Ricci flow,Courant algebroids,and renormalization of Poisson–Lie T-duality

We use a generalized Ricci tensor, defined for generalized metrics in Courant algebroids, to show that Poisson–Lie T-duality is compatible with the 1-loop renormalization group flow.     

Mirror symmetry and generalized complex manifolds: Part II. Integrability and the transform for torus bundles

In this paper we continue the development of a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. We discuss the integrability of the transform from Part I in terms of data on the base manifold. We work with semi-flat generalized complex structures on real n-torus bundles with section over an n-dimensional base and use the transform on vector bundles developed in Part I of this paper to discuss the bijective correspondence between semi-flat generalized complex structures on pairs of dual torus bundles. We give interpretations of these results in terms of relationships between the cohomology of torus bundles and their duals. We comment on the ways in which our results generalize some well established aspects of mirror symmetry. Along the way, we give methods of constructing generalized complex structures on the total spaces of the bundles we consider.     

Poisson Quasi-Nijenhuis Manifolds

We introduce the notion of Poisson quasi-Nijenhuis manifolds generalizing Poisson-Nijenhuis manifolds of Magri-Morosi. We also investigate the integration problem of Poisson quasi-Nijenhuis manifolds. In particular, we prove that, under some topological assumption, Poisson (quasi)-Nijenhuis manifolds are in one-one correspondence with symplectic (quasi)-Nijenhuis groupoids. As an application, we study generalized complex structures in terms of Poisson quasi-Nijenhuis manifolds. We prove that a generalized complex manifold corresponds to a special class of Poisson quasi-Nijenhuis structures. As a consequence, we show that a generalized complex structure integrates to a symplectic quasi-Nijenhuis groupoid, recovering a theorem of Crainic. Francqui fellow of the Belgian American Educational Foundation. Research supported by NSF grant DMS03-06665 and NSA grant 03G-142.     

Supersymmetric WZ-Poisson Sigma Model and Twisted Generalized Complex Geometry

It has been shown recently that extended supersymmetry in twisted first-order sigma models is related to twisted generalized complex geometry in the target. In the general case there are additional algebraic and differential conditions relating the twisted generalized complex structure and the geometrical data defining the model. We study in the Hamiltonian formalism the case of vanishing metric, which is the supersymmetric version of the WZ-Poisson sigma model. We prove that the compatibility conditions reduce to an algebraic equation, which represents a considerable simplification with respect to the general case. We also show that this algebraic condition has a very natural geometrical interpretation. In the derivation of these results the notion of contravariant connections on twisted Poisson manifolds turns out to be very useful.     

Generalized holomorphic structures

We define the notion of generalized holomorphic principal bundles and establish that their associated vector bundles of holomorphic representations are generalized holomorphic vector bundles defined by M. Gualtieri. Motivated by our definition, several examples of generalized holomorphic structures are constructed. A reduction theorem of generalized holomorphic structures is also included.     

Double field formulation of Yang-Mills theory

Based on our previous work on the differential geometry for the closed string double field theory, we construct a Yang-Mills action which is covariant under O(D,D) T-duality rotation and invariant under three-types of gauge transformations: non-Abelian Yang-Mills, diffeomorphism and one-form gauge symmetries. In double field formulation, in a manifestly covariant manner our action couples a single O(D,D) vector potential to the closed string double field theory. In terms of undoubled component fields, it couples a usual Yang-Mills gauge field to an additional one-form field and also to the closed string background fields which consist of a dilaton, graviton and a two-form gauge field. Our resulting action resembles a twisted Yang-Mills action.     

T-duality for string in Hořava–Lifshitz gravity

We continue our study of the Lorentz-breaking string theories. These theories are defined as string theory with modified Hamiltonian constraint which breaks the Lorentz symmetry of target space-time. We analyze the properties of this theory in the target space-time that possesses isometry along one direction. We also derive the T-duality rules for Lorentz-breaking string theories and show that they are the same as that of Buscher’s T-duality for the relativistic strings.     

Brane orbits

We complete the classification of half-supersymmetric branes in toroidally compactified IIA/IIB string theory in terms of representations of the T-duality group. As a by-product we derive a last wrapping rule for the space-filling branes. We find examples of T-duality representations of branes in lower dimensions, suggested by supergravity, of which none of the component branes follow from the reduction of any brane in ten-dimensional IIA/IIB string theory. We discuss the constraints on the charges of half-supersymmetric branes, determining the corresponding T-duality and U-duality orbits.     

