Schreiber symplectic ∞-Lie algebroid

A symplectic ∞-Lie algebroid is an $\infty$-Lie algebroid equipped with a nondegenerate binary invariant polynomial on it.

Idea

A symplectic $\infty$-Lie algebroid is a pair $(\mathfrak{a}, \omega)$ consisting of an ∞-Lie algebroid $\mathfrak{a}$ and a non-degenrate binary invariant polynomial $\omega$ on $\mathfrak{a}$.

This is equivalent to the definition of n-symplectic manifold (or symplectic NQ-supermanifold) as found in the literature. Here we prefer the fully $\infty$-Lie theoretic language over that of NQ-supermanifolds only because it seems to be more suggestive of the structure of the problem.

Definition

Definition

An $n$-symplectic ∞-Lie algebroid is a pair $(\mathfrak{a}, \omega(-,-))$ consisting of an $\infty$-Lie algebroid $\mathfrak{a}$ and a nondegenerate binary invariant polynomial $\omega \in inv(\mathfrak{a})$ of degree $n+2$.

The Poisson tensor of an $n$-symplectic $\infty$-Lie algebroid is a cocycle $\nu \in CE(\mathfrak{a})$ that is in transgression with $\omega$.

The corresponding L-∞-algebra of sections is the $L_\infty$-algebra of sections of the $\infty$-Lie algebra $\mathfrak{a}_\nu$ obtained by co-killing this cocycle, i.e the homotopy fiber

$$\mathfrak{a}_\nu \to \mathfrak{a} \stackrel{\nu}{\to} b^{n}\mathfrak{u}(1) \,.$$

The following paragraph recalls the meaning of an $\infty$-Lie algebroid cocycle being in transgression with an invariant polynomial.

Transgression cocycles

We recall the procedure by which to an ∞-Lie algebroid invariant polynomial $\omega$ we associate an ∞-Lie algebroid cocycle $\nu$ that is in transgression with $\omega$.

The dg-algebra of invariant polynomials is a sub-dg-alghebra of the kernel of the morphism $W(\mathfrak{a}) \to CE(\mathfrak{a})$ from the Weil algebra to the Chevalley-Eilenberg algebra of $\mathfrak{a}$

$$inv(\mathfrak{a}) \subset CE(\Sigma \mathfrak{a}) = ker(W(\mathfrak{a}) \to CE(\mathfrak{a})) \,.$$

From the short exact sequence

$$CE(\Sigma \mathfrak{a}) \to W(\mathfrak{a}) \to CE(\mathfrak{a})$$

we obtain the long exact sequence in cohomology

$$\cdots \to H^{n+1}(CE(\mathfrak{a})) \stackrel{\delta}{\to} H^{n+2}(CE(\Sigma \mathfrak{a})) \to \cdots \,.$$

We say that $\mu \in CE(\mathfrak{a})$ is in transgression with $\omega \in inv(\mathfrak{a}) \subset CE(\Sigma \mathfrak{a})$ if their classes map to each other under the connecting homomorphism $\delta$:

$$\delta : [\mu] \mapsto [\omega] \,.$$

The following spells out in detail how one finds to a given invariant polynomial $\omega$ the cocycle that it is in transgression with.

1. We first regard the invariant polynomial $\omega$ as an element of the Weil algebra $W(\mathfrak{a})$ under the inclusion $inv(\mathfrak{a}) \hookrightarrow W(\mathfrak{a})$, where, by the very definiton of invariant polynomials, it is closed: $d_{W(\mathfrak{a})} \omega = 0$.

2. then we find an element $cs_\omega \in W(\mathfrak{a})$ with the property that $d_{W(\mathfrak{a})} cs_\omega = \omega$. This is guranteed to exist because $W(\mathfrak{a})$ has trivial cohomology.

3. then we send this element $cs_\omega\in W(\mathfrak{a})$ along the restriction map $W(\mathfrak{a}) \to CS(\mathfrak{a})$ to an elemeent we call $\nu$.

