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Courant algebroid in nLab

Contents

Context

∞\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

Lie theory

∞-Lie groupoids

∞-Lie groupoid
- strict ∞-Lie groupoid
- Lie groupoid
  - differentiable stack
  - orbifold
∞-Lie group
- Lie group
  - simple Lie group, semisimple Lie group
- Lie 2-group

∞-Lie algebroids

Formal Lie groupoids

formal group, formal groupoid

Cohomology

Homotopy

homotopy groups of a Lie groupoid

Related topics

∞-Chern-Weil theory

Examples

∞\infty-Lie groupoids

∞\infty-Lie groups

∞\infty-Lie algebroids

∞\infty-Lie algebras

Symplectic geometry

symplectic geometry

higher symplectic geometry

Background

Basic concepts

Classical mechanics and quantization

Properties

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 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 is 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 Θ∈CE(𝔓)\Theta \in CE(\mathfrak{P}) such that

d CE(𝔓)={Θ,−}. 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.

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

Examples

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=TX⊕T *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,Y∈Γ(TX)X,Y \in \Gamma(T X) and ξ,η∈Γ(T *X)\xi, \eta \in \Gamma(T^* X)
- brackets
  
  [X+ξ,Y+η]=[X,Y]+ℒ Xη−ℒ 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 ℒ 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 T *[2]T[1]XT^*[2] T[1] X, the shifted cotangent bundle of the shifted tangent bundle,

where the differential is on each local coordinate patch ℝ n≃U⊂X\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 i∂∂θ i =dx i∂∂x i+p i∂∂θ 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.

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 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 notions works

circle n-bundle with (n−1)(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∈ℕn,k \in \mathbb{N} and k≤nk \leq n write

B nU(1) conn k≔DK[U(1)→dlogΩ 1→d⋯→dΩ k−1→dΩ k→0→0→⋯→0⏟ n−k] \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=nk = n we have the Deligne complex, representing the moduli ∞-stack of circle n-bundles with connection

B nU(1) conn n≃B 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 0≃B 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 1→B nU(1) conn k 2. \mathbf{B}^n U(1)_{conn^{k_1}} \to \mathbf{B}^n U(1)_{conn^{k_2}} \,.

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

∇:X→B 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

∇ n−1:X→∇B nU(1) conn≃B nU(1) conn n→B nU(1) conn n−1. \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 concretified automorphism ∞-group of that truncated connection is the n-group of of bisection of the Atiyah n-groupoid?

concAut(∇ n−1)∈Grp(Smooth∞Grpd). 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:X→BU(1)\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=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) conn→B nU(1) conn n−1\mathbf{B}^n U(1)_{conn} \to \mathbf{B}^n U(1)_{conn^{n-1}} induces an homomorphism of ∞-groups

concAut(∇)→concAut(∇ n−1) 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).

string algebroid
symplectic Lie n-algebroid
- symplectic manifold
- Poisson Lie algebroid
- Courant algebroid
Dirac structure
Courant sigma-model

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)