$n$-plectic geometry is a generalization of symplectic geometry to higher category theory.
It is closely related to multisymplectic geometry and n-symplectic manifolds.
For $n \in \mathbb{N}$, an n-plectic vector space is a vector space $V$ (over the real numbers) equipped with an $(n+1)$-linear skew function
such that regarded as a function
is has trivial kernel.
Let $X$ be a smooth manifold, $\omega \in \Omega^{n+1}(X)$ a differential form.
We say $(X,\omega)$ is a $n$-plectic manifold if
$\omega$ is closed: $d \omega = 0$;
for all $x \in X$ the map
given by contraction of vectors with forms
is injective.
See also the definition at multisymplectic geometry.
$n = 1$ – this is the case of an ordinary symplectic manifold this appears in Hamiltonian mechanics;
$n = 2$, this appears naturally in 1+1 dimensional quantum field theory.
For $X$ orientable, take $\omega$ the volume form. This is $(dim(X)-1)$-plectic.
$\wedge^n T^* X \to X$
$G$ a compact simple Lie group,
let $\nu : (x,y,z) \mapsto \langle x, [y,z]\rangle$ be the canonical Lie algebra 3-cocycle and extend it left-invariantly to a 3-form $\omega_\nu$ on $G$. Then $(G,\omega_\nu)$ is 2-plectic.
To an ordinary symplectic manifold is associated its Poisson algebra. Underlying this is a Lie algebra, whose Lie bracket is the Poisson bracket.
We discuss here how to an $n$-plectic manifold for $n \gt 1$ there is correspondingly assoociated not a Lie algebra, but a Lie n-algebra: the Poisson bracket Lie n-algebra. It is natural to call this a Poisson Lie $n$-algebra (see for instance at Poisson Lie 2-algebra).
(Not to be confused with the Lie algebra of a Poisson Lie group, which is a Lie group that itself is equipped with a compatible Poisson manifold structure. It is a bit unfortunate that there is no better established term for the Lie algebra underlying a Poisson algebra apart from “Poisson bracket”.)
Consider the ordinary case in $n=1$ for how a Poisson algebra is obtained from a symplectic manifold $(X, \omega)$.
Here
is an isomorphism.
Given $f \in C^\infty(X)$, $\exists ! \nu_f \in \Gamma(T X)$ such that $d f = - \omega(v_f, -)$
Define $\{f,g\} := \omega(v_f, v_g)$. Then $(C^\infty(X,), \{-,-\})$ is a Poisson algebra.
We can generalize this to $n$-plectic geometry.
Let $(X,\omega)$ be $n$-plectic for $n \gt 1$.
Observe that then $\hat \omega : T_x X \to \wedge^n T_x X$ is no longer an isomorphism in general.
Definition
Say
is Hamiltonian precisely if
such that
This makes $v_\alpha$ uniquely defined.
Denote the collection of Hamiltonian forms by $\Omega^{n-1}_{Hamilt}(X)$.
Define a bracket
by
This satisfies
k
$\{-,-\}$ is skew-symmetric;
$\{\alpha_1, \{\alpha_2, \alpha_3\}\}$ + cyclic permutations
$d \omega(v_{\alpha_1}, v_{\alpha_2}, v_{\alpha_3}, -, \cdots)$.
So the Jacobi dientity fails up to an exact term. This will yield the structure of an L-infinity algebra.
Proposition
Given an $n$-plectic manifold $(X,\omega)$ we get a Lie n-algebra structure on the complex
(where the rightmost term is taken to be in degree 0).
where
the unary bracket is $d_{dR}$;
the $k$-ary bracket is
This is the Poisson bracket Lie n-algebra.
This appears as (Rogers 11, theorem 3.14).
For $n = 1$ this recovers the definition of the Lie algebra underlying a Poisson algebra.
Recall for $n=1$ the mechanism of geometric quantization of a symplectic manifold.
