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Beilinson-Drinfeld algebra

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 \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 =0\hbar = 0 are Poisson 0-algebras and for 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

  1. a differential graded-commutative algebra AA (whose differential we denote by Δ\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,bAa,b \in A of homogeneous degree |a|,|b|{\vert a\vert}, {\vert b\vert} \in \mathbb{Z}

    Δ(ab)=(Δa)b+(1) |a|aΔb+{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)

Remark

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 Δ\Delta here is the differential in the underlying chain complex for an algebra over an operad in chain complexes, while in a BV-algebra Δ\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 0\hbar \neq 0 the bracket is actually trivial up to homotopy.

For more see at relation between BV and BD.

algebraic deformation quantization

dimensionclassical field theoryLagrangian BV quantum field theoryfactorization algebra of observables
general nnP-n algebraBD-n algebra?E-n algebra
n=0n = 0Poisson 0-algebraBD-0 algebra? = BD algebraE-0 algebra? = pointed space
n=1n = 1P-1 algebra = Poisson algebraBD-1 algebra?E-1 algebra? = A-∞ algebra

References

The notion was introduced in

A discussion is in section 2.4 of

See also

  • Owen Gwilliam, Factorization algebras and free field theories PhD thesis (2013) (pdf)

Last revised on December 21, 2016 at 15:40:00. See the history of this page for a list of all contributions to it.