Effective action of domain wall networks

Abstract

$U({N}_{\mathrm{C}})$ gauge theory with ${N}_{\mathrm{F}}$ fundamental scalars admits BPS junctions of domain walls. When the networks/webs of these walls contain loops, their size moduli give localized massless modes. We construct K\"ahler potential of their effective action. In the large size limit K\"ahler metric is well approximated by kinetic energy of walls and junctions, which is understood in terms of tropical geometry. K\"ahler potential can be expressed in terms of hypergeometric functions that are useful to understand small size behavior. Even when the loop shrinks, the metric is regular with positive curvature. Moduli space of a single triangle loop has a geometry between a cone and a cigar.