Poisson-Lie T-duality and Courant algebroids

Integration of Exact Courant Algebroids

In this paper, we describe an integration of exact Courant algebroids to symplectic 2-groupoids, and we show that the differentiation procedure from [26] inverts our integration.

Topological Membranes, Current Algebras and H-flux - R-flux Duality based on Courant Algebroids

We construct a topological sigma model and a current algebra based on a Courant algebroid structure on a Poisson manifold. In order to construct models, we reformulate the Poisson Courant algebroid by supergeometric construction on a QP-manifold. A new duality of Courant algebroids which transforms H-flux and R-flux is proposed, where the transformation is interpreted as a canonical transformation of a graded symplectic manifold.

扭Courant r-Lie代数与非扭Courant (r+1)-Lie代数胚之间的$L_\infty$-态射

$L_\infty$-morphisms between twisted Courant $r$-Lie algebras and untwisted Courant $(r{+}1)$-Lie algebroids

针对Marco Zambon提出的高阶闭形式扭化Courant代数胚是否存在典范$L_\infty$-态射的问题，本文给出肯定回答，提出通用构造框架并阐明其底层结构，验证框架可适配相关已有代数间的态射。

Courant algebroids 的 Kodaira-Spencer 理论

Kodaira-Spencer theory for Courant algebroids

本文提出 Courant algebroids 的 Kodaira-Spencer 理论，揭示其与 Batalin-Vilkovisky 公式及超引力背景的关联。

Transitive Courant algebroids

We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal Whitney sum $E\oplus C$ where E is a given Courant algebroid and C is a flat, pseudo- Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive Courant algebroid, i.e., a Courant algebroid with a surjective anchor, and describe a class of transitive Courant algebroids which are Whitney sums of a Courant subalgebroid with neutral metric and Courant-like bracket and a pseudo-Euclidean vector bundle with a flat, metric conn…

Courant algebroids: Cohomology and Matched Pairs

We introduce Courant algebroids, providing definitions, some historical notes, and some elementary properties. Next, we summarize basic properties of graded manifolds. Then, drawing on the work of Roytenberg and others, we introduce the graded or supergraded language demonstrating a cochain complex / cohomology for (general) Courant algebroids. We review spectral sequences and show how this tool is used to compute cohomology of regular Courant algebroids with a split base. Finally we analyze matched pairs of Courant algebroids including the complexified standard Courant algebroid of a complex …

The Algebroid Structure of Double Field Theory

By doubling the target space of a canonical Courant algebroid and subsequently projecting down to a specific subbundle, we identify the data of double field theory (DFT) and hence define its algebroid structure. We specify the properties of the DFT algebroid. We show that one of the Courant algebroid properties plays the role of the strong constraint in the context of DFT. The DFT algebroid is a special example when properties of a Courant algebroid are relaxed in a specific and dependent manner. When otherwise, we uncover additional structures.

On Jacobi quasi-Nijenhuis algebroids and Courant-Jacobi algebroid morphisms

We propose a definition of Jacobi quasi-Nijenhuis algebroid and show that any such Jacobi algebroid has an associated quasi-Jacobi bialgebroid. Therefore, also an associated Courant-Jacobi algebroid is obtained. We introduce the notions of quasi-Jacobi bialgebroid morphism and Courant-Jacobi algebroid morphism providing also some examples of Courant-Jacobi algebroid morphisms.

Dirac structures for generalized Courant and Courant algebroids

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 $M\times \R$.

AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories

We give a detailed exposition of the Alexandrov-Kontsevich-Schwarz- Zaboronsky superfield formalism using the language of graded manifolds. As a main illustarting example, to every Courant algebroid structure we associate canonically a three-dimensional topological sigma-model. Using the AKSZ formalism, we construct the Batalin-Vilkovisky master action for the model.

