of degree -2 is non-degenerate.
The concept of Courant algebroids was originally introduced by Irene Dorfman and Ted Courant to study geometric quantization in the presence of constraints. Later it was considered by Liu, Alan Weinstein and Ping Xu in the study of double Lie algebroids.
In these parts of the literature Courant algebroids are considered in the form of Lie algebroids with relaxed axioms on the bracket. Even of this type there are two different definitions:
in one there is a skew-symmetric bracket which fails to satisfy a Jacobi identity by a coherent term – this is the Courant bracket definition proper;
in the other there is a bracket which satisfies a Jacobi identity but is skew-symmetric only up to a correction term – this is the Dorfman version.
So there are several different ways to present the structure encoded in a Courant algebroid. The picture that seems to be emerging is that the true meaning of the notoin of Courant algebroids is given by the notion of 2-symplectic manifolds.
(… need to say more about the way the Courant Lie algebroid is obtained from a Lie bialgebroid by derived brackets …)
where denotes the Lie derivative of the 1-form by the vector field .
As every symplectic Lie n-algebroid the defining invariant polynomial on a Courant Lie 2-algebroid transgresses to a cocycle in ∞-Lie algebroid cohomology and this transgression is witnessed by a Chern-Simons element. The ∞-Chern-Simons theory induced by this element is the Courant sigma-model.
We discuss how the following tower of notiosn works
|circle n-bundle with -form connection||Lie n-algebra of of bisections of Atiyah Lie n-group|
|circle bundle||Lie algebra of sections of Atiyah Lie algebroid|
|circle 2-bundle with 1-form connection||Lie 2-algebra of sections of Courant Lie 2-algebroid|
For and write
for the smooth ∞-groupoid which is presented under the Dold-Kan correspondence by the sheaf of chain complexes, as indicated (see also at differential cohomology diagram – Examples – Deligne coefficients). This is such that for we have the Deligne complex, representing the moduli ∞-stack of circle n-bundles with connection
For there are evident truncation maps
modulates a circle n-bundle with connection (bundle (n-1)-gerbe), which we may think of as a prequantum circle n-bundle. Regarding this as an object in the slice (∞,1)-topos this has an automorphism ∞-group. The concretification of this (…) is the quantomorphism n-group .
But we can also first forget the -form pieces of the prequantum -bundle away and consider
For this is sometimes known in the literature as a “bundle gerbe with connective structure but without curving”.
For this is the group of bisections of the Atiyah Lie groupoid of the underlying circle principal bundle . Hence its Lie differentiation is the Lie algebra of sections of the coresponding Atiyah Lie algebroid.
hence an embedding of the quantomorphism n-group into the -group of bisections of the Atiyah n-groupoid. For and after Lie differentiation, this is an embedding of the Poisson Lie 2-algebra into the sections of the Courant Lie 2-algebroid. This embedding had been observed in (Rogers).
|higher Atiyah groupoid:||standard higher Atiyah groupoid||higher Courant groupoid||groupoid version of quantomorphism n-group|
|coefficient for cohomology:|
|type of fiber ∞-bundle:||principal ∞-bundle||principal ∞-connection without top-degree connection form||principal ∞-connection|
|symplectic Lie n-algebroid||Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of -d sigma-model||higher symplectic geometry||d sigma-model||dg-Lagrangian submanifold/ real polarization leaf||= brane||(n+1)-module of quantum states in codimension||discussed in:|
|0||symplectic manifold||symplectic manifold||symplectic geometry||Lagrangian submanifold||–||ordinary space of states (in geometric quantization)||geometric quantization|
|1||Poisson Lie algebroid||symplectic groupoid||2-plectic geometry||Poisson sigma-model||coisotropic submanifold (of underlying Poisson manifold)||brane of Poisson sigma-model||2-module = category of modules over strict deformation quantiized algebra of observables||extended geometric quantization of 2d Chern-Simons theory|
|2||Courant Lie 2-algebroid||symplectic 2-groupoid||3-plectic geometry||Courant sigma-model||Dirac structure||D-brane in type II geometry|
|symplectic Lie n-algebroid||symplectic n-groupoid||(n+1)-plectic geometry||AKSZ sigma-model|
(adapted from Ševera 00)
The original references in order of appearance are
Pavol Severa, Some title containing the words “homotopy” and “symplectic”, e.g. this one, In Travaux mathématiques. Fasc. XVI, chapter Trav. Math., XVI, pp. 121-137. Univ. Luxembourg, 2005.
Dmitry Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, pp. 169–185. in Contemporary Mathematics 315, Quantization, Poisson brackets and beyond (Manchester, 2001), Theodore Voronov, editor, Amer. Math. Soc. 2002. (arXiv:math/0203110)
Another useful summary of the theory of Courant algebroids is in section 3 of
The identification of the Lie 2-algebra of sctions of a Courant Lie 2-algebroid associated with a circle 2-bundle with connection as its Lie algebra of automorphisms after forgetting the “curving” is in
This is developed further in
The relation between the two different Lie-alebroid-like definition of Courant algebroids, one with skew, the other with non-skew brackets inspired on the level of Lie 2-algebras the treatment
The relation to ∞-Chern-Simons theory is discussed in