Caldararus conjecture and Tsygans formality

In this paper we complete the proof of Caldararu’s conjecture on the compatibility between the module structures on differential forms over poly-vector fields and on Hochschild homology over Hochschild cohomology. In fact we show that twisting with the square root of the Todd class gives an isomorphism of precalculi between these pairs of objects. Our methods use formal geometry to globalize the local formality quasi-isomorphisms introduced by Kontsevich and Shoikhet (the existence of the latter was conjectured by Tsygan). We also rely on the fact - recently proved by the first two authors - that Shoikhet’s quasi-isomorphism is compatible with cap product after twisting with a Maurer-Cartan element.

💡 Research Summary

The paper completes the proof of Caldararu’s conjecture concerning the compatibility between the module structures on differential forms over poly‑vector fields and on Hochschild homology over Hochschild cohomology. The authors show that twisting by the square root of the Todd class yields an isomorphism of precalculi between the pairs ((T_{\mathrm{poly}}(X),\Omega_X)) and ((D_{\mathrm{poly}}(X),C_{\mathrm{poly}}(X))).

The work is carried out in the general setting of a locally free Lie algebroid (L) over a ringed site ((X,\mathcal O)). After recalling the Atiyah class (A(L)) and defining the Todd class (\operatorname{td}(L)=\det!\big(A(L)(1-e^{-A(L)})^{-1}\big)), the authors introduce Gerstenhaber algebras and the notion of a precalculus (a Gerstenhaber algebra equipped with a contraction (\iota) and a Lie derivative (L)). Poly‑vector fields (T_{\mathrm{poly}}^L(X)=\bigoplus_{n\ge -1}\wedge^{n+1}L) and poly‑differential operators (D_{\mathrm{poly}}^L(X)) are shown to form such structures, as do differential forms (\Omega_L^\bullet) and Hochschild chains (C_{\mathrm{poly}}^L).

The central technical tool is a Fedosov‑type resolution adapted to Lie algebroids. By constructing an affine coordinate space for (L) and using a chosen (L)-connection, the authors produce explicit resolutions of the sheaves of poly‑vectors, poly‑differential operators, forms and chains. These resolutions allow the globalization of the local formality quasi‑isomorphisms of Kontsevich (for poly‑vectors) and Shoikhet (for chains), the latter originally conjectured by Tsygan.

A crucial recent result, proved by the first two authors, states that Shoikhet’s quasi‑isomorphism is compatible with the cap product after twisting by a Maurer–Cartan element. Using this, together with a careful analysis of twisting and descent for (L_\infty)-algebras and modules, the paper constructs global (L_\infty)-quasi‑isomorphisms \