Spectral Networks

Organizers: François Labourie and Jérémy Toulisse.

The goal of this reading seminar will be to study the fundamental article ¨Spectral Networks¨ of the physicists Davide Gaiotto, Gregory W. Moore and Andrew Neitzke (and the articles related). Their work is a source of many conjectures.

A rank n spectral network on a surface S is a combinatorial object (roughly, a decorated graph). Such a spectral network allows to construct a desabelianization, that is, to associate a representation of the fundamental group of S into GL(n,C) to a (abelian) representation into GL(1,C) of a covering of S, called the spectral curve. This construction is related to many other constructions, such as:

  • Higgs bundles theory.
  • Cluster coordinates on character varieties (constructed by Fock and Goncharov).
  • The wall-crossing formula of Kontsevich-Soibelman.
  • Compactifications of moduli spaces of representations.
  • Harmonic maps into buildings.
  • No specific background will be needed to understand the lectures.

    Sessions:

  • November 16th:
    • Jérémy Toulisse: Character varieties of surface groups (notes)
    • François Labourie: Quadratic differentials, laminations, trees and compactification (notes)
  • November 30th:
    • Jérémy Toulisse: Spectral Networks (notes)
    • François Labourie: Quadratic differentials, laminations, trees and compactification II (notes)
  • February 8th:
    • Jérémy Toulisse: Spectral Networks II
    • François Labourie: Quadratic differentials, laminations, trees and compactification III
  • March 29th:
    • Jérémy Toulisse: Rank n Spectral Networks and path lifting (notes)
    • François Labourie: Higher holomorphic differentials, spectral covers and construction of spectral networks (notes)
  • April 26th:
    • Indira Chatterji: Introduction to buildings
    • Carlos Simpson: WKB exponents and construction of a pre-building for SL(3)