Formal oscillatory distributions

TL;DR

This paper introduces oscillatory formal distributions supported at a point, characterizes when they are given by formal oscillatory integrals, and connects this to the naturality of star products on Poisson manifolds.

Contribution

It provides a characterization of formal oscillatory distributions via nondegeneracy, an algorithm to recover integral kernels, and links oscillatory distributions to natural star products.

Findings

Formal distributions are given by oscillatory integrals under a nondegeneracy condition.

An algorithm recovers the jet of the integral kernel from the formal distribution.

Star products are natural if and only if associated distributions are oscillatory.

Abstract

We introduce the notion of an oscillatory formal distribution supported at a point. We prove that a formal distribution is given by a formal oscillatory integral if and only if it is an oscillatory distribution that has a certain nondegeneracy property. We give an algorithm that recovers the jet of infinite order of the integral kernel of a formal oscillatory integral at the critical point from the corresponding formal distribution. We also prove that a star product $\star$ on a Poisson manifold $M$ is natural in the sense of Gutt and Rawnsley if and only if the formal distribution $f \otimes g \mapsto (f \star g)(x)$ is oscillatory for every $x \in M$.