| 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 |
