Representing vector fields by scalar or vector potentials can be a challenging task in domains with cavities or tunnels, due to the presence of harmonic fields which are both irrotational (curl-free) and solenoidal (divergence-free) but may have no scalar or vector potentials [3,7]. For harmonic fields normal to the boundary, which exist in domains with cavities, it is possible to define scalar potentials with Dirichlet boundary conditions fitted to the domain's cavities [5,7,1]. For harmonic fields tangent to the boundary, which exist in domains with tunnels, piecewise regular scalar potentials have been proposed based on disjoint cut surfaces, but a construction of vector potentials seemed to be lacking. In this talk we present a geometric construction of vector potentials that provide a basis for the tangent harmonic fields [4]. This construction is a natural extension of the usual scalar potentials for the harmonic fields normal to the boundary: just as scalar potentials can be associated to the cavities of a domain, our vector potentials are associated to the tunnels of the domain, in accordance with de Rham's theorem [2,6]. Specifically, our vector potentials are obtained by solving curl-curl problems with inhomogeneous tangent boundary conditions that are fitted to closed curves looping around the tunnels. Applied to structure-preserving finite elements [3,2], our approach also provides an exact geometric parametrization of discrete harmonic fields. One application is the structure-preserving approximation of magnetic equilibria in a domain with tunnel holes containing prescribed current flows. [1] Amrouche, Bernardi, Dauge and Girault, Vector potentials in three-dimensional non-smooth domains, Math Methods in the Applied Sciences, 1998 [2] Arnold, Falk and Winther, Finite element exterior calculus, homological techniques and applications, Acta Numerica, 2006 [3] Bossavit, Computational electromagnetism: variational formulations, complementarity, edge elements, Academic Press, 1998 [4] Campos Pinto and Owezarek, Harmonic potentials in the de Rham complex, arXiv:2508.16822, 2025 [5] Foias and Temam, Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation, Annali della Scuola Normale Superiore di Pisa, 1978. [6] Lee, Introduction to Smooth Manifolds, Springer, 2012. [7] Picard, On the boundary value problems of electro- and magnetostatics, Proc of the Royal Society of Edinburgh, 1982.