symmetric monoidal (∞,1)-category of spectra
For , a Poisson -algebra is a Poisson algebra in a category of chain complexes with Poisson bracket of degree (which is a bracket of degree 0 on ).
The homology of an E-n algebra for is a Poisson -algebra.
Moreover, in chain complexes over a field of characteristic zero the E-n operad is formal (see the little n-disk operad is formal), hence equivalent to its homology, and so in this context -algebras are equivalent to Poisson -algebras.
This fact is a higher analog of Kontsevich formality. It means that every higher dimensional prequantum field theory given 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.
There is a forgetful functor from Poisson -algebras to dg-Lie algebras given by forgetting the associative algebra structure and by shifting the underlying chain complex by .
Conversely, this functor has a derived left adjoint which sends a dg-Lie algebra to its universal enveloping Poisson n-algebra . (See also Gwilliam, section 4.5).
A Poisson 1-algebra is a Poisson algebra.
A Poisson 2-algebra is a Gerstenhaber algebra.
The Chevalley-Eilenberg algebra of a symplectic Lie n-algebroid (see there for details) is naturally a Poisson -algebra.
A classical BV complex is naturally (if obtained as a derived critical locus, or else by definition) a Poisson 0-algebra.
duality between algebra and geometry
in physics:
algebraic deformation quantization
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 |
An introduction to Poisson -algebras in dg-geometry/symplectic Lie n-algebroids is in section 4.2 of
For discussion in the context of perturbative quantum field theory/factorization algebras/BV-quantization see
Kevin Costello, Owen Gwilliam, Factorization algebras in perturbative quantum field theory : -operad (wikilass=‘newWikiWord’>P_0%20operad?</span>), pdf)
Owen Gwilliam, Factorization algebras and free field theories PhD thesis (pdf)
and for further references along these lines see at factorization algebra.
For general discusison of the relation to E-n algebras see
Last revised on November 7, 2018 at 18:56:42. See the history of this page for a list of all contributions to it.