nLab relation between BV and BD

Idea

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.

Details

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

BV

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

$$\array{ BV &\simeq& H_\bullet(E_2^{fr}) \\ \uparrow && \uparrow \\ P_2 &\simeq& H_\bullet(E_2) } \,.$$

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

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

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

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

If $A$ is a commutative smooth Poincaré duality algebra so that its Hochschild cohomology $HH^\bullet(A)$ is identified by the HKR theorem with the multi-derivations, then $HH^\bullet(A)$ carries a BV-algebra structure and under this identification the above Gerstenhaber bracket $\{-,-\}$ is the Schouten bracket and the BV-operator $\Delta_{BV}$ 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 $\Delta$, then more generally there are Poisson n-algebras for all $n$, which are as above but with the bracket of degree $1-n$

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

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

BD

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

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

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 “$\hbar = 0$” 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 $E_0$-operad. On the other hand, for finite $\hbar$, i.e. after tensoring with $\mathbb{R}[ [\hbar] ]/(\hbar-1)$ then it becomes the $\tilde E_0$-operad, which is equivalent to the $E_0$-operad (whose algebras are simply pointed chain complexes). See also Gwilliam-Haugseng.

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

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

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

subject to

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

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

References

• Kevin Costello, Owen Gwilliam, Factorization algebras in quantum field theory Volume 2 (pdf)

• Owen Gwilliam, Rune Haugseng, Linear Batalin-Vilkovisky quantization as a functor of ∞-categories (arXiv:1608.01290)