 
Summary: ON THE STABILITY OF MONOPOLE SOLUTIONS
GEORGE ANDROULAKIS AND STAMATIS DOSTOGLOU
Abstract: We study the Hessian of the YangMillsHiggs functional as the selfinteraction
parameter varies and examine how convergence in the configuration space controls the sign
of the first eigenvalue at the limit. As an application, we show that the spherically symmetric
solutions of 't Hooft and Polyakov are stable for in the neighborhood of 0 and that the
kernels of their Hessians are precisely the spaces generated by their spatial derivatives.
1. Introduction
In 1966 Higgs [H] introduced a Lagrangian coupling vector gauge fields with a pair of
scalar fields and inducing spontaneous symmetry breaking. In this way, the fields of the
model acquired masses, which corresponded to exponential decay. The YangMillsHiggs
functional E on R3
is the static, classical version of Higgs' functional and describes massive
particles with magnetic charge. These are the magnetic monopoles. The massive components
of the Higgs field have mass bounded by
, for a parameter in the functional.
The PrasadSommerfield limit of the theory is obtained by setting = 0. Inevitably, at
this limit some exponential decay is lost. The idea is that information at the limit ought to
propagate to the = 0 case, and trying to substantiate this has been the main motivation
