A Formality quasi-isomorphism for Hochschild cochains over rationals can be constructed recursively

TL;DR

This paper demonstrates that a formality quasi-isomorphism for Hochschild cochains over rationals can be constructed recursively without explicit knowledge of Kontsevich's coefficients, relying on the existence over reals.

Contribution

It introduces a recursive algorithm to construct formality quasi-isomorphisms over rationals, bypassing the need for explicit coefficients and Tamarkin's approach.

Findings

Recursive construction of formality quasi-isomorphism over rationals.

Algorithm works without explicit coefficients from Kontsevich's construction.

Relies on the existence of formality over reals.

Abstract

It is believed arXiv:0808.2762, arXiv:math/9904055 that, among the coefficients entering Kontsevich's formality quasi-isomorphism arXiv:q-alg/9709040, there are irrational (possibly even transcendental) numbers. In this paper, we prove that a formality quasi-isomorphism for Hochschild cochains of a polynomial algebra over rationals can be constructed recursively. The proof that the proposed recursive algorithm works, is based on the existence of formality quasi-isomorphism over reals. However, the algorithm requires no explicit knowledge of the coefficients entering Kontsevich's construction. Although this algorithm completely bypasses Tamarkin's approach arXiv:math/0003052, arXiv:math/9803025, the construction is inspired by Proposition 5.8 from the classical paper (Algebra i Analiz, 1990) by V. Drinfeld.