Some Exact Solutions for Maximally Symmetric Topological Defects in Anti de Sitter Space

TL;DR

This paper derives exact analytical solutions for maximally symmetric topological defects in anti de Sitter space within SO(l) Higgs theories, exploring the double BPS limit and providing both analytical and numerical results.

Contribution

It introduces new exact solutions for topological defects in anti de Sitter space, including the double BPS limit, expanding understanding of defect behavior in curved backgrounds.

Findings

Exact solutions for kink, vortex, and monopole defects in AdS space.

The double BPS limit simplifies equations, revealing maximally symmetric solutions.

Numerical solutions where analytical ones are not available.

Abstract

We obtain exact analytical solutions for a class of SO($l$) Higgs field theories in a non-dynamic background $n$-dimensional anti de Sitter space. These finite transverse energy solutions are maximally symmetric $p$-dimensional topological defects where $n=(p+1)+l$. The radius of curvature of anti de Sitter space provides an extra length scale that allows us to study the equations of motion in a limit where the masses of the Higgs field and the massive vector bosons are both vanishing. We call this the double BPS limit. In anti de Sitter space, the equations of motion depend on both $p$ and $l$. The exact analytical solutions are expressed in terms of standard special functions. The known exact analytical solutions are for kink-like defects ($p=0,1,2,\dotsc;\, l=1$), vortex-like defects ($p=1,2,3;\, l=2$), and the 'tHooft-Polyakov monopole ($p=0;\, l=3$). A bonus is that the double BPS limit automatically gives a maximally symmetric classical glueball type solution. In certain cases where we did not find an analytic solution, we present numerical solutions to the equations of motion. The asymptotically exponentially increasing volume with distance of anti de Sitter space imposes different constraints than those found in the study of defects in Minkowski space.