The procedure is illustarted by the following diagram

$$\array{ 0 && \omega &\leftarrow & \omega \\ \;\;\uparrow^{\mathrlap{d_{CE(\mathfrak{a})}}} && \;\;\uparrow^{\mathrlap{d_{W(\mathfrak{a})}}} \\ \nu &\leftarrow& cs(\omega) \\ \\ \\ \\ CE(\mathfrak{a}) &\leftarrow& W(\mathfrak{a}) &\leftarrow& inv(\mathfrak{a}) }$$

From the fact that all morphisms involved respect the differential and from the fact that the image of $\omega$ in $CE(\mathfrak{a})$ vanishes it follows that

• this element $\nu$ satisfies $d_{CE(\mathfrak{a})} \nu = 0$, hence that it is an $\infty$-Lie algebroid cocycle.

• any two different choices of $cs_\omega$ lead to cocylces $\mu$ that are cohomologous.

We say $\nu$ is a cocycle in transgression with $\omega$. We may call $cs_{\omega}$ here a Chern-Simons element of $\omega$. Because for $A : T X \to \mathfrak{a}$ any collection of ∞-Lie algebroid valued differential forms coming dually from a dg-morphism $\Omega^\bullet(X) \leftarrow W(\mathfrak{a}) : A$ the image $\omega(A)$ of $\omega$ will be a curvature characteristic form and the image $cs_\omega(A)$ its corresponding Chern-Simons form.

In the case where $\mathfrak{g}$ is an ordinary semisimple Lie algebra, this reduces to the ordinary study of ordinary Chern-Simons 3-forms associated with $\mathfrak{g}$-valued 1-forms. This is described in the section Semisimple Lie algebras .

Examples

We spell out examples of symplectic $n$-Lie algebroids in order or increasing $n$.

The simplest example that makes closest contact to well familiar objects and constructions is that where $\mathfrak{a} = \mathfrak{g}$ is an ordinary semisimple Lie algebra and $\omega(-,-) = \langle -,-\rangle$ its ordinary binary invariant polynomial. Since $\omega$ here is of degree 4 in $W(\mathfrak{g})$ this is really a degenerate example of a symplectic Lie 2-algebroid and is accordingly listed in that subsection. The reader in need of pedagogical examples should start with that example and then jump back to the section on Poisson manifolds.

Symplectic manifolds

When the binary $\omega$ is of degree 2 it must be locally the wedge product of two elements of degree 1. Since these must be in the shifted copy of the Weil algebra they must come from element in degree 0 of $\mathfrak{a}$. For $\omega$ to be non-degenerate, this implies that $\mathfrak{a}$ may not have generators of higher degree. Hence $\mathfrak{a}$ is a Lie 0-algebroid over a manifold $X$, so it is just that manifold $X$ itself. Its Chevalley-Eilenberg algebra is the algebra of smooth functions on $X$

$$CE(X) = C^\infty(X) \,.$$

The Weil algebra of $X$ is

$$W(X) = \Omega^\bullet(X)$$

the deRham algebra of $X$. A degree 2-invariant polynomial on $X$ is therefore a non-degenerate closed 2-form $\omega \in \Omega^2(X)$, a symplectic 2-form.

A symplectic manifold, being a pair

$$(X,\;\; \omega)$$

consisting of a smooth manifold $X$ and a symplectic 2-form $\omega$, is a symplectic Lie 0-algebroid.

Poisson manifolds – 1-symplectic manifolds

For a Poisson manifold $X$ with Poisson bivector $\pi \in \Gamma(T X) \wedge \Gamma(T X)$ the Chevalley-Eilenberg algebra $CE(\mathfrak{a})$ of the corresponding Poisson Lie algebroid

$$\mathfrak{a} := \mathfrak{P}(X,\pi)$$

is that of multi-vector fields on $X$, equipped with the differential $d_{CE(\mathfrak{a})} = [\pi, -]_{Sch}$ given by the Schouten bracket.

Let’s be traditional and work locally in coordinates so that we can recognize the following expressions as familiar friends from the literature.

