Quantum Teichmüller and Thurston Theories

Introduction

Based on papers with V.Fock and R.Penner, we propose the way to quantize Teichmüller and Thurston theories for Riemann surfaces with holes (punctures). These surfaces admit, in the Poincare uniformization, a graph description due to Penner and Fock. The corresponding parameters are the coordinates on the Teichmüller space, and the mapping class group (modular) transformations can be explicitly constructed. Introducing the Poincaré structure compatible with the Goldman Poisson brackets for geodesic functions that follow from 2+1-dimensional Chern-Simons theory, we were able to quantize the structure thus producing the quantum mapping class group transformations and quantum geodesic functions. The arising algebras, in some particular cases, are related to Nelson-Regge algebras of geodesics, or to algebras of Stokes parameters in isomonodromic deformations. In the second part of the talk, we consider the Thurston theory of measured geodesic laminations and show that, under the proper definition pertaining to the so-called tropical limit of mapping class group transformations, we can define the (classical and quantum) limits of the geodesic functions for arbitrary measured lamination and prove the existence of these limiting geodesic lengths, or of the corresponding operators describing quantum geodesic lengths.

Timetable

Introductory Seminar

Thursday 4th May - 4:00 pm, Room G15, Newman Building

Transparencies (3 MB)

Lecture 1

Tuesday 9th May - 2:00 pm, Room M12, MSS Building

Picture 1 - Picture 2 - Picture 3

Lecture 2

Thursday 11th May - 2:00 pm, Room M12, MSS Building

Picture 1 - Picture 2 - Picture 3

Lecture 3

Tuesday 23rd May - 2:00 pm, Room M12, MSS Building

Lecture 4

Thursday 25th May - 2:00 pm, Room M12, MSS Building

Abstracts

Lecture 1

Introduction to Teichmuller spaces and Penner--Fock coordinates for moduli spaces of Riemann surfaces with holes. Relation to 2+1 dimensional theories. Graph description and geodesics. Algrebras of geodesics. Mapping class group transformation and pentagon identity.

Lecture 2

Special cases of geodesic algebras related to integrable systems (in the sense of Stockes parameters. Laminations. Quantization and quantum geodesic algebras. Quantum ordering. Spectrum of quantum Dehn twists (following Kashaev).

Lectures 3-4

Thurston theory, train tracks, and "tropicalization" of laminations in the Zelevinsly sense. Proper lengths and theorems by Bass--Hashimoto on the graph L-functions and by G. McShine on L-function-like infinite products. Proper geodesic lengths and graph lengths Theorems about continuous limit of proper length/graph length ratio on the boundary of the Teichmuller space, quantization, and quantum continued fractions (apres Ch. and Penner)

Generalizations

Cluster algebras of Fock and Goncharov; complexified version describing Schottly uniformized smooth Riemann surfaces and relation to the Liouville theory (L.Ch. and J.Teschner's papers).

Directions

The Newman building is number 88 whereas the MSS Building is number 21 on the University Campus map. Detailed directions are available on the general information pages.

Further information

For Further information please contact Marta Mazzocco (Marta.Mazzocco[at]manchester.ac.uk)