Given a 2-form $\omega$ and the corresponding complex line bundle $P$, consider the Atiyah Lie algebroid sequence
The smooth sections of $T P/U(1) \to X$ are the $U(1)$ invariant vector fields on the total space of $P$.
Using a connection $\nabla$ on $P$ we may write such a section as
for $v \in \Gamma(T X)$ a vector field downstairs, $s(v)$ a horizontal lift with respect to the given connection and $f \in C^\infty(X)$.
Locally on a suitable patch $U \subset X$ we have that $s(V)|_U = v|_U + \iota_v \theta_i|_U$ .
We say that $\tilde v = s(v) + f \partial_t$ preserves the splitting iff $\forall u \in \Gamma(X)$ we have
One finds that this is the case precisely if
This gives a homomorphism of Lie algebras
We consder now prequantization of 2-plectic manifolds.
Let $(X,\omega)$ be a 2-plectic manifold such that the de Rham cohomology class $[\omega]/2 \pi i$ is in the image of integral cohomology (Has integral periods.)
We can form a cocycle in Deligne cohomology from this, encoding a bundle gerbe with connection.
On a cover $\{U_i \to X\}$ of $X$ this is given in terms of Cech cohomology by data
$(g_{i j k} : U_{i j k} \to U(1)) \in C^\infty(U_{i j k}, U(1))$
$A_{i j} \in \Omega^1(U_{i j})$;
$B_i \in \Omega^2(U_i)$
satisfying a cocycle condition.
Now recall that an exact Courant algebroid is given by the following data:
a vector bundle $E \to X$;
an anchor morphism $\rho : E \to T X$ to the tangent bundle;
an inner product $\langle -,-\rangle$ on the fibers of $E$;
a bracket $[-,-]$ on the sections of $E$.
Satisfying some conditions.
The fact that the Courant algebroid is exact means that
is an exact sequence.
The standard Courant algebroid is the example where
$E = T X \oplus T^* X$;
$\langle v_1 + \alpha_1, v_2 + \alpha_2\rangle = \alpha_2(v_1) + \alpha_1(v_2)$;
the bracket is the skew-symmetrization of the Dorfman bracket
Now with respect to the above Deligne cocycle, build a Courant algebroid as follows:
on each patch $U_i$ is is the standard Courant algebroid $E_i := T U_i \oplus T^* U_i$;
glued together on double intersections using the $d A_{i j}$
This gives an exact Courant algebroid $E \to X$ as well as a splitting $s : T X \to E$ given by the $\{B_i\}$.
The bracket on this $E$ is given by the skew-symmetrization of
Here a section $e = s(v) + ...$ preserves the splitting precisely if
for all $u \in \Gamma(T X)$ we have
exactly if $\alpha$ is Hamiltonian and $v = v_\alpha$.
Theorem
Recall that to every Courant algebroid $E$ is associated a Lie 2-algebra $L_\infty(E)$.
Then: we have an embedding of L-infinity algebras
given by $\phi(\alpha) = s(v_\alpha) + \alpha$.
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 $X$)
duality between algebra and geometry in physics:
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
The observation that the would-be Poisson bracket induced by a higher degree closed form extends to the Poisson bracket Lie n-algebra is due to
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)
with first discussion of application to prequantization in
Chris Rogers, 2-plectic geometry, Courant algebroids, and categorified prequantization , arXiv:1009.2975.
Chris Rogers, Higher geometric quantization, talk at Higher Structures 2011 in Göttingen (pdf slides)
Discussion in the more general context of higher differential geometry/extended prequantum field theory is in
Domenico Fiorenza, Chris Rogers, Urs Schreiber,
Higher geometric prequantum theory,
L-∞ algebras of local observables from higher prequantum bundles
See also the references at multisymplectic geometry and n-symplectic manifold.
A higher differential geometry-generalization of contact geometry in line with multisymplectic geometry/$n$-plectic geometry is discussed in
Some more references on application, on top of those mentioned in the articles above.
A survey of some (potential) applications of 2-plectic geometry in string theory and M2-brane models is in section 2 of
and in