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.
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).
Write
for the subspace of the direct sum of vector fields $v$ on $X$ and differential (n-1)-forms $J$ on $X$ satisfying
We call these the pairs of Hamiltonian forms with their Hamiltonian vector fields.
The L-∞ algebra $L_\infty(X,\omega)$ has as underlying chain complex the truncated and modified de Rham complex
with the Hamiltonian pairs, def. 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:
$l_1(J) = \mathbf{d}J$
$l_{k \geq 2}(v_1 + J_1, \cdots, v_k + J_k) = - (-1)^{\left(k+1 \atop 2\right)} \iota_{v_1 \wedge \cdots \wedge v_k}\omega$.
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
$[v_1 + \overline{\theta}_1, v_2 + \overline{\theta}_2] = [v_1, v_2] + \mathcal{L}_{v_1}\overline{\theta}_2 - \mathcal{L}_{v_2}\overline{\theta}_1$;
$[v+ \overline{\theta}, \overline{\eta}] = - [\eta, v + \overline{\theta}] = \mathcal{L}_v \overline{\eta}$.
There is an equivalence in the homotopy theory of L-∞ algebras
between the $L_\infty$-algebras of def. 2 and def. 3 (in particular def. 3 does not depend on the choice of $\overline{A}$) whose underlying chain map satisfies
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. 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
where the cocycle $\omega[\bullet]$, when realized on the model of def. 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
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)
(extension are listed for sufficiently connected $X$)
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 geometric prequantum theory (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)
See also