vi CONTENTS

1. The local renormalization group flow 113

2. The Kadanoff-Wilson picture and asymptotic freedom 122

3. Universality 125

4. Calculations in

φ4

theory 126

5. Proofs of the main theorems 131

6. Generalizations of the main theorems 135

Chapter 5. Gauge symmetry and the Batalin-Vilkovisky formalism 139

1. Introduction 139

2. A crash course in the Batalin-Vilkovisky formalism 141

3. The classical BV formalism in infinite dimensions 155

4. Example: Chern-Simons theory 160

5. Example : Yang-Mills theory 162

6. D-modules and the classical BV formalism 164

7. BV theories on a compact manifold 170

8. Effective actions 173

9. The quantum master equation 175

10. Homotopies between theories 178

11. Obstruction theory 186

12. BV theories on

Rn

189

13. The sheaf of BV theories on a manifold 196

14. Quantizing Chern-Simons theory 203

Chapter 6. Renormalizability of Yang-Mills theory 207

1. Introduction 207

2. First-order Yang-Mills theory 207

3. Equivalence of first-order and second-order formulations 210

4. Gauge fixing 213

5. Renormalizability 214

6. Universality 217

7. Cohomology calculations 218

Appendix 1: Asymptotics of graph integrals 227

1. Generalized Laplacians 227

2. Polydifferential operators 229

3. Periods 229

4. Integrals attached to graphs 230

5. Proof of Theorem 4.0.2 233

Appendix 2 : Nuclear spaces 243

1. Basic definitions 243

2. Examples 244

3. Subcategories 245

4. Tensor products of nuclear spaces from geometry 247

5. Algebras of formal power series on nuclear Fr´ echet spaces 247