So then $CE(\mathfrak{a})$ is generated from degree 0 elements $x^i$ and degree 1 elements $\partial_i$. The differential is

$$d_{CE(\mathfrak{a})} = [\pi, -]_{Sch} \,.$$

The Poisson tensor is $\nu := \pi = \pi^{i j} \partial_i \wedge \partial_j$ and that this is a Lie algebroid cocycle is the fact that

$$d_{CE(\mathfrak{a})} \pi = [\pi,\pi]_{Sch} = 0 \,.$$

By definition the Weil algebra $W(\mathfrak{a})$ is generated from the $x^i$, the $\partial_i$ and their shifted partners $\mathbf{d}x^i$ and $\mathbf{d}\partial_i$. The differential here is

$$d_{W(\mathfrak{a})} = [\pi , - ] + \mathbf{d} \,.$$

Proposition

We claim the invariant polynomial $\omega$ that is in transgression with the cocycle $\nu = \pi$ is

$$\omega = (\mathbf{d} x^i) \wedge (\mathbf{d} \partial_i) \;\;\; \in inv(\mathfrak{a}) \,.$$

Proof

One checks directly that the element

$$cs_\omega = \pi^{i j} \partial_i \wedge \partial_j + x^i \wedge \mathbf{d} \partial_i$$

is a Chern-Simons transgression element for $\nu$ and $\omega$,

i.e. $d_{W(\mathfrak{a})} cs(\omega) = \omega$. The restriction of $cs_\omega$ to $CE(\mathfrak{a})$ is evidently the Poisson tensor $\pi$.

For a Poisson manifold $X$ with Poisson tensor $\pi = \pi^{i j} \partial_i \wedge \partial_j$, the pair

$$(\mathfrak{P}(X,\pi), \;\;\; \omega = (\mathbf{d} x^i) \wedge (\mathbf{d} \partial_i))$$

consisting of the Poisson lie algebroid $\mathfrak{P}(X,\pi)$ and of the invariant polynomial $\omega$ that is in transgression with its canonical 2-cocycle $\nu = \pi$ (the Poisson tensor) is a symplectic Lie algebroid.

Courant algebroids - 2-symplectic manifolds

see e.g. Roytenberg’s discussion of Courant algebroids and proceed entirely analogously, with $\pi$ replaced by Dmitry’s $\Theta$

For a Courant algebroid $\mathfrak{P}(X,\Theta)$ over a base manifold $X$ the pair

$$(\mathfrak{P}(X,\Theta), \;\;\; \omega$$

for $\omega$ the invariant polynomial that is in transgression with the canonical 3-cocycle $\nu = \Theta$ is a symplectic Lie 2-algebroid.

Semisimple Lie algebras – 2-symplectic points

The most classical and possibly most familiar case of all the examples here is possibly that of a Courant algebroid whose base space is a point. As a symplectic 2-Lie algebroid this is nothing but

• an ordinary Lie algebra $\mathfrak{a} = \mathfrak{g}$

• equipped with a non-degenrate binary invariant polynomial $\omega(-,-) = \langle - , - \rangle$.

For instance a semisimple Lie algebra equipped with its Killing form.

In that case one finds that the cocycle in transgression with $\omega$ is the canonical 3-cocycle

$$\nu = \langle -, [-,-]\rangle$$

of $\mathfrak{g}$. Co-killing this cocycle yields the String Lie 2-algebra $\mathfrak{g}_\nu$, which is at the same time the Lie 2-algebra of sections of this example.

It may be worthwhile to spell this example out in terms of components of a chosen basis.

So let $\{t_a\}_a$ be a basis for the vector space underlying the Lie algebra $\mathfrak{g}$, which we assume for simplicity to be finite-dimensional. Let $\{t^a\}_a$ be a corresponding dual basis of the dual vector space $\mathfrak{g}^*$.

