The tensor HCIZ integral and its relation to enumerative geometry and free probability

 

In a recent paper with Collins and Gurau, we introduced a generalization of the celebrated Harish-Chandra—Itzykson—Zuber (HCIZ) integral to tensors, which finds applications for instance to random geometry, quantum information, and the two topics I will focus on in this talk: enumerative geometry and free probability.

♦ This integral can be shown to be a generating function for a generalization of Hurwitz numbers involving certain transitive factorizations of multiplets of permutations. For the original HCIZ integral, these numbers are known to count certain connected ramified coverings of the 2-sphere, and the question thus naturally arises of what our generalization enumerates. I will give two answers, one in terms of ramified coverings of a collection of 2-spheres that touch at one common node, the other in terms of coverings of the 3-sphere branched over certain subgraphs of the complete bipartite graph K_3,p.