Formality theorem for Hochschild cochains via transfer

We construct a 2-colored operad G^+ which, on the one hand, extends the operad G governing homotopy Gerstenhaber algebras and, on the other hand, extends the 2-colored operad governing open-closed homotopy algebras (OCHA). We show that Tamarkin’s G-structure on the Hochschild cochain complex C(A) of an A-infinity algebra A extends naturally to a G^+ structure on the pair (C(A), A). We show that a formality quasi-isomorphism for the Hochschild cochains of the polynomial algebra can be obtained via transfer of this G^+ structure to the cohomology of the pair (C(A), A). We show that G^+ is a sub DG operad of the first sheet E^1(SC) of the homology spectral sequence for the Fulton-MacPherson version SC of Voronov’s Swiss Cheese operad. Finally, we prove that the DG operads G^+ and E^1(SC) are non-formal.

💡 Research Summary

The paper introduces a novel two‑colored operad, denoted G⁺, which simultaneously extends the single‑colored operad G governing homotopy Gerstenhaber algebras and the two‑colored operad that controls open‑closed homotopy algebras (OCHA). The construction is carried out by assigning the first colour (the Hochschild cochain complex C(A) of an A∞‑algebra A) the full G‑structure, while the second colour (the underlying A∞‑algebra A itself) carries the OCHA‑structure, together with carefully designed mixed operations that respect the A∞‑relations on both sides.

A central achievement of the work is the observation that Tamarkin’s G‑structure on C(A) does not remain isolated on the cochain level; it lifts naturally to a G⁺‑structure on the pair (C(A), A). This lift is proved by checking compatibility of the differential, the higher Gerstenhaber brackets, and the mixed operations with the A∞‑multiplications of A.

The authors then address the formality problem for the Hochschild cochains of the polynomial algebra k