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Index Theory in Geometry and Physics Magnus Goffeng

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Index Theory in Geometry and Physics Magnus Goffeng THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Index theory in geometry and physics Magnus Goffeng Department of Mathematical Sciences Division of Mathematics CHALMERS UNIVERSITY OF TECHNOLOGY AND UNIVERSITY OF GOTHENBURG G¨oteborg, Sweden 2011 Index theory in geometry and physics Magnus Goffeng c Magnus Goffeng, 2011 ! ISBN 978-91-628-8281-5 Department of Mathematical Sciences Division of Mathematics Chalmers University of Technology and University of Gothenburg SE-412 96 G¨oteborg Sweden Telephone +46 (0)31-772 1000 Printed in G¨oteborg, Sweden 2011 Abstract This thesis contains three papers in the area of index theory and its ap- plications in geometry and mathematical physics. These papers deal with the problems of calculating the charge deficiency on the Landau levels and that of finding explicit analytic formulas for mapping degrees of H¨older continuous mappings. Paper A deals with charge deficiencies on the Landau levels for non-interacting particles in 2 under a constant magnetic field, or equivalently, one particle moving in a constant magnetic field in even-dimensional Euclidian space. The K-homology class that the charge of a Landau level defines is calculated in two steps. The first step is to show that the charge deficiencies are the same on ev- ery particular Landau level. The second step is to show that the lowest Landau level, which is equivalent to the Fock space, defines the same class as the K- homology class on the sphere defined by the Toeplitz operators in the Bergman space of the unit ball. Paper B and Paper C uses regularization of index formulas in cyclic cohomol- ogy to produce analytic formulas for the degree of H¨older continuous mappings. In Paper B Toeplitz operators and Henkin-Ramirez kernels are used to find an- alytic formulas for the degree of a function f : ∂ Ω Y , where Ω is a relatively compact strictly pseudo-convex domain in a Stein manifold→ and Y is a compact connected oriented manifold. In Paper C analytic formulas for H¨older continu- ous mappings between general even-dimensional manifolds are produced using a pseudo-differential operator associated with the signature operator. Keywords: Index theory, cyclic cohomology, regularized index formulas, Toeplitz operators, pseudo-differential operators, quantum Hall effect. 2000 Mathematics Subject Classification: 19KXX, 46L80, 19L64, 47N50, 58J40. i Preface The purpose of this thesis is to obtain the degree of Doctor for its author. The work in this thesis is based on three papers written from material gathered by the author under time spent as a PhD-student in Gothenburg and during visits to the University of Copenhagen under the time period June 2007 to May 2011. The thesis is divided up into two parts. The first part is an introduction with a summary of the results. The second part of the thesis consists of the following papers: A. ”Index formulas and charge deficiencies on the Landau levels”, Journal of Mathematical Physics 51 (2010). B. ”Analytic formulas for topological degree of non-smooth mappings: the odd-dimensional case”, submitted. C. ”Analytic formulas for topological degree of non-smooth mappings: the even-dimensional case”, submitted. Only minor modifications on these papers have been made for this thesis. These minor modifications include correcting typos and changing of notations for a homogeneous notation throughout the thesis. In addition to the above, there are four other papers by the author. These, however, will not be included in the thesis: * ”Projective pseudo-differential operators on infinite-dimensional Azumaya bundles”, submitted. * ”The Pimsner-Voiculescu sequence for coactions of compact Lie groups”, to appear in Mathematica Scandinavica. * ”A remark on twists and the notion of torsion-free discrete quantum groups”, to appear in Algebras and Representation Theory. * ”Equivariant extensions of -algebras”, New York Journal of Mathematics 16 (2010), p. 369–385. ∗ ii Acknowledgements First of all the author would like to thank his thesis advisor Grigori Rozen- blioum for his constant patience, for his impeccable grammar, for sharing his mathematical style and philosophy with the author and for suggesting the prob- lems discussed in this thesis. The author also owe a big thanks to his co-advisor Ryszard Nest who was a great source of inspiration and ideas. Thanks to the department of Mathematical Sciences at Chalmers/GU and all its employees for giving the author the opportunity to write his thesis and a good time doing it. A great thanks to the HAPDE-group at the department. The author thanks Alexander Stolin for his support. Thanks to the platform MP2 who helped the author through many problems. A big thanks also goes to Lyudmila Tur- owska who introduced the author to