Accessory parameters are mathematical objects which appear in linear differential equations, unknown functions of differential equations associated to isomonodromic deformation. Moreover, they are related to number theory, algebraic geometry, differential geometry, representation theory and mathematical physics. This workshop is organized for the purpose of exchanging research on accessory parameters and the related topics. In honor of retirement of Professor Yoshishige Haraoka, who has organized this workshop every year since 2006, it was decided to hold Workshop on Accessory Parameters in 2023 as a meeting to celebrate his retirement. All interested persons are welcome to attend.

Invited Speakers

• Shunya Adachi (Kumamoto University)
• Mikhail Bershtein (HSE, IPMU)
• Daniel Bertrand (Paris VI)
• Yoshishige Haraoka (Kumamoto University)
• Kazuki Hiroe (Chiba University)
• Kohei Iwaki (University of Tokyo)
• Fumiharu Kato (Tokyo Institute of Technology / Kadokawa Dwango educational institute)
• Makoto Kawashima (Nihon University)
• Oleg Lisovyy (University of Tours)
• Atsuhira Nagano (Kanazawa University)
• Toshio Oshima (Josai University)
• Marcin Piątek (University of Szczecin)

Program

20 March (Mon)

• 13:00 — 14:00   Atsuhira Nagano   "On a sequence of families of K3 surfaces attached to complex reflection groups"
• 14:15 — 15:15   Mikhail Bershtein   "Quantum spectral problems and isomonodromic deformations"
• (Coffee Break)
• 15:45 — 16:45   Oleg Lisovyy   "Painlevé and Heun functions from Liouville conformal blocks"

21 March (Tue)

•   9:30 — 10:30   Fumiharu Kato   "Rational points of rigid-analytic sets: a Pila-Wilkie type theorem"
• 10:45 — 11:45   Daniel Bertrand   "Torsion points and elementary integrability in families of differential forms"
• (Lunch Break)
• 13:00 — 14:00   Toshio Oshima   "Integral transformations of hypergeometric functions with several variables"
• 14:15 — 15:15   Shunya Adachi   "Unitary monodromies of Fuchsian differential equations"
• (Coffee Break)
• 15:45 — 16:45   Yoshishige Haraoka   "Shift operators and Riemann-Hilbert problem"
• 18:00 — 20:00   Retirement Party (KKR Hotel Kumamoto "AMAKUSA")

22 March (Wed)

•   9:30 — 10:30   Makoto Kawashima   "Rodligues type formulae and linear independence"
• 10:45 — 11:45   Kohei Iwaki   "Series expansion of accessory parameters of confluent Heun equations with a ramified irregular singular point"
• (Lunch Break)
• 13:00 — 14:00   Kazuki Hiroe   "A generalization of Haraoka's multiplicative middle convolution"
• 14:15 — 15:15   Marcin Piątek   "On accessory parameters, conformal blocks and some applications"

Abstracts

• Atsuhira Nagano
  Title: On a sequence of families of K3 surfaces attached to complex reflection groups
  Abstract: K3 surfaces are particular complex surfaces. We can regard them as 2-dimensional generalizations of elliptic curves. Also, finite complex reflection groups are important in geometry and combinatorics, because they are natural generalizations of Weyl groups of root systems. In this talk, the speaker will introduce a sequence of families of K3 surfaces, which are closely related to complex reflection groups of exceptional type. Namely, the periods of the families are exactly described in terms of theta functions and invariants of reflection groups. Moreover, if time allows, he will make a comparison between the above-mentioned surfaces and K3 surfaces attached with hypergeometric equations of type (3,6), which are studied by Matsumoto-Sasaki-Yoshida (1992).
  
  
• Mikhail Bershtein
  Title: Quantum spectral problems and isomonodromic deformations
  Abstract: We study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of 2 × 2 linear systems (Riemann-Hilbert correspondence). The main examples for the talk will be Mathieu operator and Lamé operator. By using the Kyiv formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions. Through blowup relations, we also find Nekrasov-Shatashvili type of quantizations. Based on joint work with A. Grassi and P. Gavrylenko
  
  
• Oleg Lisovyy
  Title: Painlevé and Heun functions from Liouville conformal blocks
  Abstract: I will review connections between the problem of construction of linear ordinary differential equations with prescribed monodromy and the 2D conformal field theory. This correspondence leads to a number of conjectures in the theory of Painlevé and Heun equations some of which have already been proven rigorously and some remain open. The two main applications I will focus on are the construction of the general solution of Painlevé VI equation and the computation of accessory parameter and connection formulas for Heun's equation in terms of Liouville conformal blocks.
  
  
• Fumiharu Kato
  Title: Rational points of rigid-analytic sets: a Pila-Wilkie type theorem
  Abstract: Joint work with Gal Binyamini (Weizmann). We will present a rigid-analytic analog of the Pila-Wilkie counting theorem, giving sub-polynomial upper bounds for the number of rational points in the transcendental part of a Q_p-analytic set, and the number of rational functions in a F_q((t))-analytic set. For Z[[t]]-analytic sets we prove such bounds uniformly for the specialization to every non-archimedean local field.
  
  
• Daniel Bertrand
  Title: Torsion points and elementary integrability in families of differential forms
  Abstract: The relative Manin-Mumford conjecture expresses a local to global principle for torsion sections on group schemes. In relative dimension 2, it fails in essentially only one case, which involves an elliptic curve with complex multiplications and two divisors satisfying certain conditions. The recent work of Masser and Zannier on elementary integrability also expresses a local to global principle, and it fails under the very same conditions. We present analytic proofs of these results, where Hermite's solutions of Lamé equations with integral index provide a unifying thread.
  
