On homogeneous CR manifolds and their CR algebras

In this paper we show some results on homogeneous CR manifolds, proved by introducing their associated CR algebras. In particular, we give different notions of nondegeneracy (generalizing the usual notion for the Levi form) which correspond to geometrical properties for the corresponding manifolds. We also give distinguished equivariant CR fibrations for homogeneous CR manifolds. In the second part of the paper we apply these results to minimal orbits for the action of a real form of a semisimple Lie group \^G on a flag manifold \^G/Q.Comment: 14 pages. AMS-LaTeX v2: minor revisio

Holomorphic Extension from Weakly Pseudoconcave CR Manifolds

Let M be a smooth locally embeddable CR manifold, having some CR dimension m and some CR codimension d. We find an improved local geometric condition on M which guarantees, at a point p on M, that germs of CR distributions are smooth functions, and have extensions to germs of holomorphic functions on a full ambient neighborhood of p. Our condition is a form of weak pseudoconcavity, closely related to essential pseudoconcavity as introduced in [HN1]. Applications are made to CR meromorphic functions and mappings. Explicit examples are given which satisfy our new condition,but which are not pseudoconcave in the strong sense. These results demonstrate that for codimension d > 1, there are additional phenomena which are invisible when d = 1

Non-factorizable long distance contributions in color suppressed decays of B mesons

$\bar B \to D\pi$, $D^*\pi$, $J/\psi\bar K$ and $J/\psi\pi$ decays are studied. Their amplitude is given by a sum of factorized and non-factorizable ones. The latter which is estimated by using a hard pion approximation is rather small in color favored $\bar B \to D\pi$ and $D^*\pi$ decays but still can efficiently interfere with the main amplitude given by the factorization. In the color suppressed $\bar B \to J/\psi\bar K$ and $J/\psi\pi$ decays, the non-factorizable contribution is very important. The sum of the factorized and non-factorizable amplitudes can reproduce well the existing experimental data on the branching ratios for the color favored $\bar B \to D\pi$ and $D^*\pi$ and the color suppressed $\bar B \to J/\psi \bar K$ and $J/\psi\pi$ decays by taking reasonable values of unknown parameters involved.Comment: 19 pages, Revte

Spencer Cohomology and 11-Dimensional Supergravity

We recover the classification of the maximally supersymmetric bosonic backgrounds of eleven-dimensional supergravity by Lie algebraic means. We classify all filtered deformations of the $\mathbb Z$-graded subalgebras $\mathfrak{h}=\mathfrak{h}_{-2}\oplus\mathfrak{h}_{-1}\oplus\mathfrak{h}_{0}$ of the Poincar\'e superalgebra $\mathfrak{g}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}=V\oplus S\oplus \mathfrak{so}(V)$ which differ only in zero degree, that is $\mathfrak{h}_0\subset\mathfrak{g}_0$ and $\mathfrak{h}_j=\mathfrak{g}_j$ for $j<0$. Aside from the Poincar\'e superalgebra itself and its $\mathbb Z$-graded subalgebras, there are only three other Lie superalgebras, which are the symmetry superalgebras of the non-flat maximally supersymmetric backgrounds. In passing we identify the gravitino variation with (a component of) a Spencer cocycle.Comment: 32 pages (v2: minor comments, reference added, final version to be published in CMP

Supersymmetric Yang-Mills theory on conformal supergravity backgrounds in ten dimensions

We consider bosonic supersymmetric backgrounds of ten-dimensional conformal supergravity. Up to local conformal isometry, we classify the maximally supersymmetric backgrounds, determine their conformal symmetry superalgebras and show how they arise as near-horizon geometries of certain half-BPS backgrounds or as a plane-wave limit thereof. We then show how to define Yang-Mills theory with rigid supersymmetry on any supersymmetric conformal supergravity background and, in particular, on the maximally supersymmetric backgrounds. We conclude by commenting on a striking resemblance between the supersymmetric backgrounds of ten-dimensional conformal supergravity and those of eleven-dimensional Poincar\'e supergravity.Comment: 30 page

