Courant algebroid


\infty-Lie theory

∞-Lie theory


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids




\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Symplectic geometry



A Courant algebroid – or better: Courant Lie 2-algebroid – (named after Theodore Courant) is precisely a symplectic Lie 2-algebroid (Roytenberg):

it is a Lie 2-algebroid 𝔓\mathfrak{P} whose Chevalley-Eilenberg algebra CE(𝔓)CE(\mathfrak{P}) is equipped with the structure of a Poisson 3-algebra whose Poisson bracket

{,}:CE(𝔓)CE(𝔓)CE(𝔓) \{-,-\} : CE(\mathfrak{P})\otimes CE(\mathfrak{P}) \to CE(\mathfrak{P})

of degree -2 in non-degenerate.

Therefore the differential d CE(𝔓)d_{CE(\mathfrak{P})} on CE(𝔓)CE(\mathfrak{P}) has a Hamiltonian with respect to this bracket in that there is an element ΘCE(𝔓)\Theta \in CE(\mathfrak{P}) such that

d CE(𝔓)={Θ,}. d_{CE(\mathfrak{P})} = \{\Theta, -\} \,.


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.

Moreover, the way Lie algebroids may be expressed in terms of Lie-Rinehart algebras, Courant algebroids yield Courant-Dorfman algebras.

(… need to say more about the way the Courant Lie algebroid is obtained from a Lie bialgebroid by derived brackets …)


Lie algebras of compact type

A Courant Lie 2-algebroid over the point is precisely an ordinary Lie algebra 𝔤\mathfrak{g} that is equipped with a quadratic and non-degenerate invariant polynomial.

Standard Courant algebroid and U(1)U(1)-gerbes

The standard Courant algebroid of a manifold XX is the one which

  • as a vector bundle with extra structure is E=TXT *XE = T X\oplus T^* X, the fiberwise direct sum of the tangent bundle and the cotangent bundle; with

    • bilinear form

      X+ξ,Y+η=η(X)+ξ(Y) \langle X + \xi , Y +\eta \rangle = \eta(X) + \xi(Y)

    for X,YΓ(TX)X,Y \in \Gamma(T X) and ξ,ηΓ(T *X)\xi, \eta \in \Gamma(T^* X)

    • brackets

      [X+ξ,Y+η]=[X,Y]+ Xη Yξ+12d(η(X)ξ(Y)) [X + \xi, Y + \eta] = [X,Y] + \mathcal{L}_X \eta - \mathcal{L}_Y \xi + \frac{1}{2} d (\eta(X) - \xi(Y))

      where Xη={d,ι X}η\mathcal{L}_X \eta = \{d,\iota_X\} \eta denotes the Lie derivative of the 1-form η\eta by the vector field XX.

  • as a dg-manifold is ΠT *ΠTX\Pi T^* \Pi T X, the shifted cotangent bundle of the shifted tangent bundle,

    where the differential is on each local coordinate patch nUX\mathbb{R}^n \simeq U \subset X with coordinates {x i}\{x^i\} in degree 0, {dx i}\{d x^i\} and {θ i}\{\theta_i\} in degree 1 and {p i}\{p_i\} in degree 2 given by

    d C =d dR+p iθ i =dx ix i+p iθ i. \begin{aligned} d_C &= d_{dR} + p_i \frac{\partial}{\partial \theta_i} \\ &= dx^i \frac{\partial}{\partial x^i } + p_i \frac{\partial}{\partial \theta_i} \end{aligned} \,.

Such a standard Courant algebroid may be understood as the higher analog of the Atiyah Lie algebroid of a line bundle. See below in Relation to Atiyah groupoids.


Generalized complex geometry

The study of Courant algebroids is to a large extent known as generalized complex geometry, where the Courant algebroid appears as the generalized tangent bundle.

Chern-Simons element and Courant σ\sigma-model

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.

Lagrangian submanifolds and Dirac structures

The Lagrangian dg-submanifolds of a Courant Lie 2-algebroid corespond to its Dirac structures.

Relation to Atiyah Lie 2-algebroid and quantomorphism 2-group

We discuss how the following tower of notiosn works

circle n-bundle with (n1)(n-1)-form connectionLie n-algebra of of bisections of Atiyah Lie n-group
circle bundleLie algebra of sections of Atiyah Lie algebroid
circle 2-bundle with 1-form connectionLie 2-algebra of sections of Courant Lie 2-algebroid

For n,kn,k \in \mathbb{N} and knk \leq n write

B nU(1) conn kDK[U(1)dlogΩ 1ddΩ k1dΩ k000 nk] \mathbf{B}^n U(1)_{conn^k} \coloneqq DK\left[ U(1) \stackrel{d log}{\to} \Omega^1 \stackrel{d}{\to} \cdots \stackrel{d}{\to} \Omega^{k-1} \stackrel{d}{\to} \Omega^k \to \underbrace{ 0 \to 0 \to \cdots \to 0 }_{n-k} \right]

for the smooth ∞-groupoid which is presented under the Dold-Kan correspondence by the sheaf of chain complexes, as indicated. This is such that for k=nk = n we have the Deligne complex, representing the moduli ∞-stack of circle n-bundles with connection

B nU(1) conn nB nU(1) conn \mathbf{B}^n U(1)_{conn^n} \simeq \mathbf{B}^n U(1)_{conn}

and for k=0k = 0 we have the moduli \infty-stack for plain circle n-group principal ∞-bundles

B nU(1) conn 0B nU(1). \mathbf{B}^n U(1)_{conn^0} \simeq \mathbf{B}^n U(1) \,.