  
• Toshio Oshima
  Title: Integral transformations of hypergeometric functions with several variables
  Abstract: As a generalization of Riemann-Liouville integral, we introduce integral transformations of convergent power series which can be applied to hypergeometric functions with several variable.
  
  
• Shunya Adachi
  Title: Unitary monodromies of Fuchsian differential equations
  Abstract: The monodromy group G of a Fuchsian ordinary differential equation is said to be unitary if and only if it admits a non-trivial G-invariant Hermitian form. It is known that the monodromy of the Gauss hypergeometric equation is unitary if the local exponents are all real numbers. For several other rigid Fuchsian equations (e.g. Generalized hypergeometric, Jordan-Pochhammer, Yokoyama’s list, etc.), the unitarity of the monodromies has also been studied. In this talk, we consider the unitarity of monodromies of non-rigid Fuchsian equations. In particular, we will give constructive characterizations of the unitarity monodromies of spectral type (111,111,111) and (11,11, ...,11). A part of this talk is based on a joint work with Yoshishige Haraoka (Kumamoto University).
  
  
• Yoshishige Haraoka
  Title: Shift operators and Riemann-Hilbert problem
  Abstract: A shift operator is an operator which sends a differential equation to another differential equation with the exponents shifted by integers. We will study the existence of shift operators for Fuchsian systems of differential equations. For the purpose, we use the theory of Lappo-Danilevsky, which we recall in the first part of this talk.
  
  
• Makoto Kawashima
  Title: Rodligues type formulae and linear independence
  Abstract: The Padé approximants of logarithm function are given by the Legendre polynomials. The Legendre polynomials have a simple representation known as Rodrigues' formula. In this talk, let us introduce a generalization of Rodrigues' formula for Padé approximants of a certain class of holonomic Laurent series. Our generalization yields new examples of explicit Padé approximants of hypergeometric functions. These examples enable us to give new linear independence of the values of hypergeometric functions at algebraic points. Part of this talk is a joint work with S.David and N.Hirata-Kohno.
  
  
• Kohei Iwaki
  Title: Series expansion of accessory parameters of confluent Heun equations with a ramified irregular singular point
  Abstract: In 2021, Lisovyy-Naidiuk developed a framework to compute a series expansion of the accessory parameter of several confluent Heun equations. They also formulated an irregular version of the Zamolodchikov conjecture which claims that the accessory parameter is given as a classical limit of a certain irregular conformal block. In this talk, I will discuss an analogous conjecture in the presence of a ramified irregular singular point. This talk is based on a joint work with Hajime Nagoya.
  
  
• Kazuki Hiroe
  Title: A generalization of Haraoka's multiplicative middle convolution
  Abstract: In [Math. Z. 2020], Haraoka introduced the middle convolution functor for local systems on complements of braid arrangements, as a natural generalization of the Katz middle convolution for local systems on P¹\{n-points}. This will be a breakthrough approach to construct a new theory of hypergeometric functions of several variables in terms of rigid local systems. In this talk, it will be explained that Haraoka's middle convolution is closely related to the functor on representations of braid groups called the Long-Moody functor which is know to be an efficient method for constructing braid representations, for example, Burau representation, Gassner representation, Lawrence-Krammer-Bigelow representation, and etc. are obtained through this method. This similarity between these functors of Haraoka and Long-Moody allows us to define a new functor which extends the framework of Katz algorithm to local systems on various topological spaces, for example B_n bundles associated to simple Weierstrass polynomials in the sense of Hansen, Møller, and Cohen-Suciu. This is a joint work with Haru Negami in Chiba University.
  
  
• Marcin Piatek
  Title: On accessory parameters, conformal blocks and some applications
  Abstract: The problem of accessory parameters arises in a context of Fuchschian uniformization of hyperbolic Riemann surfaces and is closely related to the Liouville field theory. In the first part of my talk, I will show how one can calculate accessory parameters for a four-punctured sphere and a one-punctured torus by taking the semi-classical limit of the quantum Liouville theory. Within this approach one can calculate also other important geometric quantities such as geodesic length functions and the Weil-Petersson metric on the moduli space of the four-punctured sphere M(0,4). The latter allows a numerical study of the Laplacian spectrum on M(0,4). The relationship between accessory parameters and the classical limit of the quantum Liouville theory has recently been given a beautiful mathematically rigorous explanation in the so-called probabilistic approach by H.Lacoin, R.Rhodes and V.Vargas. I will mention these results in my talk. The basic building blocks in the above calculations are the so-called classical conformal blocks. These are new special functions that have generated a lot of interest in the last decade. In the second part of my talk, I will show how the four-point classical block on the sphere solves the monodromy problem for the Heun equation, and how some solutions of this equation and Hill's-type equations can be calculated using conformal field theory formalism. Finally, I will mention various contexts in which classical conformal blocks also appear. In particular, I will review a connection between the classical limit of conformal blocks, gauge theory and quantum integrable systems.
  
  

Photos and Slides (Google drive)

Organizers

Saiei-Jaeyeong Matsubara-Heo

Faculty of Advanced Science and Technology, Kumamoto University

E-mail: saiei@educ.kumamoto-u.ac.jp

Hiroshi Ogawara

Mathematical Science Education Center, Kumamoto University

E-mail: hogawara@kumamoto-u.ac.jp

Scientific Committee

Yoshishige Haraoka

Faculty of Advanced Science and Technology, Kumamoto University

Saiei-Jaeyeong Matsubara-Heo

Faculty of Advanced Science and Technology, Kumamoto University

Hajime Nagoya

School of Mathematics and Physics, Kanazawa University