Beilinson-Drinfeld algebra (changes) in nLab
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Context
Algebra
Algebraic theories
Algebras and modules
Higher algebras
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symmetric monoidal (∞,1)-category of spectra
Model category presentations
Geometry on formal duals of algebras
Theorems
Contents
Idea
A Beilinson-Drinfeld algebra (or BD-algebra for short) is like a BV-algebra with nilpotent BV-operator, but over formal power series in one formal parameter ℏ\hbar, and such that the Gerstenhaber bracket is proportional to that parameter.
This means that one may think of a BD-algebra as an ℏ\hbar-parameterized formal family of algebras which for ℏ=0\hbar = 0 are Poisson 0-algebras and for ℏ≠0\hbar \neq 0 they look superficially like BV-algebras. But see remark 1 below.
Such BD-algbras are used to formalize formal deformation quantization in the context of the BV-BRST formalism (in Costello-Gwilliam).
Definition
Definition
A quantum BV complex or Beilinson-Drinfeld algebra is
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a differential graded-commutative algebra AA (whose differential we denote by Δ\Delta) over the ring ℝ[[ℏ]]\mathbb{R} [ [ \hbar ] ] of formal power series over the real numbers in a formal constant ℏ\hbar,
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equipped with a Poisson bracket {−,−}\{-,-\} of the same degree as the differential
such that
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the following equation holds for all elements a,b∈Aa,b \in A of homogeneous degree |a|,|b|∈ℤ{\vert a\vert}, {\vert b\vert} \in \mathbb{Z}
Δ(a⋅b)=(Δa)⋅b+(−1) |a|aΔb+ℏ{a,b} \Delta( a \cdot b) = (\Delta a) \cdot b + (-1)^{\vert a\vert} a \Delta b + \hbar \{a,b\}
(e.g. Costello-Gwilliam, def. 1.4.0.1, Gwilliam 13, def. 2.2.5)
algebraic deformation quantization
| dimension | classical field theory | Lagrangian BV quantum field theory | factorization algebra of observables |
|---|---|---|---|
| general nn | P-n algebra | BD-n algebra? | E-n algebra |
| n=0n = 0 | Poisson 0-algebra | BD-0 algebra? = BD algebra | E-0 algebra? = pointed space |
| n=1n = 1 | P-1 algebra = Poisson algebra | BD-1 algebra? | E-1 algebra? = A-∞ algebra |
References
The notion was introduced in
A discussion is in section 2.4 of
- Kevin Costello, Owen Gwilliam, Factorization algebras in quantum field theory Volume 2 (pdf)
See also
- Owen Gwilliam
,
Owen Gwilliam, Factorization algebras and free field theories PhD thesis (2013) (pdf)
Factorization algebras and free field theories PhD thesis (2013) (pdf) -
Martin Doubek, Branislav Jurčo, Lada Peksová, Ján Pulmann, Connected sum for modular operads and Beilinson-Drinfeld algebras, arXiv:2210.06517
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Constantin-Cosmin Todea, BD algebras and group cohomology, Comptes Rendus. Mathématique 359 (2021) no. 8, 925–937 doi
Last revised on October 18, 2022 at 12:55:30. See the history of this page for a list of all contributions to it.