The Numerical Sausage

TL;DR

This paper investigates the numerical solutions of the renormalization group equations for non-linear sigma models with a two-dimensional target space, revealing that sausage-shaped solutions form an attracting manifold in the U(1) symmetric case at one-loop level.

Contribution

It introduces numerical techniques combining spectral methods and variational analysis to study the evolution of metrics in non-linear sigma models, specifically focusing on sausage solutions.

Findings

Sausage solutions form an attracting manifold in the U(1) symmetric case.

Spectral methods effectively estimate the spectrum of zero-modes.

Algorithms successfully reconstruct embedded surfaces in R^3.

Abstract

The renormalization group equation describing the evolution of the metric of the non linear sigma models poses some nice mathematical problems involving functional analysis, differential geometry and numerical analysis. We describe the techniques which allow a numerical study of the solutions in the case of a two-dimensional target space (deformation of the $O(3)\; \sigma$--model. Our analysis shows that the so-called sausages define an attracting manifold in the U(1) symmetric case, at one-loop level. The paper describes i) the known analytical solutions, ii) the spectral method which realizes the numerical integrator and allows to estimate the spectrum of zero--modes, iii) the solution of variational equations around the solutions, and finally iv) the algorithms which reconstruct the surface as embedded in $R^3$.