Towards resurgence of Joyce structures

This paper demonstrates that the gauge transformations used to normalize non-linear connections in Joyce structures, as well as the resulting formal twistor Darboux coordinates, possess convergent Borel transforms, thereby establishing their resurgence.

The Gist

Imagine you are a master restorer of ancient, complex clockwork mechanisms. You are presented with a collection of clocks that are incredibly intricate—so complex that their gears and springs are governed by invisible, shifting rules.

This paper is essentially a mathematical "restoration manual" for a very specific, highly advanced type of mathematical "clockwork" called a Joyce Structure.

Here is the breakdown of the paper using that analogy:

1. The Problem: The "Messy" Clocks

In high-level geometry and physics, mathematicians study "Joyce structures." Think of these as a set of rules that tell a space how to curve and twist. However, in their natural state, these rules are often presented in a "messy" way. They are written in a complicated, non-linear language that makes it nearly impossible to see the underlying pattern. It’s like trying to study a clock where the gears are made of jelly and the hands move in unpredictable waves.

2. The Solution: The "Standard" Blueprint

The author, Iván Tulli, asks a fundamental question: "Can we clean this up?"

He proves that no matter how messy or "jelly-like" the connection (the rules of the clock) appears, you can perform a mathematical "cleaning" (called a Gauge Transformation) to turn it into a "Standard Form."

Imagine taking that messy, jelly-gear clock and, through a clever series of adjustments, turning it into a perfectly precise, rigid, Swiss-made timepiece. Once it is in this "Standard Form," the math becomes much easier to handle.

3. The "Resurgence" Mystery: Finding the Hidden Rhythm

The most exciting part of the paper involves a concept called Resurgence.

In mathematics, many important series are "divergent"—meaning if you try to add up their terms, the sum flies off to infinity and becomes useless. It’s like a song that gets louder and louder until it shatters the windows.

However, "Resurgent" series are special. They look like they are exploding toward infinity, but they actually contain a hidden, beautiful rhythm. If you use a mathematical "lens" (called a Borel Transform), the explosion disappears, and you are left with a clear, beautiful melody that can be studied.

Tulli shows that the "cleaning process" he discovered isn't just a trick; the resulting mathematical series actually possesses this "hidden rhythm." This is a huge deal because it suggests that these complex structures are not just random chaos, but are deeply connected to fundamental patterns in physics (specifically something called DT-invariants, which relate to how particles and strings behave in higher dimensions).

4. The Examples: Testing the Tools

To prove his manual works, the author tests it on two specific "models":

• The A1 Quiver: A simple, elegant clock. He shows that he can find both a "clean" version and a "messy" version, and he proves that even the messy one follows the hidden "resurgent" rhythm.
• The A2 Quiver: A much more terrifyingly complex machine. He performs the heavy lifting to show exactly how the first few gears of this machine would look once they are "cleaned up."

Summary for the Non-Mathematician

If the universe is a giant, complex machine, the rules governing its smallest parts are often written in a language that looks like gibberish. This paper provides a way to translate that gibberish into a standard, readable language and proves that, underneath the apparent chaos, there is a predictable, rhythmic heartbeat that mathematicians can finally track.

Technical Summary

Technical Summary: Towards Resurgence of Joyce Structures

Author: Iván Tulli
Affiliation: University of Sheffield
Subject Area: Differential Geometry, Complex Geometry, Resurgence Theory, Mathematical Physics (DT-invariants)

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1. Problem Statement

The paper addresses the geometric and analytic properties of Joyce structures, a framework introduced by Tom Bridgeland to describe the geometry of the space of stability conditions in 3D Calabi-Yau categories.

A Joyce structure on a holomorphic symplectic manifold $(M, \Omega)$ is characterized by a family of non-linear, flat, and symplectic connections $A_\epsilon$ on the tangent bundle $TM$, parametrized by $\epsilon \in \mathbb{C}^*$. These structures are conjectured to encode Donaldson-Thomas (DT) invariants. The central mathematical challenge addressed in this paper is twofold:

1. Formal Classification: Can any Joyce structure be formally gauged to a "trivial" or "standard" form using formal power series in $\epsilon$?
2. Resurgence: Are the formal objects (gauge transformations and Darboux coordinates) that arise from this classification resurgent? Specifically, do their Borel transforms admit analytic continuations that reveal the singularity structure associated with DT-invariants?

2. Methodology

The author employs a combination of formal geometry, gauge theory, and resurgence theory:

• Formal Completion: The author replaces the complex parameter $\epsilon$ with a formal disk $\hat{\mathbb{D}} = \text{Spf}(\mathbb{C}[[\epsilon]])$, treating the connections as meromorphic relative connections on formal analytic spaces.
• Gauge Theory: The author utilizes the Lie algebra of infinitesimal gauge transformations $\text{Lie}(\hat{G})$ to act on these connections. The problem of gauging is framed as solving a sequence of recursive equations for the coefficients of the infinitesimal gauge transformation $\dot{g} = \sum \dot{g}_k \epsilon^k$.
• Borel-Laplace Analysis: To investigate resurgence, the author applies the Borel transform to the formal series. The convergence of these transforms is analyzed using Cauchy estimates and combinatorial lemmas to bound the growth of the coefficients of the vector fields.
• Twistor Theory: The author uses the correspondence between Joyce structures and complex hyperkähler (C-HK) structures to construct formal twistor Darboux coordinates.

3. Key Contributions and Results

A. Formal Classification (Theorems 3.4 & 3.8)
The paper proves that any Joyce structure can be uniquely gauged to a trivial Joyce structure (where the connection is induced by the flat linear connection $\nabla$) via a formal gauge transformation $\hat{g}$, provided we impose the boundary condition that the infinitesimal transformation $\dot{g}$ vanishes on the zero section $M \subset TM$. Furthermore, this $\hat{g}$ is used to construct formal twistor Darboux coordinates for the associated twistor space.

B. Convergence of Borel Transforms (Theorems 4.3 & 4.5)
A major technical result is the proof that the Borel transform $\mathcal{B}[\dot{g}](\xi)$ of the infinitesimal gauge transformation converges in a neighborhood of $\xi = 0$. This is a prerequisite for resurgence. The author also proves that the Borel transforms of the formal twistor Darboux coordinates converge.

C. Explicit Examples (Section 5)
The author validates the theory through two concrete quiver-related examples:

• $A_1$ Quiver: The author demonstrates two distinct behaviors. One gauge transformation is convergent (entire Borel transform), while the other (related to the Riemann-Hilbert problem) is divergent but resurgent. The latter's Borel transform has poles at $nz$($n \in \mathbb{Z} \setminus {0}$), and its Stokes automorphisms match the expected jumps in the DT-invariants of the $A_1$ quiver.
• $A_2$ Quiver: The author provides explicit algebraic computations for the first two terms ($\dot{g}_1, \dot{g}_2$) of the gauge transformation. This connects the Joyce structure to the isomonodromy flows of the Painlevé I equation.

4. Significance

This work represents a significant step toward a rigorous analytic understanding of the relationship between stability conditions and resurgence.

1. Bridging Geometry and Physics: It provides a mathematical bridge between the geometric construction of Joyce structures and the analytic theory of resurgence, which is central to modern topological string theory.
2. Foundational Step for Resurgence: While the paper does not prove "endless analytic continuation" (the second requirement of resurgence), it establishes the convergence of the Borel transform, which is the most difficult first step.
3. Computational Tool: The explicit formulas for the $A_1$ and $A_2$ cases provide a template for future researchers to compute $\tau$-functions and other invariants using the formal expansions derived here.