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Joyce Structures from Quadratic Differentials on the Sphere.

Timothy Moy1

  • 1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA UK.

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Summary
This summary is machine-generated.

This study constructs Joyce structures on moduli spaces of meromorphic quadratic differentials using isomonodromic deformations. It offers a geometric description of hyper-Kähler structures and a complex hyper-Kähler metric with homothetic symmetry.

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Area of Science:

  • Differential Geometry
  • Mathematical Physics
  • Complex Analysis

Background:

  • Joyce structures are known on spaces of meromorphic quadratic differentials.
  • Isomonodromic deformations are crucial for understanding linear ODEs with rational potentials.

Purpose of the Study:

  • To construct Joyce structures on moduli spaces of meromorphic quadratic differentials.
  • To provide a geometric description of hyper-Kähler structures.
  • To investigate isomonodromic deformations of second-order linear ODEs.

Main Methods:

  • Analyzing infinitesimal isomonodromic deformations.
  • Utilizing the kernel of a closed 2-form from an algebraic curve's intersection pairing.
  • Constructing Joyce structures on moduli spaces.

Main Results:

  • Demonstrated that infinitesimal isomonodromic deformations form the kernel of a specific closed 2-form.
  • Successfully constructed Joyce structures on a class of moduli spaces.
  • Obtained a complex hyper-Kähler metric with homothetic symmetry for differentials with odd-order poles.
  • Presented an example leading to the Painlevé VI equation.

Conclusions:

  • The study establishes a novel geometric framework for Joyce structures and hyper-Kähler metrics.
  • The findings offer new insights into the relationship between isomonodromic deformations and differential geometry.
  • The results extend the understanding of moduli spaces of quadratic differentials.