A Characterization of CR Quadrics with a Symmetry Property

We study CR quadrics satisfying a symmetry property ( ˜ S) which is slightly weaker than the symmetry property (S), recently introduced by W. Kaup, which requires the existence of an automorphism reversing the gradation of the Lie algebra of infinitesimal automorphisms of the quadric. We characterize quadrics satisfying the ( ˜ S) property in terms of their Levi–Tanaka algebras. In many cases the ( ˜ S) property implies the (S) property; this holds in particular for compact quadrics. We also give a new example of a quadric such that the dimension of the algebra of positive-degree infinitesimal automorphisms is larger than the dimension of the quadric

The CR structure of minimal orbits in complex flag manifolds

Let bG be a complex semisimple Lie group, Q a parabolic subgroup and G a real form of bG. The flag manifold bG/Q decomposes into finitely many G-orbits; among them there is exactly one orbit of minimal dimension, which is compact. We study these minimal orbits from the point of view of CR geometry. In particular we characterize those minimal orbits that are of finite type and satisfy various nondegeneracy conditions, compute their fundamental group and describe the space of their global CR functions. Our main tool are parabolic CR algebras, which give an infinitesimal description of the CR structure of minimal orbits

On homogeneous and symmetric CR manifolds

In this paper, some questions about CR homogeneous structures are studied. In particular, conditions for which an abstract CR structure (related to a Lie algebra) can be realized as a true homogeneous CR structure are given. A main tool is the Levi-Malʹtsev and the Jordan-Chevalley fibrations on homogeneous spaces. The authors study these fibrations for a CR homogeneous space. Examples and counterexamples are produced in this situation. The authors study in more detail the CR-homogeneous structures for two specific cases. 1. The case of a semi-simple Lie group of odd dimension. In the compact case, a classification in algebraic terms was given by J.-Y. Charbonnel and H. Ounaïes-Khalgui [J. Lie Theory 14 (2004), no. 1, 165--198; MR2040175 (2005b:22012)]. 2. The so-called CR-symmetric spaces introduced by W. Kaup and D. Zaitsev [Adv. Math. 149 (2000), no. 2, 145--181; MR1742704 (2000m:32044)] and which generalize to the CR situation the classical definition of Élie Cartan. At the end of the paper, the authors classify such structures for complete flag manifolds. This is done first for the classical simple groups and then for the exceptional groups. There are also results about the real algebraic case, in particular about the embedding problem in a complex space. The paper uses previous results given by Medori and Nacinovich

REDUCTIVE COMPACT HOMOGENEOUS CR MANIFOLDS

We define and investigate a class of compact homogeneous CR manifolds, that we call -reductive. They are orbits of minimal dimension of a compact Lie group K (0) in algebraic affine homogeneous spaces of its complexification K. For these manifolds we obtain canonical equivariant fibrations onto complex flag manifolds, generalizing the Hopf fibration . These fibrations are not, in general, CR submersions, but satisfy the weaker condition of being CR-deployments; to obtain CR submersions we need to strengthen their CR structure by lifting the complex stucture of the base

Orbits of real forms in complex flag manifolds

The authors study the structure and the CR geometry of the orbits $M$ of a real form $G_0$ of a complex semisimple Lie group $G$ in a complex flag manifold $X = G/Q$. It is shown that any such orbit $M$ has a tower of fibrations over a canonically associated real flag manifold $M_e$ with fibers that are products of Euclidean complex spaces and open orbits in complex flag manifolds. This result is used to investigate some topological properties of $M$. For example, it is proved that the fundamental group $\pi_1(M)$ depends only on $M_e$ and on the conjugacy class of the maximally noncompact Cartan subgroups of the isotropy of the action of $G_0$ on $M$. In particular, the fundamental group of a closed orbit $M$ is isomorphic to that of $M_e$. Many other deep results about properties of the CR structure of the orbits and its invariants and about $G_0$-equivarant maps between orbits are obtained