Spectral curves for hypergeometric Hurwitz numbers

Abstract

We consider multi-matrix models that are generating functions for the numbers of branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,\dots,n$, (generalized Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profiles at two points, $z_1$ and $z_n$. Ramifications at other $n-2$ points enter the sum with the length of the profile at $z_2$ and with the total length of profiles at the remaining $n-3$ points. We find the spectral curve of the model for $n=5$ using the loop equation technique for the above generating function represented as a chain of Hermitian matrices with a nearest-neighbor interaction of the type tr$M_iM_{i+1}^{-1}$. The obtained spectral curve is algebraic and provides all necessary ingredients for the topological recursion procedure producing all-genus terms of the asymptotic expansion of our model in $1/N^2$. We discuss braid-group symmetries of our model and perspectives of the proposed method.