symmetric monoidal (∞,1)-category of spectra
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
The concepts of
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 $d$ of degree +1.
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:
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:
where the BV-operator $\Delta$ exists on homotopy BV-algebras but not on Poisson 2-algebras, satisfying the relation
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 $1-n$
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.
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
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.
In any case, an algebra over an operad in chain complexes over the BD-operad has operations
subject to
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.
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)
Last revised on January 3, 2018 at 10:04:41. See the history of this page for a list of all contributions to it.