A Mathematical Introduction to String Theory Book contents
- Frontmatter
- Contents
- Preface
- Dedication
- I.0 Introduction
- Acknowledgements
- I.1 The two-dimensional Plateau problem
- I.2 Topological and metric structures on the space of mappings and metrics
- I.3 Harmonic maps and global structures
- I.4 Cauchy–Riemann operators
- I.5 Zeta-function and heat-kernel determinants of an operator
- I.6 The Faddeev–Popov procedure
- I.7 Determinant bundles
- I.8 Chern classes of determinant bundles
- I.9 Gaussian measures and random fields
- I.10 Functional quantization of the Høegh-Krohn and Liouville models on a compact surface
- I.11 Small time asymptotics for heat-kernel regularized determinants
- II.1 Quantization by functional integrals
- II.2 The Polyakov measure
- II.3 Formal Lebesgue measures on Hilbert spaces
- II.4 The Gaussian integration on the space of embeddings
- II.5 The Faddeev–Popov procedure for bosonic strings
- II.6 The Polyakov measure in noncritical dimension and the Liouville measure
- II.7 The Polyakov measure in the critical dimension d=26
- II.8 Correlation functions
- References
- Index
Preface
Published online by Cambridge University Press: 01 June 2011
- Frontmatter
- Contents
- Preface
- Dedication
- I.0 Introduction
- Acknowledgements
- I.1 The two-dimensional Plateau problem
- I.2 Topological and metric structures on the space of mappings and metrics
- I.3 Harmonic maps and global structures
- I.4 Cauchy–Riemann operators
- I.5 Zeta-function and heat-kernel determinants of an operator
- I.6 The Faddeev–Popov procedure
- I.7 Determinant bundles
- I.8 Chern classes of determinant bundles
- I.9 Gaussian measures and random fields
- I.10 Functional quantization of the Høegh-Krohn and Liouville models on a compact surface
- I.11 Small time asymptotics for heat-kernel regularized determinants
- II.1 Quantization by functional integrals
- II.2 The Polyakov measure
- II.3 Formal Lebesgue measures on Hilbert spaces
- II.4 The Gaussian integration on the space of embeddings
- II.5 The Faddeev–Popov procedure for bosonic strings
- II.6 The Polyakov measure in noncritical dimension and the Liouville measure
- II.7 The Polyakov measure in the critical dimension d=26
- II.8 Correlation functions
- References
- Index
Summary
This book is intended as an introduction to certain global analytic and probabilistic aspects of string theory. Nowadays string theory is a domain where mathematics and physics meet, and proceed together concerning certain aspects. However, the theory itself is far from being complete, in fact it is suspended between purely heuristic Ansätze with little hope of mathematical justification and very advanced mathematical ideas. Our aim has been to bring together as far as presently possible the differential–geometric aspects (related to theory of harmonic maps, infinite dimensional differential geometry, Riemann surfaces) and the measure theoretical and probabilistic aspects one encounters when trying to give a sense to the heuristic “Feynman path integrals”, so often used not only by physicists but also by mathematicians “to get started”.
One of us (J. Jost) worked out a theory of strings with boundary as a quantization of Plateau's problem for minimal surfaces and lectured at several conferences on the geometric aspects of the theory. Two of us, S. Paycha and S. Scarlatti, have been working on relating these aspects with probabilistic ones, in connection with Ph.D. theses in Bochum/Paris and Rome respectively, under the direction of S. Albeverio [Pa1], [Sc]. The probabilistic aspects are connected with the study of a mass zero Høegh-Krohn model, and the first basic study of these aspects was undertaken by S. Albeverio, S. Paycha and S. Scarlatti in collaboration with the late R. Høegh-Krohn.
- Type
- Chapter
- Information
- A Mathematical Introduction to String TheoryVariational Problems, Geometric and Probabilistic Methods, pp. viiPublisher: Cambridge University PressPrint publication year: 1997