nLab Courant algebroid

Idea

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

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

of degree -2 is non-degenerate.

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

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

History

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.

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

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Examples

Lie algebras of compact type

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

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

The standard Courant algebroid of a manifold $X$ is the one which

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

  • bilinear form

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

  for $X,Y \in \Gamma(T X)$ and $\xi, \eta \in \Gamma(T^* X)$

  • brackets

    $$[X + \xi, Y + \eta] = [X,Y] + \mathcal{L}_X \eta - \mathcal{L}_Y \xi + \frac{1}{2} d (\eta(X) - \xi(Y))$$

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

• as a dg-manifold is $T^*[2] T[1] X$, the shifted cotangent bundle of the shifted tangent bundle,

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

  $$\begin{aligned} d_C &= d_{dR} + p_i \frac{\partial}{\partial \theta_i} \\ &= dx^i \frac{\partial}{\partial x^i } + p_i \frac{\partial}{\partial \theta_i} \end{aligned} \,.$$

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.

Properties

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 $\sigma$-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 notions works

circle n-bundle with $(n-1)$-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 $n,k \in \mathbb{N}$ and $k \leq n$ write

$$\mathbf{B}^n U(1)_{conn^k} \coloneqq DK\left[ U(1) \stackrel{d log}{\to} \Omega^1 \stackrel{d}{\to} \cdots \stackrel{d}{\to} \Omega^{k-1} \stackrel{d}{\to} \Omega^k \to \underbrace{ 0 \to 0 \to \cdots \to 0 }_{n-k} \right]$$

for the smooth ∞-groupoid which is presented under the Dold-Kan correspondence by the sheaf of chain complexes, as indicated (see also at differential cohomology diagram – Examples – Deligne coefficients). This is such that for $k = n$ we have the Deligne complex, representing the moduli ∞-stack of circle n-bundles with connection

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

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

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

For $k_2 \lt k_1$ there are evident truncation maps

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

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

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

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

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

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

$$\nabla_{n-1} \;\colon\; X \stackrel{\nabla}{\to} \mathbf{B}^n U(1)_{conn} \simeq \mathbf{B}^n U(1)_{conn^n} \to \mathbf{B}^n U(1)_{conn^{n-1}} \,.$$

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

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

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

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

For $n = 2$ 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 $\mathbf{B}^n U(1)_{conn} \to \mathbf{B}^n U(1)_{conn^{n-1}}$ induces an homomorphism of ∞-groups

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

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

Related concepts

higher Atiyah groupoid

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

• string algebroid

• symplectic Lie n-algebroid

  • symplectic manifold

  • Poisson Lie algebroid

  • Courant algebroid

• Dirac structure

• Courant sigma-model

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

$n \in \mathbb{N}$	symplectic Lie n-algebroid	Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-model	higher symplectic geometry	$(n+1)$d sigma-model	dg-Lagrangian submanifold/ real polarization leaf	= brane	(n+1)-module of quantum states in codimension $(n+1)$	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
$n$	symplectic Lie n-algebroid	symplectic n-groupoid	(n+1)-plectic geometry	$d = n+1$ AKSZ sigma-model

(adapted from Ševera 00)

References

The original references in order of appearance are

• Pavol Ševera, Letters to A. Weinstein (web, arXiv:1707.00265)

• 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 Ševera, 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)

• Dmitry Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds , Letters in Mathematical

  Physics, 61(2):123-137 (2002) (arXiv:math/0112152)

Another useful summary of the theory of Courant algebroids is in of

• Marco Gualtieri, section 3 of: Generalized complex geometry (arXiv:math/0401221)

On the morphisms between Courant algebroids:

• Jan Vysoky, Hitchhiker’s Guide to Courant Algebroid Relations (arXiv:1910.05347)

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

• Paul Bressler, Alexander Chervov, Courant algebroids (hep-th/0212195)

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

• Braxton Collier, Infinitesimal Symmetries of Dixmier-Douady Gerbes (arXiv:1108.1525)

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

• Chris Rogers, Courant algebroids from categorified symplectic geometry, (arXiv:1001.0040)

This is developed further in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, L-∞ algebras of local observables from higher prequantum bundles (arXiv:1304.6292)

See also

• Wikipedia, Courant algebroid

The relation between the two different Lie-algebroid-like definition of Courant algebroids, one with skew, the other with non-skew brackets inspired on the level of Lie 2-algebras the treatment

• Dmitry Roytenberg, On weak Lie 2-algebras (arXiv/0712.3461)

• Chris Rogers, 2-plectic geometry, Courant algebroids, and categorified prequantization, (arxiv:1009.2975)

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

• Marco Zambon, $L_\infty$-algebras and higher analogues of Dirac structures and Courant algebroids, J. Symplectic Geometry, J. Sympl. Geom. 10:4 (2012) 563–599 arXiv:1003.1004 Zbl:1260.53134 doi

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

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher Chern-Weil Derivation of AKSZ Sigma-Models

Discussion of Riemannian geometry on Courant algebroids and relation to supergravity equations of motion is in

• Branislav Jurco, Jan Vysoky, Courant Algebroid Connections and String Effective Actions, Proceedings of Tohoku Forum for Creativity, Special volume: Noncommutative Geometry and Physics IV (arXiv:1612.01540)

See also

• Xu Xiaomeng, Twisted Courant algebroids and coisotropic Cartan geometries (arXiv:1206.2282)