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Introduction to Chern-Simons Theories

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Preprint typeset in JHEP style - HYPER VERSION Introduction To Chern-Simons Theories Gregory W. Moore Abstract: These are lecture notes for a series of talks at the 2019 TASI school. They contain introductory material to the general subject of Chern-Simons theory. They are meant to be elementary and pedagogical. ************* THESE NOTES ARE STILL IN PREPARATION. CONSTRUCTIVE COMMENTS ARE VERY WELCOME. (In par- ticular, I have not been systematic about trying to include references. The literature is huge and I know my own work best. I will certainly be interested if people bring reference omissions to my attention.) **************** June 7, 2019 Contents 1. Introduction: The Grand Overview 5 1.1 Assumed Prerequisites 8 2. Chern-Simons Theories For Abelian Gauge Fields 9 2.1 Topological Terms Matter 9 2.1.1 Charged Particle On A Circle Surrounding A Solenoid: Hamiltonian Quantization 9 2.1.2 Charged Particle On A Circle Surrounding A Solenoid: Path Integrals 14 2.1.3 Gauging The Global SO(2) Symmetry And Chern-Simons Terms 16 2.2 U(1) Chern-Simons Theory In 3 Dimensions 19 2.2.1 Some U(1) Gauge Theory Preliminaries 19 2.2.2 From θ-term To Chern-Simons 20 2.2.3 3D Maxwell-Chern-Simons For U(1) 22 2.2.4 The Formal Path Integral Of The U(1) Chern-Simons Theory 24 2.2.5 First Steps To The Hilbert Space Of States 25 2.2.6 General Remarks On Quantization Of Phase Space And Hamiltonian Reduction 27 2.2.7 The Space Of Flat Gauge Fields On A Surface 31 2.2.8 Quantization Of Flat Connections On The Torus: The Real Story 39 2.2.9 Quantization Of Flat Connections On The Torus: The Coherent Story 41 2.2.10 Quantization Of Flat Connections On The Torus: Landau's Story 47 2.2.11 Quantization Of Flat Connections On The Torus: Heisenberg's Story 51 2.2.12 Symplectic Transformations, Large Diffeomorphisms, And Modular Transformations 55 2.2.13 The Bundle Of Physical States Over The Moduli Space Of Curves 59 2.2.14 Correlation Functions Of Line Defects And Framing Anomalies 60 2.2.15 Digression On The Gauss Linking Number 61 2.2.16 The Framing Anomaly And Ribbons 64 2.2.17 A Topological Anomaly: The Normalization Factor: N 68 2.2.18 Uniting The Path Integral And Hamiltonian Viewpoints: States Cre- ated By Wilson Lines 70 2.3 Relation To Rational Conformal Field Theory 72 2.3.1 Quantization When @Σ Is Nonempty: Emergence Of Edge States 72 2.3.2 Quantization With Wilson Lines Piercing Spatial Surface 74 2.4 The Gaussian Model For R2 Rational 74 2.4.1 Gluing The Same Chiral Algebra To Get Different Radii 79 2.4.2 The Free Fermion Radius: Level 1=2 Theta Functions And Spin Structures On The Torus 80 { 1 { 2.5 Spin Theories 82 2.6 Generalization: Torus Gauge Group 84 2.6.1 Applications To the FQHE 85 2.6.2 Coupling To External Abelian Gauge Fields 85 3. Chern-Simons Theories For NonAbelian Gauge Fields 85 3.1 Some Chern-Weil Theory 85 3.1.1 The Chern-Simons Action On M3 With A Nonempty Boundary 88 3.2 The Semiclassical Approximation To The Chern-Simons Path Integral 88 3.2.1 Semiclassical Approximation: The Framing Anomaly Revisited 89 3.2.2 Higher Loops 92 3.3 Quantization In Some Important Cases 92 3.3.1 Quantization On Σ2 × R With Σ A Compact Surface 92 3.3.2 Quantization On Punctured Surfaces 96 3.3.3 Quantization On D2 × R, With D2 A Disk: Nonabelian Edge States 97 3.4 Quantization Of The Moduli Space Of Flat Gauge Fields And Algebraic Geometry 98 3.4.1 Generalization To Conformal Blocks With Vertex Operator Insertions 99 4. WZW Model And RCFT 100 4.1 The Principal Chiral Model And The WZ Term 101 4.1.1 The Wess-Zumino Term 101 4.1.2 The Case Of Two Dimensions And The Conformal Point 107 4.2 Quantization: Virasoro And Affine Lie Algebra Symmetry 110 4.2.1 Stress-Energy Tensor 114 4.3 The Central Extension Of The Loop Group And The WZ Term 116 4.4 Relation To Anomalies 118 4.4.1 Coupling To External Gauge Fields 118 4.4.2 Anomalies Of Weyl Fermions 118 4.5 Highest Weight Representations Of Loop Groups 120 4.5.1 Integrable highest weight representations of Lfg 120 4.5.2 Integrable Highest Weight Representations Of The Group 127 4.5.3 Characters of integrable highest weight representations 129 4.5.4 Modular properties 131 4.6 The Hilbert Space Of WZW And The One-Loop Partition Function 133 4.7 Primary Fields 135 4.7.1 Semisimple Algebras 135 4.7.2 An Example Of Semisimple Fusion