The tensor Harish-Chandra–Itzykson–Zuber integral I: Weingarten calculus and a generalization of monotone Hurwitz numbers

We study a generalization of the Harish-Chandra-Itzykson-Zuber integral to tensors and its expansion over trace-invariants of the two external tensors. This gives rise to natural generalizations of monotone double Hurwitz numbers, which count certain families of constellations. We find an expression of these numbers in terms of monotone Hurwitz numbers, thereby also providing expressions for monotone double Hurwitz numbers of arbitrary genus in terms of the single ones. We give an interpretation of the different combinatorial quantities at play in terms of enumeration of nodal surfaces.