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Kohei Iwaki

Publications and source records attributed to Kohei Iwaki.

At least 19 recordsLinked to original sources
arXiv2026

Les Houches Lectures on Exact WKB Analysis and Painlevé Equations

The first part of these lecture notes is devoted to an introduction to the theory of exact WKB analysis for second-order Schrödinger-type ordinary differential equations. It reviews the construction of the WKB solution, Borel summability, connection formulas, and their application to direct monodromy problems. In the second part, we discuss recent developments in applying exact WKB analysis to the study of Painlevé equations. By combining exact WKB analysis with topological recursion, it becomes possible to explicitly compute the monodromy of linear differential equations associated with Painlevé equations, assuming Borel summability and other conditions. Furthermore, by using isomonodromy deformations (integrability of the Painlevé equations), the resurgent structure of the $τ$-function and partition function is analyzed. These lecture notes accompanied a series of lectures at the Les Houches school, ``Quantum Geometry (Mathematical Methods for Gravity, Gauge Theories and Non-Perturbative Physics)'' in Summer 2024.

math-ph↗
arXiv2026

Accessory Parameter of Confluent Heun Equations, Voros Periods and classical irregular conformal blocks

For the Heun differential equation and all of its confluent equations, we derive formal series expansions of the accessory parameters using the Voros periods. We then compare these expansions with the classical conformal blocks recently obtained by Bonelli--Shchechkin--Tanzini, and examine the Zamolodchikov-type conjecture expected to hold between them, allowing for irregular singularities. In particular, as an extension of the previous works of Mironov--Morozov, Piatek--Pietrykowski and Lisovyy--Naidiuk, we provide a detailed prescription for choosing cycles on the spectral curve that yield the Voros period which corresponds to the classical (regular or irregular) conformal blocks through the accessory parameter.

math-ph↗
arXiv2025

Many-faced Painlevé I: irregular conformal blocks, topological recursion, and holomorphic anomaly approaches

In recent years, the Fourier series (Zak transform) structure of the Painlevé I tau function has emerged in multiple contexts. Its main building block admits several conjectural interpretations, such as the partition function of an Argyres-Douglas gauge theory, the topological recursion partition function for the Weierstrass elliptic curve, and a 1-point conformal block on the Riemann sphere with an irregular insertion of rank $\frac52$. We review and further develop a mathematical framework for these constructions, and formulate conjectures on their equivalence. In particular, we give a simple explanation of the Fourier series representation of the tau function based on the Jimbo-Miwa-Ueno differential extended to the space of Stokes data. We provide an algebraic construction of the rank $\frac52$ Whittaker state for the Virasoro algebra embedded into a rank $2$ Whittaker module, prove its existence and uniqueness, and fix its descendant structure. We also prove the conifold gap property of the relevant topological recursion partition function, which, on one hand, enables its efficient computation within the holomorphic anomaly approach and, on the other, establishes the existence of solution for the latter.

math-ph↗
arXiv2025

Note on the Singularity Reduction of Isomonodromy Systems Associated with Garnier Systems

In this paper, we study the isomonodromy systems associated with the Garnier systems of type 9/2 and type 5/2+3/2. We show that the both of isomonodromy systems admit the singularity reduction (restriction to a movable pole), and the resulting linear differential equations are isomonodromic with respect to the second variable of the Garnier systems. Furthermore, we find two fourth-order nonlinear ordinary differential equations that describe the isomonodromy deformation but lack the Painlevé property. One of these equations has been already found by Dubrovin--Kapaev in 2014, while the other provides a new example.

math.CA↗
arXiv2024

Resurgent Structure of the Topological String and the First Painlevé Equation

We present an explicit formula for the Stokes automorphism acting on the topological string partition function. When written in terms of the dual partition function, our formula implies that flat coordinates in topological string theory transform as quantum periods, and according to the Delabaere-Dillinger-Pham formula. We first show how the formula follows from the non-linear Stokes phenomenon of the Painlevé I equation, together with the connection between its $τ$-function and topological strings on elliptic curves. Then, we show that this formula is also a consequence of a recent conjecture on the resurgent structure of the topological string, based on the holomorphic anomaly equations, and it is in fact valid for arbitrary Calabi-Yau threefolds.

hep-th↗
arXiv2024

Topological recursion and uncoupled BPS structures II: Voros symbols and the $τ$-function