Let $\{C^a{}_{b c}\}_{a,b,c}$ be the structure constants of the Lie bracket in this basis, given by

$$[t_b,t_c] = C^a{}_{b c} t_a \,.$$

The Chevalley-Eilenberg algebra of $\mathfrak{g}$ is

$$CE(\mathfrak{g}) = (\wedge^\bullet \mathfrak{g}^*, d_{CE(\mathfrak{g})})$$

where

$$\wedge^\bullet \mathfrak{g}^* = \mathbb{R} \oplus \mathfrak{g}^* \oplus \mathfrak{g}^* \wedge \mathfrak{g}^* \oplus \mathfrak{g}^* \wedge \mathfrak{g}^*\wedge \mathfrak{g}^* \oplus \cdots$$

is the Grassmann algebra of $\mathfrak{g}^*$. It is the graded-commutative $\mathbb{R}$-algebra whose elements are linear combinations of wedge product expressions $t^a \in \mathfrak{g}^*$, $t^a \wedge t^b = - t^b \wedge t^a \in \mathfrak{g}^* \wedge \mathfrak{g}^*$, and so onw, where $t^a$ is of degree 1.

The differential $d_{CE(\mathfrak{g})}$ is fully specified by specifying it on the generators $t^a \in \mathfrak{g}^*$, because from there it extends uniquely by the graded Leibnitz rule

$$d_{CE(\mathfrak{g})} (a \wedge b) = (d_{CE(\mathfrak{g})} a) \wedge b + (-1)^{deg a} a \wedge d_{CE(\mathfrak{g})} b$$

to the rest of the algebra. On $\mathfrak{g}^*$ the differential is simply the linear-dual of the Lie bracket

$$d_{CE(\mathfrak{g})} |_{\mathfrak{g}^*} = -[-,-]^* \,,$$

where

$$[-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^* \,.$$

In terms of our chosen basis with the structure constants as above this means that

$$d_{CE(\mathfrak{g})} t^a = -\frac{1}{2} C^a{}_{b c} \,.$$

The minus sign here is just convention and could be omitted without changing anything of the discussion. Its purpose is to make these expressions come out nicer when we pass next to the Weil algebra.

That this squares to 0

$$\begin{aligned} 0 &= d_{CE(\mathfrak{g})} \circ d_{CE(\mathfrak{g})} t^a \\ &= d_{CE(\mathfrak{g})} (-\frac{1}{2}C^a{}_{b c} t^b \wedge t^c) \\ &= -\frac{1}{2} C^a_{b c} C^b{}_{d e} t^c \wedge t^d \wedge t^e \\ &= -\frac{1}{2} C^a_{b [c} C^b{}_{d e]} t^c \wedge t^d \wedge t^e \end{aligned}$$

is equivalent to the vaishing of all the components

$$C^a_{b [c} C^b{}_{d e]} = 0 \,,$$

where the square brackets indicate antisymmetrization over the corresponding indices. This is indeed precisely equivalent to the Jacobi identity satisfied by the Lie bracket $[-,-]$, as one can check.

Write now $\{g_{a b}\}$ for the components of the invariant polynomial $\langle -,-\rangle$ on $\mathfrak{g}$ in that

$$g_{a b} := \langle t_a, t_b \rangle \,.$$

Write

$$C_{a b c} := g_{a d} C^d{}_{b c} \,.$$

It is a standard fact in the general theory of Lie algebra cohomology that the element

$$\nu := \frac{1}{6} C_{a b c} t^a \wedge t^b \wedge t^c \in CE(\mathfrak{g})$$

is closed

$$d_{CE(\mathfrak{g})} \nu = 0$$

and hence a Lie algebra 3-cocycle (for which the constant $\frac{1}{6}$ is irrelevant). We can easily see this fact systematically by passing now to the Weil algebra.

The Weil algebra $W(\mathfrak{g})$ is the dg-algebra

$$W(\mathfrak{g}) = (\wedge^\bullet \mathfrak{g}^* \oplus \mathfrak{g}^*[1], d_{W(\mathfrak{g})}) \,,$$

where $\mathfrak{g}^*[1]$ is a copy of $\mathfrak{g}^*$ but shifted up in degree by one. We have

$$\wedge^\bullet \mathfrak{g}^* \oplus \mathfrak{g}^*[1] = \mathbb{R} \oplus \mathfrak{g}^* \oplus (\mathfrak{g}^* \wedge \mathfrak{g}^* \oplus \mathfrak{g}^*[1]) \oplus \cdots .$$

Write $\{r^a\}$ for the basis of $\mathfrak{g}^*[1]$ obatained from the basis $\{t^a\}$ of $\mathfrak{g}^*$.

