This workshop is organised in conjunction with the Short Course on Nonabelian Hodge Theory by Prof. Florent Schaffhauser, 1-22 July 2021, at Waseda University.

The Short Course and Workshop are an activity of the Mathematics and Physics Unit "Multiscale Analysis, Modelling and Simulation" Top Global University Project, Waseda University

Speakers:

Martin Guest (Waseda Universiy)

Hisashi Kasuya (Osaka University)

Georgios Kydonakis (University of Heidelberg)

Qiongling Li (Chern Insitute of Mathematics, Nankai University)

Claudio Meneses (Kiel University)

Florent Schaffhauser (University of Los Andes and University of Strasbourg)

Yota Shamoto (Waseda University)

Szilard Szabo (Budapest University of Technology)

Schedule:

Tuesday 5 October

16:00-16:45 Schaffhauser

17:00-17:45 Kydonakis

break

18:15-19:00 Szabo

19:15-20:00 Shamoto

Wednesday 6 October

16:00-16:45 Guest

17:00-17:45 Kasuya

break

18:15-19:00 Li

19:15-20:00 Meneses

(Japan time)

Titles and abstracts:

Martin Guest

Topological-antitopological fusion and Higgs bundles

Examples of harmonic bundles/Higgs bundles given by solutions of the tt* equations have been studied from various points of view over the past 30 years, starting with the work of Cecotti-Vafa on topological-antitopological fusion in the 1990's. Existence results for solutions are known from work of McCoy-Tracy-Wu, Its-Novokshenov, Tracy-Widom, Guest-Its-Lin, Mochizuki, amongst others. We shall give a brief introduction to this topic and then describe some new mathematical aspects. In particular, beyond the question of existence, it is the properties of solutions related to geometry and physics which are of primary interest in our work.

Hisashi Kasuya

Higgs bundles in odd-dimensions

On a compact Kahler manifold, there is an equivalence between semi-simple flat vector bundles and polystable Higgs bundles with vanishing Chern classes via harmonic metrics (Corlette, Simpson). I gave an analogue of this equivalence on compact Sasakian manifolds which are an odd-dimensional counterpart to Kahler manifolds (joint work with I. Biswas , CMP (2021)). The main purpose of this talk is to observe this equivalence on $3$-dimensional manifolds and compare with Riemann surfaces.

Georgios Kydonakis

Counting connected components of parabolic $G$-Higgs bundle moduli spaces

We will describe a method for counting the number of connected components of moduli spaces of parabolic $G$-Higgs bundles, in the cases of Hermitian symmetric Lie groups $G$ of tube-type. When the parabolic weights are rational, an equivalence between parabolic bundles and holomorphic bundles over $V$-surfaces (that is, 2-dimensional orbifolds), allows one to introduce new topological invariants as Stiefel-Whitney classes of associated $V$-bundles. The number of these invariants provides a lower bound for the number of connected components, which can be improved to an exact component count by studying the minima of a moment map on the moduli space, which is a Morse-Bott function. This is joint work with Hao Sun and Lutian Zhao.

Qiongling Li

Complete solutions of Toda equations and cyclic Higgs bundles over non-compact surfaces

A Higgs bundle over a Riemann surface is a pair consisting of a holomorphic vector bundle E and a Higgs bundle as a End(E)-valued holomorphic 1-form. We want to study the Hitchin-Kobayashi correspondence for Higgs bundles over general non-compact Riemann surfaces. The most important direction of the correspondence is looking for a Hermitian-Yang-Mills metric for a given Higgs bundle. On a Riemann surface with a holomorphic r-differential, one can naturally define a Toda equation and a cyclic Higgs bundle with a grading. A solution of the Toda equation is equivalent to a Hermitian-Yang-Mills metric of the Higgs bundle for which the grading is orthogonal. In this talk, we focus on a general non-compact Riemann surface with an $r$-differential which is not necessarily meromorphic at infinity. We introduce the notion of complete solution of the Toda equation, and we prove the existence and uniqueness of a complete solution/metric. This is joint work with Takuro Mochizuki (RIMS).

Claudio Meneses

A glimpse of the Kahler geometry of moduli of parabolic bundles in mathematical physics

Since their introduction in the 1980s, moduli spaces of parabolic bundles have arisen in a surprisingly large and ever-increasing number of occasions at the interface of geometry, topology, and mathematical physics. The natural Kahler structure carried by these moduli spaces constitutes a primary piece in the broad puzzle of relations between these subjects. In this talk I will present a condensed overview of the beautiful history of these interactions, focusing on the peculiarities of the genus 0 case.

Yota Shamoto

Stokes filtered quasi-local systems and equivariant analogue of gamma conjecture

In this talk, I would like to explain an attempt to formulate an equivariant analogue of the gamma conjecture.
H. Iritani introduced a characteristic class called the gamma class. He used it to give an integral structure for the quantum D-module (QDM) of a complex projective manifold under some conditions. A consequence of the gamma conjecture formulated by Galkin-Golyshev-Iritani is that the integral structure is compatible with the Stokes structure. Here, the Stokes structure is a structure that describes the asymptotic behavior of solutions to the QDM around an irregular singular point.
In the equivariant case, we already have the equivariant gamma class and equivariant QDM. Then the generalization seems to be straightforward at a glance. However, there are two new and interesting points:
1. In the equivariant case, another ingredient, the shift operator, appears. The shift operator is a difference operator acting on equivariant QDM. The analogous conjecture should describe the Stokes structure of the solutions to the equivariant QDM as a differential-difference module.
2. Moreover, in the equivariant case, we can also treat a non-projective manifold if it is equipped with a good torus action. A typical example is the complex plane with its natural multiplicative group action. In this case, the solution of the equivariant QDM is essentially the gamma function.
To describe the relevant Stokes structure, we introduced the notion of a Stokes filtered quasi-local system. In the talk, I will explain the motivation sketched above and ideas for this structure. Part of this talk is on joint work in progress with F. Sanda.

Florent Schaffhauser

Twisted local systems and Higgs bundles for nonconstant groups

Complex analytic orbi-curves give rise to natural examples of twisted local systems, for which the fundamental group acts non-trivially on the coefficients. In this talk, we construct moduli varieties of twisted local systems, and prove that these are affine varieties over the complex numbers, whose (strong) topology can be studied through an appropriate version of the non-Abelian Hodge correspondence to the case of nonconstant coefficients. This partially answers a question of Carlos Simpson on the meaning of the Dolbeault moduli space in the nonconstant case.

Szilard Szabo

WKB-analysis of Hitchin system in rank 2

After an introduction to tame non-abelian Hodge theory and Riemann--Hilbert correspondence, we turn to the study of the large scale behaviour of these maps. Their description involves on the one hand asymptotic models of solutions of Hitchin's equations given by T. Mochizuki and R. Mazzeo et al, and on the other hand explicit co-ordinates on the character variety introduced by C. Simpson in the spirit of Teichmueller theory.

Organising Committee: Martin Guest (Waseda) and Florent Schaffhauser (Los Andes)

* Participants will receive Zoom log in information by e-mail. In order to receive Zoom log in information, please send an e-mail to Prof. Martin Guest (martin at waseda.jp) stating your name, university affiliation, and position/student status.

This workshop is also supported by the Institute for Mathematical Science, Waseda University and University of Strasbourg Institute of Advanced Study