Multi-Skyrmions on $AdS_2 \times S_2$, Rational maps and Popcorn Transitions

Abstract

By combining two different techniques to construct multi-soliton solutions of the (3+1)-dimensional Skyrme model, the generalized hedgehog and the rational map ansatz, we find multi-Skyrmion configurations in $AdS_{2}\times S_{2}$. We construct Skyrmionic multi-layered configurations such that the total Baryon charge is the product of the number of kinks along the radial $AdS_{2}$ direction and the degree of the rational map. We show that, for fixed total Baryon charge, as one increases the charge density on $\partial\left( AdS_{2}\times S_{2}\right) $, it becomes increasingly convenient energetically to have configurations with more peaks in the radial $AdS_{2}$ direction but a lower degree of the rational map. This has a direct relation with the so-called holographic popcorn transitions in which, when the charge density is high, multi-layered configurations with low charge on each layer are favored over configurations with few layers but with higher charge on each layer. The case in which the geometry is $M_{2}\times S_{2}$ can also be analyzed.