A fiber structure of Teichmller space and conformal field theory - PowerPoint PPT Presentation

A fiber structure of Teichmüller space and conformal field theory David Radnell 1 Eric Schippers 2 1 Department of Mathematics and Statistics American University of Sharjah Sharjah, UAE 2 Department of Mathematics University of Manitoba Winnipeg, Canada Varna, June 11, 2008 David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 1 / 15

Table of contents Introduction 1 Overview Conformal Field Theory Teichmüller Theory 2 Quasiconformal Maps Riemann surfaces Definition and Facts Sewing 3 Definition Conformal Welding Quasisymmetric Sewing Fiber Theorem 4 David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 2 / 15

Introduction Overview Introduction Conformal Field Theory (CFT): Special class of 2D quantum field theories. Mathematical definition (G. Segal, Kontevich ≈ 1986) Deeply connected to algebra, topology and analysis. David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 3 / 15

Introduction Overview Introduction Conformal Field Theory (CFT): Special class of 2D quantum field theories. Mathematical definition (G. Segal, Kontevich ≈ 1986) Deeply connected to algebra, topology and analysis. Complex analysis/geometry: ( ∞ -dim) moduli space of Riemann surfaces Sewing=gluing=welding Quasiconformal mappings David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 3 / 15

Introduction Overview Introduction Conformal Field Theory (CFT): Special class of 2D quantum field theories. Mathematical definition (G. Segal, Kontevich ≈ 1986) Deeply connected to algebra, topology and analysis. Complex analysis/geometry: ( ∞ -dim) moduli space of Riemann surfaces Sewing=gluing=welding Quasiconformal mappings Our General Aim: Provide a natural analytic setting for the rigorous definition of CFT in higher genus. Definitions and Theorems. Use CFT ideas (especially sewing) to prove new results in Teichmüller theory and geometric function theory. David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 3 / 15

Introduction Conformal Field Theory Motivation/Application: Conformal Field Theory Objects Morphism Objects out in S 1 ψ o 1 ψ in S 1 1 S 1 Σ ψ o 2 ψ in S 1 2 S 1 ψ o 3 Projective Functor A ([Σ , ψ ]) H ⊗ H ⊗ H H ⊗ H David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 4 / 15

Teichmüller Theory Quasiconformal Maps Quasiconformal Maps I f : Ω ⊂ C → C . Homeomorphism. Orientation Preserving. Jacobian ( f ) = · · · = | f z | 2 − | f ¯ z | 2 > 0. So, | f ¯ z / f z | < 1. David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 5 / 15

Teichmüller Theory Quasiconformal Maps Quasiconformal Maps I f : Ω ⊂ C → C . Homeomorphism. Orientation Preserving. Jacobian ( f ) = · · · = | f z | 2 − | f ¯ z | 2 > 0. So, | f ¯ z / f z | < 1. Complex Dilatation = µ ( z ) = f ¯ z / f z . David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 5 / 15

Teichmüller Theory Quasiconformal Maps Quasiconformal Maps I f : Ω ⊂ C → C . Homeomorphism. Orientation Preserving. Jacobian ( f ) = · · · = | f z | 2 − | f ¯ z | 2 > 0. So, | f ¯ z / f z | < 1. Complex Dilatation = µ ( z ) = f ¯ z / f z . Jacobian(f) : minor axis = 1 + | µ | Circular Dilatation = major axis 1 −| µ | David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 5 / 15

Teichmüller Theory Quasiconformal Maps Quasiconformal Maps I f : Ω ⊂ C → C . Homeomorphism. Orientation Preserving. Jacobian ( f ) = · · · = | f z | 2 − | f ¯ z | 2 > 0. So, | f ¯ z / f z | < 1. Complex Dilatation = µ ( z ) = f ¯ z / f z . Jacobian(f) : minor axis = 1 + | µ | Circular Dilatation = major axis 1 −| µ | Note: f ( z ) conformal ⇐ ⇒ f ¯ z = 0 ⇐ ⇒ µ ( z ) = 0 ⇐ ⇒ Circ . Dil . = 1. David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 5 / 15

Teichmüller Theory Quasiconformal Maps Quasiconformal Maps II Circular Dilatation = 1 + | µ | f : Ω ⊂ C → C . µ ( z ) = f ¯ z / f z . 1 −| µ | . David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 6 / 15

Teichmüller Theory Quasiconformal Maps Quasiconformal Maps II Circular Dilatation = 1 + | µ | f : Ω ⊂ C → C . µ ( z ) = f ¯ z / f z . 1 −| µ | . Geometric Definition: f is K -quasiconformal if its circular dilatation is globally bounded by K . (i.e. Infinitesimally, circles map to ellipses of bounded eccentricity). David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 6 / 15