Rules: Finite Group Theory 138 4.8 Tensor Product Of Highest Weight Representations 140 4.8.1 Fusion Rules For The WZW Model 142 4.9 Conformal Blocks And Monodromy 143 4.10 Open String WZW Model: Symmetry Preserving Boundary Conditions 143 { 2 { 5. Topological Field Theories And Categories 144 5.1 Basic Ideas 144 5.1.1 More Structure 149 5.2 The Definition Of Topological Field Theory 149 5.2.1 Some General Properties 150 5.2.2 Unitarity 156 5.3 One Dimensional Field Theories 156 5.4 Two-Dimensional Topological Field Theories 158 5.5 Extended Topological Field Theories 159 5.6 Three-Dimensional Topological Field Theories 159 5.7 Some Source Material For Topological Field Theories 159 6. Computing Nonabelian Chern-Simons Correlators Using RCFT 160 6.1 3D Interpretation Of Braiding And Fusing Matrices 160 6.2 Surgery Manipulations 160 6.3 The Jones polynomial and WRT Invariants 161 6.4 Beyond knot polynomials: The idea of categorification and Knot homology. 161 7. Finite Group Chern-Simons Theory 161 7.1 The General Compact Group G 161 7.1.1 Chern-Simons Theory When The Gauge Group Is A Finite Group 161 8. p-form Generalizations And Differential Cohomology 161 9. More About Topological Field Theories 161 10. Anyons, nonabelions, and quantum computation 162 10.1 Anyons in 2+1 dimensions 162 11. A Survey Of Further Generalizations And Applications 165 11.1 Applications To Supergravity 165 11.2 Noncompact Groups 165 11.3 Applications to Holography 165 11.4 Many Other Applications And Extensions 165 A. Symplectic Geometry 166 A.1 Kahler quotients: A lightning review 166 A.1.1 The moment map 166 A.1.2 Symplectic Quotient 168 A.1.3 K¨ahlerquotient 169 B. Lightning Review Of K¨ahlerQuantization 171 B.1 The Bargmann Representation 171 B.2 Generalization: Berezin-Toeplitz Quantization 172 B.2.1 Example: Quantization Of The Sphere And Representations Of SU(2)174 { 3 { B.3 The Borel-Weil-Bott Theorem 175 C. A Few Details About SL(2; Z) 176 C.1 Generators And A Fundamental Domain For P SL(2; Z) 176 C.2 Expressing Elements Of SL(2; Z) As Words In S And T 179 D. Level k Theta Functions 182 D.1 Definition 182 D.2 Modular Transformation Laws For Level k Theta Functions 182 E. Theta Functions And Holomorphic Line Bundles On Tori 184 E.1 Heisenberg Groups As Principal U(1) Bundles 185 E.2 A Natural Connection 187 E.3 Putting A Complex Structure On The Symplectic Tori 187 E.4 Lagrangian Decompositions 188 E.5 The Space Of Complex Structures On The Torus: The Siegel Upper Half- Plane 188 E.6 Lagrangian Decomposition: Group-Theoretic Interpretation 192 E.7 Hermitian Holomorphic Line Bundles 192 E.7.1 Computing The First Chern Class 194 E.7.2 A Basis Of Holomorphic Sections: Level κ Theta Functions 194 E.8 Modular Transformation Laws 195 E.9 The Heat Equation 195 F. Hodge Theory And Analytic Torsion 195 F.1 Some Hodge Theory 195 F.2 Analytic Torsion 197 G. Orientation, Spin, Spinc, P in±, And Pinc Structures On Manifolds 198 G.1 Reduction Of Structure Group: General Discussion 198 G.2 Obstructions To Spin And Pin Structures 200 G.3 Spinc Structures On Four-Manifolds 202 G.4 't Hooft Flux 203 H. Connections, Parallel Transport, And Holonomy 203 I. Chern-Weil Theory And Chern-Simons Forms 211 I.1 Characteristic classes 211 I.2 Basic forms 212 I.3 Invariant polynomials on the Lie algebra 212 I.3.1 Examples 213 I.4 The Chern-Weil homomorphism 214 I.5 Dependence on connection: Construction of characteristic classes 215 I.6 The Borel Theorem 217 { 4 { I.7 Secondary characteristic classes: The Chern-Simons forms 217 I.7.1 Other paths to Chern-Simons 223 J. Some Material On Lie Groups And Lie Algebras 224 J.1 The Structure Of The General Compact Lie Group 224 J.2 Dual Coxeter Number 225 K. Categories: Groups and Groupoids 226 K.1 Groupoids 236 K.2 Tensor Categories 238 K.2.1 Z2-graded vector spaces 239 K.2.2 Category Of Representations Of A Group 239 L. Bordism 240 L.1 Unoriented Bordism: Definition And Examples 240 L.2 The Bordism Category Bordhd−1;di 241 SO L.3 The Oriented Bordism Category Bordhd−1;di 242 L.4 Other Bordism Categories 243 M. Some Sources From Course Notes 244 \This whole book is but a draught nay, but the draught of a draught." - Herman Melville 1. Introduction: The Grand Overview Chern-Simons theory is a quantum gauge theory involving a rather subtle action principle. It leads to quantum field theory in which many, many, natural questions can be explicitly answered.
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