Dirac structures, momentum maps, and quasi-Poisson manifolds

Poisson-generalized geometry and R-flux

International Journal of Modern Physics A ◽

10.1142/s0217751x15500979 ◽

2015 ◽

Vol 30 (17) ◽

pp. 1550097 ◽

Cited By ~ 10

Author(s):

Tsuguhiko Asakawa ◽

Hisayoshi Muraki ◽

Shuhei Sasa ◽

Satoshi Watamura

Keyword(s):

Exact Sequence ◽

Semidirect Product ◽

Tangent Bundle ◽

Cotangent Bundle ◽

Poisson Manifolds ◽

Dirac Structures ◽

Courant Algebroid ◽

Starting Point ◽

Generalized Geometry ◽

Alternative Version

We study a new kind of Courant algebroid on Poisson manifolds, which is a variant of the generalized tangent bundle in the sense that the roles of tangent and the cotangent bundle are exchanged. Its symmetry is a semidirect product of [Formula: see text]-diffeomorphisms and [Formula: see text]-transformations. It is a starting point of an alternative version of the generalized geometry based on the cotangent bundle, such as Dirac structures and generalized Riemannian structures. In particular, [Formula: see text]-fluxes are formulated as a twisting of this Courant algebroid by a local [Formula: see text]-transformations, in the same way as [Formula: see text]-fluxes are the twist of the generalized tangent bundle. It is a three-vector classified by Poisson three-cohomology and it appears in a twisted bracket and in an exact sequence.

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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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