Contents

# Contents

## Idea

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 vaccum 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 $\mathbb{R} \times (\mathbb{R}^3 - D^2 \times \mathbb{R}^1)$, hence with a 1-dimensional (“string”-like) piece taken out, then homotopy classes of maps $X \to \Pi(G/H)$ are classified by the fundamental group $\pi_1(G/H)$. For a given nontrivial element here the correponding vacuum is said to contain a cosmic string defect. (“Cosmic” just because this effect is thought to be most relevant on scales of cosmology.)

In other words this means that the vacuum strcture changes continuously as one moves around the string, but has a singularity on the locus of the string itself.

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

There is no direct relation to strings in the sense of perturbative string theory; maybe to D1-branes.

## References

• Alexander Vilenkin, E.P.S. Shellard, Cosmic strings and other topological defects, Cambridge University Press (1994)

Last revised on December 20, 2020 at 12:45:33. See the history of this page for a list of all contributions to it.