Harmonic maps to the circle with higher dimensional singular set

In a closed, oriented ambient manifold $(M^n,g)$ we consider the problem of finding ${\mathbb{S}}^1$-valued harmonic maps with prescribed singular set. We show that the boundary of any oriented $(n-1)$-submanifold can be realised as the singular set of an ${\mathbb{S}}^1$-valued map, which is classically harmonic away from the singularity and distributionally harmonic across. If the singular set $\mathrm{\Gamma }$ is also embedded and $C^{1,1}$, we consider three variational relaxations of the same problem and show that the energy of minimisers converges, after renormalisation, to the volume ${\mathcal{H}}^{n-2}(\mathrm{\Gamma })$ plus a lower-order “renormalised energy” – common to all relaxations – describing an energetic interaction between different components of the singular set.