Prof. Dr. Katrin Wendland

Professor in Pure Mathematics
at Trinity College Dublin

Diploma (Mathematics)


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Analytic Torsion and Critical Metrics

Ray-Singer Analytic Torsion, Quillenmetric and Regularized Determinants

Table of contents

Introduction 1
General preliminaries 6
1 Calculations with local holomorphic coordinates 6
1.1 Differential forms and metrics on complex manifolds 6
1.2 Definition of the Dolbeaut Laplacian 11
1.3 Variation of a Kähler metric within its Kähler class 14
2 Basics from differential geometry 16
2.1 Manifolds of constant holomorphic curvature 16
2.2 Chern classes 18
2.3 K3-surfaces 21
3 Functional determinants and Ray-Singer analytic torsion 26
3.1 The regularized determinant of the Laplacian 26
3.2 Definition of the Ray-Singer analytic torsion 30
3.3 Determinant line bundles and Quillenmetric 33
Mathematical results 42
4 Known results for compact Riemann surfaces 42
5 Extremal metrics under conformal variation 45
5.1 Noncompact Riemann surfaces 45
5.2 Known results in higher dimensions 60
6 Extremal metrics under variation within the Kähler class 65
6.1 Preliminaries for the computation of a variational formula 65
6.2 The anomaly formula 68
6.3 Evaluation of the anomaly formula on Riemann surfaces 76
6.4 Manifolds of constant holomorphic curvature 80
6.5 Elementary proof of a variational formula 84
6.6 Twimanifolds 87
6.7 Analytic torsion and Quillenmetric on twimanifolds 91
6.8 Complex tori 98
6.9 Compact K3-surfaces 103
7 Quillenmetric for Dirac operators on forms with values in tensor products of ample line bundles 107
Applications in theoretical physics 119
8 Regularized determinants in path integrals 119
8.1 Path integrals 119
8.2 Perturbation series as a means of approximating path integrals 125
8.3 Dimensional regularization 130
8.4 Scaling of the partition function 136
8.5 Anomalies 138
9 Ray-Singer (analytic) torsion in theoretical physics 143
9.1 Ray-Singer torsion in topological quantum field theory 143
9.2 Ray-Singer analytic torsion in string theory 148
Summary 154
References 157
Acknowledgements 164