John Alexander Cruz Morales, Alexander Torres-Gomez
In this note we introduce the concept of F-algebroid, and give its elementary properties and some examples. We provide a description of the almost duality for Frobenius manifolds, introduced by Dubrovin, in terms of a composition of two anchor maps of a unique cotangent F-algebroid.
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This work continues the study of $F$--manifolds $(M,\circ)$, first defined by Hertling and Manin and investigated in [He]. The notion of a compatible flat structure $\nabla$ is introduced, and it is shown that many constructions known for Frobenius manifolds do not in fact require invariant metrics and can be developed for all such triples $(M,\circ ,\nabla).$ In particular, we extend and generalize recent Dubrovin's duality.
A vector field E on an F-manifold (M, o, e) is an eventual identity if it is invertible and the multiplication X*Y := X o Y o E^{-1} defines a new F-manifold structure on M. We give a characterization of such eventual identities, this being a problem raised by Manin. We develop a duality between F-manifolds with eventual identities and we show that is compatible with the local irreducible decomposition of F-manifolds and preserves the class of Riemannian F-manifolds. We find necessary and sufficient conditions on the eventual identity which insure that harmonic Higgs bundles and DChk-structure…
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.
Dubrovin duality for open Hurwitz flat F-manifolds
本文研究了开放Hurwitz流形的杜布罗金对偶，证明其可扩展为具有相容平坦结构的F流形，并通过实例展示了双平坦流形的构造与非齐次解的生成。
We prove a twisted Poincaré duality for (full) Hopf algebroids with bijective antipode. As an application, we recover the Hochschild twisted Poincaré duality of Van Den Bergh [VDB]. We also get a Poisson twisted Poincaré duality, which was already stated for oriented Poisson manifolds in [CLYZ].
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 …
In this paper, we extend the T-duality isomorphism by Gualtieri and Cavalcanti, from invariant exact Courant algebroids, to exotic exact Courant algebroids such that the momentum and winding numbers are exchanged, filling in a gap in the literature.
In this paper, we use reduction by extended actions to give a construction of transitive Courant algebroids from string classes. We prove that T-duality commutes with the reductions and thereby determine global conditions for the existence of T-duals in heterotic string theory. In particular we find that T-duality exchanges string structures and gives an isomorphism of transitive Courant algebroids. Consequently we derive the T-duality transformation for generalised metrics and show that the heterotic Einstein equations are preserved. The presence of string structures significantly extends the…
We introduce the notion of G-algebroid, generalising both Lie and Courant algebroids, as well as the algebroids used in $E_{n(n)}\times\mathbb{R}^+$ exceptional generalised geometry for $n\in\{3,\dots,6\}$. Focusing on the exceptional case, we prove a classification of "exact" algebroids and translate the related classification of Leibniz parallelisable spaces into a tractable algebraic problem. After discussing the general notion of Poisson-Lie duality, we show that the Poisson-Lie U-duality is compatible with the equations of motion of supergravity.
Supplementary comments about generalized Lie algebroids are presented and a new point of view over the construction of the Lie algebroid generalized tangent bundle of a (dual) vector bundle is introduced. Using the general theory of exterior differential calculus for generalized Lie algebroids, a covariant derivative for exterior forms of a (dual) vector bundle is introduced. Using this covariant derivative, the complete lift of an arbitrary section of a (dual) vector bundle is discovered. A theory of Legendre type and Legendre duality between vertical and complete lifts is presented. Finally,…
We introduce conformal Courant algebroids, a mild generalization of Courant algebroids in which only a conformal structure rather than a bilinear form is assumed. We introduce exact conformal Courant algebroids and show they are classified by pairs $(L,H)$ with $L$ a flat line bundle and $H \in H^3(M,L)$ a degree 3 class with coefficients in $L$. As a special case gerbes for the crossed module $({\rm U}(1) \to \mathbb{Z}_2)$ can be used to twist $TM \oplus T^*M$ into a conformal Courant algebroid. In the exact case there is a twisted cohomology which is 4-periodic if $L^2 = 1$. The structure o…
We reexamine the notions of generalized Ricci tensor and scalar curvature on a general Courant algebroid, reformulate them using objects natural w.r.t. pull-backs and reductions, and obtain them from the variation of a natural action functional. This allows us to prove, in a very general setup, the compatibility of the Poisson-Lie T-duality with the renormalization group flow and with string background equations. We thus extend the known results to a much wider class of dualities, including the cases with gauging (so called dressing cosets, or equivariant Poisson-Lie T-duality). As an illustra…
It is known that the topological T-duality exchanges $H$ and $F$-fluxes. In this paper, we reformulate the topological T-duality as an exchange of two Lie algebroids in the generalized tangent bundle. Then, we apply the same formulation to the Poisson-generalized geometry, which is introduced in arXiv:1408.2649 to define $R$-fluxes as field strength associated with $β$-transformations. We propose a definition of $Q$-flux associated with $β$-diffeomorphisms, and show that the topological T-duality exchanges $R$ and $Q$-fluxes.
