nLab Beilinson-Drinfeld algebra

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 $\hbar$, 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 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

1. 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 $\hbar$,

2. 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 ${\vert a\vert}, {\vert b\vert} \in \mathbb{Z}$

  $$\Delta( a \cdot b) = (\Delta a) \cdot b + (-1)^{\vert a\vert} a \Delta b + \hbar \{a,b\}$$

(e.g. Costello-Gwilliam, def. 1.4.0.1, Gwilliam 13, def. 2.2.5)

Remark

If in the equation in def. 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.

Related concepts

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

References

The notion was introduced in

• Alexander Beilinson, Vladimir Drinfeld, Chiral Algebras

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, 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

• Constantin-Cosmin Todea, BD algebras and group cohomology, Comptes Rendus. Mathématique 359 (2021) no. 8, 925–937 doi