Categorifying isomonodromic deformations via Lie groupoids I: Logarithmic singularities

TL;DR

This paper extends the classical concept of isomonodromic deformations to a categorical framework using Lie groupoids, revealing new geometric and higher homotopical structures in the moduli of flat connections with logarithmic singularities.

Contribution

It introduces a functorial approach to isomonodromic deformations via Lie groupoids, encoding higher homotopical data and providing a geometric perspective through Morita equivalences.

Findings

Isomonodromy induces a map between moduli stacks of flat connections.

Higher homotopical information is encoded at the level of the fundamental 2-groupoid.

Isomonodromy functors are realized as Morita equivalences, offering a geometric interpretation.

Abstract

We upgrade the classical operation of \textit{isomonodromic deformations} along a path $\gamma$ to a functor $\mathbb{P}_{\gamma}$ between categories of flat connections with logarithmic singularities along a divisor $D$, which itself depends functorially on $\gamma$, using tools from the theory of Lie groupoids. As applications, (1) we get that isomonodromy gives a map of moduli \textit{stacks} of flat connections with logarithmic singularities, (2) we encode higher homotopical information at level 2, i.e. we get an action of the fundamental 2-groupoid of the base of our family on the categories of logarithmic flat connections on the fibres, and (3) our methods produce a geometric incarnation of the isomonodromy functors as Morita equivalences which are more primary than the isomonodromy functors themselves, and from which they can be formally extracted by passing to representation categories.