Next Seminar
1 October 2025 (Wed) 17:30～(JST)/10:30～(CEST/UTC+2)
Alba Grassi (University of Geneva and CERN)
Functional Identities and Geometry of Painlevé III3 Quantum Operators
Abstract :
In the study of the Painlevé III3 equation, two quantum mechanical
operators on L2(IR) emerge: the modified Mathieu operator and the
McCoy-Tracy-Wu operator. The first arises from the scalar equation
associated with a Painlevé 2x2 linear system. The second, instead,
appears in the analysis of special solutions to Painlevé III3; more
precisely, its Fredholm determinant computes a one-parameter family of
Painlevé tau-functions. In this talk, I will show that these two
operators share a common geometric origin: they both emerge from two
specific limit of the quantum mirror curve associated with a local
Calabi-Yau manifold known as local F0. I will then explain how this
perspective naturally leads to a simple and elegant functional relation
between the two operators.
Painlevé Seminar Topics
Recently, we have encountered many interesting links between differential
equations of Painlevé types and other fields of mathematics such
as algebraic geometry, non-abelian Hodge theory and topological recursion
and so on.
In this web seminar, we will organize research talks on Painlevé
equations and related topics on web (zoom), basically about once every
two or three weeks.
From September 2022, Regular time for seminar talk will be on Wednesday from 17:30--18:30 + discussion time
in JST(Japan) or 10:30--11:30 in UTC+1. We are looking forward to your participation in
the seminar. In order to get zoom access, please register through the form.
List of Talks
You can find the titles and abstracts of all seminars. ⇒ List of Talks
Registration
Online. Please register at the form, we will send you about web seminar address.
Organizers
Arata Komyo (Hyogo), Frank Loray (Rennes 1),
Ryo Ohkawa (Osaka Metropolitan), Masa-Hiko Saito (Kobe Gakuin), Takafumi Matsumoto (RIMS, Kyoto)
Scientific Committee (other than the Organizing Committee)
Supported by
French ANR-16-CE40-0008 project "Foliage" (Frank Loray)
JSPS Grant-in-aid (A) 22H00094 (PI: Masa-Hiko Saito)
Osaka Central Advanced Mathematical Institute:
MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849.
Joyce structures were introduced by T. Bridgeland in the context of the space of stability conditions of a three-dimensional Calabi-Yau category and its associated Donaldson-Thomas invariants, and have also appeared in the context of isomonodromic systems associated with Painlevé equations. Under certain non-degeneracy conditions, they encode a complex hyperkähler structure on the tangent bundle of the base of the Joyce structure. Taking inspiration from this work, I'll give a description of (real) hyperkahler metrics compatible with an integrable system structure that is very similar to the description of Joyce structures. In particular, I will give examples of such hyperkähler metrics where there is a precise analog of the Plebański function appearing in the context of Joyce structures. This is work in progress, which is a follow-up to https://arxiv.org/abs/2403.00548.