  We present a non-perturbative formalism for measuring defect free energies
(monopole mass or vortex tension) in three-dimensional SU(2)+adjoint Higgs
models. Starting from twisted, translation invariant boundary conditions, we
perform a change of variables that allows us to express the defect free
energies in terms of 't Hooft loops. We propose that the defect free energies
can be used to distinguish between phases in this model, and also more
generally in other gauge field theories where no local order parameters exist.
In the case of monopoles, our construction can also be used in four-dimensional
pure-gauge SU(2) theory, where it gives the monopole mass in the maximally
Abelian gauge without the need of actually fixing the gauge in the simulation.
Moreover, the expression is manifestly independent of the choice of the Abelian
projection as long as it is compatible with the classical 't Hooft-Polyakov
solution.