We continue our study of the correspondence between BPS structures and topological recursion in the uncoupled case, this time from the viewpoint of quantum curves. For spectral curves of hypergeometric type, we show the Borel-resummed Voros symbols of the corresponding quantum curves solve Bridgeland's "BPS Riemann-Hilbert problem". In particular, they satisfy the required jump property in agreement with the generalized definition of BPS indices $Ω$ in our previous work. Furthermore, we observe the Voros coefficients define a closed one-form on the parameter space, and show that (log of) Bridgeland's $τ$-function encoding the solution is none other than the corresponding potential, up to a constant. When the quantization parameter is set to a special value, this agrees with the Borel sum of the topological recursion partition function $Z_{\rm TR}$, up to a simple factor.

math-ph↗
arXiv2020

Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies

For the hypergeometric spectral curve and its confluent degenerations (spectral curves of "hypergeometric type"), we obtain a simple formula expressing the topological recursion free energies as a sum over BPS states (degenerate spectral networks) for a corresponding quadratic differential $φ$. In doing so, we generalize Gaiotto-Moore-Neitzke's construction of BPS structures to include the case where $φ$ has simple poles or supports a degenerate ring domain. For the nine spectral curves of hypergeometric type, we provide a complete description of the corresponding BPS structures over a generic locus in the relevant parameter space; in particular, we prove the existence of saddles trajectories at the expected parameter values. We determine the corresponding BPS cycles, central charges, and BPS invariants, and verify our formula in each case. We conjecture that a similar relation should hold more generally whenever the corresponding BPS structure is uncoupled, and provide experimental evidence in two simple higher rank examples.

math-ph↗
arXiv2020

Witten-Reshetikhin-Turaev function for a knot in Seifert manifolds

In this paper, for a Seifert loop (i.e., a knot in a Seifert three-manifold), first we give a family of an explicit function $Φ(q; N)$ whose special values at roots of unity are identified with the Witten-Reshetikhin-Turaev invariants of the Seifert loop for the integral homology sphere. Second, we show that the function $Φ(q; N)$ satisfies a $q$-difference equation whose classical limit coincides with a component of the character varieties of the Seifert loop. Third, we give an interpretation of the function $Φ(q; N)$ from the view point of the resurgent analysis.

math.GT↗
arXiv2019

Reconstructing GKZ via topological recursion

In this article, a novel description of the hypergeometric differential equation found from Gel'fand-Kapranov-Zelevinsky's system (referred to GKZ equation) for Givental's $J$-function in the Gromov-Witten theory will be proposed. The GKZ equation involves a parameter $\hbar$, and we will reconstruct it as the WKB expansion from the classical limit $\hbar\to 0$ via the topological recursion. In this analysis, the spectral curve (referred to GKZ curve) plays a central role, and it can be defined as the critical point set of the mirror Landau-Ginzburg potential. Our novel description is derived via the duality relations of the string theories, and various physical interpretations suggest that the GKZ equation is identified with the quantum curve for the brane partition function in the cohomological limit. As an application of our novel picture for the GKZ equation, we will discuss the Stokes matrix for the equivariant $\mathbb{C}\textbf{P}^{1}$ model and the wall-crossing formula for the total Stokes matrix will be examined. And as a byproduct of this analysis we will study Dubrovin's conjecture for this equivariant model.

math-ph↗
arXiv2019

2-parameter $τ$-function for the first Painlevé equation -Topological recursion and direct monodromy problem via exact WKB analysis-

We show that a 2-parameter family of $τ$-functions for the first Painlevé equation can be constructed by the discrete Fourier transform of the topological recursion partition function for a family of elliptic curves. We also perform an exact WKB theoretic computation of the Stokes multipliers of associated isomonodromy system assuming certain conjectures.

math-ph↗
arXiv2018

Voros Coefficients for the Hypergeometric Differential Equations and Eynard-Orantin's Topological Recursion - Part I : For the Weber Equation

We develop the theory of quantization of spectral curves via the topological recursion. We formulate a quantization scheme of spectral curves which is not necessarily admissible in the sense of Bouchard and Eynard. The main result of this paper and the second part [IKoT] establishes a relation between the Voros coefficients for the quantum curves and the free energy for spectral curves associated with the confluent family of Gauss hypergeometric differential equations. We focus on the Weber equation in this article, and generalize the result for the other members of the confluent family in the second part. We also find explicit formulas of free energy for those spectral curves in terms of the Bernoulli numbers.

