Courant Algebroids and Poisson Geometry

We present a graded-geometric approach to modular classes of Lie algebroids and their generalizations, introducing in this setting an idea of relative modular class of a Dirac structure for certain type of Courant algebroids, called projectable. This novel approach puts several concepts related to Poisson geometry and its generalizations in a new light and simplifies proofs. It gives, in particular, a nice geometric interpretation of modular classes of twisted Poisson structures on Lie algebroids.

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Geometry and Physics: Volume II

10.1093/oso/9780198802020.001.0001 ◽

2018 ◽

Keyword(s):

Moduli Spaces ◽

Mirror Symmetry ◽

Geometric Analysis ◽

Mathematical Physics ◽

Poisson Geometry ◽

Special Holonomy ◽

Low Dimensional Topology ◽

Generalized Complex Structures ◽

Wide Range ◽

Low Dimensional

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.

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Exotic Courant algebroids and T-duality

Journal of Geometry and Physics ◽

10.1016/j.geomphys.2021.104155 ◽

2021 ◽

Vol 163 ◽

pp. 104155

Author(s):

Jaklyn Crilly ◽

Varghese Mathai

Keyword(s):

Courant Algebroids

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Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures

Journal of High Energy Physics ◽

10.1007/jhep11(2020)173 ◽

2020 ◽

Vol 2020 (11) ◽

Author(s):

Athanasios Chatzistavrakidis ◽

Grgur Šimunić

Keyword(s):

Sigma Models ◽

Sigma Model ◽

Target Space ◽

Two Dimensional ◽

Courant Algebroids ◽

Dirac Structures ◽

Courant Algebroid ◽

Abelian Case ◽

Jacobi Manifolds ◽

Jacobi Structure

Abstract We study aspects of two-dimensional nonlinear sigma models with Wess-Zumino term corresponding to a nonclosed 3-form, which may arise upon dimensional reduction in the target space. Our goal in this paper is twofold. In a first part, we investigate the conditions for consistent gauging of sigma models in the presence of a nonclosed 3-form. In the Abelian case, we find that the target of the gauged theory has the structure of a contact Courant algebroid, twisted by a 3-form and two 2-forms. Gauge invariance constrains the theory to (small) Dirac structures of the contact Courant algebroid. In the non-Abelian case, we draw a similar parallel between the gauged sigma model and certain transitive Courant algebroids and their corresponding Dirac structures. In the second part of the paper, we study two-dimensional sigma models related to Jacobi structures. The latter generalise Poisson and contact geometry in the presence of an additional vector field. We demonstrate that one can construct a sigma model whose gauge symmetry is controlled by a Jacobi structure, and moreover we twist the model by a 3-form. This construction is then the analogue of WZW-Poisson structures for Jacobi manifolds.

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