Effective Action of Domain Wall Networks

TL;DR

This paper derives the effective action for domain wall networks in a U(Nc) gauge theory, revealing their moduli space structure, metric properties, and geometric interpretations, including tropical geometry and hypergeometric functions.

Contribution

It constructs the Kahler potential for the moduli space of domain wall networks, providing new insights into their geometric and metric properties, especially for looped configurations.

Findings

Kahler metric approximated by wall and junction kinetic energy in large size limit

Metric remains regular and positively curved even when loops shrink

Moduli space of a triangle loop exhibits geometry between a cone and a cigar

Abstract

U(Nc) gauge theory with Nf 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 Kahler potential of their effective action. In the large size limit Kahler metric is well approximated by kinetic energy of walls and junctions, which is understood in terms of tropical geometry. Kahler potential can be expressed in terms of hypergeometric functions which 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.