arXiv subjects

Liana David

Publications and source records attributed to Liana David.

At least 19 recordsLinked to original sources

arXiv2026

Integrability of generalized structures on odd exact Courant algebroids using generalized connections

Odd exact Courant algebroids constitute a simple class of transitive Courant algebroids. Their underlying vector bundle is of odd rank and differs from a generalized tangent bundle by the addition of a line bundle. In this article we study natural analogues of almost complex and almost pseudo-Hermitian structures on such Courant algebroids, which are called B_n-generalized almost complex/pseudo-Hermitian structures. The corresponding integrable structures are known as B_n-generalized complex structures and B_n-generalized pseudo-K\"{a}hler structures, respectively. We characterize the integrability of B_n-generalized almost complex/pseudo-Hermitian structures on odd exact Courant algebroids in terms of existence of adapted generalized connections. We describe the affine spaces of adapted generalized connections for such integrable generalized structures.

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arXiv2025

Components of generalised complex structures on transitive Courant algebroids

Generalised almost complex structures $\mathcal J$ on transitive Courant algebroids $E$ are studied in terms of their components with respect to a splitting $E\cong TM \oplus T^*M \oplus \mathcal G$, where $M$ denotes the base of $E$ and $\mathcal G$ its bundle of quadratic Lie algebras. Necessary and sufficient integrability equations for $\mathcal J$ are established in this formalism. As an application, it is shown that the integrability of $\mathcal J$ implies that one of the components defines a Poisson structure on $M$. Then the structure (normal form) of generalised complex structures for which the Poisson structure is non-degenerate is determined. It is shown that it is fully encoded in a pair $(\omega , \rho )$ consisting of a symplectic structure $\omega$ on $M$ and a representation $\rho : \pi_1(M) \to \mathrm{Aut}(\mathfrak g, \langle \cdot ,\cdot \rangle_{\mathfrak{g}}, J_{\mathfrak{g}})$ by automorphism of a quadratic Lie algebra $(\mathfrak g, \langle \cdot ,\cdot \rangle_{\mathfrak{g}})$ commuting with an integrable (in the sense of Lie algebras) skew-symmetric complex structure $J_{\mathfrak{g}}$. Examples of such representations and obstructions for the existence of non-degenerate generalised complex structures are discussed. Finally, a construction of generalised complex structures on transitive Courant algebroids over complex manifolds for which the Poisson structure degenerates along a complex analytic hypersurface is presented.

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arXiv2025

Fiber-wise linear F-manifolds, compatible flat connections and Euler fields

We define a fiber-wise linear (shortly, FWL) F-manifold as an F-manifold on the total space of a vector bundle \pi : E \rightarrow M for which the multiplication and unit field are FWL tensor fields. We develop a duality between FWL F-manifolds on E and the total space E^{*} of the dual vector bundle and we enrich it with FWL Euler fields and FWL compatible connections. We present examples in dimension two and three. We construct prolongations of F-manifolds and we prove that if the initial F-manifold admits a suitable Euler field or compatible flat connection then all prolongations inherit FWL Euler fields and FWL compatible flat connections.

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arXiv2025

Darboux theorem for generalized complex structures on transitive Courant algebroids

Under natural assumptions we find local normal forms for generalized complex structures on transitive Courant algebroids, which extend Gualtieri's Darboux theorem for generalized complex structures on manifolds. When the base of the Courant algebroid is a point, they reduce to Wang's description of invariant complex structures on compact semisimple Lie groups.

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arXiv2023

Classification of odd generalized Einstein metrics on 3-dimensional Lie groups

An odd generalized metric E_{-} on a Lie group G of dimension n is a left-invariant generalized metric on a Courant algebroid E_{H, F} of type B_n over G with left-invariant twisting forms H and F. Given an odd generalized metric E_{-} on G we determine the affine space of left invariant Levi-Civita generalized connections of E_ {-}. Given in addition a left-invariant divergence operator δwe show that there is a left-invariant Levi-Civita generalized connection of E_{-} with divergence δand we compute the corresponding Ricci tensor Ricci^δ of the pair (E_{-}, δ). The odd generalized metric E_{-} is called odd generalized Einstein with divergence δif Ricci^δ =0. We describe all odd generalized Einstein metrics of arbitrary left-invariant divergence on all 3-dimensional Lie groups.

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arXiv2022

B_n-generalized pseudo-Kahler structures

We define the notions of B_n-generalized pseudo-Hermitian and B_n-generalized pseudo-Kahler structures on an odd exact Courant algebroid E. When E is in the standard form (or of type B_n) we express these notions in terms of classical tensor fields on the base of E. This is analogous to the bi-Hermitian viewpoint on generalized Kahler structures on exact Courant algebroids. We describe left-invariant B_n-generalized pseudo-Kahler structures on Courant algebroids of type B_n over Lie groups of dimension two, three and four.

