nLab Poisson bracket Lie n-algebra

1. 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.

2. 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 2.1. 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.

(FRS 13b, def. 2.1.3)

Definition 2.2. 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. 2.1, in degree 0 and with the 0-forms (smooth functions) in degree $n-1$, and its non-vanishing $L_\infty$-brackets are as follows:

• $l_1(J) = \mathbf{d}J$

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

(FRS 13b, prop. 3.1.2)

Definition 2.3. 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

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

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

(FRS 13b, def./prop. 4.2.1)

Proposition 2.4. 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. 2.2 and def. 2.3 (in particular def. 2.3 does not depend on the choice of $\overline{A}$) whose underlying chain map satisfies

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

(FRS 13b, theorem 4.2.2)

3. Properties

The extension theorem

Proposition 3.1. 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. 2.1 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. 2.2, 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 3.2. These kinds of extensions are known traditionally form current algebras.

4. Related concepts

• higher Poisson structure

  • Poisson n-algebra

  • Nambu bracket

• current algebra

• geometry of physics – prequantum geometry

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)

$$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$$

$n$	geometry	structure	unextended structure	extension by	quantum extension
$\infty$	higher prequantum geometry	cohesive ∞-group	Hamiltonian symplectomorphism ∞-group	moduli ∞-stack of $(\Omega \mathbb{G})$-flat ∞-connections on $X$	quantomorphism ∞-group
1	symplectic geometry	Lie algebra	Hamiltonian vector fields	real numbers	Hamiltonians under Poisson bracket
1		Lie group	Hamiltonian symplectomorphism group	circle group	quantomorphism group
2	2-plectic geometry	Lie 2-algebra	Hamiltonian vector fields	line Lie 2-algebra	Poisson Lie 2-algebra
2		Lie 2-group	Hamiltonian 2-plectomorphisms	circle 2-group	quantomorphism 2-group
$n$	n-plectic geometry	Lie n-algebra	Hamiltonian vector fields	line Lie n-algebra	Poisson Lie n-algebra
$n$		smooth n-group	Hamiltonian n-plectomorphisms	circle n-group	quantomorphism n-group

(extension are listed for sufficiently connected $X$)

5. References

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

• Chris Rogers, $L_\infty$ algebras from multisymplectic geometry, Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50 (arXiv:1005.2230, journal).

• Chris Rogers, Higher symplectic geometry PhD thesis (2011) (arXiv:1106.4068)

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

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher $U(1)$-gerbe connections in geometric prequantization, Rev. Math. Phys., Vol. 28, Issue 06, 1650012 (2016) (arXiv:1304.0236)

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, L-∞ algebras of local observables from higher prequantum bundles, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 (arXiv:1304.6292)

• Nestor Leon Delgado, Lagrangian field theories: ind/pro-approach and L-infinity algebra of local observables (arXiv:1805.10317)

See also

• Patricia Ritter, Christian Sämann, Automorphisms of Strong Homotopy Lie Algebras of Local Observables (arXiv:1507.00972)