N-manifolds of degree 2 and metric double vector bundles

This paper shows the equivalence of the categories of $N$-manifolds of degree $2$ with the category of double vector bundles endowed with a linear metric. Split Poisson $N$-manifolds of degree $2$ are shown to be equivalent to self-dual representations up to homotopy. As a consequence, the equivalence above induces an equivalence between so called metric VB-algebroids and Poisson $N$-manifolds of degree $2$. Then a new description of split Lie $2$-algebroids is given, as well as their "duals", the Dorfman $2$-representations. We show that Dorfman $2$-representations are equivalent in a simple …

Hitchhiker's Guide to Courant Algebroid Relations

Courant algebroids provide a useful mathematical tool (not only) in string theory. It is thus important to define and examine their morphisms. To some extent, this was done before using an analogue of canonical relations known from symplectic geometry. However, it turns out that applications in physics require a more general notion. We aim to provide a self-contained and detailed treatment of Courant algebroid relations and morphisms. A particular emphasis is placed on providing enough motivating examples. In particular, we show how Poisson-Lie T-duality and Kaluza-Klein reduction of supergrav…

库朗代数簇的同伦约化

Homological reduction of Courant algebroids

该研究构建了与辛几何BFV公式体系平行的库朗代数簇哈密顿约化同伦模型，通过关联微分分次库朗-多夫曼代数，在正则假设下用其同调得到约化库朗代数簇的对应代数。

On higher-dimensional Courant algebroids

We define the transgression functor which associates to a (higher-dimensional) Courant algebroid on a manifold a Lie algebroid on the shifted tangent bundle of the manifold.

Odd transgression for Courant algebroids

The "odd transgression" introduced by the authors in an earlier article is applied to construct and study the inverse image functor in the theory of Courant algebroids.

2-plectic geometry, Courant algebroids, and categorified prequantization

A 2-plectic manifold is a manifold equipped with a closed nondegenerate 3-form, just as a symplectic manifold is equipped with a closed nondegenerate 2-form. In 2-plectic geometry we meet higher analogues of many structures familiar from symplectic geometry. For example, any 2-plectic manifold has a Lie 2-algebra consisting of smooth functions and Hamiltonian 1-forms. This is equipped with a Poisson-like bracket which only satisfies the Jacobi identity up to `coherent chain homotopy'. Over any 2-plectic manifold is a vector bundle equipped with extra structure called an exact Courant algebroid…

Courant-Nijenhuis 代数胚

Courant-Nijenhuis algebroids

本文通过引入 Courant 1-导子建立 Courant 代数胚与线性 (1,1)-张量场的相容性，提出 Courant-Nijenhuis 代数胚，并统一描述全纯 Courant 代数胚、Dirac-Nijenhuis 结构与 Lie-Nijenhuis 双代数胚。

Nijenhuis and Compatible Tensors on Lie and Courant algebroids

We show that well known structures on Lie algebroids can be viewed as Nijenhuis tensors or pairs of compatible tensors on Courant algebroids. We study compatibility and construct hierarchies of these structures.

L-infinity algebras and higher analogues of Dirac structures and Courant algebroids

We define a higher analogue of Dirac structures on a manifold M. Under a regularity assumption, higher Dirac structures can be described by a foliation and a (not necessarily closed, non-unique) differential form on M, and are equivalent to (and simpler to handle than) the Multi-Dirac structures recently introduced in the context of field theory by Vankerschaver, Yoshimura and Marsden. We associate an L-infinity algebra of "observables" to every higher Dirac structure, extending work of Baez, Hoffnung and Rogers on multisymplectic forms. Further, applying a recent result of Getzler, we associa…

Courant Algebroids. A Short History

The search for a geometric interpretation of the constrained brackets of Dirac led to the definition of the Courant bracket. The search for the right notion of a "double" for Lie bialgebroids led to the definition of Courant algebroids. We recount the emergence of these concepts.

Remarks on the Definition of a Courant Algebroid

The notion of Courant algebroid was introduced by Liu, Weinstein and Xu in 1997. Its definition consists of five axioms and an assumption for a derivation. It is shown that two of the axioms and the assumption for the derivation follow from the rest of the axioms.