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arXiv2021

T-duality for transitive Courant algebroids

We develop a theory of T-duality for transitive Courant algebroids. We show that T-duality between transitive Courant algebroids E\rightarrow M and \tilde{E}\rightarrow \tilde{M} induces a map between the spaces of sections of the corresponding canonical weighted spinor bundles \mathbb{S}_{E} and \mathbb{S}_{\tilde{E}} intertwining the canonical Dirac generating operators. The map is shown to induce an isomorphism between the spaces of invariant spinors, compatible with an isomorphism between the spaces of invariant sections of the Courant algebroids. The notion of invariance is defined after lifting the vertical parallelisms of the underlying torus bundles M\rightarrow B and \tilde{M} \rightarrow B to the Courant algebroids and their spinor bundles. We prove a general existence result for T-duals under assumptions generalizing the cohomological integrality conditions for T-duality in the exact case. Specializing our construction, we find that the T-dual of an exact or a heterotic Courant algebroid is again exact or heterotic, respectively.

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arXiv2021

Generalized connections, spinors, and integrability of generalized structures on Courant algebroids

We present a characterization, in terms of torsion-free generalized connections, for the integrability of various generalized structures (generalized almost complex structures, generalized almost hypercomplex structures, generalized almost Hermitian structures and generalized almost hyper-Hermitian structures) defined on Courant algebroids. We develop a new, self-contained, approach for the theory of Dirac generating operators on regular Courant algebroids with scalar product of neutral signature. As an application we provide a criterion for the integrability of generalized almost Hermitian structures (G, \mathcal J) and generalized almost hyper-Hermitian structures (G, \mathcal J_{1}, \mathcal J_{2}, \mathcal J_{3}) defined on a regular Courant algebroid E with scalar product of neutral signature, in terms of canonically defined differential operators on spinor bundles associated to E_{\pm} (the subbundles of E determined by the generalized metric G).

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arXiv2020

(TE)-structures over the irreducible 2-dimensional globally nilpotent F-manifold germ

We find formal and holomorphic normal forms for a class of meromorphic connections (the so-called $(TE)$-structures) over the irreducible $2$-dimensional globally nilpotent $F$-manifold germ $\mathcal N_{2}$. We find normal forms for Euler fields on $\mathcal N_{2}$ and we characterize the Euler fields on $\mathcal N_{2}$ which are induced by a $(TE)$-structure.

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arXiv2019

Meromorphic connections over F-manifolds

This paper review one construction of Frobenius manifolds (and slightly weaker structures). It splits it into several steps and discusses the freedom and the constraints in these steps. The steps pass through holomorphic bundles with meromorphic connections. A conjecture on existence and uniqueness of certain such bundles, a proof of the conjecture in the 2-dimensional cases, and some other new results form a research part of this paper.

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arXiv2019

(T)-structures over 2-dimensional F-manifolds: formal classification

A $(TE)$-structure $\nabla$ over a complex manifold $M$ is a meromorphic connection defined on a holomorphic vector bundle over $\mathbb{C}\times M$, with poles of Poincaré rank one along $\{ 0 \} \times M.$ Under a mild additional condition (the so called unfolding condition), $\nabla$ induces a multiplication on $TM$ and a vector field on $M$ (the Euler field), which make $M$ into an $F$-manifold with Euler field. By taking the pull-backs of $\nabla$ under the inclusions $\{ z\} \times M \rightarrow \mathbb{C}\times M$ we obtain a family of flat connections on vector bundles over $M$, parameterized by $z\in \mathbb{C}^{*}$. The properties of such a family of connections give rise to the notion of $(T)$-structure. Therefore, any $(TE)$-structure underlies a $(T)$-structure but the converse is not true. The unfolding condition can be defined also for $(T)$-structures. A $(T)$-structure with the unfolding condition induces on its parameter space the structure of an $F$-manifold (without Euler field). After a brief review on the theory of $(T)$ and $(TE)$-structures, we determine normal forms for the equivalence classes, under formal isomorphisms, of $(T)$-structures which induce a given irreducible germ of $2$-dimensional $F$-manifolds.

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arXiv2017

Twist, elementary deformation, and KK correspondence in generalized complex geometry

We define the operations of conformal change and elementary deformation in the setting of generalized complex geometry. Then we apply Swann's twist construction to generalized (almost) complex and Hermitian structures obtained by these operations and establish necessary and sufficient conditions for the Courant integrability of the resulting twisted structures. In particular, we associate to any appropriate generalized Kahler manifold (M, G, \mathcal J ) with a Hamiltonian Killing vector field a new generalized Kahler manifold, depending on the choice of a pair of non-vanishing functions and compatible twist data. We study this construction when (M, G, \mathcal J) is (diagonal) toric, with emphasis on the four dimensional case. In particular, we apply it to deformations of the standard flat Kahler metric on C^{n}, the Fubini-Study Kahler metric on CP^{2} and the so called admissible Kahler metrics on Hirzebruch surfaces. As a further application, we recover the KK (Kahler-Kahler) correspondence, which is obtained by specializing to the case of an ordinary Kahler manifold.

