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

TL;DR

This paper extends the Harish-Chandra-Itzykson-Zuber integral to tensors, introduces generalized monotone Hurwitz numbers, and connects them to enumeration of branched coverings and nodal surfaces, broadening mathematical understanding of these structures.

Contribution

It provides a tensor generalization of the integral, relates new monotone Hurwitz numbers to existing ones, and offers combinatorial interpretations involving nodal surface enumeration.

Findings

Derived expressions for generalized monotone Hurwitz numbers in terms of simpler cases.

Connected tensor integrals to enumeration of branched coverings of spheres.

Provided combinatorial interpretations involving nodal surfaces and isomorphism classes.

Abstract

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 simple 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. In particular, our generalization of Hurwitz numbers is shown to enumerate certain isomorphism classes of branched coverings of a bouquet of $D$ 2-spheres that touch at one common non-branch node.