We prove a $$\varGamma $$
Γ
-convergence result for a class of Ginzburg–Landau type functionals with $${\mathscr {N}}$$
N
-well potentials, where $${\mathscr {N}}$$
N
is a closed and $$(k-2)$$
(
k
-
2
)
-connected submanifold of $${\mathbb {R}}^m$$
R
m
, in arbitrary dimension. This class includes, for instance, the Landau-de Gennes free energy for nematic liquid crystals. The energy density of minimisers, subject to Dirichlet boundary conditions, converges to a generalised surface (more precisely, a flat chain with coefficients in $$\pi _{k-1}({\mathscr {N}})$$
π
k
-
1
(
N
)
) which solves the Plateau problem in codimension k. The analysis relies crucially on the set of topological singularities, that is, the operator $${\mathbf {S}}$$
S
we introduced in the companion paper [17].