Geometry of superconformal manifolds

The main facts about complex curves are generalized to superconformal manifolds. The results thus obtained are relevant to the fermion string theory and, in particular, they are useful for computation of determinants of super laplacians which enter the string partition function.To the memory of our friend and colleague Vadim Knizhnik     

Magnetic backgrounds from generalised complex manifolds

The magnetic backgrounds that physically give rise to spacetime noncommutativity are generally treated using noncommutative geometry. In this paper we prove that also the theory of generalised complex manifolds contains the necessary elements to generate B-fields geometrically. As an example, the Poisson brackets of the Landau model (electric charges on a plane subject to an external, particularly applied magnetic field) are rederived using the techniques of generalised complex manifolds.     

Geometry of paraquaternionic Kähler manifolds with torsion

We study the geometry of PQKT connections. We find conditions for the existence of a PQKT connection and prove that if it exists it is unique. We show that PQKT geometry persists in a conformal class of metrics.     

Geometry of paraquaternionic Kähler manifolds with torsion

We study the geometry of PQKT connections. We find conditions for the existence of a PQKT connection and prove that if it exists it is unique. We show that PQKT geometry persists in a conformal class of metrics.     

Supersymmetric backgrounds from generalized Calabi–Yau manifolds

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 Kähler form e^iJ 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: e^iJ is closed under the action of the twisted exterior derivative in IIA theory, and similarly Ω is closed in IIB. This means that supersymmetric SU(3)-structure manifolds are always complex in IIB while they are twisted symplectic in IIA. Modulo a different action of the B-field, these are all generalized Calabi–Yau manifolds, as defined by Hitchin. To cite this article: M. Graña et al., C. R. Physique 5 (2004).

Résumé

On montre que les transformations de supersymétrie pour les théories des cordes de type II peuvent être traduites dans des équations différentielles pour une paire de spineurs purs, l'exponentiel de la forme de Kähler e^iJ et la forme holomorphe Ω. Ces équations sont symétriques sous l'échange des deux spineurs purs et des formes de RR de rang pair ou impair. Cette propriété est la symétrie miroir pour les variétés avec torsion. On voit aussi que les fluxes de RR entrent seulement dans une des deux équations : e^iJ est fermé sous l'action de la dérivée extérieure « twisted » dans la corde de type IIA, et de la même manière Ω est fermé en type IIB. Cela implique que les variétés supersymétriques de structure SU(3) sont toujours complexes en type IIB ou bien symplectiques « twisted » en IIA. Ces variétés sont donc des variétés des Calabi–Yau généralisées selon la définition de Hitchin, mais avec une action du champ B différente. Pour citer cet article : M. Graña et al., C. R. Physique 5 (2004).     

Solution of the Dirac equation in plane wave metric backgrounds

We present a new solution of the Dirac equation in the background of a plane wave metric. We examine the relation between sections of the exterior and Clifford bundles of a (pseudo-)Riemannian manifold. A spinor calculus is established and used to investigate a new solution of the Dirac equation lying in a minimal left ideal characterized by a certain idempotent projector.     

Laplacian eigenvalue functionals and metric deformations on compact manifolds

In this paper, we investigate critical points of the eigenvalues of the Laplace operator considered as functionals on the space of Riemannian metrics or a conformal class of metrics on a compact manifold. We introduce a natural notion of the critical metric of such a functional and obtain necessary and sufficient conditions for a metric to be critical. We derive specific consequences concerning possible locally maximizing metrics. We also characterize critical metrics of the ratio of two consecutive eigenvalues.     

Geometry of inertial manifolds probed via a Lyapunov projection method

A method for determining the dimension and state space geometry of inertial manifolds of dissipative extended dynamical systems is presented. It works by projecting vector differences between reference states and recurrent states onto local linear subspaces spanned by the Lyapunov vectors. A sharp characteristic transition of the projection error occurs as soon as the number of basis vectors is increased beyond the inertial manifold dimension. Since the method can be applied using standard orthogonal Lyapunov vectors, it provides a possible way to also determine experimentally inertial manifolds and their geometric characteristics.     

Hyper-parahermitian manifolds with torsion

Necessary and sufficient conditions for the existence of a hyper-parahermitian connection with totally skew-symmetric torsion (HPKT-structure) are presented. It is shown that any HPKT-structure is locally generated by a real (potential) function. An invariant first order differential operator is defined on any almost hyper-paracomplex manifold showing that it is two-step nilpotent exactly when the almost hyper-paracomplex structure is integrable. A local HPKT-potential is expressed in terms of this operator. Examples of (locally) invariant HPKT-structures with closed as well as non-closed torsion 3-form on a class of (locally) homogeneous hyper-paracomplex manifolds (some of them compact) are constructed.     

Microgeon with a Kerr metric

The structure of electromagnetic and gravitational fields near the singular ring of the Kerr metric is analyzed. An approximate joint solution of the system of Maxwell-Einstein equations is given for a metric with the parameters of an electron; this solution describes a microgeon, a model of an electron. A new interpretation of the two-sheet nature of the Kerr metric is given.     

Symplectic manifolds with Lagrangian fibration

A structure theorem is presented for certain kinds of symplectic manifold with a Lagrangian fibration. As a corollary, the class of cotangent bundles is characterized up to the appropriat equivalence, as the type of symplectic manifold considered in the theorem for which in addition, a certain cohomology class vanishes. These results and techniques are then applied to two situations in classical mechanics where symplectic manifolds foliated by Lagrangian submanifolds arise, namely, the Legendre transformation and Hamilton-Jacobi theory.     

Reduction of symplectic manifolds with symmetry

We give a unified framework for the construction of symplectic manifolds from systems with symmetries. Several physical and mathematical examples are given; for instance, we obtain Kostant’s result on the symplectic structure of the orbits under the coadjoint representation of a Lie group. The framework also allows us to give a simple derivation of Smale's criterion for relative equilibria. We apply our scheme to various systems, including rotationally invariant systems, the rigid body, fluid flow, and general relativity.     

Exceptional Calabi–Yau spaces: the geometry of backgrounds with flux

In this paper we define the analogue of Calabi–Yau geometry for generic , flux backgrounds in type II supergravity and M‐theory. We show that solutions of the Killing spinor equations are in one‐to‐one correspondence with integrable, globally defined structures in generalised geometry. Such “exceptional Calabi–Yau” geometries are determined by two generalised objects that parametrise hyper‐ and vector‐multiplet degrees of freedom and generalise conventional complex, symplectic and hyper‐Kähler geometries. The integrability conditions for both hyper‐ and vector‐multiplet structures are given by the vanishing of moment maps for the “generalised diffeomorphism group” of diffeomorphisms combined with gauge transformations. We give a number of explicit examples and discuss the structure of the moduli spaces of solutions. We then extend our construction to and flux backgrounds preserving eight supercharges, where similar structures appear, and finally discuss the analogous structures in generalised geometry.