Waseda Special Lectures

Special Lecturer (series of lectures)

Professor Emeritus Martin Guest (Waseda University) 

Theme: Introduction to the Dubrovin conjecture

Abstract: 

These lectures will be an introduction to the Dubrovin Conjecture, and to the subsequent work of Galkin-Golyshev-Iritani and Cotti-Dubrovin-Guzzetti on the verification of the conjecture in several cases.

The Dubrovin Conjecture relates (1) the monodromy data (Stokes and connection matrices) of the quantum differential equation of a Fano manifold X, to (2) the Euler form and exceptional sequences of vector bundles (or coherent sheaves) on X.  The starting point for this relation was Givental's theory of quantum differential equations, which was developed further by Dubrovin using methods from differential geometry and the theory of integrable systems. However, the Dubrovin Conjecture brought a new aspect into this picture, namely a rather surprising relation with the algebraic geometry of vector bundles on X. The key concept in this relation is a topological invariant called the gamma class of X, which is a multiplicative characteristic class in the sense of Hirzebruch.  

We shall give an introduction to this theory, and explain some of the progress which has been made on establishing (and understanding) the conjecture. 

Invited Lecturers

Atsushi Kanazawa (Waseda University)
Title: TBA

Tadashi Udagawa (Waseda University)
Title: On tt*-structures from ADE-type Stokes data 

Abstract: 

Cecotti and Vafa introduced topological–anti-topological fusion (tt*) structures in the study of N=2 supersymmetric field theories. In the context of quantum cohomology, tt*-structures give rise to the Dubrovin connection. In mathematics, Dubrovin observed that tt*-structures can be described as isomonodromic deformations of certain linear systems and encoded by upper unitriangular matrices, called Stokes matrices. In this framework, Cecotti and Vafa proposed an ADE classification of tt*-structures.

In this talk, we focus on a class of tt*-structures on C∗ for which the Higgs field has distinct eigenvalues and the Hermitian metric is radial. Under these assumptions,  we address two main issues: the ambiguity of Stokes matrices and the solvability of the associated Riemann–Hilbert problem. We show that the ambiguity is governed by an action of the braid group and that tt*-structures are classified by equivalence classes of Stokes matrices.

Finally, we prove that matrices related to ADE-type Cartan matrices give rise to tt*-structures by solving an associated Riemann–Hilbert problem. This provides a direct analytic realization of the ADE classification and clarifies the geometric mechanism underlying the interplay between Stokes phenomena and the positivity of Cartan-type matrices.

Timetable

April 21 (Tue)  13:10 - 14:50  Martin Guest
Introduction to the Dubrovin conjecture, Part I

April 21 (Tue)  15:05 - 16:45  Martin  Guest
Introduction to the Dubrovin conjecture, Part II

April 22 (Wed)  13:10 - 14:50  Martin Guest
special seminar (Quantum cohomology and related topics)

April 22 (Wed)  15:05 - 16:45  Martin Guest  
special seminar (Quantum cohomology and related topics)

Organizers

Yoshihiro Ohnita (chair, Waseda University & OCAMI)

Yuichiro Sato (Waseda University)

Martin Guest (professor emeritus of Waseda University)

Support

- JSPS Grant-in-Aid for Scientific Research (C) No.22K03293 (Principal Investigator: Katsuhiro Moriya)

- JSPS Grant-in-Aid for Scientific Research (A) No.23H00083 (Principal Investigator: Martin Guest)

- JSPS Grant-in-Aid for Scientific Research (A) No.22H00094 (Principal Investigator: Masa-hiko Saito)

- Waseda Institute for Mathematical Science (WIMS) 

- Osaka Central Advanced Mathematical Institute (OCAMI), Osaka Metropolitan University  

Contact

Yoshihiro Ohnita (Waseda U. & OCAMI) ohnita @omu.ac.jp

Links

Waseda Special Lectures 2024 "Integrabilities in Differential Geometry, and their Applications"

Waseda Special Lectures 2025 "Exceptional Symmetric Spaces and Related Topics"

Waseda Special Lectures 2025 in Fall "Geometric Analysis of Hypersurfaces - Rigidity, Uniqueness and Geometric Inequalities -"