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Resurgence of Tritronquées Solutions of the Deformed Painlevé I Equation

Mohamad Alameddine, Olivier Marchal, Nikita Nikolaev, Nicolas Orantin

TL;DR

This work provably establishes the resurgence of the formal $\hbar$-power series solution to the deformed Painlevé I equation, showing that its Borel transform converges and admits endless analytic continuation on a geometric Borel space isomorphic to a Fermat quintic modulo an involution. The authors develop a global geometric framework using holomorphic Lie groupoids to encode Borel data, with a central charge map $Z$ yielding a fivefold Borel cover over each fibre and ten Borel singularities, constraining Stokes phenomena to two opposite rays. They prove the existence of five deformed tritronquée holomorphic solutions $q_\alpha$ corresponding to directions in Stokes regions, and provide explicit Stokes-jump formulas in terms of Borel transform variations. By relating the deformed equation to the usual Painlevé I via a rescaling, they connect the resurgence structure to classic transcendent behavior while outlining a path toward extending to tronquées and more general transseries. The results give a detailed, computable picture of the global resurgent geometry governing the deformed PI equation, with potential implications for exact perturbation theory and isomonodromic systems in mathematical physics.

Abstract

We prove that the formal $\hbar$-power series solution of the deformed Painlevé I equation is resurgent, which means it is generically Borel summable and its Borel transform admits endless analytic continuation. In particular, we find that the Borel transform defines a global multivalued holomorphic function on a singular algebraic surface isomorphic to the Fermat quintic surface $x^5 + y^5 + z^5 = 0$ modulo an involution. This surface is an algebraic fibration over the complex plane of the differential equation with generic fibre a smooth quintic curve. Each fibre is equipped with a fivefold covering map over another complex plane (the Borel plane) with ten ramification points (the Borel singularities) spread equally over two branch points giving two opposite Stokes rays.

Resurgence of Tritronquées Solutions of the Deformed Painlevé I Equation

TL;DR

This work provably establishes the resurgence of the formal -power series solution to the deformed Painlevé I equation, showing that its Borel transform converges and admits endless analytic continuation on a geometric Borel space isomorphic to a Fermat quintic modulo an involution. The authors develop a global geometric framework using holomorphic Lie groupoids to encode Borel data, with a central charge map yielding a fivefold Borel cover over each fibre and ten Borel singularities, constraining Stokes phenomena to two opposite rays. They prove the existence of five deformed tritronquée holomorphic solutions corresponding to directions in Stokes regions, and provide explicit Stokes-jump formulas in terms of Borel transform variations. By relating the deformed equation to the usual Painlevé I via a rescaling, they connect the resurgence structure to classic transcendent behavior while outlining a path toward extending to tronquées and more general transseries. The results give a detailed, computable picture of the global resurgent geometry governing the deformed PI equation, with potential implications for exact perturbation theory and isomonodromic systems in mathematical physics.

Abstract

We prove that the formal -power series solution of the deformed Painlevé I equation is resurgent, which means it is generically Borel summable and its Borel transform admits endless analytic continuation. In particular, we find that the Borel transform defines a global multivalued holomorphic function on a singular algebraic surface isomorphic to the Fermat quintic surface modulo an involution. This surface is an algebraic fibration over the complex plane of the differential equation with generic fibre a smooth quintic curve. Each fibre is equipped with a fivefold covering map over another complex plane (the Borel plane) with ten ramification points (the Borel singularities) spread equally over two branch points giving two opposite Stokes rays.

Paper Structure

This paper contains 8 sections, 1 theorem, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results.
  3. Context.
  4. Relation to the usual Painlevé I equation.
  5. Organisation.
  6. Acknowledgements.
  7. The Formal Power Series Solution
  8. Existence and Uniqueness

Key Result

Proposition 1.1

The deformed Painlevé I equation 250204124541 has a unique formal power series solution $\widehat{q} (t,\hbar)$ of the form 250205091941 whose coefficients are holomorphic but multivalued functions on the punctured $t$-plane $\mathbb{C} \smallsetminus \mathopen{}\mathclose{\left\{ 0 \right\}$. Namel where the coefficients $a_{2n}^\pm, b_{2n+1}^\pm \in \mathbb{C}$ are defined by the recurrence rela

Figures (1)

  • Figure 1:

Theorems & Definitions (3)

  • Proposition 1.1: Formal Existence and Uniqueness Theorem
  • Definition 1.2: formal solutions
  • Remark 1.3