Hom-Poisson-Nijenhuis structures on Hom-Lie algebroids and Hom-Dirac structures on Hom-Courant algebroids

In this paper, we develop the theory of Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids introduced by Cai, Liu and Sheng. Specifically, we introduce the notions of Hom-Poisson, Hom-Nijenhuis and Hom-Poisson-Nijenhuis structures on a Hom-Lie algebroid and the notion of Hom-Dirac structures on a Hom-Courant algebroid. We show that these structures satisfy similar properties to structures non "Hom-"version. For example, there exists the hierarchy of a Hom-Poisson-Nijenhuis structure and we have a relation between Hom-Dirac structures and Maurer-Cartan type equation. Moreover w…

Courant类型代数胚、余伴随轨道及相关可积流

The Courant type algebroids, the coadjoint orbits and related integrable flows

本文分析了仿射Courant类型代数胚相关的Poisson结构，构造了环路微分同胚群余伴随作用不变的Lie-Poisson括号，并描述了推广heavenly类型微分系统的可积Hamilton流。

Courant algebroids, derived brackets and even symplectic supermanifolds

In this dissertation we study Courant algebroids, objects that first appeared in the work of T. Courant on Dirac structures; they were later studied by Liu, Weinstein and Xu who used Courant algebroids to generalize the notion of the Drinfeld double to Lie bialgebroids. As a first step towards understanding the complicated properties of Courant algebroids, we interpret them by associating to each Courant algebroid a strongly homotopy Lie algebra in a natural way. Next, we propose an alternative construction of the double of a Lie bialgebroid as a homological hamiltonian vector field on an even…

传递Courant代数胚与双重辛群胚

Transitive Courant algebroids and double symplectic groupoids

该研究将双重辛群胚的Lu-Weinstein构造推广至传递Courant代数胚且Atiyah代数胚可积的李双代数胚，分类了李群胚上的正合扭曲Courant代数胚，证明了扭曲CA-群胚基上的叶状结构，推广了现有双重辛群胚例子。

Double Field Theory and Membrane Sigma-Models

We investigate geometric aspects of double field theory (DFT) and its formulation as a doubled membrane sigma-model. Starting from the standard Courant algebroid over the phase space of an open membrane, we determine a splitting and a projection to a subbundle that sends the Courant algebroid operations to the corresponding operations in DFT. This describes precisely how the geometric structure of DFT lies in between two Courant algebroids and is reconciled with generalized geometry. We construct the membrane sigma-model that corresponds to DFT, and demonstrate how the standard T-duality orbit…

Cohomology of Courant algebroids with split base

We study the (standard) cohomology $H^\bullet_{st}(E)$ of a Courant algebroid $E$. We prove that if $E$ is transitive, the standard cohomology coincides with the naive cohomology $H_{naive}^\bullet(E)$ as conjectured by Stienon and Xu. For a general Courant algebroid we define a spectral sequence converging to its standard cohomology. If $E$ is with split base, we prove that there exists a natural transgression homomorphism $T_3$ (with image in $H^3_{naive}(E)$) which, together with the naive cohomology, gives all $H^\bullet_{st}(E)$. For generalized exact Courant algebroids, we give an explic…

Symplectic structures on the integration of exact Courant algebroids

We construct an infinite-dimensional symplectic 2-groupoid as the integration of an exact Courant algebroid. We show that every integrable Dirac structure integrates to a "Lagrangian" sub-2-groupoid of this symplectic 2-groupoid. As a corollary, we recover a result of Bursztyn-Crainic-Weinstein-Zhu that every integrable Dirac structure integrates to a presymplectic groupoid.

