derived smooth geometry
It involves two kinds of generalizations:
In the full higher symplectic geometry both of these aspects are unified: a multisymplectic -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 -multisymplectic form is the invariant polynomial that defines the theory.
an invariant polynomial on which is -linear (takes arguments):
such that the contraction morphism
If is a Lie 0-algebroid (over a smooth manifold) then it is simply that smooth manifold, . In this case is the ordinary de Rham complex and an invariant polynomial is a closed differential form of positive degree.
In this case an -plectic form on is a closed -form on such that for every vector field we have
The variation of that functional is
Therefore the condition that the invariant polynomial is -plectic amounts to saying that has no spurious global symmetries.
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|
|symplectic Lie n-algebroid||Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of -d sigma-model||higher symplectic geometry||d sigma-model||dg-Lagrangian submanifold/ real polarization leaf||= brane||(n+1)-module of quantum states in codimension||discussed in:|
|0||symplectic manifold||symplectic manifold||symplectic geometry||Lagrangian submanifold||–||ordinary space of states (in geometric quantization)||geometric quantization|
|1||Poisson Lie algebroid||symplectic groupoid||2-plectic geometry||Poisson sigma-model||coisotropic submanifold (of underlying Poisson manifold)||brane of Poisson sigma-model||2-module = category of modules over strict deformation quantiized algebra of observables||extended geometric quantization of 2d Chern-Simons theory|
|2||Courant Lie 2-algebroid||symplectic 2-groupoid||3-plectic geometry||Courant sigma-model||Dirac structure||D-brane in type II geometry|
|symplectic Lie n-algebroid||symplectic n-groupoid||(n+1)-plectic geometry||AKSZ sigma-model|
(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.