|
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 |