symmetric monoidal (∞,1)-category of spectra
This fact is a higher analog of Kontsevich formality. It means that every higher dimensional prequantum field theory giveny by a algebra does have a deformation quantization (as factorization algebras) and that the space of choice of these a torsor over the automorphism infinity-group of , a higher analog of the Grothendieck-Teichmüller group.
See also tho MO discussion linked to below.
A Poisson 1-algebra is a Poisson algebra.
A Poisson 2-algebra is a Gerstenhaber algebra.
|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|
|dimension||classical field theory||Lagrangian BV quantum field theory||factorization algebra of observables|
|general||P-n algebra||BD-n algebra?||E-n algebra|
|Poisson 0-algebra||BD-0 algebra? = BD algebra||E-0 algebra? = pointed space|
|P-1 algebra = Poisson algebra||BD-1 algebra?||E-1 algebra? = A-∞ algebra|
and for further references along these lines see at factorization algebra.
For general discusison of the relation to E-n algebras see