rough notes from a talk by Bruno Valette – raw material to be polished
see also framed little 2-disk operad
homotopy theory for $A_\infty$-algebras
for $V$ a complex with the structure of an $A_\infty$-algebra and for $V \to W$ a morphism of chain cmoplexes, we get an induced $A_\infty$-structure on $W$.
Application: for $V$ a differential graded algebra its chain cohomology inherits the structure of an $A_\infty$-algebra. The product operations are the Massey products.
There is a cofibrantly generated model category structure on the category of differential graded operads (operads in the monoidal cateory of chain complexes or cochain complexes).
If $P_\infty$ is a cofibrant dg-operad, then under some assumptions $P_\infty$-algebra for an operad structures are preserved by weak equivalences.
A quasi-free dg-operad is one that is free after forgetting the differential
Cofibrant operads are the retracts of quasi-free operads.
In particular quasi-free operads are cofibrant.
So we look for quasi-free resolutions.
Gerstenhaber algebra: essentially a Poisson algebra in the dg context.
How to define a Gerstenhaber algebra up to homotopy?
Extend the operations dfined by Gerstenhaber on $CH(A,A)$ whjich induce the Gerstenhaber algebra on $HH(A,A)$ to a Gerstenhaber algebra up to homotopy.
So we have a strict structure on Hochschild homology $HH(A,A)$ and are asking for from which homotopy structure it may come on Hochschild chains in $CH(A,A)$.
So we define the Gerstenhaber algebra operad? whose algebra for an operad are Gertsenhaber algebra?s.
It is generated of course from the product operation and the bracket operation modulo the associativity constraint for the product, the Jacobi identity for the bracket and the Leibnitz property for their interaction.
These relations are always encoded in quadratic expressions.
the dual notion of operad is that of cooperad (reverse all arrows)
There exist adjoint functors
given by bar and cobar construction
under this equivalence
with $P_\infty := \Omega P' \to P$ morphism of dg-operads.
$P$ is Koszul if $P_\infty = \Omega P' \stackrel{\simeq}{\to} P$ is a quasi-isomorphism, i.e. a cofibrant replacement.
If $P = F(V)/(R)$ be a quadratic operad with $dim(V) \lt + \infty$ The linear dual $P'^*$ is a quadratic operad aabd the suspension of $P'^* \simeq ' := F(V^* \otimes sgn)/(R^{\perp})$.
Compute $P'$ and its operadic structure with this formula, then dualize everything to get this formula.
The Gerstenhaber operad $G$ is Koszul.
This is a Gerstenhaber algebra $A$ with a unary operator $\Delta : A \to A$ of degree +1 such that
$\Delta^2 = 0$
$[-,-]$ measures failure of $\Delta$ to be a derivation with respect to the product $\cdot$.
This also is the algebra over an operad. But this operad is no longer a quadratic operad?.
So we define:
As above but now demand $\Delta$ a derivation of both the product $\cdot$ and the bracket $[-,-]$.
This is a first step in the resolution process.
Now the corresponding operad is Koszul. So we get
as a quasi-free resolution.
In terms of this we have a homotopy BV-algebra structure on $CH(A,A)$.
Add a suitable differential $d_1 : qBV' \to qBV'$.
Let $P = F(V)/(R)$ be a quadratic and linear presentation of a dg-operad
let $q : F(V) \to F(V)^{(2)}$ the quadratic projection
and $qP := F(V)/(qR)$ the quadratic analogue of $P$
$P = F(V)/(R)$ is a Koszul operad if
$R \cap V = \{0\} \Leftrightarrow V is minimal$
(more …)
$q P$ is Koszul
This now yields a good machinery for cofibrant resolutions for Koszul operads.
When $P$ is a quadratic linear Koszul operad then
This applies to the BV operad.
A $BV_\infty$-algebra is an algebra over this cofibrant replacement for the BV operad.
$A$ a Frobenius algebra.
Its Hochschild cohomology $HH(A,A)$ is a BV-algebra.
…
Another definition of homotopy BV-algebra by Kravchenko – this turns out to be a special case of the definition here by setting some operations to $0$ (her algebra is not an algebra over a cofibrant operad).
Another definition by Tamarkin–Tsygan: this is more general than the one here. TT have many more operations, namely operations with sevearl outputs.
The notion here also difers from that in Beilinson–Drinfeld.
The free operad $F(V)$ is filtered $\Rightarrow P = F(V)/(R)$ is filtered. There is then a morphism of operads $q P \to gr P$.
For $P$ Koszul we have
For instance
Let $dD$ be the framed little disk operad; then:
Getzler: $H_\bullet(fD) \simeq BV$.
$fD$ is formal, i.e. quasi-isomorphic by zig-zags to its homology
A topological conformal field theory is an algebra over the PROP $C_\bullet(\mathcal{R})$ of Riemann surfaces.
Any TCFT carries a homotopy BV-algebra structure which lifts the BV-algebra structure of Getzler in its homology.
In some sense homotopy BV is a formal extension of homotopy Gerstenhaber.
For any topological vertex algebra? $A$ with $\mathbb{N}$-graded conformal weight there exists an expliciit $BV_\infty$-algebra structure on $A$ which extends Lian–Zuckermann operations on $A$ and which lifts the BV-algebra structure on $H(A)$.
The above material probably roughly follows the talk slides
The corresponding article is
See also