Courant algebroids, derived brackets and even symplectic supermanifolds

Abstract

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 symplectic supermanifold. The classical BRST complex and the Weil algebra arise as special cases. We recover the Courant algebroid via the derived bracket construction and give a simple proof of the doubling theorem of Liu, Weinstein and Xu. We also introduce a generalization, quasi-Lie bialgebroids, analogous to Drinfeld's quasi-Lie bialgebras; we show that the derived bracket construction in this case also yields a Courant algebroid. Finally, we compute the Poisson cohomology of a one-parameter family of SU(2)- covariant Poisson structures on S^2. As an application, we show that these structures are non-trivial deformations of each other, and that they do not admit rescaling.

A slightly updated version of my UC Berkeley Ph.D. thesis; 76 pages

Cited by in corpus (140)

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• Supergroupoids, double structures, and equivariant cohomology
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• Mirror Symmetry and Generalized Complex Manifolds
• Bianchi Identities for Non-Geometric Fluxes - From Quasi-Poisson Structures to Courant Algebroids
• Graded contact manifolds and contact Courant algebroids
• A Unique Connection for Born Geometry
• Non-geometric Backgrounds and the First Order String Sigma Model
• On Regular Courant Algebroids
• Lectures on Generalized Complex Geometry and Supersymmetry
• Lectures on AKSZ Sigma Models for Physicists
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• $Q$-manifolds and Higher Analogs of Lie Algebroids
• BV Quantization of Topological Open Membranes
• Letters to Alan Weinstein about Courant algebroids
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• Higher Symplectic Geometry
• Quasi-Hamiltonian groupoids and multiplicative Manin pairs
• Unified picture of non-geometric fluxes and T-duality in double field theory via graded symplectic manifolds
• Leibniz 2-algebras and twisted Courant algebroids
• Quasi-Lie bialgebroids and twisted Poisson manifolds
• Equivalences of Higher Derived Brackets
• Topological Open Membranes
• Lie algebroid modules and representations up to homotopy
• Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids
• Deformations of Closed Strings and Topological Open Membranes
• Integration of Exact Courant Algebroids
• Gravity from Lie algebroid morphisms
• On the structure of graded symplectic supermanifolds and Courant algebroids
• LA-Courant algebroids and their applications
• Modular classes of skew algebroid relations
• Courant Algebroids. A Short History
• The Pontryagin Class for Pre-Courant Algebroids
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• Graded Geometry, $Q$-Manifolds, and Microformal Geometry
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• Doubled Aspects of Vaisman Algebroid and Gauge Symmetry in Double Field Theory
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• Double field theory for the A/B-models and topological S-duality in generalized geometry
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• Matched pairs of Courant algebroids
• Symplectic structures on the integration of exact Courant algebroids
• Gravity theory on Poisson manifold with $R$-flux
• Courant-Nijenhuis tensors and generalized geometries
• Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroids and Gauge Theory
• Star products on graded manifolds and ${\alpha }^{\prime }$-corrections to Courant algebroids from string theory
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• Global Aspects of Doubled Geometry and Pre-rackoid
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• Notions of double for Lie algebroids
• Courant cohomology, Cartan calculus, connections, curvature, characteristic classes
• Deformation spaces and normal forms around transversals
• AKSZ constructions for topological membranes on $G_2$-manifolds
• Q-manifolds and Mackenzie theory: an overview
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• Twisted chiral de Rham complex, generalized geometry, and T-duality
• Momentum section on Courant algebroid and constrained Hamiltonian mechanics
• Graded manifolds and Drinfeld doubles for Lie bialgebroids
• Topological T-duality for torus bundles with monodromy
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• Tulczyjew triples and higher Poisson/Schouten structures on Lie algebroids
• Courant algebroids from categorified symplectic geometry
• Dirac Geometry of the Holonomy Fibration
• Weyl quantization of degree 2 symplectic graded manifolds
• Metric-Connection Geometries on Pre-Leibniz Algebroids: A Search for Geometrical Structure in String Models
• Hyperstructures on Lie algebroids
• Metric algebroid and Dirac generating operator in Double Field Theory
• Current algebras from QP-manifolds in general dimensions
• Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach
• Hitchhiker's Guide to Courant Algebroid Relations
• On quasi-Jacobi and Jacobi-quasi bialgebroids
• The gauging of BV algebras
• Topological G${}_2$ and Spin(7) strings at 1-loop from double complexes
• Poisson groupoids and moduli spaces of flat bundles over surfaces
• On homotopy Lie bialgebroids
• BFV-complex and higher homotopy structures
• H-twisted Courant algebroids
• Graded Generalized Geometry
• Lie algebroids, non-associative structures and non-geometric fluxes
• Dorfman connections and Courant algebroids
• Moser Lemma in Generalized Complex Geometry
• Shifted Poisson structures on differentiable stacks
• Dirac geometry, quasi-Poisson actions and D/G-valued moment maps
• Courant morphisms and moment maps
• From Hypercomplex to Holomorphic Symplectic Structures
• Deformation Theory of Courant Algebroids via the Rothstein Algebra
• Variation of Hodge structure for generalized complex manifolds
• Basic curvature and the Atiyah cocycle in gauge theory
• The Algebroid Structure of Double Field Theory
• Doubled Formalism, Complexification and Topological Sigma-Models
• Projective Connections and Schwarzian Derivatives for Supermanifolds, and Batalin-Vilkovisky Operators
• Classical worldvolumes as generalised geodesics
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• Deformations of pre-symplectic structures and the Koszul $L_{\infty }$-algebra
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• Twisting on pre-Lie algebras and quasi-pre-Lie bialgebras
• Transitive Courant algebroids
• The Free Courant Algebroid
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• Vertex Poisson algebras associated with Courant algebroids and their deformations; I
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• AKSZ-type Topological Quantum Field Theories and Rational Homotopy Theory
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