The homotopy braces formality morphism

Abstract

We extend M. Kontsevich's formality morphism to a homotopy braces morphism and to a homotopy Gerstenhaber morphism. We show that this morphism is homotopic to D. Tamarkin's formality morphism, obtained using formality of the little disks operad, if in the latter construction one uses the Alekseev-Torossian associator. Similar statements can also be shown in the "chains" case, i. e., on Hochschild homology instead of cohomology. This settles two well known and long standing problems in deformation quantization and unifies the several known graphical constructions of formality morphisms and homotopies by Kontsevich, Shoikhet, Calaque, Rossi, Alm, Cattaneo, Felder and the author. Contents 65 Appendix C. Operadic twisting 67 Appendix D. Colored operadic twists of several operads 71 Appendix E. Proof of Proposition 75 72 Appendix F. The cohomology of fSGraphs and SGraphs 78 References 82

Formality and Kontsevich–Duflo type theorems for Lie pairs

“…An explicit formula for chain formality was first constructed by Shoikhet [64]. Formality for chains is related to Tsygan's program of noncommutative calculus [70,67], which is itself closely related to the algebraic index theorem [53,8,27,56,55,57,74]. In a separate publication, we will study Tsygan's formality for Lie pairs and its application to the index theorem.…”

Multiple zeta values in deformation quantization

“…Our approach also applies with minimal modification to the integrals appearing in various extensions of the formality morphism (e.g. [53,62]), to the calculation of the coefficients of the Alekseev-Torossian connection/associator [2,29], and presumably also to the calculation of correlation functions in other two-dimensional field theories, although we have not made an effort to explicitly pursue these further applications here.…”

Gravity formality

“…Concretely, it is the suboperad of Tw Gra consisting of graphs containing no connected components without external vertices and all internal vertices have valence at least 3. The construction of this operad using operadic twisting was first done in [Wil16].…”

“…The natural question to ask is whether Kontsevich's formality morphism can be extended to a BV ∞ quasi-isomorphism. Tamarkin [Tam98,Hin03] constructed a non-explicit Ger ∞ (homotopy Gerstenhaber) quasi-isomorphism T poly → D poly depending on a solution of Deligne's conjecture whose underlying Lie ∞ morphism was later shown by Willwacher [Wil16] to be homotopy equivalent to Kontsevich's map if one uses the Alekseev-Torossian associator to construct a solution to Deligne's conjecture. Furthermore, Willwacher shows that the original formality morphism can be strictly extended to a Ger ∞ morphism.…”