para-Hermitian Manifold Research Articles

Yano F structures and extended supersymmetry

It is shown how extended supersymmetry realised directly on the (2, 2) semichiral superfields of a symplectic sigma model gives rise to a geometry on the doubled tangent bundle consisting of two Yano F structures on an almost para-hermitian manifold. Closure of the algebra and invariance of the action is discussed in this framework and integrability of the F structures is defined and shown to hold. The reduction to the usual (1, 1) sigma model description and identification with the bi-quaternionic set of complex structures and their properties is elucidated. The F structure formulation should be applicable to many other models and will have an equivalent formulation in Generalised Geometry.

D-Branes in Para-Hermitian Geometries

We introduce T-duality invariant versions of D-branes in doubled geometry using a global covariant framework based on para-Hermitian geometry and metric algebroids. We define D-branes as conformal boundary conditions for the open string version of the Born sigma-model, where they are given by maximally isotropic vector bundles which do not generally admit the standard geometric picture in terms of submanifolds. When reduced to the conventional sigma-model description of a physical string background as the leaf space of a foliated para-Hermitian manifold, integrable branes yield D-branes as leaves of foliations which are interpreted as Dirac structures on the physical spacetime. We define a notion of generalised para-complex D-brane, which realises our D-branes as para-complex versions of topological A/B-branes. We illustrate how our formalism recovers standard D-branes in the explicit example of reductions from doubled nilmanifolds.

Some relations between almost paracontact metric manifolds and almost parahermitian manifolds

In this study, almost paracontact metric manifolds and almost para-Hermitian manifolds are considered. The relations between almost paracontact metric manifolds and almost para-Hermitian manifolds are investigated and certain results are acquired. Examples of almost para-Hermitian manifolds are presented using the determined relations.

Harmonicity of $(\varphi,\varphi^{\prime})$-holomorphic sections of a (semi-riemannian) almost paracontact fiber bundle of type II

In this paper (ϕ, ϕ0 )-holomorphic maps from an almost paraHermitian manifold to an almost paracontact metric manifold are studied and a criterion for the harmonicity of such (ϕ, ϕ0 )-holomorphic maps is obtained. Also (ϕ, ϕ0 )-holomorphic sections of (semi−Riemannian) almost paracontact fiber bundles of type II are studied and a criterion for the harmonicity of such (ϕ, ϕ0 )-holomorphic sections is obtained.

Harmonicity of $(\varphi,\varphi^{\prime})$-holomorphic sections of a (semi-riemannian) almost paracontact fiber bundle of type II

In this paper (ϕ, ϕ0 )-holomorphic maps from an almost paraHermitian manifold to an almost paracontact metric manifold are studied and a criterion for the harmonicity of such (ϕ, ϕ0 )-holomorphic maps is obtained. Also (ϕ, ϕ0 )-holomorphic sections of (semi−Riemannian) almost paracontact fiber bundles of type II are studied and a criterion for the harmonicity of such (ϕ, ϕ0 )-holomorphic sections is obtained.

Born sigma-models for para-Hermitian manifolds and generalized T-duality

We give a covariant realization of the doubled sigma-model formulation of duality-symmetric string theory within the general framework of para-Hermitian geometry. We define a notion of generalized metric on a para-Hermitian manifold and discuss its relation to Born geometry. We show that a Born geometry uniquely defines a worldsheet sigma-model with a para-Hermitian target space, and we describe its Lie algebroid gauging as a means of recovering the conventional sigma-model description of a physical string background as the leaf space of a foliated para-Hermitian manifold. Applying the Kotov–Strobl gauging leads to a generalized notion of T-duality when combined with transformations that act on Born geometries. We obtain a geometric interpretation of the self-duality constraint that halves the degrees of freedom in doubled sigma-models, and we give geometric characterizations of non-geometric string backgrounds in this setting. We illustrate our formalism with detailed worldsheet descriptions of closed string phase spaces, of doubled groups where our notion of generalized T-duality includes non-abelian T-duality, and of doubled nilmanifolds.

