Short course on

Semi-classical approximations in conformal field theory and analytic geometry of moduli spaces

by Dr. Claudio Meneses

(DFG Principal Investigator, University of Kiel)

Waseda University

January 2022

real time Zoom lectures* (please see below for registration information)



Two-dimensional conformal field theories are examples of quantum field theories encoding incredibly rich algebraic and geometric structures which have captivated the interest of many mathematicians in the last 40 years. From a geometric perspective, they have been formulated as a branch in moduli theory, and even at the semi-classical level, they have shed new light into an extensive series of classical problems in the theory of complex ODE. The aim of these lectures is to present an overview of examples of this phenomenon in the analytic geometry of moduli spaces of Riemann surfaces and parabolic bundles.

Each lecture will include ample time for discussion and questions.

Graduate students and researchers in mathematics, physics, and engineering are welcome.

Schedule (all times are Japan time):

Monday 17 January 17:00-18:30
Tuesday 18 January 17:00-18:30
Wednesday 19 January 17:00-18:30
Thursday 20 January 17:00-18:30

Monday 24 January 17:00-18:30
Tuesday 25 January 17:00-18:30
Wednesday 26 January 17:00-18:30
Thursday 27 January 17:00-18:30


Lecture 1: Riemann surfaces and their moduli

Lecture 2: Parabolic vector bundles on a Riemann surface and their moduli

Lecture 3: Analytic geometry of moduli spaces

Lecture 4: The idea of a conformal field theory. A geometric interpretation

Lecture 5: The compact WZNW model and isomonodromy

Lecture 6: The Verlinde formula and relation to 2d quantum Yang-Mills theory

Lecture 7: Liouville theory, Fuchsian uniformization, and the Weil-Petersson metric

Lecture 8: A non-compact WZNW action, Fuchsian systems, and the Narasimhan-Atiyah-Bott metric

Bibliography A detailed list of references for material covered in the lectures.

* 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 stating your name, university affiliation, and position/student status.

A related workshop was held 5-6 October 2021: Toda equations, parabolic Higgs bundles, and related topics

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