Courant Bracket

In a field of mathematics known as differential geometry, the Courant bracket is a generalization of the Lie bracket from an operation on the tangent bundle to an operation on the direct sum of the tangent bundle and the vector bundle of p-forms.

The case p = 1 was introduced by Theodore James Courant in his 1990 doctoral dissertation as a structure that bridges Poisson geometry and presymplectic geometry, based on work with his advisor Alan Weinstein. The twisted version of the Courant bracket was introduced in 2001 by Pavol Severa, and studied in collaboration with Weinstein.

Today a complex version of the p=1 Courant bracket plays a central role in the field of generalized complex geometry, introduced by Nigel Hitchin in 2002. Closure under the Courant bracket is the integrability condition of a generalized almost complex structure.

Read more about Courant BracketDefinition, Properties, Dirac and Generalized Complex Structures, Dorfman Bracket, Courant Algebroid

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Twisted Courant Bracket - Integral Twists and Gerbes
... geometric interpretation of the twisted p=0 Courant bracket only exists when H represents an integral class ... Similarly at higher values of p the twisted Courant brackets can be geometrically realized as untwisted Courant brackets twisted by gerbes when H is an integral cohomology class ...