For k 2<k 1k_2 \lt k_1 there are evident truncation maps

B nU(1) conn k 1B nU(1) conn k 2. \mathbf{B}^n U(1)_{conn^{k_1}} \to \mathbf{B}^n U(1)_{conn^{k_2}} \,.

Now for XX \in SmthMfd \hookrightarrow Smooth∞Grpd a smooth manifold, a map

:XB nU(1) conn \nabla \;\colon\; X \to \mathbf{B}^n U(1)_{conn}

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 H /B nU(1)\mathbf{H}_{/\mathbf{B}^n U(1)} this has an automorphism ∞-group. The concretification of this (…) is the quantomorphism n-group QuantMorph()QuantMorph(\nabla).

QuantMorph()concAut()={X ϕ X B nU(1) conn}. \mathbf{QuantMorph}(\nabla) \coloneqq conc\mathbf{Aut}(\nabla) = \left\{ \array{ X && \underoverset{\simeq}{\phi}{\to} && X \\ & {}_{\mathllap{\nabla}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}^n U(1)_{conn} } \right\} \,.

But we can also first forget the nn-form pieces of the prequantum nn-bundle away and consider

n1:XB nU(1) connB nU(1) conn nB nU(1) conn n1. \nabla_{n-1} \;\colon\; X \stackrel{\nabla}{\to} \mathbf{B}^n U(1)_{conn} \simeq \mathbf{B}^n U(1)_{conn^n} \to \mathbf{B}^n U(1)_{conn^{n-1}} \,.

For n=2n = 2 this is sometimes known in the literature as a “bundle gerbe with connective structure but without curving”.

The concretifid automorphism ∞-group of that truncated connection is the n-group of of bisection of the Atiyah n-groupoid?

concAut( n1)Grp(SmoothGrpd). conc \mathbf{Aut}(\nabla_{n-1}) \in Grp(Smooth \infty Grpd) \,.

For n=1n = 1 this is the group of bisections of the Atiyah Lie groupoid of the underlying circle principal bundle 0:XBU(1)\nabla_0 \colon X \to \mathbf{B} U(1). Hence its Lie differentiation is the Lie algebra of sections of the coresponding Atiyah Lie algebroid.

For n=2n = 2 the Lie differentiation of this Lie 2-group is the Lie 2-algebra of sections of the corresponding Courant Lie 2-algebroid. With a little bit of translation, this is what is shown in (Collier).

Finally notice that the forgetful map B nU(1) connB nU(1) conn n1\mathbf{B}^n U(1)_{conn} \to \mathbf{B}^n U(1)_{conn^{n-1}} induces an homomorphism of ∞-groups

concAut()concAut( n1) conc \mathbf{Aut}(\nabla) \to conc \mathbf{Aut}(\nabla_{n-1})

hence an embedding of the quantomorphism n-group into the nn-group of bisections of the Atiyah n-groupoid. For n=2n = 2 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

higher Atiyah groupoid:standard higher Atiyah groupoidhigher Courant groupoidgroupoid version of quantomorphism n-group
coefficient for cohomology:B𝔾\mathbf{B}\mathbb{G}B(B𝔾 conn)\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})B𝔾 conn\mathbf{B} \mathbb{G}_{conn}
type of fiber ∞-bundle:principal ∞-bundleprincipal ∞-connection without top-degree connection formprincipal ∞-connection

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

nn \in \mathbb{N}symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of (n+1)(n+1)-d sigma-modelhigher symplectic geometry(n+1)(n+1)d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension (n+1)(n+1)discussed in:
0symplectic manifoldsymplectic manifoldsymplectic geometryLagrangian submanifoldordinary space of states (in geometric quantization)geometric quantization
1Poisson Lie algebroidsymplectic groupoid2-plectic geometryPoisson sigma-modelcoisotropic submanifold (of underlying Poisson manifold)brane of Poisson sigma-model2-module = category of modules over strict deformation quantiized algebra of observablesextended geometric quantization of 2d Chern-Simons theory
2Courant Lie 2-algebroidsymplectic 2-groupoid3-plectic geometryCourant sigma-modelDirac structureD-brane in type II geometry
nnsymplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometryd=n+1d = n+1 AKSZ sigma-model

(adapted from Ševera 00)


The original references in order of appearance are

  • Pavol Severa, Letters to A. Weinstein (web)

  • Dmitry Roytenberg, Alan Weinstein, Courant algebroids and strongly homotopy Lie algebras, Lett. Math. Physics 46(1):81-93, 1998.

  • Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds PhD thesis, University of California, Berkeley, 1999. (math.DG/9910078)

  • 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

A discussion of Courant algebroids with an eye towards the relation of the standard Courant algebroid to bundle gerbes is

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

The embedding of the Poisson Lie 2-algebra of a given 2-plectic geometry into the Lie 2-algebra of sections of the Courant Lie 2-algebroid of the corresponding prequantum 2-bundle is observed in

This is developed further in

See also

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

A proposal for a higher analog of the standard Courant algebroid with the generalized tangent bundle TXT *XT X \oplus T^* X replaced by TX nT *XT X \oplus \wedge^n T^* X – for a notion of standard higher Courant Lie algebroid? – is discussed in

  • Marco Zambon, L L_\infty-algebras and higher analogues of Dirac structures and Courant algebroids, J. Symplectic Geometry, (arXiv:1003.1004)

The relation to ∞-Chern-Simons theory is discussed in

Revised on December 11, 2013 04:11:19 by Urs Schreiber (