Exact WKB analysis, cluster algebras and topological recursion

• Project/Area Number: 16K17613
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: The University of Tokyo (2020); Nagoya University (2016-2019)
• Principal Investigator: Iwaki Kohei 東京大学, 大学院数理科学研究科, 准教授 (00750598)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 完全WKB解析 / 位相的漸化式 / Painleve方程式 / 団代数 / BPS構造 / Voros係数 / 量子曲線 / パンルヴェ方程式

Outline of Final Research Achievements

We investigated a relationship between exact WKB analysis, cluster algebras, and topological recursion. The results we obtained are the following: (1) We proved that the discrete Fourier transform of topological recursion partition function gives a Painleve tau-function. (2) For a class spectral curves, we established a formula which allows us to describe the free energy of topological recursion in terms of the corresponding BPS structure.

Academic Significance and Societal Importance of the Research Achievements

微分方程式論の研究手法である完全WKB解析と、代数学の研究対象である団代数やBPS構造、さらには幾何学や数理物理学と密接に関わる位相的漸化式との関係性に関する研究を行ったことで、全く異なる分野の知見を共有する足掛かりを与えることができた。これが本研究成果の学術的意義である。

Research Products

• [Int'l Joint Research] Lyon/Tours(France)
• [Int'l Joint Research] Virginia(米国)
• [Int'l Joint Research] Universite de Lyon(France)
• [Int'l Joint Research] UC Davis(米国)
• [Int'l Joint Research] University of Toronto(Canada)
• [Journal Article] Witten－Reshetikhin－Turaev Function for a Knot in Seifert Manifolds (2021)
  • Author(s): Hiroyuki Fuji, Kohei Iwaki, Hitoshi Murakami and Yuji Terashima
  • Journal Title: Communications in Mathematical Physics Volume: － Issue: 1 Pages: 225-251
  • DOI: 10.1007/s00220-021-03953-y
  • Peer Reviewed / Open Access
• [Journal Article] 2-Parameter tau-Function for the First Painleve Equation: Topological Recursion and Direct Monodromy Problem via Exact WKB Analysis (2020)
  • Author(s): Kohei Iwaki
  • Journal Title: Commun. Math. Phys. Volume: 未定 Issue: 2 Pages: 1047-1098
  • DOI: 10.1007/s00220-020-03769-2
  • Peer Reviewed
• [Journal Article] 2-parameter τ-function for the first Painleve equation --Topological recursion and direct monodromy problem via exact WKB analysis-- (2020)
  • Author(s): Kohei Iwaki
  • Journal Title: Communications in Mathematical Physics (to appear) Volume: 未定
  • Peer Reviewed
• [Journal Article] Voros coefficients for the hypergeometric differential equations and Eynard?Orantin’s topological recursion: Part II: For confluent family of hypergeometric equations (2019)
  • Author(s): Iwaki Kohei、Koike Tatsuya、Takei Yumiko
  • Journal Title: Journal of Integrable Systems Volume: 4 Issue: 1
  • DOI: 10.1093/integr/xyz004
  • Peer Reviewed / Open Access
• [Journal Article] Reconstructing GKZ via topological recursion (2019)
  • Author(s): Hiroyuki Fuji, Kohei Iwaki, Masahide Manabe, Ikuo Satake
  • Journal Title: Communications in Mathematical Physics Volume: 371 Issue: 3 Pages: 839-920
  • DOI: 10.1007/s00220-019-03590-6
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Duality/sum formulas for iterated integrals and their application to multiple zeta values (2019)
  • Author(s): Hirose Minoru、Iwaki Kohei、Sato Nobuo、Tasaka Koji
  • Journal Title: Journal of Number Theory Volume: 195 Pages: 72-83
  • DOI: 10.1016/j.jnt.2018.05.019
  • Peer Reviewed / Open Access
• [Journal Article] Painlev\'e equations, topological type property and reconstruction by the topological recursion (2018)
  • Author(s): Iwaki K.、Marchal O.、Saenz A.
  • Journal Title: Journal of Geometry and Physics Volume: 124 Pages: 16-54
  • DOI: 10.1016/j.geomphys.2017.10.009
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Exact WKB analysis of Schr\"odinger equations with a Stokes curve of loop type (2018)
  • Author(s): Takashi Aoki、Kohei Iwaki、Toshinori Takahashi
  • Journal Title: Funkcialaj Ekvacioj (掲載予定) Volume: 未定
  • Peer Reviewed
• [Journal Article] Painleve 2 Equation with Arbitrary Monodromy Parameter, Topological Recursion and Determinantal Formulas (2017)
  • Author(s): Kohei Iwaki and Olivier Marchal
  • Journal Title: Annales Henri Poincare Volume: 未定 Issue: 8 Pages: 1-40
  • DOI: 10.1007/s00023-017-0576-z
