We consider Courant and Courant-Jacobi brackets on the stable tangent bundle $TM\times\mathds{R}^h$ of a differentiable manifold and corresponding Dirac, Dirac-Jacobi and generalized complex structures. We prove that Dirac and Dirac-Jacobi structures on $TM\times\mathds{R}^h$ can be prolonged to $TM\times\mathds{R}^k$, $k>h$, by means of commuting infinitesimal automorphisms. Some of the stable, generalized, complex structures are a natural generalization of the normal, almost contact structures; they are expressible by a system of tensors $(P,θ,F,Z_a,ξ^a)$ $(a=1,...,h)$, where $P$ is a bivector field, $θ$ is a 2-form, $F$ is a $(1,1)$-tensor field, $Z_a$ are vector fields and $ξ^a$ are 1-forms, which satisfy conditions that generalize the conditions satisfied by a normal, almost contact structure $(F,Z,ξ)$. We prove that such a generalized structure projects to a generalized, complex structure of a space of leaves and characterize the structure by means of the projected structure and of a normal bundle of the foliation. Like in the Boothby-Wang theorem about contact manifolds, principal torus bundles with a connection over a generalized, complex manifold provide examples of this kind of generalized, normal, almost contact structures.
We establish the conditions for the induced generalized metric F structure of an oriented hypersurface of a generalized Kähler manifold to be a generalized CRFK structure. Then, we discuss a notion of generalized almost contact structure on a manifold $M$ that is suggested by the induced structure of a hypersurface. Such a structure has an associated generalized almost complex structure on $M\times\mathds{R}$. If the latter is integrable, the former is normal and we give the corresponding characterization. If the structure on $M\times\mathds{R}$ is generalized Kähler, the structure on $M$ is s…
A generalized F-structure is a complex, isotropic subbundle $E$ of $T_cM\oplus T^*_cM$ ($T_cM=TM\otimes_{\mathds{R}}\mathds{C}$ and the metric is defined by pairing) such that $E\cap\bar E^{\perp}=0$. If $E$ is also closed by the Courant bracket, $E$ is a generalized CRF-structure. We show that a generalized F-structure is equivalent with a skew-symmetric endomorphism $Φ$ of $TM\oplus T^*M$ that satisfies the condition $Φ^3+Φ=0$ and we express the CRF-condition by means of the Courant-Nijenhuis torsion of $Φ$. The structures that we consider are generalizations of the F-structures defined by Y…
We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, have a symplectic structure. Particularly, if compatibility with a generalized Riemannian metric is required, the symplectic structure is of the Kaehler type.
We establish some fundamental relations between Dirac subbundles $L$ for the generalized Courant algebroid $(A\oplus A^{\ast}, ϕ+W)$ over a differentiable manifold $M$ and the associated Dirac subbubndles $\tilde{L}$ for the corresponding Courant algebroid $\tilde{A} \oplus \tilde{A}^{\ast}$ over $M\times \R$.
We first recall some basic definitions and facts about Jacobi manifolds, generalized Lie bialgebroids, generalized Courant algebroids and Dirac structures. We establish an one-one correspondence between reducible Dirac structures of the generalized Lie bialgebroid of a Jacobi manifold $(M,Λ,E)$ for which 1 is an admissible function and Jacobi quotient manifolds of $M$. We study Jacobi reductions from the point of view of Dirac structures theory and we present some examples and applications.
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.
We introduce linear Dirac and generalized complex structures on Cartan geometries and give criteria for Dirac subalgebras of $\frkg\ltimes\frkg^*$ representing Dirac structures on a Cartan geometry. We prove that there is a bijection between the linear generalized structures on a torsion free Cartan geometry and the equivariant generalized structures on its corresponding homogeneous space.
Under two boundary conditions, the generalized Atiyah-Patodi-Singer boundary condition and the modified generalized -Atiyah-Patodi-Singer boundary condition, we get the lower bounds for the eigenvalues of the fundamental Dirac operator on compact spin manifolds with nonempty boundary.
The regular reduction of a Dirac manifold acted upon freely and properly by a Lie group is generalized to a nonfree action. For this, several facts about $G$-invariant vector fields and one-forms are shown.
We partially describe equivariant Dirac and generalized complex structures on a homogeneous space $G/K$ by giving equivalent data involving only the Lie algebra. We consider real semisimple adjoint orbits in any semisimple Lie algebra over $\mathbb R$ and real nilpotent orbits in $sl_n (\mathbb R)$. We give a complete classification for Riemannian symmetric spaces and for a compact group modulo a closed, connected subgroup containing a Cartan subgroup.
We study the Dirac spectrum on compact Riemannian spin manifolds $M$ equipped with a metric connection $\nabla$ with skew torsion $T\inΛ^3 M$ in the situation where the tangent bundle splits under the holonomy of $\nabla$ and the torsion of $\nabla$ is of `split' type. We prove an optimal lower bound for the first eigenvalue of the Dirac operator with torsion that generalizes Friedrich's classical Riemannian estimate.
A representation of generalized Weierstrass formulae for an immersion of generic surfaces into a 4-dimensional complex space in terms of spinors treated as minimal left ideals of Clifford algebras is proposed. The relation between integrable deformations of surfaces via mVN-hierarchy and integrable deformations of spinor fields on the surface is also discussed.
We survey the correct definition of a generalized Dirac operator on a Space--Time and the classical result about propagation of singularities. This says that light travels along light--like geodesics. Finally we show this is also true for generalized Dirac operators.
