  A fermionic representation is given for all the quantities entering in the
generating function approach to weighted Hurwitz numbers and topological
recursion. This includes: KP and 2D Toda $\tau$-functions of hypergeometric
type, which serve as generating functions for weighted single and double
Hurwitz numbers; the Baker function, which is expanded in an adapted basis
obtained by applying the same dressing transformation to all vacuum basis
elements; the multipair correlators and the multicurrent correlators.
Multiplicative and differential recursion relations are deduced for the adapted
bases and their duals, and a Christoffel-Darboux type formula is derived for
the pair correlator. The quantum and classical spectral curves linking this
theory with the topological recursion programme are derived, as well as the
generalized cut and join equations. The results are detailed for four special
cases: the simple single and double Hurwitz numbers, the weakly monotone case,
corresponding to signed enumeration of coverings, the strongly monotone case,
corresponding to Belyi curves and the simplest version of quantum weighted
Hurwitz numbers.