Teichmüller Theory Quasiconformal Maps Quasiconformal Maps II Circular Dilatation = 1 + | µ | f : Ω ⊂ C → C . µ ( z ) = f ¯ z / f z . 1 −| µ | . Geometric Definition: f is K -quasiconformal if its circular dilatation is globally bounded by K . (i.e. Infinitesimally, circles map to ellipses of bounded eccentricity). Analytic Definition: f is K -quasiconformal if it satisfies the Beltrami Equation ∂ f z = µ ( z ) ∂ f ∂ ¯ ∂ z for some µ ( z ) with || µ || ∞ = k < 1. K = ( 1 + k ) / ( 1 − k ) . Note: Technical conditions skipped. QC maps are only differentiable almost everywhere etc. David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 6 / 15

Teichmüller Theory Riemann surfaces Basic Objects Riemann Surfaces with boundary David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 7 / 15

Teichmüller Theory Riemann surfaces Basic Objects Riemann Surfaces with boundary Fix: g = genus, n = # of boundary components. The moduli space is the space of conformal equivalence classes of surfaces. David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 7 / 15

Teichmüller Theory Riemann surfaces Basic Objects Riemann Surfaces with boundary Fix: g = genus, n = # of boundary components. The moduli space is the Σ B space of conformal equivalence classes of surfaces. Quasiconformal map f Σ B 1 David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 7 / 15

Teichmüller Theory Riemann surfaces Basic Objects Riemann Surfaces with boundary Fix: g = genus, n = # of boundary components. The moduli space is the Σ B space of conformal equivalence classes of surfaces. Quasiconformal map f Quasisymmetric map Definition: h : S 1 → S 1 Σ B h has quasiconformal extensions to C . 1 David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 7 / 15

Teichmüller Theory Riemann surfaces Basic Objects Riemann Surfaces with boundary Fix: g = genus, n = # of boundary components. The moduli space is the Σ B space of conformal equivalence classes of surfaces. Quasiconformal map f Quasisymmetric map Quasisymmetric boundary parametrization Σ B 1 qs map David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 7 / 15

Teichmüller Theory Definition and Facts Teichmüller Space = space of Riemann surfaces Fix a base Riemann surface Σ . Given Σ 1 and quasiconformal f : Σ → Σ 1 , write (Σ , f , Σ 1 ) . David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 8 / 15

Teichmüller Theory Definition and Facts Teichmüller Space = space of Riemann surfaces Fix a base Riemann surface Σ . Given Σ 1 and quasiconformal f : Σ → Σ 1 , write (Σ , f , Σ 1 ) . Definition (Teichmüller space:) T (Σ) = { (Σ , f , Σ 1 ) } / ∼ . (Σ , f , Σ 1 ) ∼ (Σ , g , Σ 2 ) ⇐ ⇒ ∃ conformal σ : Σ 1 → Σ 2 such that g − 1 ◦ σ ◦ f ≈ id (rel. boundary) David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 8 / 15

Teichmüller Theory Definition and Facts Teichmüller Space = space of Riemann surfaces Fix a base Riemann surface Σ . Given Σ 1 and quasiconformal f : Σ → Σ 1 , write (Σ , f , Σ 1 ) . Definition (Teichmüller space:) T (Σ) = { (Σ , f , Σ 1 ) } / ∼ . (Σ , f , Σ 1 ) ∼ (Σ , g , Σ 2 ) ⇐ ⇒ ∃ conformal σ : Σ 1 → Σ 2 such that g − 1 ◦ σ ◦ f ≈ id (rel. boundary) Teichmüller metric: f , g log ( circular dilatation of g ◦ f − 1 ) distance ([Σ , f , Σ 1 ] , [Σ , g , Σ 2 ]) = inf This measures how close (in the quasiconformal sense) to a conformal map there is from Σ 1 to Σ 2 . David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 8 / 15

Teichmüller Theory Definition and Facts Teichmüller space facts Fix Σ . f : Σ → Σ 1 . T (Σ) = Teichmüller space. Why? µ ( f ) = f ¯ z / f z is a differential form on the base surface. Study the Teichmüller space by studying certain spaces of forms. This is classical work from the 50’s and 60’s of Ahlfors and Bers et al. Well developed theory. David Radnell (AUS) Fiber structure of Teichmüller space Varna, June 11, 2008 9 / 15