  Higher-order networks encode the many-body interactions existing in complex
systems, such as the brain, protein complexes, and social interactions.
Simplicial complexes are higher-order networks that allow a comprehensive
investigation of the interplay between topology and dynamics. However,
simplicial complexes have the limitation that they only capture undirected
higher-order interactions while in real-world scenarios, often there is a need
to introduce the direction of simplices, extending the popular notion of
direction of edges. On graphs and networks the Magnetic Laplacian, a special
case of Connection Laplacian, is becoming a popular operator to treat edge
directionality. Here we tackle the challenge of treating directional simplicial
complexes by formulating Higher-order Connection Laplacians taking into account
the configurations induced by the simplices' directions. Specifically, we
define all the Connection Laplacians of directed simplicial complexes of
dimension two and we discuss the induced higher-order diffusion dynamics by
considering instructive synthetic examples of simplicial complexes. The
proposed higher-order diffusion processes can be adopted in real scenarios when
we want to consider higher-order diffusion displaying non-trivial frustration
effects due to conflicting directionalities of the incident simplices.
