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## Hurwitz number from Feynman diagrams

Theoretical and Mathematical Physics. 2020. Vol. 204. No. 3. P. 1166-1194.
Natanzon S. M., Orlov A. Y.

To obtain a generating function of the most general form for Hurwitz numbers with arbitrary base surfaceand arbitrary ramification profiles, we consider a matrix model constructed according to a graph on anoriented connected surfaceΣwith no boundary. The vertices of this graph, called stars, are small discs,and the graph itself is a clean dessin d’enfants. We insert source matrices in boundary segments of eachdisc. Their product determines the monodromy matrix for a given star, whose spectrum is called the starspectrum. The surfaceΣconsists of glued maps, and each map corresponds to the product of randommatrices and source matrices. Wick pairing corresponds to gluing the surface from the set of maps, and anadditional insertion of a special tau function in the integration measure corresponds to gluing the M ̈obiusbands. We calculate the matrix integral as a Feynman power series in which the star spectrul data playthe role of coupling constants, and the coefficients of this power series are just Hurwitz numbers. Theydetermine the number of coverings ofΣ(or its extensions to a Klein surface obtained by inserting M ̈obiusbands) for any given set of ramification profiles at the vertices of the graph. We focus on a combinatorialdescription of the matrix integral. The Hurwitz number is equal to number of Feynman diagrams of acertain type divided by the order of the automorphism group of the graph