higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Higher symplectic geometry is the generalization of symplectic geometry to the context of higher geometry.
It involves two kinds of generalizations:
the symplectic form generalizes from a 2-form to a form of arbitrary arity. This aspect is called multisymplectic geometry.
the base manifold is generalized to a smooth ∞-groupoid or ∞-Lie algebroid. For binary symplectic forms this is called a symplectic Lie n-algebroid.
In the full higher symplectic geometry both of these aspects are unified: a multisymplectic $\infty$-groupoid is a smooth ∞-groupoid equipped with a differential n-form on smooth ∞-groupoids satisfying some condition.
Examples of higher symplectic geometries arise naturally as the covariant phase spaces “over the point” or “in top codimension” (in the sense of extended topological quantum field theory) in systems of ∞-Chern-Simons theory: their $\infty$-multisymplectic form is the invariant polynomial that defines the theory.
Let $\mathfrak{a}$ be an L-∞ algebroid. For $n \in \mathbb{N}$, an n-plectic form or multisymplectic form of $n$ arguments on $\mathfrak{a}$ is
an invariant polynomial $\omega$ on $\mathfrak{a}$ which is $n$-linear (takes $n$ arguments):
such that the contraction morphism
in injective.
If $\mathfrak{a}$ is a Lie 0-algebroid (over a smooth manifold) then it is simply that smooth manifold, $\mathfrak{a} = X$. In this case $W(\mathfrak{a}) = \Omega^{\bullet}(X)$ is the ordinary de Rham complex and an invariant polynomial is a closed differential form of positive degree.
In this case an $n$-plectic form on $\mathfrak{a}$ is a closed $n$-form $\omega(-,-,.\dots, -)$ on $X$ such that for every vector field $v \in \Gamma(T X)$ we have
Every invariant polynomial $\omega$ induces an ∞-Chern-Simons theory action functional
The variation of that functional is
Therefore the condition that the invariant polynomial is $n$-plectic amounts to saying that $S_{CS}$ has no spurious global symmetries.
(…)
duality between algebra and geometry in physics:
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
Discussion of what here we call “higher symplectic geometry over Lie 0-algebroids” (multisymplectic geometry) is in
For more references see multisymplectic geometry.