Nonlinear models in \(2+\epsilon\) dimensions
This paper, the author's Ph. D. thesis, is a study of the renormalization of the general nonlinear model in two and \(2+\epsilon\)-dimensions. In these models, a field \(\phi\) (x) takes values in some manifold M. The coupling is given by a Riemannian metric \(g_{ij}(\phi)\) on M. These models generalize the classical Heisenberg model, where M is a sphere with the standard O(N) invariant metric. The main current interest in this work comes from another interpretation which arises in string theory. There the manifold M is spacetime, and the field \(\phi\) (x) describes the world surface of the string. The metric on spacetime is the condensate of the gravitational mode of the string. The renormalization of these models is of interest in string theory because only conformally invariant models correspond to condensates which solve the string equation of motion. The renormalization of the metric coupling \(g_{ij}\) is described by the beta-function, which in this case is a vector field on the infinite dimensional space of all metrics on M. The conformally invariant models are the fixed points of the renormalization group flow, i.e., the zeros of the beta-function. To the one loop approximation, the beta-function is \(R_{ij}\), the Ricci tensor of the metric, so the fixed point equation is Einstein's equation \(R_{ij}=0\). The two loop correction is calculated, and it is shown that the only fixed points which persist beyond one loop are the flat metrics. The generalization of these calculations to supersymmetric nonlinear models, by Alvarez-Gaume and Freedman, showed the possibility of nontrivial fixed points. The supersymmetric models are used in fermionic string theory to describe nontrivial compactifications of spacetime. Beyond these basic results, there is a systematic proof of renormalizability for the general nonlinear models in two dimensions; effective algorithms are developed for calculating to all orders in the loop expansion; and a mathematical study is made of the topology of the renormalization group flow, especially near the fixed points, which can be nontrivial in \(2+\epsilon\) dimensions.
nonlinear models in quantum field theory
scientific article; zbMATH DE number 3934065
scientific article