If we write $\sigma : \mathfrak{g}^* \to \mathfrak{g}^* [1]$ for the degree 1 morphism of graded vector spaces that is the identity on the underlying un-graded vector spaces, then

$$r^a = \sigma t^a \,.$$

The differential $d_{W(\mathfrak{g})}$ on the Weil algebra is again fixed by its action generators. Restricted to the original generators we have

$$d_{W(\mathfrak{g})} |_{\mathfrak{g}^*} = d_{CE(\mathfrak{g})} + \sigma^* \,.$$

In terms of our chosen basis this reads

$$d_{W(\mathfrak{g})} t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a \,.$$

One finds that from this and using $(d_{W(\mathfrak{g})})^2 = 0$ the action of $d_{W(\mathfrak{g})}$ is already fixed. It is

$$d_{W(\mathfrak{g})}|_{\mathfrak{g}^*[1]} = - \sigma^* \circ d_{CE(\mathfrak{g})}\circ (\sigma^*)^{-1} \,.$$

In terms of our basis this is

$$d_{W(\mathfrak{g})} r^a = - C^a{}_{b c} t^b \wedge r^c \,.$$

Now, the invariant polynomial $\langle-,-\rangle$ is to be regarded as an element in the shifted part of $W(\mathfrak{g})$ which in terms of our basis reads

$$\omega = g_{a b} r^a \wedge r^b \,.$$

Notice that since the $r^a$ are in degree 2, we have $r^a \wedge r^b = + r^b \wedge r^a$.

The statement that this is a closed element of $W(\mathfrak{g})$ is now

$$\begin{aligned} 0 &= d_{W(\mathfrak{g})} r^a \\ &= - g_{a d}C^{b c} r^a \wedge r^b \wedge t^c \\ &= C_{a b c} r^a \wedge r^b \wedge t^c \\ &= C_{(a b) c} r^a \wedge r^b \wedge t^c \end{aligned}$$

and hence that all the components

$$C_{(a b) c} = 0$$

vanish. Indeed, we see that this means that $\{C_{a b c}\}$ is completely antisymmetric in all its indices.

Now we want to find the Chern-Simons element corresponding to $\omega$. There is a systematic formula for finding that, but in the simple case at hand here one sees that it has to be of the form

$$cs_\omega = \frac{1}{6}C_{a b c} t^a \wedge t^b \wedge t^c + g_{a b} t^a \wedge r^b$$

Then indeed

$$\begin{aligned} d_{W(\mathfrak{g})} cs_\omega &= d_{W(\mathfrak{g})} ( \frac{1}{6} C_{a b c} t^a \wedge t^b \wedge t^c + g_{a b} t^a \wedge r^b ) \\ &= g_{a b} r^a \wedge r^b = \omega \,. \end{aligned}$$

Finally, the restriction of $cs_\omega$ along the projection $W(\mathfrak{g}) \to CE(\mathfrak{g})$ is simply obtained by setting all the shifted generators $r^a$ to 0. So this does yield $\nu = \frac{1}{6}C_{a b c}t^a \wedge t^b \wedge t^c$ as the transgression cocycle

The pair

$$(\mathfrak{g}, \;\; \omega = \langle -,-\rangle)$$

consisting of a semisimple Lie algebra $\mathfrak{g}$ and the invariant polynomial given by its invariant bilinear form (with some choice of normalization) is a symplectic Lie 2-algebra.

References

For reference on the notion of n-symplectic manifold in general and Poisson Lie algebroid and Courant algebroid in particular, please see these entries.

A discussion of cocycles and invariant polynomials in transgression on an $\infty$-Lie algebroid is in

• Sati, Schreiber, Stasheff, $L_\infty$-connections in: B. Fauser, J. Tolksdorf, E. Zeidler (eds.) Recent Progress in

  Quantum Field Theory – Competetive Models_ Birkhäuser, Basel (2009) (arXiv)