# Alternative to Kontsevich formality

Has anyone considered an alternative approach to Kontsevich formality in which the DGLA of poly-vector fields is deformed to an $$L_\infty$$-algebra?

Some vocabulary:

• DGLA = Differential Graded Lie Algebra
• $$L_\infty$$-algebra = SHLA = Strong Homotopy Lie Algebra
• DGCA? Could you elaborate on your idea? Feb 11, 2014 at 12:16
• forget the formality for the moment, has there been any study of L∞-algebra deformations of the GCA of poly-vector fields is deformed to an L∞-algebra? (d=0 initially) Mar 27, 2014 at 14:29
• I guess DGCA should be DGLA (that is, Differential Graded Lie Algebra). Oct 6, 2021 at 19:45

I'm not sure I understand the question, we have a graded commutative algebra --- how can you deform it to an $L(\infty)$ algebra? However, one can simply regard poly-vector fields as a Lie algebra and ask what are the $L(\infty)$ deformations of that. I'm not sure this has a nice answer.

Tamarkin's approach to proving Kontsevich formality is to upgrade this idea and realize that poly-vector fields has a $G(\infty)$ structure, which is essentially the GCA structure together with the Lie structure. And then one should study the deformations of that $G(\infty)$ algebra. I'm not sure if this has a clean answer for a general variety or manifold either.

However, this deformation theory is indeed trivial for say $\mathbb{C}[x_1,...,x_n]$. From there, there is a local to global patching argument of the local formality isomorphisms. This is the part of the proof that uses formal geometry.

This idea works in fact for proving the analogue of Kontsevich formality for $\mathbb{C}[x_1,...,x_n]$ thought of as an $E_n$ algebra for any $n$, "generalized Kontsevich formality". I don't know if this argument patches nicely and the known proof of generalized Kontsevich formality proceeds via some reduction scheme.

You might want to have a look at §2.2 of An $$L^\infty$$ algebra structure on polyvector fields by Boris Shoikhet, where Boris constructs an exotic $$L_\infty$$-structure on poly-vector fields on a (possibly infinite dimensional) affine space, which deforms the usual Schouten–Nijenhuis graded Lie structure.

This exotic $$L_\infty$$-structure has been used in a wise way in An explicit two step quantization of Poisson structures and Lie bialgebras by Merkulov and Willwacher in order to split Kontsevich's quantization procedure into two independent steps (only one of them depending on the choice of an associator).

Furthermore, this exotic $$L_\infty$$-structure has been shown (by Willwacher in The oriented graph complexes) to be the unique non-trivial $$L_\infty$$-deformation of the Schouten-Nijenhuis graded Lie structure that is universal (in the sense that it doesn't depend on the dimension $$n$$ of the affine space one is working with, including $$n=+\infty$$).

For poly-vector fields on a finite dimensional affine space this exotic $$L_\infty$$-structure seems to be quasi-isomorphic to the Schouten–Nijenhuis one (Merkulov–Willwacher write in their paper that this is a folklore conjecture; but it seems to be proven in Shoikhet's paper, see the Corollary of the Main Theorem in §2.3).