Renormalization and Galois Theories (Irma Lectures in Mathematics and Theoretical Physics)

Preface......Page 5
Foreword......Page 7
Contents......Page 9
Introduction......Page 11
Classical motives: an overview......Page 13
A first approach to motives......Page 15
Grothendieck\'s pure motives......Page 16
Fundamental structures......Page 21
Examples of pure motives......Page 25
Endomotives: an overview......Page 26
The abelian category of cyclic modules......Page 27
Bimodules and KK-theory......Page 29
Geometric correspondences......Page 31
Algebraic correspondences and K-theory......Page 32
Algebraic endomotives......Page 34
Examples of algebraic endomotives......Page 37
Analytic endomotives......Page 38
The endomotive of the BC system......Page 41
Applications: the geometry of the space of adèles classes......Page 43
Introduction......Page 49
The quantum Hall effect......Page 52
String theory in background field......Page 53
Field theory on Moyal space......Page 54
The Grosse–Wulkenhaar breakthrough......Page 59
The non-commutative Gross–Neveu model......Page 61
A dynamical matrix model......Page 63
Multi-scale analysis......Page 65
Propagators on non-commutative space......Page 69
Propagators and renormalisability......Page 72
Short and long variables......Page 74
Routing, Filk moves......Page 75
Renormalisation......Page 79
Non-commutative hyperbolic polynomials......Page 83
References......Page 88
Mould expansions for the saddle-node and resurgence monomials......Page 93
The saddle-node and its formal normalisation......Page 94
Mould-comould expansions for the saddle-node......Page 96
The algebra of moulds......Page 102
Alternality and symmetrality......Page 106
General mould-comould expansions......Page 116
Contraction into a cosymmetral comould......Page 122
Resurgence of the normalising series......Page 126
The \'s as resurgence monomials – introduction to alien calculus......Page 140
The Bridge Equation for the saddle-node......Page 150
Relation with Martinet–Ramis\'s invariants......Page 158
The resurgence monomials _a\'s and the freeness of alien derivations......Page 164
Other applications of mould calculus......Page 168
References......Page 173
The basic question......Page 175
A naive approach......Page 176
Periods and motives......Page 177
Grothendieck\'s period conjecture......Page 181
Galois theory of periods......Page 183
Relationship with differential Galois theory......Page 186
Introduction......Page 189
Connected graded bialgebras......Page 190
Connected filtered bialgebras......Page 192
The convolution product......Page 193
Characters and infinitesimal characters......Page 195
Renormalization in connected filtered Hopf algebras......Page 196
The Baker–Campbell–Hausdorff recursion......Page 198
Application to perturbative renormalization I......Page 200
Rota–Baxter and dendriform algebras......Page 201
Application to perturbative renormalization II......Page 209
Non-commutative Bohnenblust–Spitzer formulas......Page 210
The matrix representation......Page 212
The matrix form of Connes–Kreimer\'s Birkhoff decomposition......Page 213
Introduction......Page 219
Graded connected Hopf algebras......Page 220
Symmetric and quasi-symmetric functions......Page 222
Hopf algebras of rooted trees......Page 224
Hopf algebras of planar rooted trees......Page 227
Lifting to the Foissy Hopf algebra......Page 231
Some particular families of rooted trees......Page 232
Combinatorial Dyson–Schwinger equations......Page 233
A toy model for some differential equations......Page 239
A generalization......Page 240
Identity-tangent diffeomorphisms and character in the Faà di Bruno Hopf algebra......Page 241
The renormalization scheme......Page 242
Diffeomorphisms and substitutions automorphisms......Page 243
Mould–comould expansions for the conjugacy problem......Page 244
Reminder on moulds......Page 245
The case d N......Page 247
The shuffle Hopf algebra sh_==========N......Page 248
Divergences for the moulds (or characters) U_d and V_d......Page 249
Ramified conjugacy......Page 250
 The logarithmic-alogarithmic factorization of R_0 and its interpretation......Page 251
Proofs......Page 252
Conclusion......Page 255
Introduction......Page 257
Feynman integrals......Page 258
The Mellin–Barnes transformation......Page 262
Shuffle algebras......Page 265
Multiple polylogarithms......Page 269
Laurent expansion of Feynman integrals......Page 272
Conclusions......Page 275