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Categorifying isomonodromic deformations via Lie groupoids I: Logarithmic singularities

Waleed Qaisar

TL;DR

The paper develops a categorified view of isomonodromic deformations for flat connections, replacing traditional monodromy-based deformations with a 2-functor constructed from Morita equivalences of holomorphic Lie groupoids. The core construction yields a 2-local system on the base S valued in holomorphic stacks, realized via a Morita-equivalence-based geometric mechanism that naturally induces functors between categories of flat connections and natural transformations along homotopies. In the logarithmic setting, twisted fundamental groupoids replace ordinary fundamental groupoids, enabling a parallel categorification that preserves divisor data; this leads to a 2-functor boldsymbol{P} categorifying logarithmic isomonodromy and connecting to a generalized Riemann–Hilbert correspondence. The work further reveals a π2(S,s)-action on the fiber categories through the first Hirzebruch surface and outlines future extensions to irregular singularities and symplectic structures on moduli stacks, broadening the geometric understanding of isomonodromy as Morita data rather than merely monodromy representations.

Abstract

We upgrade the classical operation of \textit{isomonodromic deformations} along a path $γ$ to a functor $\mathbb{P}_γ$ between categories of flat connections with logarithmic singularities along a divisor $D$, which itself depends functorially on $γ$, 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.

Categorifying isomonodromic deformations via Lie groupoids I: Logarithmic singularities

TL;DR

The paper develops a categorified view of isomonodromic deformations for flat connections, replacing traditional monodromy-based deformations with a 2-functor constructed from Morita equivalences of holomorphic Lie groupoids. The core construction yields a 2-local system on the base S valued in holomorphic stacks, realized via a Morita-equivalence-based geometric mechanism that naturally induces functors between categories of flat connections and natural transformations along homotopies. In the logarithmic setting, twisted fundamental groupoids replace ordinary fundamental groupoids, enabling a parallel categorification that preserves divisor data; this leads to a 2-functor boldsymbol{P} categorifying logarithmic isomonodromy and connecting to a generalized Riemann–Hilbert correspondence. The work further reveals a π2(S,s)-action on the fiber categories through the first Hirzebruch surface and outlines future extensions to irregular singularities and symplectic structures on moduli stacks, broadening the geometric understanding of isomonodromy as Morita data rather than merely monodromy representations.

Abstract

We upgrade the classical operation of \textit{isomonodromic deformations} along a path to a functor between categories of flat connections with logarithmic singularities along a divisor , which itself depends functorially on , 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.

Paper Structure

This paper contains 18 sections, 16 theorems, 24 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. The problem of producing functorial isomonodromy
  4. Relation to other work
  5. Review of contents
  6. Acknowledgements
  7. Conventions and notation
  8. Morita equivalences
  9. Categorification of nonsingular isomonodromy
  10. Construction of the complex manifold underlying the Morita equivalence
  11. The groupoid actions on $Q_{\gamma}$
  12. Constructing the morphism of Morita equivalences
  13. Categorification of logarithmic isomonodromy
  14. Review of twisted fundamental groupoids
  15. Construction of the Morita equivalence in the logarithmic case
  16. ...and 3 more sections

Key Result

Theorem 1.1

a) Let $(X,D) \to S$ be a family of complex curves with divisor, given by a holomorphic submersion $p:X \to S$ with 1-dimensional fibres which is a topological fibre bundle (for example a proper submersion), and $D$ a reduced smooth divisor in $X$ such the restriction $p|_D: D \to S$ is a covering m

Theorems & Definitions (41)

  • Theorem 1.1
  • Definition 1
  • Definition 2
  • Remark 1
  • Proposition 1
  • Proposition 2
  • proof
  • Definition 3
  • Proposition 3
  • Lemma 1
  • ...and 31 more