Order and disorder in gauge systems and magnets

We show how phase transitions in Abelian two-dimensional spin and four-dimensional gauge systems can be understood in terms of condensation of topological objects. In the spin systems these objects are kinks and in the gauge systems either magnetic monopoles or fluxoids (quantized lines of magnetic flux). Four models are studied: two-dimensional Ising and $\mathrm{XY}$ models and four-dimensional $Z_2$ and U(1) gauge systems.