Expansion for the solutions of the Bogomolny equations on the torus

TL;DR

This paper develops a power series expansion method for solutions of the Bogomolny equations on a torus, enabling precise analysis of their shape for various fluxes and areas, and suggests the series converges even at infinite area.

Contribution

It introduces a high-order expansion technique for Bogomolny equations on a torus, providing detailed solutions across different fluxes and areas, including the large-area limit.

Findings

Expansion carried out to 51 orders for key cases

Solutions approach plane solutions as area increases

Series likely converges at infinite area

Abstract

We show that the solutions of the Bogomolny equations for the Abelian Higgs model on a two-dimensional torus, can be expanded in powers of a quantity epsilon measuring the departure of the area from the critical area. This allows a precise determination of the shape of the solutions for all magnetic fluxes and arbitrary position of the Higgs field zeroes. The expansion is carried out to 51 orders for a couple of representative cases, including the unit flux case. We analyse the behaviour of the expansion in the limit of large areas, in which case the solutions approach those on the plane. Our results suggest convergence all the way up to infinite area.