Foliated Lie and Courant Algebroids

If $A$ is a Lie algebroid over a foliated manifold $(M,\mathcal{F})$, a foliation of $A$ is a Lie subalgebroid $B$ with anchor image $T\mathcal{F}$ and such that $A/B$ is locally equivalent with Lie algebroids over the slice manifolds of $\mathcal{F}$. We give several examples and, for foliated Lie algebroids, we discuss the following subjects: the dual Poisson structure and Vaintrob's super-vector field, cohomology and deformations of the foliation, integration to a Lie groupoid. In the last section, we define a corresponding notion of a foliation of a Courant algebroid $A$ as a bracket-close…

Lie rackoids integrating Courant algebroids

We construct an infinite dimensional Lie rackoid Y which hosts an integration of the standard Courant algebroid. As a set, Y = C $\infty$ ([0, 1], T * M) for a compact manifold M. The rackoid product is by automorphisms of the Dorfman bracket. The first part of the article is a study of the Lie rackoid Y and its tangent Leibniz algebroid a quotient of which is the standard Courant algebroid. In a second part, we study the equivalence relation related to the quotient on the rackoid level and restrict then to an integrable Dirac structure. We show how our integrating object contains the correspo…

Anchored vector bundles and algebroids

Inspired by recent works of Zang Liu, Alan Weinstein and Ping Xu, we introduce the notions of CC algebroids and non asymmetric Courant algebroids and study these structures. It is shown that CC algebroids of rank greater than 3 are the same as Courant algebroids up to a constant factor, though the definition of CC algebroids is much simpler than that of Courant algebroids,requiring only 2 axioms instead of 5. The situation is similar to that of Lie algebroids, where in the usual definition used by all of he experts there is a redundant axiom, e.g.[GG,KO1,KO2,MK,PL]. Non asymmetric Courant alge…

Reduction of Courant algebroids and generalized complex structures

We present a theory of reduction for Courant algebroids as well as Dirac structures, generalized complex, and generalized Kähler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction. The enhanced symmetry group of a Courant algebroid leads us to define \emph{extended} actions and a generalized notion of moment map. Key examples of generalized Kähler reduced spaces include new explicit bi-Hermitian metrics on $\CC P^2$.

多变重流形上的观测量与高阶 Courant 代数胚

Observables on multisymplectic manifolds and higher Courant algebroids

本文研究了多变重流形上由闭非退化微分形式关联的可观测量的 $L_{\infty}$-代数与扭曲的高阶 Courant algebroid 截面的 $L_{\infty}$-代数，证明了前者的 $L_{\infty}$-嵌入，给出了涉及伯努利数的显式公式，并证明其与规范变换相容。

Non-commutative Courant algebroids and Quiver algebras

In this paper, we develop a differential-graded symplectic (Batalin-Vilkovisky) version of the framework of Crawley-Boevey, Etingof and Ginzburg on noncommutative differential geometry based on double derivations to construct non-commutative analogues of the Courant algebroids introduced by Liu, Weinstein and Xu. Adapting geometric constructions of Ševera and Roytenberg for (commutative) graded symplectic supermanifolds, we express the BRST charge, given in our framework by a `homological double derivation', in terms of Van den Bergh's double Poisson algebras for graded bi-symplectic non-commu…

On Poisson quasi-Nijenhuis Lie algebroids

We propose a definition of Poisson quasi-Nijenhuis Lie algebroids as a natural generalization of Poisson quasi-Nijenhuis manifolds and show that any such Lie algebroid has an associated quasi-Lie bialgebroid. Therefore, also an associated Courant algebroid is obtained. We introduce the notion of a morphism of quasi-Lie bialgebroids and of the induced Courant algebroids morphism and provide some examples of Courant algebroid morphisms. Finally, we use paired operators to deform doubles of Lie and quasi-Lie bialgebroids and find an application to generalized complex geometry.

On the structure of graded symplectic supermanifolds and Courant algebroids

This paper is devoted to a study of geometric structures expressible in terms of graded symplectic supermanifolds. We extend the classical BRST formalism to arbitrary pseudo-Euclidean vector bundles (E\to M_{0}) by canonically associating to such a bundle a graded symplectic supermanifold ((M,Ω)), with (\textrm{deg}(Ω)=2). Conversely, every such manifold arises in this way. We describe the algebra of functions on (M) in terms of (E) and show that ``BRST charges'' on (M) correspond to Courant algebroid structures on (E), thereby constructing the standard complex for the latter as a generalizati…

