Cohomologie De Hochschild Des Surfaces De Klein

TL;DR

This paper computes the Hochschild cohomology of the algebra of functions on Klein surfaces and singular plane curves, aiding deformation quantization of these geometric structures.

Contribution

It provides a new, detailed calculation of Hochschild cohomology for Klein surfaces, extending previous results and employing Kontsevich's complex and Gr"obner bases.

Findings

Hochschild cohomology for Klein surfaces determined

Revisits and refines Fronsdal's results on singular curves

Uses Kontsevich's complex and Gr"obner bases for calculations

Abstract

Given a mechanical system $(M, \mathcal{F}(M))$, where $M$ is a Poisson manifold and $\mathcal{F}(M)$ the algebra of regular functions on $M$, it is important to be able to quantize it, in order to obtain more precise results than through classical mechanics. An available method is the deformation quantization, which consists in constructing a star-product on the algebra of formal power series $\mathcal{F}(M)[[\hbar]]$. A first step toward study of star-products is the calculation of Hochschild cohomology of $\mathcal{F}(M)$. The aim of this article is to determine this Hochschild cohomology in the case of singular curves of the plane -- so we rediscover, by a different way, a result proved by Fronsdal and make it more precise -- and in the case of Klein surfaces. The use of a complex suggested by Kontsevich and the help of Gr\"obner bases allow us to solve the problem.