Global aspects of doubled geometry and pre-rackoid

The integration problem of a C-bracket and a Vaisman (metric, pre-DFT) algebroid that are geometric structures of double field theory (DFT) is analyzed. We introduce a notion of a pre-rackoid as a global group-like object for an infinitesimal algebroid structure. We propose several realizations of pre-rackoid structures: One realization is that elements of a pre-rackoid are defined by cotangent paths along doubled foliations in a para-Hermitian manifold. Another realization is proposed as a formal exponential map of the algebroid of DFT. We show that the pre-rackoid reduces to a rackoid that is the integration of the Courant algebroid when the strong constraint of DFT is imposed. Finally, for a physical application, we show an implementation of the (pre-)rackoid in a three-dimensional topological sigma model.

More on doubled aspects of algebroids in double field theory

We continue to study doubled aspects of algebroid structures equipped with the C-bracket in double field theory (DFT). We find that a family of algebroids, the Vaisman (metric or pre-DFT), the pre- and the ante-Courant algebroids are constructed by the analogue of the Drinfel’d double of Lie algebroid pairs. We examine geometric implementations of these algebroids in the para-Hermitian manifold, which is a realization of the doubled space–time in DFT. We show that the strong constraint in DFT is necessary to realize the doubled and non-trivial Poisson structures but can be relaxed for some algebroids. The doubled structures of twisted brackets and those associated with group manifolds are briefly discussed.

Born sigma model for branes in exceptional geometry

Abstract In double field theory, the physical space has been understood as a subspace of the doubled space. Recently, the doubled space has been defined as the para-Hermitian manifold and the physical space is realized as a leaf of a foliation of the doubled space. This construction naturally introduces the fundamental 2-form, which plays an important role in a reformulation of string theory known as the Born sigma model. In this paper, we present the Born sigma model for $p$-branes in M-theory and type IIB theory by extending the fundamental 2-form into $U$-duality-covariant $(p+1)$-forms.

Generalized Kinematics on ρ-Commutative Algebras

The geometric framework of Double Field Theory (DFT) can be constructed on a para-Hermitian manifold. The canonical, generalized-metric connections and the global expression of the corresponding covariant derivative, a generalization of the kinematical structure of DFT, generalized curvature, a corresponding generalized Lie derivative for the Leibniz algebroid on the tangent bundle are constructed. In the present paper, we construct the similar action of DFT and para-Hermitian manifolds on ρ-commutative algebras and extend the above notions on it.

Doubled aspects of Vaisman algebroid and gauge symmetry in double field theory

The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs the gauge symmetry algebra generated by the C-bracket in double field theory (DFT). We show that the Vaisman algebroid is obtained by an analog of the Drinfel’d double of Lie algebroids. Based on a geometric realization of doubled space-time as a para-Hermitian manifold, we examine exterior algebras and a para-Dolbeault cohomology on DFT and discuss the structure of the Drinfel’d double behind the DFT gauge symmetry. Similar to the Courant algebroid in the generalized geometry, Lagrangian sub-bundles (L,L̃) in a para-Hermitian manifold play Dirac-like structures in the Vaisman algebroid. We find that an algebraic origin of the strong constraint in DFT is traced back to the compatibility condition needed for (L,L̃) to be a Lie bialgebroid. The analysis provides a foundation toward the “coquecigrue problem” for the gauge symmetry in DFT.

Para-Hermitian Geometry and Doubled Aspects of Vaisman Algebroid

The geometry of double field theory (DFT) is described by a para-Hermitian manifold M. A tangent bundle of the para-Hermitian manifold T M is decomposed into two eigenbundles L and ∼L associated with the eigenvalues of the para-complex structure K. We define a Lie algebroid structure on the eigenbundles L, ∼L. The gauge symmetry algebra of DFT is governed by the C-bracket. The algebraic structure based on the C-bracket is not a Courant algebroid, but a metric algebroid proposed by Vaisman (the Vaisman algebroid). We show that the Vaisman algebroid in DFT is naturally defined on T M by an analogue of the Drinfel’d double of L, ∼L. We also find that an algebraic origin of the strong constraint is the condition for (L, ∼L) to become a Lie bialgebroid.

