
This workshop is organised in conjunction with the Short Course on Nonabelian Hodge Theory by Prof. Florent Schaffhauser, 122 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
16:0016:45 Schaffhauser
17:0017:45 Kydonakis
break
18:1519:00 Szabo
19:1520:00 Shamoto
Wednesday 6 October
16:0016:45 Guest
17:0017:45 Kasuya
break
18:1519:00 Li
19:1520:00 Meneses
(Japan time)
Titles and abstracts:
Martin Guest
Topologicalantitopological 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 CecottiVafa on topologicalantitopological fusion in the 1990's.
Existence results for solutions are known from work of McCoyTracyWu, ItsNovokshenov,
TracyWidom, GuestItsLin, 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 odddimensions
On a compact Kahler manifold, there is an equivalence between semisimple 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 odddimensional 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 tubetype. When the parabolic weights are rational, an equivalence
between parabolic bundles and holomorphic bundles over $V$surfaces (that is,
2dimensional orbifolds), allows one to introduce new topological invariants
as StiefelWhitney 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 MorseBott function. This is joint work with
Hao Sun and Lutian Zhao.
Qiongling Li
Complete solutions of Toda equations and cyclic Higgs bundles over
noncompact 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 1form. We
want to study the HitchinKobayashi correspondence for Higgs bundles over general
noncompact Riemann surfaces. The most important direction of the correspondence
is looking for a HermitianYangMills metric for a given Higgs bundle. On a
Riemann surface with a holomorphic rdifferential, 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 HermitianYangMills metric of the Higgs bundle
for which the grading is orthogonal. In this talk, we focus on a general noncompact
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 everincreasing 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 quasilocal 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 Dmodule (QDM) of a complex
projective manifold under some conditions. A consequence of the gamma conjecture
formulated by GalkinGolyshevIritani 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 differentialdifference module.
2. Moreover, in the equivariant case, we can also treat a nonprojective 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 quasilocal 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 orbicurves give rise to natural examples of twisted local
systems, for which the fundamental group acts nontrivially 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 nonAbelian 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
WKBanalysis of Hitchin system in rank 2
After an introduction to tame nonabelian Hodge theory and RiemannHilbert 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 coordinates 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 email. In order to receive Zoom log in information, please send an email 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