List of contributors ........................................... ix
Introduction .................................................... 1
1 A brief introduction to Dirac manifolds ...................... 4
HENRIQUE BURSZTYN
1.1 Introduction ............................................ 4
1.2 Presymplectic and Poisson structures .................... 6
1.3 Dirac structures ....................................... 11
1.4 Properties of Dirac structures ......................... 14
1.5 МофЫзтз of Dirac manifolds ............................. 16
1.6 Submanifolds of Poisson manifolds and constraints ...... 26
1.7 Brief remarks on further developments .................. 33
References .................................................. 36
2 Differential geometry of holomorphic vector bundles on
a curve ..................................................... 39
FLORENT SCHAFFHAUSER
2.1 Но1отофЫс vector bundles on Riemann surfaces ........... 39
2.2 Holomorphic structures and unitary connections ......... 53
2.3 Moduli spaces of semi-stable vector bundles ............ 67
References .................................................. 79
3 Paths towards an extension of Chem-Weil calculus to
a class of infinite dimensional vector bundles .............. 81
SYLVIE PAYCHA
3.1 The gauge group of a bundle ............................ 85
3.2 The diffeomorphism group of a bundle ................... 87
3.3 The algebra of zero-order classical
pseudodifferential operators ........................... 88
3.4 The group of invertible zero-order ψdos ................ 94
3.5 Traces on zero-order classical ψdos ................... 100
3.6 Logarithms and central extensions ..................... 102
3.7 Linear extensions of the L2-trace ..................... 107
3.8 Chem-Weil calculus in finite dimensions ............... 115
3.9 A class of infinite dimensional vector bundles ........ 117
3.10 Frame bundles and associated ψdo-algebra bundles ...... 119
3.11 Logarithms and closed forms ........................... 123
3.12 Chem-Weil forms in infinite dimensions ................ 125
3.13 Weighted Chem-Weil forms; discrepancies ............... 127
3.14 Renormalised Chem-Weil forms on ψdo Grassmannians ..... 132
3.15 Regular Chem-Weil forms in infinite dimensions ........ 135
References ................................................. 139
4 Introduction to Feynman integrals .......................... 144
STEFAN WEINZIERL
4.1 Introduction .......................................... 144
4.2 Basics of perturbative quantum field theory ........... 146
4.3 Dimensional regularisation ............................ 154
4.4 Loop integration in D dimensions ...................... 157
4.5 Multi-loop integrals .................................. 163
4.6 How to obtain finite results .......................... 165
4.7 Feynman integrals and periods ......................... 170
4.8 Shuffle algebras ...................................... 171
4.9 Multiple polylogarithms ............................... 176
4.10 From Feynman integrals to multiple polylogarithms ..... 178
4.11 Conclusions ........................................... 184
References ................................................. 185
5 Iterated integrals in quantum field theory ................. 188
FRANCIS BROWN
5.1 btroduction ........................................... 188
5.2 Definition and first properties of iterated
integrals ............................................. 190
5.3 The case  1{0, 1, ∞} and polylogarithms ............. 198
5.4 The KZ equation and the monodromy of polylogarithms ... 203
5.5 A brief overview of multiple zeta values .............. 208
5.6 Iterated integrals and homotopy invariance ............ 214
5.7 Feynman integrals ..................................... 222
References ................................................. 239
6 Geometric issues in quantum field theory and string
theory ..................................................... 241
LUIS J. BOYA
6.1 Differential geometry for physicists .................. 241
6.2 Holonomy .............................................. 253
6.3 Strings and higher dimensions ......................... 260
6.4 Some issues on compactification ....................... 267
Exercises .................................................. 272
References ................................................. 273
7 Geometric aspects of the Standard Model and the mysteries
of matter .................................................. 274
FLORIAN SCHECK
7.1 Radiation and matter in gauge theories and General
Relativity ............................................ 274
7.2 Mass matrices and state mixing ........................ 284
7.3 The space of connections and the action functional .... 289
7.4 Constructions within noncommutative geometry .......... 293
7.5 Further routes to quantization via BRST symmetry ...... 297
7.6 Some conclusions and outlook .......................... 301
Exercises .................................................. 302
Appendix: Proof of relation (7.11a) ........................ 303
References ................................................. 305
8 Absence of singular continuous spectrum for some
geometric Laplacians ....................................... 307
LEONARDO A. CANO GARCIA
8.1 Меromorphiс extension of the resolvent and singular
continuous spectrum ................................... 309
8.2 Analytic dilation on complete manifolds with comers
of codimension 2 ...................................... 313
References ................................................. 320
9 Models for formal groupoids ................................ 322
IVÁN CONTRERAS
9.1 Motivation and plan ................................... 322
9.2 Definitions and examples .............................. 323
9.3 Algebraic structure for formal groupoids .............. 328
9.4 The symplectic case ................................... 336
References ................................................. 339
10 Elliptic PDEs and smoothness of weakly Einstein metrics
of Holder regularity ....................................... 340
ANDRÉS VARGAS
10.1 Introduction .......................................... 340
10.2 Basics on function spaces ............................. 341
10.3 Elliptic operators and PDEs ........................... 347
10.4 Riemannian regularity and harmonic coordinates ........ 355
10.5 Ricci curvature and the Einstein condition ............ 360
References ................................................. 365
11 Regularized traces and the index formula for manifolds
with boundary .............................................. 366
ALEXANDER CARDONA AND CÉSAR DEL CORRAL
11.1 General heat kernel expansions and zeta functions ..... 368
11.2 Weighted traces, weighted trace anomalies and index
terms ................................................. 371
11.3 Eta-invariant and super-traces ........................ 376
Acknowledgements ........................................... 379
References ................................................. 379
Index ......................................................... 381
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