this paper is to investigate the geometric structures admitting by the Godel spacetime which produces a new class of semi-Riemannian manifolds and also consider some extension of Godel metric.
Abstract: The main aim of this paper is to investigate the geometric structures admitting by the Godel spacetime which produces a new class of semi-Riemannian manifolds. We also consider some extension of Godel metric.
TL;DR: In this paper, it was shown that although Palatini-theories are equivalent to Brans-Dicke theories, still the first pass the Mercury precession of perihelia test, while the second do not.
Abstract: We shall show that although Palatini -theories are equivalent to Brans–Dicke theories, still the first pass the Mercury precession of perihelia test, while the second do not. We argue that the two models are not physically equivalent due to different assumptions about free fall. We shall also go through perihelia test without fixing a conformal gauge (clocks or rulers) in order to highlight what can be measured in a conformal invariant way and what cannot. We shall argue that the conformal gauge is broken by choosing a definition of clock, rulers or, equivalently, of masses.
TL;DR: In this paper, geometric techniques are elaborated and applied for constructing generic off-diagonal exact solutions in f(R, T)-modified gravity for systems of gravitational-Yang-Mills-Higgs equations.
Abstract: We show that geometric techniques can be elaborated and applied for constructing generic off-diagonal exact solutions in f(R, T)-modified gravity for systems of gravitational-Yang–Mills–Higgs equations. The corresponding classes of metrics and generalized connections are determined by generating and integration functions which depend, in general, on all space and time coordinates and may possess, or not, Killing symmetries. For nonholonomic constraints resulting in Levi-Civita configurations, we can extract solutions of the Einstein–Yang–Mills–Higgs equations. We show that the constructions simplify substantially for metrics with at least one Killing vector. Some examples of exact solutions describing generic off-diagonal modifications to black hole/ellipsoid and solitonic configurations are provided and analyzed.
TL;DR: The well-formulation and the well-posedness of the Cauchy problem for hybrid metric-Palatini gravity are discussed in this paper, where it is shown that the initial value problem can always be well-formed and furthermore can be wellposed depending on the adopted matter sources.
Abstract: The well-formulation and the well-posedness of the Cauchy problem are discussed for hybrid metric-Palatini gravity, a recently proposed modified gravitational theory consisting of adding to the Einstein–Hilbert Lagrangian an f(R)-term constructed a la Palatini. The theory can be recast as a scalar-tensor one predicting the existence of a light long-range scalar field that evades the local Solar System tests and is able to modify galactic and cosmological dynamics, leading to the late-time cosmic acceleration. In this work, adopting generalized harmonic coordinates, we show that the initial value problem can always be well-formulated and, furthermore, can be well-posed depending on the adopted matter sources.
TL;DR: Using the hard Lefschetz theorem for Sasakian manifolds, the authors found two examples of compact K-contact nilmanifolds with no compatible SASakian metric in dimensions 5 and 7.
Abstract: Using the hard Lefschetz theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions 5 and 7, respectively.
TL;DR: In this article, a thermodynamic approach based on Onsager's classical theory of irreversible processes and Prigogine's non-unitary transformation theory is proposed to incorporate time irreversibility into quantum mechanics.
Abstract: It has been argued that gravity acts dissipatively on quantum-mechanical systems, inducing thermal fluctuations that become indistinguishable from quantum fluctuations. This has led some authors to demand that some form of time irreversibility be incorporated into the formalism of quantum mechanics. As a tool toward this goal, we propose a thermodynamical approach to quantum mechanics, based on Onsager's classical theory of irreversible processes and Prigogine's nonunitary transformation theory. An entropy operator replaces the Hamiltonian as the generator of evolution. The canonically conjugate variable corresponding to the entropy is a dimensionless evolution parameter. Contrary to the Hamiltonian, the entropy operator is not a conserved Noether charge. Our construction succeeds in implementing gravitationally-induced irreversibility in the quantum theory.
TL;DR: In this paper, it was shown that current M2-brane models fit this expectation: they can be reformulated as higher gauge theories on such categorised bundles, and they thus add to the still very sparse list of physically interesting higher-gauge theories.
Abstract: M2-branes couple to a 3-form potential, which suggests that their description involves a non-abelian 2-gerbe or, equivalently, a principal 3-bundle. We show that current M2-brane models fit this expectation: they can be reformulated as higher gauge theories on such categorified bundles. We thus add to the still very sparse list of physically interesting higher gauge theories.
TL;DR: In this paper, the authors study the scalar field cosmology (SFC) using the geometric language of the phase space and show that scaling solutions are represented by unstable separatrices of the saddle points.
