Magnetic bags in hyperbolic space

A magnetic bag is an Abelian approximation to a large number of coincident $SU(2)$ Bogomol’nyi-Prasad-Sommerfield monopoles. In this paper we consider magnetic bags in hyperbolic space and derive their Nahm transform from the large-charge limit of the discrete Nahm equation for hyperbolic monopoles. An advantage of studying magnetic bags in hyperbolic space, rather than Euclidean space, is that a range of exact charge $N$ hyperbolic monopoles can be constructed, for arbitrarily large values of $N$, and compared with the magnetic bag approximation. We show that a particular magnetic bag (the magnetic disc) provides a good description of the axially symmetric $N$-monopole. However, an Abelian magnetic bag is not a good approximation to a roughly spherical $N$-monopole that has more than $N$ zeros of the Higgs field. We introduce an extension of the magnetic bag that does provide a good approximation to such monopoles and involves a spherical non-Abelian interior for the bag, in addition to the conventional Abelian exterior.

Figure 1

Energy density isosurfaces, in the ball model of ${\mathbb{H}}^3$, for hyperbolic monopoles with $N=371$. The hyperbolic monopole in the left image has axial symmetry and is cherry flavor, whereas the one in the right image has icosahedral symmetry and is strawberry flavor. The blue sphere represents the boundary of hyperbolic space.

Figure 2

The red curves display the length of the Higgs field $|\mathrm{\Phi }|$ as a function of geodesic distance from the origin $\rho$, along an axis that is perpendicular to the symmetry axis of the axial hyperbolic $N$-monopole with $N=100$ and $N=10000$. The blue curves show the corresponding magnetic disc approximation.

Figure 3

The length of the Higgs field, $|\mathrm{\Phi }|$, for an icosahedrally symmetric strawberry-flavor monopole with charge $N=371$. The plot shows $|\mathrm{\Phi }|$ as a function of geodesic distance from the origin $\rho$ along a radial half line that passes through a vertex (black curve) and a face center (yellow curve) of the associated polyhedron. The blue curve is the spherical average of $|\mathrm{\Phi }|$, obtained by integrating over the angular coordinates. The red curve is the new magnetic bag approximation.