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 in terms trace invariants two external tensors. This gives rise natural generalizations monotone double Hurwitz numbers, which count certain families constellations. find an expression these numbers simple thereby also providing expressions for arbitrary genus single ones. give interpretation different combinatorial quantities at play enumeration nodal surfaces. In particular, our is shown isomorphism classes branched coverings bouquet $D$ 2-spheres that touch one common non-branch node.