The geodesic rule for higher codimensional global defects

Details The geodesic rule for higher codimensional global defects The geodesic rule for higher codimensional global defects

2008-05-15

arxiv.org/abs/0805.2353v1

Mod.Phys.Lett.A23:2053-2066,2008

Physics

High Energy Physics

High Energy Physics - Theory

17 pages, no figures. To be published in Mod. Phys. Lett. A

Scientific paper

10.1142/S0217732308027242

We generalize the geodesic rule to the case of formation of higher codimensional global defects. Relying on energetic arguments, we argue that, for such defects, the geometric structures of interest are the totally geodesic submanifolds. On the other hand, stochastic arguments lead to a diffusion equation approach, from which the geodesic rule is deduced. It turns out that the most appropriate geometric structure that one should consider is the convex hull of the values of the order parameter on the causal volumes whose collision gives rise to the defect. We explain why these two approaches lead to similar results when calculating the density of global defects by using a theorem of Cheeger and Gromoll. We present a computation of the probability of formation of strings/vortices in the case of a system, such as nematic liquid crystals, whose vacuum is $\mathbb{R}P^2$.

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