Short course on

Nonabelian Hodge theory

by Prof. Florent Schaffhauser

(Universidad de Los Andes (Bogotá); Max Planck Institute (Bonn); University of Strasbourg)

Waseda University

July 2021

real time Zoom lectures*

 

 

The course will be an introduction to nonabelian Hodge theory over compact Riemann surfaces, with a view towards the topology of character varieties. A significant part of the course will be devoted to the ideas and techniques used in Simpson's famous proof of the nonabelian Hodge correspondence. The central notion of the course will be that of a Higgs bundle, which was introduced by Hitchin, and can be seen as a degenerate version of a holomorphic bundle with connection.

Each lecture will include ample time for discussion and questions. There will also be discussion sessions following the lectures, in which students can present their own research and receive advice from the lecturer.

Graduate students and researchers in mathematics, physics, and engineering are welcome (see below for registration information).


Related workshop: Toda equations, parabolic Higgs bundles, and related topics 5-6 October 2021 (Zoom workshop)


Schedule:

Thursday, 1 July, Lecture 16:30-18:00

Monday, 5 July, Lecture 16:30-18:00, Discussion Session 1: 18:30-19:30 including talk by Marwan Benyoussef (Freie Universitaet Berlin)

Thursday, 8 July, Lecture 16:30-18:00, Discussion Session 2: 18:30-19:30 including talk by Yoshiki Kaneko (Waseda University)

Monday, 12 July, Lecture 16:30-18:00, Discussion Session 3: 18:30-19:30 including talk by Ryosuke Odoi (Waseda University)

Thursday, 15 July, Lecture 16:30-18:00, Discussion Session 4: 18:30-19:30 including talk by Soma Ohno (Waseda University)

Monday, 19 July, Lecture 16:30-18:00, Discussion Session 5: 18:30-19:30 including talk by Juan Martin Perez (Universite de Nice)

Thursday, 22 July, Lecture 16:30-18:00


Topics

1. character varieties
2. Higgs bundles
3. nonabelian Hodge correspondence
4. Hitchin fibration
6. higher Teichmueller spaces


Student presentations (at Discussion Sessions):

Monday, 5 July "Computing E-Polynomials for certain character varieties", Marwan Benyoussef (Freie Universitaet Berlin)

ABSTRACT: We will discuss arithmetic techniques introduced by Hausel and Rodriguez-Villegas for computing the E-polynomial of certain families of algebraic varieties, called character varieties of surface groups. Construction of such varieties follows from Geometric Invariant Theory. The arithmetic techniques are based on a fundamental theorem of Katz, that reduces the computation of E-polynomials to point counting over finite fields.

Thursday, 8 July "Solutions of the tt*-Toda equations and quantum cohomology of flag manifolds", Yoshiki Kaneko (Waseda University)

ABSTRACT: We introduce the definition of the tt*-Toda equations for general complex Lie groups. and the construction of solutions from a certain DPW potential. Guest, Its and Lin found that the Dubrovin connection from the quantum cohomology of complex projective space is of this form. We obtain a DPW potential of the tt*-Toda equations from the quantum cohomology of minuscule flag manifolds.

Monday, 12 July "Symplectic aspects of the tt*-Toda equations", Ryosuke Odoi (Waseda University)

ABSTRACT: Solutions of the tt*-Toda equations can be considered as special kinds of harmonic maps from an open subset of ¥mathbb{C}^{*} to the symmetric space SL(n+1,¥mathbb{R})/SO(n+1), or as certain isomonodromic deformations of meromorphic connections. Solutions can be parametrized by two kinds of data, and they correspond to each other via the Riemann-Hilbert correspondence. This correspondence can be considered as a transformation between two charts of the moduli space of the solutions. We will see that the transformation is symplectic and we will give an application of this.

Thursday, 15 July "Rarita-Schwinger fields on nearly Kähler manifolds", Soma Ohno (Waseda University)

ABSTRACT: We study Rarita-Schwinger fields on 6-dimensional compact strict nearly Kähler manifolds. In order to investigate them, we clarify the relationship between some differential operators for the Hermitian connection and the Levi-Civita connection. As a result, we show that the space of the Rarita-Schwinger fields coincides with the space of the harmonic 3-forms. Applying the same technique to a deformation theory, we also find that the space of the infinitesimal deformations of Killing spinors coincides with the direct sum of a certain eigenspace of the Laplace operator and the space of the Killing spinors.

Monday, 19 July "An introduction to opers", Juan Martin Perez (Universite de Nice)

ABSTRACT: An oper is a holomorphic connection on a vector bundle which induces a filtration on it. This connection satisfies some special properties which allow for the construction of a differential operator between two line bundles, related to the filtration.


* 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.


The course is an activity of the

Mathematics and Physics Unit "Multiscale Analysis, Modelling and Simulation" Top Global University Project, Waseda University

Waseda students may register to obtain credit for this course (MATX72ZL Advanced Study of Nonlinear Mechanics).


These lectures are also supported by the Institute for Mathematical Science, Waseda University