relation between BV and BD



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The concepts of

  1. BV-algebras

  2. BD-algebras

have a superficial similarity and are both related to BV-BRST formalism in physics. Indeed the “BV-operad” was named such because its algebras resemble the structure seen on the BV-complexes in physics. However in the formal deformation quantization picture of BV-formalism used in Costello-Gwilliam, the quantum BV-complexes crucially are BD-algebras and not BV-algebras. On the other hand, if one does not insist on a formal deformation parameter, one may also formalize quantum BV complexes using actual BV-algebras.

The following note means to indicate how all these structures are similar and yet different.


Consider all chain complexes in the following with differential dd of degree +1.


Write E 2E_2 for the little disk operad and E 2 frE_2^{fr} for the framed little disk operad. The chain homology of these is the BV-operad BVBV and the operad P 2P_2 for Poisson 2-algebras, respectively:

BV H (E 2 fr) P 2 H (E 2). \array{ BV &\simeq& H_\bullet(E_2^{fr}) \\ \uparrow && \uparrow \\ P_2 &\simeq& H_\bullet(E_2) } \,.

An algebra over these in chain complexes – a homotopy BV-algebra or Poisson 2-algebra, respectively – is a chain complex (V ,d)(V^\bullet, d) equipped with the following operations of the degree shown:

operation d ()() {,} Δ degree +1 0 1 1, \array{ operation && d && (-)\cdot(-) & & \{-,-\} && \Delta \\ degree && +1 && 0 && -1 && -1 } \,,

where the BV-operator Δ\Delta exists on homotopy BV-algebras but not on Poisson 2-algebras, satisfying the relation

Δ(ab)=(Δa)b+(1) |a|aΔb{a,b}. \Delta (a \cdot b) = (\Delta a) \cdot b + (-1)^{\vert a \vert} a \cdot \Delta b - \{a,b\} \,.

If AA is a commutative smooth Poincaré duality algebra so that its Hochschild cohomology HH (A)HH^\bullet(A) is identified by the HKR theorem with the multi-derivations, then HH (A)HH^\bullet(A) carries a BV-algebra structure and under this identification the above Gerstenhaber bracket {,}\{-,-\} is the Schouten bracket and the BV-operator Δ BV\Delta_{BV} is the image of the de Rham differential dualized. Hence this is then indeed the standard BV-complex as discussed there in the section Multivector fields dual to differential forms.

If we disregard the BV-operator Δ\Delta, then more generally there are Poisson n-algebras for all nn, which are as above but with the bracket of degree 1n1-n

operation {,} degree 1n. \array{ operation && \{-,-\} \\ degree && 1-n } \,.

Notice that this means that P 2P_2 algebras are very similar to P 0P_0-algebras: for the former the backet has degree -1, for the latter it has degree +1.


Now the BD-operad, which is a chain complex of [[]]\mathbb{R}[ [\hbar ] ]-modules (formal power series ring in one variable \hbar) has as chain complex of binary operations

BD(2)( 0 () d operad {,} 0 ) 0 1. BD(2) \;\coloneqq\; \array{ ( \cdots &\to& 0 &\to& \left\langle(-\cdot -)\right\rangle &\stackrel{d_{operad}}{\to}& \left\langle \hbar \{-,-\} \right\rangle &\to& 0 &\to& \cdots ) \\ && && 0 && 1 } \,.

Notice: The differential takes the single generator in degree 0 to the single generator in degree 1. As a differential in chain complexes of \mathbb{R}-modules this would be the null complex, but since the bracket carries a prefactor of \hbar, we get something non-null if we regard this suitably filtered over \hbar… In particular, setting “=0\hbar = 0” by forming the tensor product with [[]]/()\mathbb{R} \simeq \mathbb{R}[ [\hbar] ]/(\hbar) turns it into the chain complex concetrated on a single generator in degree 0. This is the E 0E_0-operad. On the other hand, for finite \hbar, i.e. after tensoring with [[]]/(1)\mathbb{R}[ [\hbar] ]/(\hbar-1) then it becomes the E˜ 0\tilde E_0-operad, which is equivalent to the E 0E_0-operad (whose algebras are simply pointed chain complexes). See also Gwilliam-Haugseng.

E˜ 0 E 0 =1 BD =0 P 0. \array{ && \tilde E_0 & \simeq E_0 \\ & {}^{\mathllap{\hbar = 1}}\nearrow \\ BD \\ & {}_{\mathllap{\hbar = 0}}\searrow \\ && P_0 } \,.

In any case, an algebra over an operad in chain complexes over the BD-operad has operations

operation d {,} degree +1 +1 \array{ operation && d && \hbar \{-,-\} \\ degree && +1 && +1 }

subject to

d(ab)=(da)b+(1) |a|adb{a,b}. d (a \cdot b) = (d a) \cdot b + (-1)^{\vert a \vert} a \cdot d b - \hbar\{a,b\} \,.

This has the same structure as the equation above for BV-algebras if we syntactically replace dd by Δ\Delta and remove the factor of \hbar. But since dd and Δ\Delta are on a different footing, the similarity between these two equations is only superficial.


Last revised on January 3, 2018 at 05:04:41. See the history of this page for a list of all contributions to it.