束 gerbe 与 Courant 代数胚的无穷小对称性

Infinitesimal symmetries of bundle gerbes and Courant algebroids

本文证明了光滑流形上 χ-扭曲 Courant 代数胚截面的 Lie 2-代数与 3-曲率为 χ 的 S^1-束 gerbe 上保联络乘法向量场的 Lie 2-代数拟同构，建立了二者的无穷小对称性对应。

Canonical metrics on holomorphic Courant algebroids

The solution of the Calabi Conjecture by Yau implies that every Kähler Calabi-Yau manifold $X$ admits a metric with holonomy contained in $\textrm{SU}(n)$, and that these metrics are parametrized by the positive cone in $H^{1,1}(X,\mathbb{R})$. In this work we give evidence of an extension of Yau's theorem to non-Kähler manifolds, where $X$ is replaced by a compact complex manifold with vanishing first Chern class endowed with a holomorphic Courant algebroid $Q$ of Bott-Chern type. The equations that define our notion of best metric correspond to a mild generalization of the Hull-Strominger sy…

Group actions on dg-manifolds and Exact Courant algebroids

Let G be a Lie group acting by diffeomorphisms on a manifold M and consider the image of T[1]G and T[1]M, of G and M respectively, in the category of differential graded manifolds. We show that the obstruction to lift the action of T[1]G on T[1]M to an action on a R[n]-bundle over T[1]M is measured by the G equivariant cohomology of M. We explicitly calculate the differential graded Lie algebra of the symmetries of the R[n]-bundle over T[1]M and we use this differential graded Lie algebra to understand which actions are hamiltonian. We show how split Exact Courant algebroids could be obtained …

广义复结构和旋量关系

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。

Omni-Lie algebroids

A generalized Courant algebroid structure is defined on the direct sum bundle D(E) +J(E), where D(E) and J(E) are the gauge Lie algebroid and the jet bundle of a vector bundle E respectively. Such a structure is called an omni-Lie algebroid since it is reduced to the omni-Lie algebra introduced by A.Weinstein if the base manifold is a point. We prove that any Lie algebroid structure on E is characterized by a Dirac structure as the graph of a bundle map from J(E) to D(E).

Pseudo-Dirac Structures

A Dirac structure is a Lagrangian subbundle of a Courant algebroid, $L\subset\mathbb{E}$, which is involutive with respect to the Courant bracket. In particular, $L$ inherits the structure of a Lie algebroid. In this paper, we introduce the more general notion of a pseudo-Dirac structure: an arbitrary subbundle, $W\subset\mathbb{E}$, together with a pseudo-connection on its sections, satisfying a natural integrability condition. As a consequence of the definition, $W$ will be a Lie algebroid. Allowing non-isotropic subbundles of $\mathbb{E}$ incorporates non-skew tensors and connections into D…

Constant Symplectic 2-groupoids

We propose a definition of symplectic 2-groupoid which includes integrations of Courant algebroids that have been recently constructed. We study in detail the simple but illustrative case of constant symplectic 2-groupoids. We show that the constant symplectic 2-groupoids are, up to equivalence, in one-to-one correspondence with a simple class of Courant algebroids that we call constant Courant algebroids. Furthermore, we find a correspondence between certain Dirac structures and Lagrangian sub-2-groupoids.

Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory

Motivated by questions from quantum group and field theories, we review structures on manifolds that are weaker versions of Poisson structures, and variants of the notion of Lie algebroid. We give a simple definition of the Courant algebroids and introduce the notion of a deriving operator for the Courant bracket of the double of a proto-bialgebroid. We then describe and relate the various quasi-Poisson structures, which have appeared in the literature since 1991, and the twisted Poisson structures studied by Severa and Weinstein.

