Singular propagators in deformation quantization and Shoikhet-Tsygan formality

TL;DR

This paper extends the interpolation family of Kontsevich formality maps to Shoikhet-Tsygan formality, exploring their properties, relations, and integral weights, with implications for deformation quantization and number theory.

Contribution

It demonstrates the extension of Kontsevich formality maps to Shoikhet-Tsygan formality with a complex parameter, providing new relations and computations of integral weights.

Findings

Extension of formality maps to Shoikhet-Tsygan formalism

Derived elementary relations for the polynomials involved

Computed Kontsevich integral weights and analyzed their number theoretic significance

Abstract

This paper adds some details to the seminal approach to logarithmic formality \cite{AWRT} and interpolation formality \cite{WR} by Alekseev, Rossi, Torossian and Willwacher: We prove that the interpolation family of Kontsevich formality maps extends to Shoikhet-Tsygan formality and a complex interpolation parameter. We show some elementary relations satisfied by this polynomials. We also compute some Kontsevich integral weights and reason on the number theoretic meaning of the invariance of Kontsevich's propagator under real translations and scalings in the case of the Merkulov $n$-wheels.