Geometric and topological methods for quantum field theory (Cambridge, 2013). - ОГЛАВЛЕНИЕ / CONTENTS

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ОбложкаGeometric and topological methods for quantum field theory: proc. of the 2009 Villa de Leyva Summer school / ed. by A.Cardona, I.Contreras, A.F.Reyes-Lega - Cambridge: Cambridge university press, 2013. - ix, 383 p.: ill. - Incl. bibl. ref. - Ind.: p.381-383. - ISBN 978-1-107-02683-4
 

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Оглавление / Contents
 
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 fig.21{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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