In this paper, we introduce the notion of modular class of a Lie algebroid $A$ equipped with a Nambu structure satisfying some suitable hypothesis. We also introduce cohomology and homology theories for such Lie algebroids and prove that these theories are connected by a duality isomorphism when the modular class is null.
T-Dualities and Courant Algebroid Relations
本文基于库朗代数关系提出一种新的T-对偶性方法，包含通常T-对偶性及其推广，引入广义等距弱化概念构造T-对偶背景，证明存在唯一性并重现标准关系，还应用于几乎仿厄米特流形的广义T-对偶性变换。
T-duality for transgressive fibrations
本研究将拓扑T-对偶从定向球面丛推广到更广义的transgressive纤维化，证明其对偶具有同构扭曲上同调与Clifford–Courant代数胚，拓展了非阿贝尔T-对偶的研究范畴。
In some previous papers, a Legendre duality between Lagrangian and Hamiltonian Mechanics has been developed. The (ρ,η)-tangent application of the Legendre bundle morphism associated to a Lagrangian L or Hamiltonian H is presented. Using that, a Legendre description of Lagrangian Mechanics and Hamiltonian Mechanics is developed. Duality between Lie algebroids structure, adapted (ρ,η)-basis, distinguished linear (ρ,η)-connections and mechanical (ρ,η)-systems is the scope of this paper. In the particular case of Lie algebroids, new results are presented. In the particular case of the usual Lie al…
We give a systematic derivation of the local expressions of the NS H-flux, geometric F- as well as non-geometric Q- and R-fluxes in terms of bivector beta- and two-form B-potentials including vielbeins. They are obtained using a supergeometric method on QP-manifolds by twist of the standard Courant algebroid on the generalized tangent space without flux. Bianchi identities of the fluxes are easily deduced. We extend the discussion to the case of the double space and present a formulation of T-duality in terms of canonical transformations between graded symplectic manifolds. Finally, the constr…
In this paper we establish a duality between etale Lie groupoids and a class of non-necessarily commutative algebras with a Hopf algebroid structure. For any etale Lie groupoid G over a manifold M, the groupoid algebra C_c(G) of smooth functions with compact support on G has a natural coalgebra structure over C_c(M) which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over C_c(M) we construct the associated spectral etale Lie groupoid G_sp(A) over M such that G_sp(C_c(G)) is naturally isomorphic to G. Both these constructions are functorial, and C_c is fully faithful left…
Non-principal T-duality, generalized complex geometry and blow-ups
本研究将T-对偶推广到非主环面作用的流形，借助椭圆切丛李代数胚控制奇点、定义环面作用联络，实现不变广义复结构的T-对偶迁移，为Haefliger-Salem的环面作用分类提供新视角。
In this note a functorial approach to the integration problem of an LA-groupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibred products in the categories of Lie groupoids and Lie algebroids, giving necessary and sufficient conditions for the existence of such. In particular, it turns out, that the fibred product of Lie algebroids along integrable morphisms is always integrable by a fibred product of Lie groupoids. We show that to every LA-groupoid with integrable top structure one can associate a differentiable graph in the category of Lie groupoids, which is…
The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study the geometry underlying these Maurer-Cartan elements in the light of Lie algebroid theory. In particular, we extend Lichnerowicz-Poisson cohomology and Koszul-Brylinski homology to the realm of extended Poisson manifolds; we establish a sufficient criterion for these to be finite…
Generalised Complex and Spinor Relations
本文研究了广义复结构与旋量之间的关系，探讨了旋量关系如何转化为Courant叠代结构关系，并证明了T对偶关系如何诱导旋量关系，推广了拓扑T对偶相关的扭曲上同调同构。
We begin by presenting a symmetric version of the circle equivariant T-duality result in a joint work of the second author with Siye Wu, thereby generalising the results there. We then initiate the study of twisted equivariant Courant algebroids and equivariant generalised geometry and apply it to our context. As before, T-duality exchanges type II A and type II B string theories. In our theory, both spacetime and the T-dual spacetime can be singular spaces when the fixed point set is non-empty; the singularities correspond to Kaluza-Klein monopoles. We propose that the Ramond-Ramond charges o…
We give a self-contained survey of some approaches aimed at a global description of the geometry underlying double field theory. After reviewing the geometry of Courant algebroids and their incarnations in the AKSZ construction, we develop the theory of metric algebroids including their graded geometry. We use metric algebroids to give a global description of doubled geometry, incorporating the section constraint, as well as an AKSZ-type construction of topological doubled sigma-models. When these notions are combined with ingredients of para-Hermitian geometry, we demonstrate how they reprodu…
We give a simplified definition of topological T-duality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for T-duals are shown to follow. We determine necessary and sufficient conditions for existence of a T-dual in the case of affine torus bundles. This is general enough to include all principal torus bundles as well as torus bundles with arbitrary monodromy representations. We show that isomorphisms in twisted cohomology, twisted K-theory and of Co…
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