Let $D_{poly}$ be the differential graded Lie algebra (dgLA) of differentiable Hochschild cochains on a manifold $\mathscr M$, endowed with the usual Gerstenhaber bracket $[-,-]_G$ and Hochschild differential $\delta_\mu:=[\mu,-]$, where $\mu$ stands for the pointwise product of functions on $\mathscr M$. The cohomology of $D_{poly}$ is the graded Lie algebra (gLA) of polyvector fields, denoted $T_{poly}$, and endowed with the Schouten bracket $[-,-]_S$. This can be equivalently stated by saying that the HKR map $HKR:T_{poly} \to D_{poly}$ is a quasi-isomorphism of complexes.

In An homotopy formula for the Hochschild cohomology, De Wilde and Lecomte introduced an homotopy retract $h:D^\bullet_{poly}\to D^{\bullet-1}_{poly}$ for the HKR map.

Using this retract and the homotopy transfer theorem, one can transfer the dgLA structure of $D_{poly}$ into a $L_\infty$-algebra structure $\lambda$ on $T_{poly}$ and also obtain a $L_\infty$ quasi-isomorphism $\mathcal U:(T_{poly},\lambda)\to (D_{poly},\delta_\mu,[-,-]_G)$, such that the first map identifies with the HKR map i.e. $\mathcal U_1=HKR$.

If it were that the latter $L_\infty$-algebra structure $\lambda$ coincided with the Schouten gLA (i.e. if all the higher brackets $\lambda_{>2}$ of the transferred $L_\infty$-algebra were to vanish), then $\mathcal U$ would be a formality quasi-isomorphism akin to Kontsevich's.

Since neither the homotopy transfer nor De Wilde and Lecomte's homotopy involve a Drinfel'd associator, I assume that it is not the case and that rather the transferred $L_\infty$-algebra $\lambda$ on $T_{poly}$ coincides with B. Shoikhet's exotic (and unique) $L_\infty$-algebra structure on polyvector fields, see here.

Question: If the above is correct, is there an explicit statement/proof of this fact available in the literature?

  • 2
    $\begingroup$ We did something similar in this paper in the algebraic setting (so for polynomial vector fields on $\mathbb R^d$ if you like). We use a different homotopy which works in a more general setting, but homotopy transfer indeed gives an L∞ structure with nonvanishing higher brackets. (We also give a combinatorial interpretation of the Maurer-Cartan elements of the transferred L∞ algebra structure on polyvector fields. Given such a Maurer-Cartan element, one also obtains a graphical calculus for describing the associated star product.) $\endgroup$ May 4 at 12:15


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.