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arXiv2016

Prolongation of Tanaka structures: an alternative approach

The classical theory of prolongation of G-structures was generalized by N. Tanaka to a wide class of geometric structures (Tanaka structures), which are defined on a non-holonomic distribution. Examples of Tanaka structures include subriemannian, subconformal, CR-structures, structures associated to second order differential equations and structures defined by gradings of Lie algebras (in the setting of parabolic geometry). Tanaka's prolongation procedure associates to a Tanaka structure of finite order a manifold with an absolute parallelism. It is a very fruitful method for the description of local invariants, investigation of the automorphism group and the equivalence problem. In this paper we develop an alternative constructive approach for Tanaka's prolongation procedure, based on the theory of quasi-gradations in filtered vector spaces, G-structures and their torsion functions.

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arXiv2016

Regular F-manifolds: initial conditions and Frobenius metrics

A regular F-manifold is an F-manifold (with Euler field) (M, \circ, e, E), such that the endomorphism {\mathcal U}(X) := E \circ X of TM is regular at any p\in M. We prove that the germ ((M,p), \circ, e, E) is uniquely determined (up to isomorphism) by the conjugacy class of {\mathcal U}_{p} : T_{p}M \rightarrow T_{p}M. We obtain that any regular F-manifold admits a preferred system of local coordinates and we find conditions, in these coordinates, for a metric to be Frobenius. We study the Lie algebra of infinitesimal symmetries of regular F-manifolds. We show that any regular F-manifold is locally isomorphic to the parameter space of a Malgrange universal connection. We prove an initial condition theorem for Frobenius metrics on regular F-manifolds.

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arXiv2016

Hermitian metrics on F-manifolds

An $F$-manifold is complex manifold with a multiplication on the holomorphic tangent bundle with a certain integrability condition. Important examples are Frobenius manifolds and especially base spaces of universal unfoldings of isolated hypersurface singularities. This paper reviews the construction of hermitian metrics on $F$-manifolds from $tt^*$ geometry. It clarifies the logic between several notions. It also introduces a new {\it canonical} hermitian metric. Near irreducible points it makes the manifold almost hyperbolic. This holds for the singularity case and will hopefully lead to applications there.

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arXiv2015

Tanaka structures (non holonomic G-structures) and Cartan connections

Let \gh = \gh_{-k}\oplus \cdots \oplus \gh_{l} (k >0, l \geq 0) be a finite dimensional real graded Lie algebra, with a Euclidian metric \langle \cdot , \cdot \rangle adapted to the gradation. The metric \langle\cdot , \cdot \rangle is called admissible if the codifferentials \partial^{*} : C^{k+1}(\gh_{-}, \gh ) \ra C^{k} (\gh_{-}, \gh) (k\geq 0) are Ad_{Q}-invariant (Lie(Q) = \gh_{0}\oplus \gh_{+}). We find necessary and sufficient conditions for a Euclidian metric, adapted to the gradation, to be admissible, and we develop a theory of normal Cartan connections, when these conditions are satisfied. We show how the treatment by A. Cap and J. Slovak (Parabolic Geometry I, Mathematical Surveys and Monographs, vol. 154, 2009), about normal Cartan connections of semisimple type, fits into our theory. We also consider in some detail the case when \gh = t^{*} (\gg ) is the cotangent Lie algebra of a non-positively graded Lie algebra \gg.

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arXiv2014

On cotangent manifolds, complex structures and generalized geometry

We develop various properties of symmetric generalized complex structures (in connection with their holomorphic space and B-field transformations), which are analogous to the well-known results of Gualtieri on skew-symmetric generalized complex structures. Given a symmetric or skew-symmetric generalized complex structure \mathcal J and a connection D on a manifold M, we construct an almost complex structure J^{\mathcal J,D} on the cotangent manifold T^{*}M and we study its integrability. For \mathcal J skew-symmetric, we relate the Courant integrability of \mathcal J with the integrability of J^{\mathcal J, D}. We consider in detail the case when M=G is a Lie group and \mathcal J , D are left-invariant. We also show that our approach generalizes various well-known results from special complex geometry.

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arXiv2013

tt*-Geometry on the big phase space

The big phase space, the geometric setting for the study of quantum cohomology with gravitational descendents, is a complex manifold and consists of an infinite number of copies of the small phase space. The aim of this paper is to define a Hermitian geometry on the big phase space. Using the approach of Dijkgraaf and Witten, we lift various geometric structures of the small phase space to the big phase space. The main results of our paper state that various notions from tt*-geometry are preserved under such liftings.

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