Abstract: We study the Scalar Field Cosmology (SFC) using the geometric language of the phase space. We define and study an ensemble of dynamical systems as a Banach space with a Sobolev metric. The metric in the ensemble is used to measure a distance between different models. We point out the advantages of visualization of dynamics in the phase space. It is investigated the genericity of some class of models in the context of fine tuning of the form of the potential function in the ensemble of SFC. We also study the symmetries of dynamical systems of SFC by searching for their exact solutions. In this context, we stressed the importance of scaling solutions. It is demonstrated that scaling solutions in the phase space are represented by unstable separatrices of the saddle points. Only critical point itself located on two-dimensional stable submanifold can be identified as scaling solution. We have also found a class of potentials of the scalar fields forced by the symmetry of differential equation describing the evolution of the Universe. A class of potentials forced by scaling (homology) symmetries was given. We point out the role of the notion of a structural stability in the context of the problem of indetermination of the potential form of the SFC. We characterize also the class of potentials which reproduces the ΛCDM model, which is known to be structurally stable. We show that the structural stability issue can be effectively used is selection of the scalar field potential function. This enables us to characterize a structurally stable and therefore a generic class of SFC models. We have found a nonempty and dense subset of structurally stable models. We show that these models possess symmetry of homology.
TL;DR: In this article, an extension of General Relativity with an explicit non-minimal coupling between matter and curvature is examined, and the implications of the latter to various contexts, ranging from astrophysical matter distributions to a cosmological setting.
Abstract: We examine an extension of General Relativity with an explicit non-minimal coupling between matter and curvature. The purpose of this work is to present an overview of the implications of the latter to various contexts, ranging from astrophysical matter distributions to a cosmological setting. Various results are discussed, including the impact of this non-minimal coupling on the choice of Lagrangian density, on a mechanism to mimic galactic and cluster dark matter, on the possibility of accounting for the accelerated expansion of the Universe, energy density fluctuations and modifications to post-inflationary reheating. The equivalence between a model exhibiting a non-minimal coupling and multi-scalar-theories is also discussed.
TL;DR: In this paper, M-theory and D-brane quantum partition functions for microscopic black hole ensembles were studied in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes.
Abstract: We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lies in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi–Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology.
TL;DR: In this article, the authors studied spectral triples over noncommutative principal U(1)-bundles of arbitrary dimension and a compatibility condition between the connection and the Dirac operator on the total space and on the base space of the bundle.
Abstract: We study spectral triples over noncommutative principal U(1)-bundles of arbitrary dimension and a compatibility condition between the connection and the Dirac operator on the total space and on the base space of the bundle. Examples of low-dimensional noncommutative tori are analyzed in more detail and all connections found that are compatible with an admissible Dirac operator. Conversely, a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection is exhibited. These examples are extended to the theta-deformed principal U(1)-bundle .
TL;DR: In this article, it was shown that a W2-flat spacetime is conformally flat and hence it is of Petrov type O, and if the perfect fluid spacetime with vanishing W 2-curvature tensor obeys Einstein's field equation without cosmological constant, then the spacetime has vanishing acceleration vector and expansion scalar and the ideal fluid always behaves as a cosmologically constant.
Abstract: The object of this paper is to study spacetimes admitting W2-curvature tensor. At first we prove that a W2-flat spacetime is conformally flat and hence it is of Petrov type O. Next, we prove that if the perfect fluid spacetime with vanishing W2-curvature tensor obeys Einstein's field equation without cosmological constant, then the spacetime has vanishing acceleration vector and expansion scalar and the perfect fluid always behaves as a cosmological constant. It is also shown that in a perfect fluid spacetime of constant scalar curvature with divergence-free W2-curvature tensor, the energy-momentum tensor is of Codazzi type and the possible local cosmological structure of such a spacetime is of type I, D or O.
TL;DR: In this article, the authors put forward a systematic and unifying approach to construct gauge invariant composite fields out of connections, which relies on the existence in the theory of a group-valued field with a prescribed gauge transformation.
Abstract: In this paper, we put forward a systematic and unifying approach to construct gauge invariant composite fields out of connections. It relies on the existence in the theory of a group-valued field with a prescribed gauge transformation. As an illustration, we detail some examples. Two of them are based on known results: the first one provides a reinterpretation of the symmetry breaking mechanism of the electroweak part of the Standard Model of particle physics; the second one is an application to Einstein's theory of gravity described as a gauge theory in terms of Cartan connections. The last example depicts a new situation: starting with a gauge field theory on Atiyah Lie algebroids, the gauge invariant composite fields describe massive vector fields. Some mathematical and physical discussions illustrate and highlight the relevance and the generality of this approach.
TL;DR: In this article, the authors review the theory of characteristics and bicharacteristics of PDEs, with a special emphasis on intrinsic aspects, i.e., those aspects which are invariant under general changes of coordinates.
Abstract: Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic Cauchy data. Characteristics of PDEs play an important role both in Mathematics and in Physics. I will review the theory of characteristics and bicharacteristics of PDEs, with a special emphasis on intrinsic aspects, i.e. those aspects which are invariant under general changes of coordinates. After a basically analytic introduction, I will pass to a modern, geometric point of view, presenting characteristics within the jet space approach to PDEs. In particular, I will discuss the relationship between characteristics and singularities of solutions and observe that: "wave-fronts are characteristic surfaces and propagate along bicharacteristics". This remark may be understood as a mathematical formulation of the wave/particle duality in optics and/or quantum mechanics. The content of the paper reflects the three-hour mini-course that I gave at the XXII International Fall Workshop on Geometry and Physics, September 2–5, 2013, Evora, Portugal.