K-theory, the K-theory classification of AF-algebras by Elliott, that to an undergrad seems magical. Thanks to Jana Madjarova for her encouragement. Thanks to Vilhelm Adolfsson for many inter- esting discussions about everything between heaven and earth except mapping degrees. The author owes much to his beloved Marie for her constant support and for always believing in all the author’s endeavors, for that the author thanks her. The author thanks his brother Bj¨orn, his parents and the rest of his family for encouraging him to pursue with mathematics and more. And of course the author wouldn’t be here without Jacob M¨ollstam with whom he discussed the big things just before starting as a PhD-student. The author also thanks Ulrica Dahlberg for giving him the final push into mathematics. Thanks also to Middagsg¨anget Jacob Sznajdman, Peter Hegarty, Dennis Eriksson, Johan Tykesson and Oskar Sandberg where much of the author’s mathematical moral and need for mathematical gossip was created. Let us not forget AGMP-g¨anget Johan Oinert,¨ Kalle R¨okaeus, Qimh Xantcha and Chris- tian Svensson with whom the author have spent many great times with. Thanks also to Bram Mesland with whom the author discussed the depths of KK, to Micke Persson for being a good fadder, Jonas Hartwig for all his wisdom on the Serre relations and to Oskar Hamlet for being a good and understanding room mate. Thanks to all other PhD-students at the department for making my stay comfortable. Finally, a big thanks to the entire complex analysis group at Chalmers/GU for helping the author to keep it real. Bo Berndtsson for teaching the author Cauchy’s formula and about Fredholm operators. Mats Andersson for helping the author with all the integral representations. Robert Berman for reminding the author about the elementary. Rickard L¨ark¨ang for all strong discussions on weak stuff. Contents Preface i Acknowledgements ii I Introduction 1 Introductory remarks 3 1 K-theory and K-homology 7 1.1 Even K-theory . 7 1.2 Odd K-theory . 12 2 Magnetism and K-theory 19 2.1 Quantum Hall effect . 20 2.2 Hall conductance . 23 3 Index formulas 27 3.1 Atiyah-Singer’s index theorem . 28 3.2 Cyclic homology . 31 3.3 H¨older continuous symbols . 37 II Research papers 45 A Index formulas and charge deficiencies on the Landau levels 47 A.1 Particular Landau levels . 50 A.2 Toeplitz operators on the Landau levels . 51 2n 1 A.3 Pulling symbols back from S − ..................... 55 A.4 The special cases and 2 ........................ 59 A.5 Index formula on the particular Landau levels . 63 iii iv CONTENTS B Analytic formulas for degree of non-smooth mappings: the odd- dimensional case 67 B.1 The volume form as a Chern character . 71 B.2 Toeplitz operators and their index theory . 78 B.3 The Toeplitz pair on the Hardy space . 85 B.4 The index- and degree formula . 87 C Analytic formulas for degree of non-smooth mappings: the even-dimensional case 91 C.1 K-theory and Connes’ index formula . 93 C.2 The projection with Chern character being the volume form . 99 C.3 Index theory for pseudo-differential operators . 102 C.4 Index of a H¨older continuous twist . 105 C.5 Degrees of H¨older continuous mappings . 110 C.6 Example on S2n ................................ 111 Bibliography 115 Part I Introduction 1 Introductory remarks ”Where there is matter, there is geometry.” Kepler The starting point for index theory and K-theory was the Riemann-Roch theorem which originates from Riemann and Roch in the 1850:s, see [73] and [74]. The Riemann-Roch theorem relates an analytic quantity, the holomorphic Euler characteristic, with a topological quantity associated with a surface. The holomorphic Euler characteristic is the index of a twisted Dolbeault operator on the Riemann surface. The generalizations were many and came in differ- ent shapes. Hirzebruch made a generalization to complex manifolds allowing a calculation of the holomorphic Euler characteristic in terms of topological quan- tities in [52] and Grothendieck found the place for Riemann-Roch’s theorem in the realm of algebraic geometry. Grothendieck’s formulation was made in terms of his K-theory, a group of formal differences of locally free sheafs. The ideas of Grothendieck were transformed by Atiyah, see [2], into topo- logical K-theory and used in the proof of the Atiyah-Singer index theorem. The Atiyah-Singer index theorem was a large step from the Riemann-Roch theorem in that it gave an explicit method to calculate the index of any elliptic differ- ential operator in terms of topological data from the manifold and the highest order symbol of the differential operator. On a vague level the index theorem related an analytic, or for that matter a global, invariant such as the Fredholm index with a geometric or local invariant such as the topological index. More generally, finding index formulas deals often with going from global to local or from analytic to geometric.
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