Courant morphisms and moment maps

We study Hamiltonian spaces associated with pairs (E,A), where E is a Courant algebroid and A\subset E is a Dirac structure. These spaces are defined in terms of morphisms of Courant algebroids with suitable compatibility conditions. Several of their properties are discussed, including a reduction procedure. This set-up encompasses familiar moment map theories, such as group-valued moment maps, and it provides an intrinsic approach from which different geometrical descriptions of moment maps can be naturally derived. As an application, we discuss the relationship between quasi-Poisson and pres…

Leibniz algebroid associated with a Nambu-Poisson structure

The notion of Leibniz algebroid is introduced, and it is shown that each Nambu-Poisson manifold has associated a canonical Leibniz algebroid. This fact permits to define the modular class of a Nambu-Poisson manifold as an appropiate cohomology class, extending the well-known modular class of Poisson manifolds.

Shifted symplectic Lie algebroids

Shifted symplectic Lie and $L_\infty$ algebroids model formal neighbourhoods of manifolds in shifted symplectic stacks, and serve as target spaces for twisted variants of classical AKSZ topological field theory. In this paper, we classify zero-, one- and two-shifted symplectic algebroids and their higher gauge symmetries, in terms of classical geometric "higher structures", such as Courant algebroids twisted by $Ω^2$-gerbes. As applications, we produce new examples of twisted Courant algebroids from codimension-two cycles, and we give symplectic interpretations for several well known features …

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted Poisson structure, and of Poisson-Nijenhuis manifolds. A review of the spinor approach to the modular class concludes the paper.

Reduction of Poisson-Nijenhuis Lie algebroids to symplectic-Nijenhuis Lie algebroids with nondegenerate Nijenhuis tensor

We show how to reduce, under certain regularities conditions, a Poisson-Nijenhuis Lie algebroid to a symplectic-Nijenhuis Lie algebroid with nondegenerate Nijenhuis tensor. We generalize the work done by Magri and Morosi for the reduction of Poisson-Nijenhuis manifolds. The choice of the more general framework of Lie algebroids is motivated by the geometrical study of some reduced bi-Hamiltonian systems. An explicit example of reduction of a Poisson-Nijenhuis Lie algebroid is also provided.

Twisted Jacobi manifolds, twisted Dirac-Jacobi structures and quasi-Jacobi bialgebroids

We study twisted Jacobi manifolds, a concept that we had introduced in a previous Note. Twisted Jacobi manifolds can be characterized using twisted Dirac-Jacobi, which are sub-bundles of Courant-Jacobi algebroids. We show that each twisted Jacobi manifold has an associated Lie algebroid with a 1-cocycle. We introduce the notion of quasi-Jacobi bialgebroid and we prove that each twisted Jacobi manifold has a quasi-Jacobi bialgebroid canonically associated. Moreover, the double of a quasi-Jacobi bialgebroid is a Courant-Jacobi algebroid. Several examples of twisted Jacobi manifolds and twisted D…

Tulczyjew triples and higher Poisson/Schouten structures on Lie algebroids

We show how to extend the construction of Tulczyjew triples to Lie algebroids via graded manifolds. We also provide a generalisation of triangular Lie bialgebroids as higher Poisson and Schouten structures on Lie algebroids.

Levi decomposition of analytic Poisson structures and Lie algebroids

We prove the existence of a local analytic Levi decomposition for analytic Poisson structures and Lie algebroids.

Q-algebroids and their cohomology

A Q-algebroid is a Lie superalgebroid equipped with a compatible homological vector field and is the infinitesimal object corresponding to a Q-groupoid. We associate to every Q-algebroid a double complex. As a special case, we define the BRST model of a Lie algebroid, which generalizes the BRST model for equivariant cohomology. We extend to this setting the Mathai-Quillen-Kalkman isomorphism of the BRST and Weil models, and we suggest a definition of a basic subcomplex which, however, requires a choice of a connection. Other examples include Roytenberg's homological double of a Lie bialgebroid…

Stability of higher order singular points of Poisson manifolds and Lie algebroids

We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first (not necessarily linear) approximation of the given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical Lichnerowicz-Poisson cohomology and the deformation cohomology for Lie algebroids recently introduced by Crainic and Moerdijk. We also provide several examples of stable singular points of order $k \geq 1$ for Poisson structures and Lie algebroids. Finally, we apply our results …