nLab domain wall

Idea

In general, a domain wall is a defect of codimension 1.

More specifically, in a gauge theory with a degenerate vacuum (such as when a Higgs mechanism applies), the moduli space of vacua is the quotient $G/H$ (the coset) of the gauge group $G$ by the stabilizer subgroup $H \hookrightarrow G$ of any of these vacua (spontaneous symmetry breaking).

This means that gauge equivalence classes of vacuum configurations on a spacetime $X$ are given by homotopy classes of maps $X \to \Pi(G/H)$ (where the notation on the right denotes the underlying homotopy type of the coset space, $\Pi$ is the shape modality).

If spacetime is locally to be taken of the form $\mathbb{R} \times (\mathbb{R}^3 - D^1 \times \mathbb{R}^2)$, hence with a 1-dimensional (“wall”-like) piece taken out, them homotopy classes of maps $X \to \Pi(G/H)$ are classified by the 0-th homotopy group $\pi_0(G/H)$. For a given nontrivial element here the corresponding vacuum is said to contain a domain wall defect.

For more see at QFT with defects the section Topological defects from spontaneously broken symmetry.

Related concepts

• cosmic string, monopole

singularity	field theory with singularities
boundary condition/brane	boundary field theory
domain wall/bi-brane	QFT with defects

• wall crossing

• defect brane

References

• Alexander Vilenkin, E. P. S. Shellard, Cosmic strings and other topological defects, Cambridge University Press (1994) [ISBN:9780521654760, spire:1384873]

See also:

• Wikipedia, Domain wall