  The methods for studying the role of vortex loops in the phase transition of
the Ginzburg-Landau theory of superconductivity using lattice Monte Carlo
simulations are discussed. Gauge-invariant observables that measure the
properties of the vortex loop distribution are defined. The exact relations
between the lattice and continuum quantities make it possible to extrapolate
the results of the simulations to the continuum limit. The relationship between
spontaneous symmetry breaking and phase transitions is also reviewed, with an
emphasis on the fact that a local symmetry cannot be broken.
