Reduction of Courant algebroids and generalized complex structures

By H. Bursztyn, G. Cavalcanti, M. Gualtieri. 2005

268 citations recorded by the source corpus.

We present a theory of reduction for Courant algebroids as well as Dirac structures, generalized complex, and generalized Kahler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction. The enhanced symmetry group of a Courant algebroid leads us to define extended actions and a generalized notion of moment map. Key examples of generalized Kahler reduced spaces include new explicit bi-Hermitian metrics on CP2.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1306d626fd24705f8b7b3ffc1c316176f8ca0e06

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. R. Mehta
2. M. Gualtieri
3. G. Cavalcanti
4. V. Apostolov
5. E. Meinrenken
6. Izu Vaisman
7. David Li-Bland
8. Anna Fino

In Mathematical Physics

1. Anna Fino
2. M. Gualtieri
3. R. Mehta
4. G. Cavalcanti
5. Adriano Tomassini
6. R. Goto
7. Izu Vaisman
8. E. Meinrenken

In Statistical and Nonlinear Physics

1. M. Gualtieri
2. Anna Fino
3. R. Mehta
4. G. Cavalcanti
5. Adriano Tomassini
6. E. Meinrenken
7. David Li-Bland
8. R. Goto

Citations in the corpus

• Q-algebroids and their cohomology (2007, methodology): “oid and Ginzburg’s [8] equivariant Lie algebroid cohomology. As an example of a Q-algebroid that does not arise from a double Lie algebroid, we construct a Q-algebroid associated to a “lifted action” [4] of a Lie algebra on a Courant algebroid. Q-algebroids are the infinitesimal objects associated to”
• Courant Algebroids and Poisson Geometry (2008, background): “See [6] and [28] for similar constructions for the case of exact Courant algebroids.”
• Reduction and submanifolds of generalized complex manifolds (2005, background): “Remark 1.2. The above definitions may be applied to vector bundles and Courant algebroids [17, 3 ]. For instance [17], if π is a Poisson bivector field on M, T bigM also has the Courant algebroid structure with anchor Id + ♯π”
• Generalized Kähler Geometry (2010, methodology): “The reduction of Dirac structures proceeds as follows [8].”
• Generalized complex geometry and T-duality (2011, background): “Conversely, given a lift of an action of a compact Lie group, one can always average TM for that lifted action to obtain a new splitting of TM⊕ T M which is preserved by the action [8].”
• A survey on strong KT structures (2009, background): “Constructions of non-trivial generalized Kähler structures are given for instance in [24, 3, 5, 31, 36, 35, 15].”
• Generalized Kähler Manifolds, Commuting Complex Structures, and Split Tangent Bundles (2007, background): “Bihermitian complex surfaces were studied in [1, 2, 8, 10, 20, 21, 24, 27, 30] and classified for even first Betti number in [2], where the classification of Poisson surfaces [3] is used, and existence is only partially proven.”
• Bihermitian metrics on Del Pezzo surfaces (2006, unknown): “D G ] 9 A ug 2 00 6 Bihermitian metricson DelPezzosurfaces NigelHitchin January 19,2022”
• Deformations of generalized complex and generalized Kahler structures (2007, background): “Interesting generalized K¨ ahler structures have been constructed on Fano surfaces and certain toric K¨ ahler manifolds by the r e duction [3], [18] which is a generalization of the symplectic reduction.”