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Moduli spaces of local systems and higher Teichmuller theory

V. V. Fock, A. B. Goncharov

TL;DR

The paper develops an algebraic-geometric framework for higher Teichmüller theory by defining and constructing positive representations of surface groups into split semisimple Lie groups with trivial centers, and by organizing these into positive moduli spaces ${\mathcal L}^+_{G,S}$, ${\mathcal X}^+_{G,S}$, and ${\mathcal A}^+_{G,S}$ endowed with positive atlases. It shows that positive representations are faithful, discrete, and positive hyperbolic, and it demonstrates how dual spaces of higher Teichmüller theory (X and A spaces) encode lamination structures via tropical points, with a rich interplay between positivity, cluster structures, and motivic data. The work connects to Hitchin components, universal higher Teichmüller spaces, and the classical Teichmüller theory (PGL2) while laying groundwork for quantization and canonical bases through cluster ensembles and motivic avatars of Weil-Petersson forms. It also outlines cutting/gluing properties, completions of spaces, and conjectural dualities with Langlands dual groups, positioning higher Teichmüller theory as a unified algebraic-geomtric extension of Thurston–Teichmüller ideas. The introduction thus sets the stage for a combinatorial, explicit approach to higher Teichmüller theory with broad connections to representation theory, cluster algebras, and motivic geometry.

Abstract

Let G be a split semisimple algebraic group with trivial center. Let S be a compact oriented surface, with or without boundary. We define {\it positive} representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmuller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil-Petersson form for one of these spaces. It is related to the motivic dilogarithm.

Moduli spaces of local systems and higher Teichmuller theory

TL;DR

The paper develops an algebraic-geometric framework for higher Teichmüller theory by defining and constructing positive representations of surface groups into split semisimple Lie groups with trivial centers, and by organizing these into positive moduli spaces , , and endowed with positive atlases. It shows that positive representations are faithful, discrete, and positive hyperbolic, and it demonstrates how dual spaces of higher Teichmüller theory (X and A spaces) encode lamination structures via tropical points, with a rich interplay between positivity, cluster structures, and motivic data. The work connects to Hitchin components, universal higher Teichmüller spaces, and the classical Teichmüller theory (PGL2) while laying groundwork for quantization and canonical bases through cluster ensembles and motivic avatars of Weil-Petersson forms. It also outlines cutting/gluing properties, completions of spaces, and conjectural dualities with Langlands dual groups, positioning higher Teichmüller theory as a unified algebraic-geomtric extension of Thurston–Teichmüller ideas. The introduction thus sets the stage for a combinatorial, explicit approach to higher Teichmüller theory with broad connections to representation theory, cluster algebras, and motivic geometry.

Abstract

Let G be a split semisimple algebraic group with trivial center. Let S be a compact oriented surface, with or without boundary. We define {\it positive} representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmuller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil-Petersson form for one of these spaces. It is related to the motivic dilogarithm.

Paper Structure

This paper contains 1 section, 24 theorems, 52 equations, 7 figures.

Table of Contents

  1. Introduction

Key Result

lemma thmcounterlemma

There is a natural bijection between the framed $G({\mathbf C})$-local systems on $\widehat{S}$ and $\rho$-equivariant maps (4.1.04.10facoma), where where $\rho: \pi_1(S) \to G({\mathbf C})$ is a representation, modulo $G({\mathbf C})$-conjugation.

Figures (7)

  • Figure 1: The Farey set: a cyclic $\pi_1(S)$-set ${\@fontswitch\mathcal{F}}_{\infty}(S)$.
  • Figure 2: The Farey triangulation of the hyperbolic disc.
  • Figure 3: Defining the edge invariant.
  • Figure 4: A positive configuration of six flags in ${\bf P}^2({\mathbf R})$.
  • Figure 5: The $4$-triangulation arising from a triangulation of a surface.
  • ...and 2 more figures

Theorems & Definitions (36)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • theorem 1
  • corollary thmcountercorollary
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • theorem 2
  • ...and 26 more