Wild Betti sheaves

TL;DR

This paper develops an enlarged category of Betti sheaves supporting exponential local systems and Fourier equivalences, unifying Tamarkin's enhanced sheaves with a universal construction that has a torsor structure over the integers.

Contribution

It introduces a universal construction of an extended Betti sheaf category supporting exponential local systems and Fourier transforms, generalizing Tamarkin's enhanced sheaves.

Findings

Constructs an enlarged category of Betti sheaves supporting exponential local systems.

Establishes a Fourier equivalence on all sheaves within this category.

Shows the category has a universal property and a torsor structure over \\mathrm{Spec}(\\mathbb{Z}).

Abstract

In this note, we consider the problem of constructing an enlargement of the category of Betti sheaves that supports an ``exponential local system'' on $\mathbb R$, and a Fourier equivalence defined on all sheaves. We show that there is a universal solution, recovering a construction of Tamarkin known also as ``enhanced sheaves''. The universality property implies that the category of coefficients of this theory is, in a suitable sense, a nontrivial $\mathbb R_{>0}$-torsor over $\mathrm{Spec}(\mathbb Z)$.