The p-cycle of Holonomic D-modules and Quantization of Exact Algebraic Lagrangians

TL;DR

This paper introduces a new invariant called the monodromy divisor for exact Lagrangians in cotangent bundles, proves a conjecture relating it to holonomic D-modules, and establishes a link between algebra autoequivalences and symplectomorphisms, confirming a conjecture of Kontsevich.

Contribution

It defines the monodromy divisor invariant for Lagrangians, proves a conjecture relating it to D-modules, and links Morita autoequivalences to symplectomorphisms, confirming a conjecture of Kontsevich.

Findings

The monodromy divisor obstructs attaching certain holonomic D-modules.

Holonomic D-modules form a torsor over finite order characters of the fundamental group.

Group of Morita autoequivalences of Weyl algebra is isomorphic to symplectomorphisms.

Abstract

Let $X=\mathbb{A}^{n}$ be complex affine space, and let $T^{*}X$ be its cotangent bundle. For any exact Lagrangian $L\subset T^{*}X$, we define a new invariant, A, living in $ \text{Div}_{\mathbb{Q}/\mathbb{Z}}(L)$. We call this invariant the monodromy divisor of $L$. We conjecture that the existence of a finite order character of $\pi_{1}(L$) whose monodromy is exactly A defines an obstruction to attaching a holonomic $\mathcal{D}_{X}$-module M associated to L - here, the association goes via positive characteristic and p-supports. In the case where $\mathbb{H}_{dR}^{1}(L)=0$, we prove this conjecture, and then go on the show that the set of such holonomic $\mathcal{D}_{X}$-modules forms a torsor over the group of finite order characters of $\pi_{1}$. This proves a version of a conjecture of Kontsevich. As a consequence, we deduce that the group of Morita autoequivalences of the n-th Weyl algebra is isomorphic to the group of symplectomorphisms of $T^{*}\mathbb{A}^{n}$. This generalizes an old theorem of Dixmier (in the case n=1) and settles a conjecture of Belov-Kanel and Kontsevich in general.