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About this book

This volume is based on four advanced courses held at the Centre de Recerca Matemàtica (CRM), Barcelona. It presents both background information and recent developments on selected topics that are experiencing extraordinary growth within the broad research area of geometry and quantization of moduli spaces. The lectures focus on the geometry of moduli spaces which are mostly associated to compact Riemann surfaces, and are presented from both classical and quantum perspectives.

Table of Contents


Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

The main idea of this work is to demonstrate the equivalence of two a priori different methods of construction and description of a wide class of integrable models, and thus, to propose a unified approach for their investigation. In the first, well-known method [24], the phase space is taken as a quotient of double Bruhat cells of a Kac–Moody Lie group, with the Poisson structure defined by a classical r-matrix, and the integrals of motion are just the Ad-invariant functions. The second method was suggested recently by A.
Vladimir V. Fock, Andrey Marshakov

Chapter 2. Lectures on Klein Surfaces and Their Fundamental Group

These notes are based on a series of three 1-hour lectures given in 2012 at the CRM in Barcelona, as part of the event Master Class and Workshop on Representations of Surface Groups, itself a part of the research program Geometry and Quantization of Moduli Spaces. The goal of the lectures was to give an introduction to the general theory of Klein surfaces, particularly the appropriate notion of fundamental groups for such surfaces, emphasizing throughout the analogy with a more algebraic perspective on fundamental groups in real algebraic geometry.
Florent Schaffhauser

Chapter 3. Five Lectures on Topological Field Theory

Topological quantum field theories – TQFTs – arose in physics as the baby (zero energy) sector of honest quantum field theories, which showed an unexpected dependence on the large scale topology of space-time. The zero energy part of the Hilbert space of states does not evolve in time, as by definition it is killed by the Hamiltonian; so, at first sight, its physics appears to be uninteresting. But this argument fails to consider a space-time with interesting topology.
Constantin Teleman

Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

These notes are based on lectures given at the Third International School on Geometry and Physics at the Centre de Recerca Matemàtica in Barcelona, March 26–30, 2012. The aim of the School’s four lecture series was to give a rapid introduction to Higgs bundles, representation varieties, and mathematical physics. While the scope of these subjects is very broad, that of these notes is far more modest.
Richard Wentworth


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