  Multiparametric families of hypergeometric $\tau$-functions of KP or Toda
type serve as generating functions for weighted Hurwitz numbers, providing
weighted enumerations of branched covers of the Riemann sphere. A graphical
interpretation of the weighting is given in terms of constellations mapped onto
the covering surface. The theory is placed within the framework of topological
recursion, with the Baker function at ${\bf t} ={\bf 0}$ shown to satisfy the
quantum spectral curve equation, whose classical limit is rational. A basis for
the space of formal power series in the spectral variable is generated that is
adapted to the Grassmannian element associated to the $\tau$-function.
Multicurrent correlators are defined in terms of the $\tau$-function and shown
to provide an alternative generating function for weighted Hurwitz numbers.
Fermionic VEV representations are provided for the adapted bases, pair
correlators and multicurrent correlators. Choosing the weight generating
function as a polynomial, and restricting the number of nonzero "second" KP
flow parameters in the Toda $\tau$-function to be finite implies a finite rank
covariant derivative equation with rational coefficients satisfied by a finite
"window" of adapted basis elements. The pair correlator is shown to provide a
Christoffel-Darboux type finite rank integrable kernel, and the WKB series
coefficients of the associated adjoint system are computed recursively, leading
to topological recursion relations for the generators of the weighted Hurwitz
numbers.
