symmetric monoidal (∞,1)-category of spectra
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 $\hbar$, and such that the Gerstenhaber bracket is proportional to that parameter.
This means that one may think of a BD-algebra as an $\hbar$-parameterized formal family of algebras which for $\hbar = 0$ are Poisson 0-algebras and for $\hbar \neq 0$ 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).
A quantum BV complex or Beilinson-Drinfeld algebra is
a differential graded-commutative algebra $A$ (whose differential we denote by $\Delta$) over the ring $\mathbb{R} [ [ \hbar ] ]$ of formal power series over the real numbers in a formal constant $\hbar$,
equipped with a Poisson bracket $\{-,-\}$ of the same degree as the differential
such that
the following equation holds for all elements $a,b \in A$ of homogeneous degree ${\vert a\vert}, {\vert b\vert} \in \mathbb{Z}$
(e.g. Costello-Gwilliam, def. 1.4.0.1, Gwilliam 13, def. 2.2.5)
If in the equation in def. 1 one replaces $\hbar$ by 1, then it takes the form characteristic of a BV-algebra. However the differential $\Delta$ here is the differential in the underlying chain complex for an algebra over an operad in chain complexes, while in a BV-algebra $\Delta$ is the unary operation encoded in the BV-operad, hence present already for algebras in plain modules/chain complexes over that operad. Accordingly, in the context of BD-algebra then for $\hbar \neq 0$ the bracket is actually trivial up to homotopy.
For more see at relation between BV and BD.
algebraic deformation quantization
dimension | classical field theory | Lagrangian BV quantum field theory | factorization algebra of observables |
---|---|---|---|
general $n$ | P-n algebra | BD-n algebra? | E-n algebra |
$n = 0$ | Poisson 0-algebra | BD-0 algebra? = BD algebra | E-0 algebra? = pointed space |
$n = 1$ | P-1 algebra = Poisson algebra | BD-1 algebra? | E-1 algebra? = A-∞ algebra |
The notion was introduced in
A discussion is in section 2.4 of
See also