Hitchhiker's Guide to Courant Algebroid Relations

Abstract

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 supergravity can be interpreted as Courant algebroid relations compatible with generalized metrics (generalized isometries).

arXiv commentSome minor things tweaked, typos corrected, references added

References in corpus (25)

• Courant algebroids, derived brackets and even symplectic supermanifolds
• Supergravity as Generalised Geometry I: Type II Theories
• Reduction of Courant algebroids and generalized complex structures
• T-duality, Generalized Geometry and Non-Geometric Backgrounds
• Large N and double scaling limits in two dimensions
• Higher vector bundles and multi-graded symplectic manifolds
• Current Algebras and Differential Geometry
• Torsion-free generalized connections and Heterotic Supergravity
• Towards a double field theory on para-Hermitian manifolds
• Towards an invariant geometry of double field theory
• A Unique Connection for Born Geometry
• Generalised Kinematics for Double Field Theory
• Poisson-Lie T-duality and Courant algebroids
• Generalized complex geometry and T-duality
• Algebroid Structures on Para-Hermitian Manifolds
• Poisson-Lie T-duality as a boundary phenomenon of Chern-Simons theory
• Poisson-Lie T-duality of String Effective Actions: A New Approach to the Dilaton Puzzle
• Courant Algebroids. A Short History
• On integrability of 2-dimensional $\sigma$-models of Poisson-Lie type
• On the relation of Lie algebroids to constrained systems and their BV/BFV formulation
• Kaluza-Klein Reduction of Low-Energy Effective Actions: Geometrical Approach
• Branes on Poisson varieties
• Effective Actions for $\sigma$-Models of Poisson-Lie Type
• Sylvester's law of inertia for quadratic forms on vector bundles
• Supergravity Fluxes and Generalised Geometry

Cited by in corpus (4)

• Born Sigma-Models for Para-Hermitian Manifolds and Generalized T-Duality
• Manin triples on multiplicative Courant algebroids
• A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids
• Odd transgression for Courant algebroids