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

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.

43 pages, 7 figures

References in corpus (5)

• Vanishing of Beta Function of Non Commutative ${\Phi }_4^4$ Theory to all orders
• Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability
• Large deviations and stochastic calculus for large random matrices
• Monotone Hurwitz numbers and the HCIZ integral II
• On the Complex Asymptotics of the HCIZ and BGW Integrals

Cited by in corpus (2)

• Universality of the Wigner-Gurau limit for random tensors
• The loop equations for noncommutative geometries on quivers