math.CA↗
arXiv2018

Voros Coefficients for the Hypergeometric Differential Equations and Eynard-Orantin's Topological Recursion - Part II : For the Confluent Family of Hypergeometric Equations

We show that the each member of the confluent family of the Gauss hypergeometric equations is realized as quantum curves for appropriate spectral curves. As an application, relations between the Voros coefficients of those equations and the free energy of their classical limit computed by the topological recursion are established. We will also find explicit expressions of the free energy and the Voros coefficients in terms of the Bernoulli numbers and Bernoulli polynomials.

math.CA↗
arXiv2017

Painlevé equations, topological type property and reconstruction by the topological recursion

In this article we prove that Lax pairs associated with $\hbar$-dependent six Painlevé equations satisfy the topological type property proposed by Bergère, Borot and Eynard for any generic choice of the monodromy parameters. Consequently we show that one can reconstruct the formal $\hbar$-expansion of the isomonodromic $τ$-function and of the determinantal formulas by applying the so-called topological recursion to the spectral curve attached to the Lax pair in all six Painlevé cases. Finally we illustrate the former results with the explicit computations of the first orders of the six $τ$-functions.

math-ph↗
arXiv2017

Painlev'e 2 equation with arbitrary monodromy parameter, topological recursion and determinantal formulas

The goal of this article is to prove that the determinantal formulas of the Painlev'e 2 system identify with the correlation functions computed from the topological recursion on their spectral curve for an arbitrary non-zero monodromy parameter. The result is established for two different Lax pairs associated to the Painlev'e 2 system, namely the Jimbo-Miwa Lax pair and the Harnad-Tracy-Widom Lax pair, whose spectral curves are not connected by any symplectic transformation. We provide a new method to prove the topological type property without using the insertion operators. In the process, taking the time parameter t to infinity gives that the symplectic invariants F(g) computed from the Hermite-Weber curve and the Bessel curve are equal to respectively. This result generalizes similar results obtained from random matrix theory in the special case where θ = 0. We believe that this approach should apply for all 6 Painlev'e equations with arbitrary monodromy parameters. Explicit computations up to g = 3 are provided along the paper as an illustration of the results.

math-ph↗
arXiv2016

Quantum Curve and the First Painlevé Equation

We show that the topological recursion for the (semi-classical) spectral curve of the first Painlevé equation $P_{\rm I}$ gives a WKB solution for the isomonodromy problem for $P_{\rm I}$. In other words, the isomonodromy system is a quantum curve in the sense of [Dumitrescu O., Mulase M., Lett. Math. Phys. 104 (2014), 635-671, arXiv:1310.6022] and [Dumitrescu O., Mulase M., arXiv:1411.1023].

math-ph↗
arXiv2014

Exact WKB analysis and cluster algebras II: Simple poles, orbifold points, and generalized cluster algebras

This is a continuation of developing mutation theory in exact WKB analysis using the framework of cluster algebras. Here we study the Schrodinger equation on a compact Riemann surface with turning points of simple-pole type. We show that the orbifold triangulations by Felikson, Shapiro, and Tumarkin provide a natural framework of describing the mutation of Stokes graphs, where simple poles correspond to orbifold points. We then show that under the mutation of Stokes graphs around simple poles the Voros symbols mutate as the variables of generalized cluster algebras introduced by Chekhov and Shapiro.

math.CA↗
arXiv2014

On WKB theoretic transformations for Painleve transcendents on degenerate Stokes segments

The WKB theoretic transformation theorem established in [KT2] implies that the first Painleve equation gives a normal form of Painleve equations with a large parameter near a simple P-turning point. In this paper we extend this result and show that the second Painleve equation (PII) and the third Painleve equation (PIII'(D7)) of type D7 give a normal form of Painleve equations on a degenerate P-Stokes segments connecting two different simple P-turning points and on a degenerate P-Stokes segment of loop-type, respectively. That is, any 2-parameter formal solution of a Painleve equation is reduced to a 2-parameter formal solution of (PII) or (PIII'(D7)) on these degenerate P-Stokes segments by our transformation.

math.CA↗