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

Cover......Page 1
IRMA Lectures in Mathematics and Theoretical Physics 15......Page 2
Publications in this series......Page 3
Title......Page 4
Copyright......Page 5
Preface......Page 6
Foreword......Page 8
Contents......Page 10
Introduction......Page 12
Classical motives: an overview......Page 14
A first approach to motives......Page 16
Grothendieck's pure motives......Page 17
Fundamental structures......Page 22
Examples of pure motives......Page 26
Endomotives: an overview......Page 27
The abelian category of cyclic modules......Page 28
Bimodules and KK-theory......Page 30
Geometric correspondences......Page 32
Algebraic correspondences and K-theory......Page 33
Algebraic endomotives......Page 35
Examples of algebraic endomotives......Page 38
Analytic endomotives......Page 39
The endomotive of the BC system......Page 42
Applications: the geometry of the space of adèles classes......Page 44
Introduction......Page 50
The quantum Hall effect......Page 53
String theory in background field......Page 54
Field theory on Moyal space......Page 55
The Grosse–Wulkenhaar breakthrough......Page 60
The non-commutative Gross–Neveu model......Page 62
A dynamical matrix model......Page 64
Multi-scale analysis......Page 66
Propagators on non-commutative space......Page 70
Propagators and renormalisability......Page 73
Short and long variables......Page 75
Routing, Filk moves......Page 76
Renormalisation......Page 80
Non-commutative hyperbolic polynomials......Page 84
References......Page 89
Mould expansions for the saddle-node and resurgence monomials......Page 94
The saddle-node and its formal normalisation......Page 95
Mould-comould expansions for the saddle-node......Page 97
The algebra of moulds......Page 103
Alternality and symmetrality......Page 107
General mould-comould expansions......Page 117
Contraction into a cosymmetral comould......Page 123
Resurgence of the normalising series......Page 127
The 's as resurgence monomials – introduction to alien calculus......Page 141
The Bridge Equation for the saddle-node......Page 151
Relation with Martinet–Ramis's invariants......Page 159
The resurgence monomials _a's and the freeness of alien derivations......Page 165
Other applications of mould calculus......Page 169
References......Page 174
The basic question......Page 176
A naive approach......Page 177
Periods and motives......Page 178
Grothendieck's period conjecture......Page 182
Galois theory of periods......Page 184
Relationship with differential Galois theory......Page 187
Introduction......Page 190
Connected graded bialgebras......Page 191
Connected filtered bialgebras......Page 193
The convolution product......Page 194
Characters and infinitesimal characters......Page 196
Renormalization in connected filtered Hopf algebras......Page 197
The Baker–Campbell–Hausdorff recursion......Page 199
Application to perturbative renormalization I......Page 201
Rota–Baxter and dendriform algebras......Page 202
Application to perturbative renormalization II......Page 210
Non-commutative Bohnenblust–Spitzer formulas......Page 211
The matrix representation......Page 213
The matrix form of Connes–Kreimer's Birkhoff decomposition......Page 214
Introduction......Page 220
Graded connected Hopf algebras......Page 221
Symmetric and quasi-symmetric functions......Page 223
Hopf algebras of rooted trees......Page 225
Hopf algebras of planar rooted trees......Page 228
Lifting to the Foissy Hopf algebra......Page 232
Some particular families of rooted trees......Page 233
Combinatorial Dyson–Schwinger equations......Page 234
A toy model for some differential equations......Page 240
A generalization......Page 241
Identity-tangent diffeomorphisms and character in the Faà di Bruno Hopf algebra......Page 242
The renormalization scheme......Page 243
Diffeomorphisms and substitutions automorphisms......Page 244
Mould–comould expansions for the conjugacy problem......Page 245
Reminder on moulds......Page 246
The case d N......Page 248
The shuffle Hopf algebra sh_==========N......Page 249
Divergences for the moulds (or characters) U_d and V_d......Page 250
Ramified conjugacy......Page 251
The logarithmic-alogarithmic factorization of R_0 and its interpretation......Page 252
Proofs......Page 253
Conclusion......Page 256
Introduction......Page 258
Feynman integrals......Page 259
The Mellin–Barnes transformation......Page 263
Shuffle algebras......Page 266
Multiple polylogarithms......Page 270
Laurent expansion of Feynman integrals......Page 273
Conclusions......Page 276
Back Cover......Page 282