Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 70 / 3 / a

Choose a vacuum order parameter ${\Phi }_0$ and let $H$ be its stabilizer in $G$. Acting with $G$ produces all symmetry-related minima; two elements produce the same minimum precisely when they differ by an element of $H$. The vacuum manifold is thereforeThis assumes a single transitive orbit of minima; additional independent vacuum sectors would require more than one orbit. For a local gauge symmetry, $G/H$ describes the vacuum directions used in finite-energy boundary conditions and defect classification, rather than distinct observable vacua after quotienting by every gauge redundancy.

Finite-energy configurations approach the vacuum manifold outside defect cores. The surrounding boundary in the transverse directions is a sphere, and noncontractible boundary data prevent a smooth deformation to the uniform vacuum.

Nontrivial ${\pi }_0(M)$ means disconnected vacuum components. Choosing different components on the two sides of an interface produces domain walls. Here ${\pi }_0$ labels components and need not itself carry a group structure. Nontrivial ${\pi }_1(M)$ means noncontractible loops: a circle surrounding a line can wind around the vacuum manifold, producing cosmic strings. Nontrivial ${\pi }_2(M)$ means noncontractible maps from an enclosing two-sphere, producing pointlike magnetic monopoles in three spatial dimensions.During the transition, initially uncorrelated vacuum choices in causally separated regions can generate these boundary classes through the Kibble mechanism. Nontrivial topology allows stable defects; it does not specify their exact formation efficiency or abundance.