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Topological recursion, quantum curves and the second Painleve equation (2017)
  • Author(s): Kohei Iwaki
  • Journal Title: RIMS Kokyuroku Bessatsu (to appear) Volume: B61
  • Peer Reviewed
• [Presentation] Topological recursion for hypergeometric type spectral curves (2020)
  • Author(s): 岩木耕平
  • Organizer: ミラー対称性の諸相2020
  • Invited
• [Presentation] Topological recursion and uncoupled BPS structure arising from spectral curves of hypergeometric type (2020)
  • Author(s): Kohei Iwaki
  • Organizer: Strings and QFTs for Eurasian time zone (online)
  • Invited
• [Presentation] Topological recursion and uncoupled BPS structure arising from spectral curves of hypergeometric type (2020)
  • Author(s): Kohei Iwaki
  • Organizer: SISSA's Integrable System and Mathematical Physics Seminar (online)
  • Invited
• [Presentation] Topological recursion, exact WKB analysis and Painleve equation (2020)
  • Author(s): Kohei Iwaki
  • Organizer: Integrable Systems 2020 (online)
  • Int'l Joint Research / Invited
• [Presentation] Exact WKB analysis and related topics (5 talks) (2019)
  • Author(s): Kohei Iwaki
  • Organizer: Graduate Mini Course on Resurgence in Mathematics and Physics
  • Int'l Joint Research / Invited
• [Presentation] Topological recursion and Painleve I τ-function (2019)
  • Author(s): Kohei Iwaki
  • Organizer: IX Workshop on Geometric Correspondences of Gauge Theories
  • Int'l Joint Research / Invited
• [Presentation] Painleve τ-function and topological recursion (2019)
  • Author(s): Kohei Iwaki
  • Organizer: Moduli Spaces, Representation Theory and Quantization
  • Int'l Joint Research / Invited
• [Presentation] Topological recursion and Painleve I τ-function (2019)
  • Author(s): 岩木耕平
  • Organizer: リーマン面に関連する位相幾何学
  • Invited
• [Presentation] Painleve τ-関数と位相的漸化式 (2019)
  • Author(s): 岩木耕平
  • Organizer: 可積分系数理の深化と展開
  • Invited
• [Presentation] 位相的漸化式と第 I 型 Painleve 方程式の τ 函数 (2019)
  • Author(s): 岩木耕平
  • Organizer: 日本数学会 2019 年 度秋季総合分科会 (函数方程式論分科会)
• [Presentation] 2-parameter tau-function for the first Painleve equation via topological recursion (2019)
  • Author(s): 岩木耕平
  • Organizer: 超幾何方程式研 究会 2019
  • Invited
• [Presentation] 完全WKB解析入門,完全WKB解析とクラスター代数 (2コマ) (2019)
  • Author(s): 岩木耕平
  • Organizer: 第72回 ENCOUNTERwithMATHEMATICS
  • Invited
• [Presentation] 完全 WKB 解析と spectral networks 1, 2 (2 コマ) (2019)
  • Author(s): 岩木耕平
  • Organizer: モジュライ空間と双曲幾何の進展
  • Invited
• [Presentation] Toward a construction of 2-parameter family of Painleve tau-function via the topological recursion (2018)
  • Author(s): Kohei Iwaki
  • Organizer: D-modules, quantum geometry, and related topics
  • Int'l Joint Research / Invited
• [Presentation] Exact WKB analysis and related topics I, II (2コマ) (2018)
  • Author(s): 岩木耕平
  • Organizer: ALTReT2018
  • Invited
• [Presentation] 位相的漸化式と WKB 解析 1,2 (2コマ) (2018)
  • Author(s): 岩木耕平
  • Organizer: 複素領域における微分方程式とその周辺
  • Invited
• [Presentation] 完全 WKB 解析とその周辺 (2018)
  • Author(s): 岩木耕平
  • Organizer: 第 65 回幾何学シンポジウム
  • Invited
• [Presentation] 完全 WKB 解析と位相的漸化式 (2018)
  • Author(s): 岩木耕平
  • Organizer: 日本数学会無限可積分系セッション特別講演
  • Invited
• [Presentation] Topological recursion and Voros coefficients (2018)
  • Author(s): Kohei Iwaki
  • Organizer: International Symposium, RIKKYO MathPhys 2018, Rikkyo Univeristy, 12-14 March, 2018.
  • Int'l Joint Research / Invited
• [Presentation] Exact WKB analysis and related topics (2017)
  • Author(s): Kohei Iwaki
  • Organizer: The 2nd OCAMI-KOBE-WASEDA Joint International Workshop on Differential Geometry and Integrable Systems
  • Place of Presentation: Osaka City University
  • Year and Date: 2017-03-14
  • Int'l Joint Research / Invited
• [Presentation] Exact WKB analysis and related topics (2017)
  • Author(s): Kohei Iwaki
  • Organizer: Irregular Connections, Character Varieties and Physics
  • Place of Presentation: Paris VII, France
  • Year and Date: 2017-03-06
  • Int'l Joint Research / Invited
• [Presentation] Exact WKB Analysis and Cluster Algebras (2017)
  • Author(s): Kohei Iwaki
