∞-Lie theory (higher 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 $\mathfrak{P}$ whose Chevalley-Eilenberg algebra $CE(\mathfrak{P})$ is equipped with the structure of a Poisson 3-algebra whose Poisson bracket
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
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 …)
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.
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
for $X,Y \in \Gamma(T X)$ and $\xi, \eta \in \Gamma(T^* X)$
brackets
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 $\Pi T^* \Pi T X$, 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
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.
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.
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.
The Lagrangian dg-submanifolds of a Courant Lie 2-algebroid corespond to its Dirac structures.
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 |
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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
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
and for $k = 0$ we have the moduli $\infty$-stack for plain circle n-group principal ∞-bundles
For $k_2 \lt k_1$ there are evident truncation maps
Now for $X \in$ SmthMfd $\hookrightarrow$ Smooth∞Grpd a smooth manifold, a map
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 $QuantMorph(\nabla)$.
But we can also first forget the $n$-form pieces of the prequantum $n$-bundle away and consider
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?
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
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).
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 |
Courant algebroid
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(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
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
The relation to ∞-Chern-Simons theory is discussed in
See also