In differential geometry, a field of mathematics, 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 ${\displaystyle p=1}$ was introduced by Theodore James Courant in his 1990 doctoral dissertation as a structure that bridges Poisson geometry and pre-symplectic 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 ${\displaystyle p=1}$ 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.

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be vector fields on an ${\displaystyle N}$-dimensional real manifold ${\displaystyle M}$ and let ${\displaystyle \xi }$ and ${\displaystyle \eta }$ be ${\displaystyle p}$-forms. Then ${\displaystyle X+\xi }$ and ${\displaystyle Y+\eta }$ are sections of the direct sum ${\displaystyle TM\oplus \wedge ^{p}T^{*}M}$ of the tangent bundle and the bundle of ${\displaystyle p}$-forms. The Courant bracket of ${\displaystyle X+\xi }$ and ${\displaystyle Y+\eta }$ is defined to be

where ${\displaystyle {\mathcal {L}}_{X}}$ is the Lie derivative along the vector field ${\displaystyle X}$, ${\displaystyle d}$ is the exterior derivative and ${\displaystyle \iota }$ is the interior product.

The Courant bracket is antisymmetric but it does not satisfy the Jacobi identity for ${\displaystyle p}$ greater than zero.

However, at least in the case ${\displaystyle p=1}$, the Jacobiator, which measures a bracket's failure to satisfy the Jacobi identity, is an exact form. It is the exterior derivative of a form which plays the role of the Nijenhuis tensor in generalized complex geometry.

The Courant bracket is the antisymmetrization of the Dorfman bracket, which does satisfy a kind of Jacobi identity.