Chapter 1. Introduction ......................................... 1
1 Overview ..................................................... 1
2 Functional integrals in quantum field theory ................. 4
3 Wilsonian low energy theories ................................ 6
4 A Wilsonian definition of a quantum field theory ............ 13
5 Locality .................................................... 13
6 The main theorem ............................................ 16
7 Renormalizability ........................................... 19
8 Renormalizable scalar field theories ........................ 21
9 Gauge theories .............................................. 22
10 Observables and correlation functions ....................... 27
11 Other approaches to perturbative quantum field theory ....... 27
Acknowledgements ............................................... 28
Chapter 2. Theories, Lagrangians and counterterms .............. 31
1 Introduction ................................................ 31
2 The effective interaction and background field functional
integrals ................................................... 32
3 Generalities on Feynman graphs .............................. 34
4 Sharp and smooth cut-offs ................................... 42
5 Singularities in Feynman graphs ............................. 44
6 The geometric interpretation of Feynman graphs .............. 47
7 A definition of a quantum field theory ...................... 53
8 An alternative definition ................................... 55
9 Extracting the singular part of the weights of Feynman
graphs ...................................................... 57
10 Constructing local counterterms ............................. 61
11 Proof of the main theorem ................................... 67
12 Proof of the parametrix formulation of the main theorem ..... 69
13 Vector-bundle valued field theories ......................... 71
14 Field theories on non-compact manifolds ..................... 80
Chapter 3. Field theories on  n ................................ 91
1 Some functional analysis .................................... 92
2 The main theorem on n ...................................... 99
3 Vector-bundle valued field theories on n .................. 104
4 Holomorphic aspects of theories on n ...................... 107
Chapter 4. Renormalizability .................................. 113
1 The local renormalization group flow ....................... 113
2 The Kadanoff-Wilson picture and asymptotic freedom ......... 122
3 Universality ............................................... 125
4 Calculations in ø4 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 n .......................................... 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 Fréchet spaces .. 247
Bibliography .................................................. 249
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