Geometric approach to path integration in string theory
Kevin M. Short · Spiral (Imperial College London) · 1988
This thesis is concerned with developing a geometric approach to path integration, particularly with respect to Polyakov string theory.Introductory remarks are in chapter one.Chapter 2 develops a non-perturbative approach to Polyakov string theory in variable dimensions.This leads to a reinterpretation of the critical dimension and an effective compactification of the theory.The variable dimension theory is based on gaussian measures, and the dimension can be taken to infinity.The limit represents a correction to [DetA]'1^2, which arises as a naive limit.The analysis involves an interplay between zero modes and sets of measure zero with respect to the gaussian measures.There is a discussion of functional Haar measures, and source terms are considered.An appendix is added concerning the definition of gaussian measures based on cylinder set measures, and the sets of measure zero which arise.The third chapter will consider the extension of string theory to curved target spaces.It is shown that only trivial classical solutions exist in euclidean space, and then discusses the existence of classical solutions for curved target spaces.A toy model of strings in S3 will be developed, which will be extended to a simple model of cosmological membranes in S3 x R.In the fourth chapter, a new formulation of string theory which is applicable to curved target spaces will be developed.The observables in the new formulation represent the deformations of the image surface.The path integral is "gauge fixed", but it retains a global gauge invariance.A new measure for the embedding integration is developed which encodes a new geometric regularization procedure.The new