  • Organizer: Quantum groups, special functions and integrable systems, LMPT Tours, France, 17-19 May 2017
  • Int'l Joint Research / Invited
• [Presentation] Exact WKB analysis and related topics (4 lectures) (2017)
  • Author(s): Kohei Iwaki
  • Organizer: School and International Conference on Geometry and Quantization (GEOQUANT 2017), Centre for Quantum Geometry of Moduli Spaces, University of Aarhus, Denmark, July 31, August 11, 2017.
  • Int'l Joint Research / Invited
• [Presentation] Exact WKB analysis and Voros symbols (2017)
  • Author(s): Kohei Iwaki
  • Organizer: Higgs Bundles, Harmonic Maps, and Integrable Systems, Hannover, Germany, August 14, 18
  • Int'l Joint Research / Invited
• [Presentation] Stokes structure of equivariant CP1 and wall-crossing (2017)
  • Author(s): Kohei Iwaki
  • Organizer: Algebraic Geometry and Symplectic Geometry at Crossroads, Kyoto University, September 25-26, 2017.
  • Int'l Joint Research / Invited
• [Presentation] Stokes structure of equivariant CP1 and wall-crossing (2017)
  • Author(s): Kohei Iwaki
  • Organizer: Differential Geometry and Differential Equations: the influence of Mirror Symmetry and Physics, Waseda University, 11-15 December, 2017.
  • Int'l Joint Research / Invited
• [Presentation] Exact WKB analysis for oscillatory integrals (2 lectures) (2017)
  • Author(s): Kohei Iwaki
  • Organizer: Winter School on Singularity and Mathematical Physics, Hotel Le Win, Jeonju, Korea, December 18-20, 2017.
  • Int'l Joint Research / Invited
• [Presentation] Topological recursion, Painleve equations and cluster algebras (2016)
  • Author(s): Kohei Iwaki
  • Organizer: Algebraic Geometry and Integrable Systems
  • Place of Presentation: Kobe University
  • Year and Date: 2016-12-05
  • Int'l Joint Research / Invited
• [Presentation] Stokes graphs for isomonodromy systems and classification of Painleve equations (2016)
  • Author(s): Kohei Iwaki
  • Organizer: Conformal Field theory, Isomonodromic tau-functions and Painleve equations
  • Place of Presentation: Kobe University
  • Year and Date: 2016-11-21
  • Int'l Joint Research / Invited
• [Presentation] 位相的漸化式と Painleve方程式 (2016)
  • Author(s): 岩木耕平
  • Organizer: 可積分系セミナー
  • Place of Presentation: 神戸大学
  • Year and Date: 2016-10-24
  • Invited
• [Presentation] Resurgence, exact WKB and wall-crossing (2016)
  • Author(s): Kohei Iwaki
  • Organizer: Geometric Representation Theory
  • Place of Presentation: 京都大学数学教室
  • Year and Date: 2016-10-10
  • Int'l Joint Research / Invited
• [Presentation] Exact WKB analysis vs spectral networks (2016)
  • Author(s): Kohei Iwaki
  • Organizer: New development of microlocal analysis and singular perturbation theory
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2016-10-03
  • Int'l Joint Research / Invited
• [Presentation] 完全 WKB 解析とその応用 1, 2 (2 コマ) (2016)
  • Author(s): 岩木耕平
  • Organizer: リーマン面に関連する位相幾何学
  • Place of Presentation: 東京大学
  • Year and Date: 2016-09-04
  • Invited
• [Presentation] Exact WKB analysis, Painleve equations and the Stokes phenomenon (2016)
  • Author(s): Kohei Iwaki
  • Organizer: AMS von Neumann Symposium: Topological Recursion and its Influence in Analysis, Geometry, and Topology
  • Place of Presentation: Hilton Charlotte University Place, Charlotte, NC
  • Year and Date: 2016-07-04
  • Int'l Joint Research / Invited
• [Presentation] Exact WKB analysis, cluster algebras and Painleve equations, (2016)
  • Author(s): Kohei Iwaki
  • Organizer: String-Math 2016
  • Place of Presentation: College de France
  • Year and Date: 2016-06-27
  • Int'l Joint Research / Invited
• [Presentation] Exact WKB analysis for Painleve equations and wall-crossing type structures (2016)
  • Author(s): Kohei Iwaki
  • Organizer: School - Wall-crossing and quiver varieties
  • Place of Presentation: Centre Interfacultaire Bernoulli, Lausanne, Switzerland
  • Year and Date: 2016-05-23
  • Int'l Joint Research
• [Presentation] Exact WKB analysis and spectral networks (2016)
  • Author(s): Kohei Iwaki
  • Organizer: String Theory Seminar
  • Place of Presentation: UC Davis, US
  • Year and Date: 2016-04-14
  • Invited