Gravitating anisotropic merons and squashed spheres in the three-dimensional Einstein-Yang-Mills-Chern-Simons theory

We construct the first analytic examples of self-gravitating anisotropic merons in the Einstein-Yang-Mills-Chern-Simons theory in three dimensions. The gauge field configurations have different meronic parameters along the three Maurer-Cartan $1$-forms and they are topologically nontrivial as the Chern-Simons invariant is nonzero. The corresponding backreacted metric is conformally a squashed three-sphere. The amount of squashing is related to the degree of anisotropy of the gauge field configurations that we compute explicitly in different limits of the squashing parameter. Moreover, the spectrum of the Dirac operator on this background is obtained explicitly for spin-1/2 spinors in the fundamental representation of $SU(2)$, and the genuine non-Abelian contributions to the spectrum are identified. The physical consequences of these results are discussed.