Fundamental solutions of Dirac type operators are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of $\mathbb{R}^n$ by arithmetic subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented together with associated Eisenstein and Poincaré type series.
We find sufficient conditions for the absence of harmonic $L^2$ spinors on spin manifolds constructed as cone bundles over a compact Kähler base. These conditions are fulfilled for certain perturbations of the Euclidean metric, and also for the generalized Taub-NUT metrics of Iwai-Katayama, thus proving a conjecture of Vi\csinescu and the second author.
A classical result in differential geometry due to Lichnerowicz [8] is concerned with the decomposition of the square of Dirac operators defined by Clifford connections on a Clifford module ${\cal E}$\ over a Riemannian manifold $M$. Recently, this formula has been generalized to arbitrary Dirac operators [2]. In this paper we prove a supersymmetric version of the generalized Lichnerowicz formula, motivated by the fact that there is a one-to-one correspondence between Clifford superconnections and Dirac operators. We extend this result to obtain a simple formula for the supercurvature of a gen…
We consider Dirac-like operators with piecewise constant mass terms on spin manifolds, and we study the behaviour of their spectra when the mass parameters become large. In several asymptotic regimes, effective operators appear: the extrinsic Dirac operator and a generalized MIT Bag Dirac operator. This extends some results previously known for the Euclidean spaces to the case of general spin geometry.
Semi-Riemannian $\text{spin}^c$ manifolds carrying generalized Killing spinors and the classification of Riemannian $\text{spin}^c$ manifolds admitting a type I imaginary generalized Killing spinor
We consider a notion of conservation for the heat semigroup associated to a generalized Dirac Laplacian acting on sections of a vector bundle over a noncompact manifold with a (possibly noncompact) boundary under mixed boundary conditions. Assuming that the geometry of the underlying manifold is controlled in a suitable way and imposing uniform lower bounds on the zero order (Weitzenböck) piece of the Dirac Laplacian and on the endomorphism defining the mixed boundary condition we show that the corresponding conservation principle holds. A key ingredient in the proof is a domination property f…
This article is one of a series of papers. For this decade, the Dirac operator on a submanifold has been studied as a restriction of the Dirac operator in $n$-dimensional euclidean space $\EE^n$ to a surface or a space curve as physical models. These Dirac operators are identified with operators of the Frenet-Serret relation for a space curve case and of the generalized Weierstrass relation for a conformal surface case and completely represent the submanifolds. For example, the analytic index of Dirac operator of a space curve is identified with its writhing number. As another example, the ope…
We study the Dirac spectrum on compact Riemannian spin manifolds $M$ equipped with a metric connection $\nabla$ with skew torsion $T\inΛ^3M$ by means of twistor theory. An optimal lower bound for the first eigenvalue of the Dirac operator with torsion is found that generalizes Friedrich's classical Riemannian estimate. We also determine a novel twistor and Killing equation with torsion and use it to discuss the case in which the minimum is attained in the bound.
We study ten-dimensional supersymmetric vacua with NSNS non-geometric fluxes, in the framework of $β$-supergravity. We first provide expressions for the fermionic supersymmetry variations. Specifying a compactification ansatz to four dimensions, we deduce internal Killing spinor equations. These supersymmetry conditions are then reformulated in terms of pure spinors, similarly to standard supergravity vacua admitting an SU(3)xSU(3) structure in Generalized Complex Geometry. The standard d-H acting on the pure spinors is traded for a generalized Dirac operator D, depending here on the non-geome…
Using the submanifold quantum mechanical scheme, the restricted Dirac operator in a submanifold is defined. Then it is shown that the zero mode of the Dirac operator expresses the local properties of the submanifold, such as the Frenet-Serret and generalized Weierstrass relations. In other words this article gives a representation of a further generalized Weierstrass relations for a general k-spin manifolds immersed in n-dimensional euclidean space (0<k<n).
The main new result here is the cancellation of global anomalies in the Type I superstring, with and without D-branes. Our argument here depends on a precise interpretation of the 2-form abelian gauge field using KO-theory; then the anomaly cancellation follows from a geometric form of the full Atiyah-Singer index theorem for families of Dirac operators. This is a refined version of the Green-Schwarz mechanism. It seems that a geometric interpretation of this mechanism-the cancellation of local and global fermion anomalies against local and global anomalies in the electric coupling of an abeli…
This article investigates local properties of the further generalized Weierstrass relations for a spin manifold $S$ immersed in a higher dimensional spin manifold $M$ from viewpoint of study of submanifold quantum mechanics. We show that kernel of a certain Dirac operator defined over $S$, which we call submanifold Dirac operator, gives the data of the immersion. In the derivation, the simple Frobenius reciprocity of Clifford algebras $S$ and $M$ plays important roles.
In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that those objects can be naturally associated to systems with constraints -- we recall the mathematical construction in the context of so called implicit Lagrangian systems. We explain how they can be used to produce new numerical methods, that we call Dirac integrators. On a test example of a simple pendulum in a gravity field we compare the Dirac integrators wit…
We reformulate the compatibility condition between a generalized metric and a small (non-maximal rank) Dirac structure in an exact Courant algebroid found in the context of the gauging of strings and formulated by means of two connections in purely Dirac-geometric terms. The resulting notion, a transverse generalized metric, is also what is needed for the dynamics on the reduced phase space of a string theory.
