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A mathematical introduction to string theory. Variational problems, geometric and probabilistic methods. (English) Zbl 0882.53056

London Mathematical Society Lecture Note Series. 225. Cambridge: Cambridge University Press. viii, 135 p. (1997).
Notwithstanding its very general title, this book deals exclusively with a certain approach to the first quantization of the bosonic relativistic string, namely the functional integral formulation introduced by A. M. Polyakov. It was the authors’ aim to give this essentially heuristic concept a firm mathematical basis, and they have been successful at it.
The book consists of two main parts. Chapter I presents all the mathematical tools needed for the rigorous definition of the Polyakov functional integral elaborated in Chapter II. It contains sections on the two-dimensional Plateau problem (the variational principle underlying a string propagating in Euclidean spacetime), the basic topological and metric structures on the space of mappings and metrics, harmonic maps and global structurs on 2-dimensional differentiable manifolds with boundary. Moreover, one finds a detailed discussion of the zeta-function and heat-kernel definition of determinants of operators (they are shown to be identical up to some constant in the present application), a mathematical definition of the Faddeev-Popov procedure (to “factor out” a certain volume term related to the “gauge group” from heuristic functional integrals), and an overview of determinant line bundles and the Quillen metric (giving rise to the notion of regularized determinants) and Chern classes associated with them. Included in Chapter I is also an introduction to Gaussian measures and associated random fields and the application of these concept ot the quantized Liouville model (alias mass zero Høegh-Krohn model) on a Riemann surface, and finally the description of the small time asymptotics for heat-kernel regularized determinants.
Chapter II introduces path integral quantization and the heuristic Polyakov measure for strings and establishes its connections with the rigorous concepts presented in Chapter I. Formal Lebesgue measures on Hilbert spaces are introduced and related to Gaussian measures on the space of string embeddings. The Faddeev-Popov procedure is specialized to bosonic strings and the Liouville model is shown to be contained in the Polyakov measure in “noncritical” dimension \((d<26)\). Also discussed are the Polyakov measure in the critical dimension \(d=26\) and string amplitudes in general. However, only Riemann surfaces of fixed genus are considered.
The book conforms to the highest level of mathematical rigor throughout. Physicist readers may find that, due to the limited size of the book, the material of Chapter I is not self-contained (with the exception of the very readable sections on the Faddeev-Popov procedure, regularized determinants and probabilistic concepts). In particular previous acquaintance with Riemann surfaces and algebraic geometry is helpful (all relevant references are given in the book).
Reviewer: H.Rumpf (Wien)

MSC:

53Z05 Applications of differential geometry to physics
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
58D30 Applications of manifolds of mappings to the sciences
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