Bi-Slant Submanifolds of Para Hermitian Manifolds

In this paper, we introduce the notion of bi-slant submanifolds of a para Hermitian manifold. They naturally englobe CR, semi-slant, and hemi-slant submanifolds. We study their first properties and present a whole gallery of examples.

Algebroid structures on para-Hermitian manifolds

We present a global construction of a so-called D-bracket appearing in the physics literature of Double Field Theory (DFT) and show that if certain integrability criteria are satisfied, it can be seen as a sum of two Courant algebroid brackets. In particular, we show that the local picture of the extended space-time used in DFT fits naturally in the geometrical framework of para-Hermitian manifolds and that the data of an (almost) para-Hermitian manifold is sufficient to construct the D-bracket. Moreover, the twists of the bracket appearing in DFT can be interpreted in this framework geometrically as a consequence of certain deformations of the underlying para-Hermitian structure.

The Ricci tensor of almost parahermitian manifolds

We study the pseudoriemannian geometry of almost parahermitian manifolds, obtaining a formula for the Ricci tensor of the Levi-Civita connection. The formula uses the intrinsic torsion of an underlying SL(n,R)-structure; we express it in terms of exterior derivatives of some appropriately defined differential forms. As an application, we construct Einstein and Ricci-flat examples on Lie groups. We disprove the parak\"ahler version of the Goldberg conjecture, and obtain the first compact examples of a non-flat, Ricci-flat nearly parak\"ahler structure. We study the paracomplex analogue of the first Chern class in complex geometry, which obstructs the existence of Ricci-flat parak\"ahler metrics.

Generalised kinematics for double field theory

We formulate a kinematical extension of Double Field Theory on a 2d-dimensional para-Hermitian manifold left(mathcal{P},eta, omega right) where the O(d, d) metric η is supplemented by an almost symplectic two-form ω. Together η and ω define an almost bi-Lagrangian structure K which provides a splitting of the tangent bundle Tmathcal{P}=Loplus tilde{L} into two Lagrangian sub-spaces. In this paper a canonical connection and a corresponding generalised Lie derivative for the Leibniz algebroid on Tmathcal{P} are constructed. We find integrability conditions under which the symmetry algebra closes for general η and ω, even if they are not flat and constant. This formalism thus provides a generalisation of the kinematical structure of Double Field Theory. We also show that this formalism allows one to reconcile and unify Double Field Theory with Generalised Geometry which is thoroughly discussed.

Slant Submanifolds of Para-Hermitian Manifolds

In this paper, we introduce the notion of slant submanifolds of a para-Hermitian manifold. We study their first properties and present a whole gallery of examples.

ON THE BASE SPACE OF AN ALMOST PARACONTACT SUBMERSION

The purpose of this note is to describe the base space of an almost paracontact submersion. Here the base space is an almost para-Hermitian manifold. So, the paper intertwines paracontact and para-Hermitian structures via the theory of submersions.

Para-Quaternionic Structures on the 3-Jet Bundle

In this paper we construct an almost para-quaternionic structure on the 3-jet bundle of an almost parahermitian manifold and we study its integrability. We give a necessary and sufficient conditions that are provided for these structures to become para-hyper-Kähler and we prove that the 3-jet bundle can not be a para-quaternionic Kähler manifold.

Anti-invariant Riemannian Submersions: A Lie-theoretical Approach

We give a construction which is Lie theoretic of anti-invariant Riemannian submersions from almost Hermitian manifolds, from quaternion manifolds, from para-Hermitian manifolds, from para-quaternion manifolds, and from octonian manifolds. This yields many compact Einstein examples.