1 1 Title & Abstract

1 2 Contents

1 5 Introduction

1 8 Overview
2 9 Finite-Dimensional Analogs
2 11 Role Of Morse Theory
2 12 Stokes Phenomena
2 13 Singularities Of The Analytically Continued Theory
2 13 Analytic Continuation For Compact Gauge Groups
3 15 Downward Flow Again
3 17 Indeterminacy of C
3 17 Knots
2 18 Comparing Compact And Complex Lie Groups
3 19 Knots Again
2 21 Other Real Forms
3 22 Comparison Of Holomorphic Volume Forms

1 24 Finite-Dimensional Oscillatory Integrals, Morse Theory, And Steepest Descent
2 24 The Airy Function
3 25 Morse Theory
3 29 Crossing A Stokes Ray
3 31 Asymptotic Behavior of Integrals Over Lefschetz Thimbles
3 31 Sum Of Asymptotic Series
3 32 Expressing The Real Cycle In Terms Of Lefschetz Thimbles
3 33 Asymptotic Behavior Of The Airy Function
2 35 Generalization To Dimension n
3 38 Essential Singularities
3 40 A Special Situation
2 42 Symmetries
3 42 Oscillatory Integrals On More General Manifolds
3 43 Manifolds Of Critical Points
3 44 Group Actions
3 47 More on Group Actions
2 49 Singularities
3 49 More on the Airy Function
3 51 An Example With SO(2) Symmetry
3 55 One More Case
2 56 Analytic Continuation In k: A Prototype
3 59 Description By Lefschetz Thimbles
3 60 Singularities At ρ = ±1

1 63 Applications to Chern-Simons Theory
2 64 The Flow Equation For Chern-Simons
3 67 The Index
3 69 Applications Of The Flow Equation
2 72 Knots And Morse Theory
3 74 Classical Equations In The Presence Of A Knot
3 75 More On The Classical Equations
3 78 Analog For Noncompact Lie Groups
3 79 Classification Of Critical Points
3 81 Variation Of The Action
3 84 Branches Of The Logarithm
3 85 Tensoring With A Flat Line Bundle

1 86 Analytic Continuation Of The Colored Jones Polynomial
2 87 Preliminaries
3 90 Local Models
3 92 Normalization Of The Path Integral
3 94 Orientation Of The Integration Cycle
2 95 The Trefoil Knot
3 96 Singularities Of The Moduli Space Of Flat Connections
3 100 The Chern-Simons Function
3 101 The Integration Cycle And The Asymptotic Behavior Of The Path Integral
3 104 Stokes Curves
2 109 The Figure-Eight Knot
3 110 Real m
3 112 m Of Modulus 1
3 114 The Real Part Of The Chern-Simons Function
3 117 The Imaginary Part Of W
3 118 The Volume Conjecture For The Figure Eight Knot

1 122 Four or Five-Dimensional Interpretation

1 123 Acknowledgements

1 123 References