Contents

Idea

The Lie n-algebra that generalizes the Poisson bracket from symplectic geometry to n-plectic geometry: the Poisson bracket $L_\infty$-algebra of local observables in higher prequantum geometry.

More discussion is here at n-plectic geometry.

Applied to the symplectic current (in the sense of covariant phase space theory, de Donder-Weyl field theory) this is the higher current algebra (see there) of conserved currents of a prequantum field theory.

Definition

Throughout, Let $X$ be a smooth manifold, let $n \geq 1$ a natural number and $\omega \in \Omega^{n+1}_{cl}(X)$ a closed differential (n+1)-form on $X$. The pair $(X,\omega)$ we may regard as a pre-n-plectic manifold.

We define two L-∞ algebras defined from this data and discuss their equivalence. Either of the two or any further one equivalent to the two is the Poisson bracket Lie $n$-albebra of $(X,\omega)$. The first definition is due to (Rogers 10), the second due to (FRS 13b). Here in notation we follow (FRS 13b).

Definition

Write

$Ham^{n-1}(X) \subset Vect(X) \oplus \Omega^{n-1}(X)$

for the subspace of the direct sum of vector fields $v$ on $X$ and differential (n-1)-forms $J$ on $X$ satisfying

$\iota_v \omega + \mathbf{d} J = 0 \,.$

We call these the pairs of Hamiltonian forms with their Hamiltonian vector fields.

Definition

The L-∞ algebra $L_\infty(X,\omega)$ has as underlying chain complex the truncated and modified de Rham complex

$\Omega^0(X) \stackrel{\mathbf{d}}{\to} \Omega^1(X) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^{n-2}(X) \stackrel{(0,\mathbf{d})}{\longrightarrow} Ham^{n-1}(X)$

with the Hamiltonian pairs, def. , in degree 0 and with the 0-forms (smooth functions) in degree $n-1$, and its non-vanishing $L_\infty$-brackets are as follows:

• $l_1(J) = \mathbf{d}J$

• $l_{k \geq 2}(v_1 + J_1, \cdots, v_k + J_k) = - (-1)^{\left(k+1 \atop 2\right)} \iota_{v_1 \wedge \cdots \wedge v_k}\omega$.

Definition

Let $\overline{A}$ be any Cech-Deligne-cocycle relative to an open cover $\mathcal{U}$ of $X$, which gives a prequantum n-bundle for $\omega$. The L-∞ algebra $dgLie_{Qu}(X,\overline{A})$ is the dg-Lie algebra (regarded as an $L_\infty$-algebra) whose underlying chain complex is

$dgLie_{Qu}(X,\overline{A})^0 = \{v+ \overline{\theta} \in Vect(X)\oplus Tot^{n-1}(\mathcal{U}, \Omega^\bullet) \;\vert\; \mathcal{L}_v \overline{A} = \mathbf{d}_{Tot}\overline{\theta}\}$;

$dgLie_{Qu}(X,\overline{A})^{i \gt 0} = Tot^{n-1-i}(\mathcal{U},\Omega^\bullet)$

with differential given by $\mathbf{d}_{Tot}$ (where $Tot$ refers to total complex of the Cech-de Rham double complex).

The non-vanishing dg-Lie bracket on this complex are defined to be

• $[v_1 + \overline{\theta}_1, v_2 + \overline{\theta}_2] = [v_1, v_2] + \mathcal{L}_{v_1}\overline{\theta}_2 - \mathcal{L}_{v_2}\overline{\theta}_1$;

• $[v+ \overline{\theta}, \overline{\eta}] = - [\eta, v + \overline{\theta}] = \mathcal{L}_v \overline{\eta}$.

Proposition

There is an equivalence in the homotopy theory of L-∞ algebras

$f \colon L_\infty(X,\omega) \stackrel{\simeq}{\longrightarrow} dgLie_{Qu}(X,\overline{A})$

between the $L_\infty$-algebras of def. and def. (in particular def. does not depend on the choice of $\overline{A}$) whose underlying chain map satisfies

• $f(v + J) = v - J|_{\mathcal{U}} + \sum_{i = 0}^n (-1)^i \iota_v A^{n-i}$.

Properties

The extension theorem

Proposition

Given a pre n-plectic manifold $(X,\omega_{n+1})$, then the Poisson bracket Lie $n$-algebra $\mathfrak{Pois}(X,\omega)$ from above is an extension of the Lie algebra of Hamiltonian vector fields $Vect_{Ham}(X)$, def. by the cocycle infinity-groupoid $\mathbf{H}(X,\flat \mathbf{B}^{n-1} \mathbb{R})$ for ordinary cohomology with real number coefficients in that there is a homotopy fiber sequence in the homotopy theory of L-infinity algebras of the form

$\array{ \mathbf{H}(X,\flat \mathbf{B}^{d-1}\mathbb{R}) &\longrightarrow& \mathfrak{Pois}(X,\omega) \\ && \downarrow \\ && Vect_{Ham}(X,\omega) &\stackrel{\omega[\bullet]}{\longrightarrow}& \mathbf{B} \mathbf{H}(X,\flat \mathbf{B}^{d-1}\mathbb{R}) } \,,$

where the cocycle $\omega[\bullet]$, when realized on the model of def. , is degreewise given by by contraction with $\omega$.

This is FRS13b, theorem 3.3.1.

As a corollary this means that the 0-truncation $\tau_0 \mathfrak{Pois}(X,\omega)$ is a Lie algebra extension by de Rham cohomology, fitting into a short exact sequence of Lie algebras

$0 \to H^{d-1}_{dR}(X) \longrightarrow \tau_0 \mathfrak{Pois}(X,\omega) \longrightarrow Vect_{Ham}(X) \to 0 \,.$
Remark

These kinds of extensions are known traditionally form current algebras.

slice-automorphism ∞-groups in higher prequantum geometry

cohesive ∞-groups:Heisenberg ∞-group$\hookrightarrow$quantomorphism ∞-group$\hookrightarrow$∞-bisections of higher Courant groupoid$\hookrightarrow$∞-bisections of higher Atiyah groupoid
L-∞ algebras:Heisenberg L-∞ algebra$\hookrightarrow$Poisson L-∞ algebra$\hookrightarrow$Courant L-∞ algebra$\hookrightarrow$twisted vector fields

higher and integrated Kostant-Souriau extensions:

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $\mathbb{G}$-principal ∞-connection)

$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$
$n$geometrystructureunextended structureextension byquantum extension
$\infty$higher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of $(\Omega \mathbb{G})$-flat ∞-connections on $X$quantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
$n$n-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
$n$smooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected $X$)

The Poisson bracket $L_\infty$-algebra $L_\infty(X,\omega)$ was introduced in

Discussion in the broader context of higher differential geometry and higher prequantum geometry is in