On holomorphic maps and Generalized Complex Geometry

We introduce a natural notion of holomorphic map between generalized complex manifolds and we prove some related results on Dirac structures and generalized Kähler manifolds.     

S-matrix elements from T-duality

Recently it has been speculated that the S-matrix elements satisfy the Ward identity associated with the T-duality. This indicates that a group of S-matrix elements is invariant under the linear T-duality transformations on the external states. If one evaluates one component of such T-dual multiplet, then all other components may be found by the simple use of the linear T-duality. The assumption that fields must be independent of the Killing coordinate, however, may cause, in some cases, the T-dual multiplet not to be gauge invariant. In those cases, the S-matrix elements contain more than one T-dual multiplet which are intertwined by the gauge symmetry.     

Stringy differential geometry for double field theory, beyond Riemann

While the fundamental object in Riemannian geometry is a metric, closed string theories call for us to put a two-form gauge field and a scalar dilaton on an equal footing with the metric. Here we propose a novel differential geometry which treats the three objects in a unified manner, manifests not only diffeomorphism and one-form gauge symmetry but also O(D, D) T-duality, and enables us to rewrite the known low energy effective action of them as a single term. We comment that the notion of cosmological constant naturally changes.     

Contact manifolds and generalized complex structures

We give simple characterizations of contact 1-forms in terms of Dirac structures. We also relate normal almost contact structures to the theory of Dirac structures.     

On Nurowski’s conformal structure associated to a generic rank two distribution in dimension five

For a generic distribution of rank two on a manifold M$M$ of dimension five, we introduce the notion of a generalized contact form. To such a form we associate a generalized Reeb field and a partial connection. From these data, we explicitly construct a pseudo-Riemannian metric on M$M$ of split signature. We prove that a change of the generalized contact form only leads to a conformal rescaling of this metric, so the corresponding conformal class is intrinsic to the distribution.     

Entropy of black-branes system and T-duality

A general black-branes system under the T-duality transformation could produce a smeared system with different dimensional black branes. We first use some simple examples to see that both systems have the same entropy and then present a rigorous method to prove this general property. Using the property we could easily know the entropy of some complex black-brane systems.     

Symplectic twistor spaces

The paper describes the geometry of the bundle (M, ω) of the compatible complex structures of the tangent spaces of an (almost) symplectic manifold (M, ω), from the viewpoint of general twistor spaces [3], [9], [1]. It is shown that M has an either complex or almost Kaehler twistor space iff it has a flat symplectic connection. Applications of the twistor space to the study of the differential forms of M, and to the study of mappings : N → M, where N is a Kaehler manifold are indicated.     

Le théorème de réduction de Marsden-Weinstein en géométrie cosymplectique et de contact

We show that the classical Marsden-Weinstein Reduction theorem for Hamiltonian systems with symmetries is still true for contact manifolds and cosympletic manifolds (i.e. canonical manifolds in the sense of A. Lichnerowicz).

In fact, we precise the notion of transitive almost contact structure, which enables us to consider the cosymplectic geometry as a limit of the contact geometry when a certain parameter goes to zero. This point of view unifies both theories.

However, we have to give two distinct proofs for the contact Reduction theorem and the cosymplectic one.     

Partial and unsharp quantum logics

The total and the sharp character of orthodox quantum logic has been put in question in different contexts. This paper presents the basic ideas for a unified approach to partial and unsharp forms of quantum logic. We prove a completeness theorem for some partial logics based on orthoalgebras and orthomodular posets. We introduce the notion of unsharp orthoalgebra and of generalized MV algebra. The class of all effects of any Hilbert space gives rise to particular examples of these structures. Finally, we investigate the relationship between unsharp orthoalgebras, generalized MV algebras, and orthomodular lattices.     

Quantum Surfaces,Special Functions,and the Tunneling Effect

The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irreducible representations of associative algebras and the corresponding trace formulas over leaves with complex polarization are obtained. The noncommutative product on the leaves incorporates a closed 2-form and a measure which (in general) are different from the classical symplectic form and the Liouville measure. The quantum objects are related to some generalized special functions. The difference between classical and quantum geometrical structures could even occur to be exponentially small with respect to the deformation parameter. This is interpreted as a tunneling effect in the quantum geometry.