Workshop on Accessory Parameters



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.
Date: 20 (Mon) — 22 (Wed), March 2023
Venue: Room D201, Kurokami South E3 (Faculty of Science Bldg. 3),
Faculty of Science, Kumamoto University
Date: 21 (Tue), March 2023, 18:00 — 20:00
Retirement Party of Prof. Yoshishige Haraoka
Venue: KKR Hotel Kumamoto "AMAKUSA"
331 Chibajomachi, Chuoku, Kumamotoshi
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 rigidanalytic sets: a PilaWilkie 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 RiemannHilbert 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 2dimensional 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 abovementioned surfaces and K3 surfaces attached with hypergeometric equations of type (3,6), which are studied by MatsumotoSasakiYoshida (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 (RiemannHilbert 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 selfdual Nekrasov functions. Through blowup relations, we also find NekrasovShatashvili 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 rigidanalytic sets: a PilaWilkie type theorem
Abstract: Joint work with Gal Binyamini (Weizmann). We will present a rigidanalytic analog of the PilaWilkie counting theorem, giving subpolynomial 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 nonarchimedean local field.

Daniel Bertrand
Title: Torsion points and elementary integrability in families of differential forms
Abstract: The relative ManinMumford 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 RiemannLiouville 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 nontrivial Ginvariant 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, JordanPochhammer, Yokoyama’s list, etc.), the unitarity of the monodromies has also been studied.
In this talk, we consider the unitarity of monodromies of nonrigid 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 RiemannHilbert 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 LappoDanilevsky, 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.HirataKohno.

Kohei Iwaki
Title: Series expansion of accessory parameters of confluent Heun equations with a ramified irregular singular point
Abstract: In 2021, LisovyyNaidiuk 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 P^{1}\{npoints}.
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 LongMoody functor which is know to be an efficient method for constructing braid representations, for example, Burau representation, Gassner representation, LawrenceKrammerBigelow representation, and etc. are obtained through this method.
This similarity between these functors of Haraoka and LongMoody 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 CohenSuciu.
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 fourpunctured sphere and a onepunctured torus by taking the semiclassical limit of the quantum Liouville theory. Within this approach one can calculate also other important geometric quantities such as geodesic length functions and the WeilPetersson metric on the moduli space of the fourpunctured 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 socalled 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 socalled 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 fourpoint classical block on the sphere solves the monodromy problem for the Heun equation, and how some solutions of this equation and Hill'stype 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.
Organizers
 SaieiJaeyeong MatsubaraHeo
 Faculty of Advanced Science and Technology, Kumamoto University
 Email: saiei@educ.kumamotou.ac.jp
 Hiroshi Ogawara
 Mathematical Science Education Center, Kumamoto University
 Email: hogawara@kumamotou.ac.jp
Scientific Committee
 Yoshishige Haraoka
 Faculty of Advanced Science and Technology, Kumamoto University
 SaieiJaeyeong MatsubaraHeo
 Faculty of Advanced Science and Technology, Kumamoto University
 Hajime Nagoya
 School of Mathematics and Physics, Kanazawa University