  An $n$-cross field is a locally-defined orthogonal coordinate system
invariant with respect to the cubic symmetry group. Cross fields are finding
wide-spread use in mesh generation, computer graphics, and materials science
among many applications. It was recently by other authors that $3$-cross fields
can be embedded into the set of symmetric $4$th-order tensors. Another
concurrent work further develops a relaxation of this tensor field via a
certain set of varieties. In this paper, we consider the problem of generating
an arbitrary $n$-cross field using a fourth-order $Q$-tensor theory that is
constructed out of tensored projection matrices. We establish that by a
Ginzburg-Landau relaxation towards a global projection, one can reliably
generate an $n$-cross field on arbitrary Lipschitz domains. Our work provides a
rigorous approach that offers several new results including porting the tensor
framework to arbitrary dimensions, providing a new relaxation method that
embeds the problem into a global steepest descent, and offering a relaxation
scheme for aligning the cross field with the boundary. Our approach is designed
to fit within the classical Ginzburg-Landau PDE theory, offering a concrete
road map for the future careful study of singularities of energy minimizers.
