Formality theorems for Hochschild complexes and their applications

We give a popular introduction to formality theorems for Hochschild complexes and their applications. We review some of the recent results and prove that the truncated Hochschild cochain complex of a polynomial algebra is non-formal.

💡 Research Summary

The paper provides a broad yet accessible overview of formality theorems for Hochschild complexes, surveys recent advances, and presents a new non‑formality result for a truncated Hochschild cochain complex of a polynomial algebra. It begins by recalling the classical formality theorem of Kontsevich, which asserts that the differential graded Lie algebra (DGLA) of Hochschild cochains on a smooth commutative algebra is L∞‑quasi‑isomorphic to its cohomology Gerstenhaber algebra. The authors then discuss extensions such as the Tamarkin‑Shoikhet theorem for multidimensional formal geometry and the Dolgushev‑Willwacher framework that incorporates modules, higher‑order operations, and global quantization. These results rely on the existence of an L∞‑transfer map that preserves the full hierarchy of higher brackets μₖ (k ≥ 2).

The core contribution of the paper is the proof that the truncated Hochschild cochain complex C⁎_{\le m}(A,A) of the polynomial algebra A = k