Related papers: Fermionic approach to weighted Hurwitz numbers and…
The KP and 2D Toda tau-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted…
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…
Double Hurwitz numbers enumerating weighted $n$-sheeted branched coverings of the Riemann sphere or, equivalently, weighted paths in the Cayley graph of $S_n$ generated by transpositions are determined by an associated weight generating…
This is an overview of recent results on the use of 2D Toda $\tau$-functions as generating functions for multiparametric families of weighted Hurwitz numbers. The Bose-Fermi equivalence composed with the characteristic map provides an…
The KP $\tau$-function of hypergeometric type serving as generating function for quantum weighted Hurwitz numbers is used to compute the Baker function and the corresponding adapted basis elements, expressed as absolutely convergent Laurent…
Multicurrent correlators associated to KP $\tau$-functions of hypergeometric type are used as generating functions for weighted Hurwitz numbers. These are expressed as formal Taylor series and used to compute generic, simple, rational and…
We generalize the determinant representation of the KP $\tau$ functions to the case of the 2D Toda $\tau$ functions. The generating functions for the weighted Hurwitz numbers are a parametric family of 2D Toda $\tau$ functions; for which we…
The construction of hypergeometric 2D Toda $\tau$-functions as generating functions for quantum Hurwitz numbers is extended here to multispecies families. Both the enumerative geometrical significance of these multispecies quantum Hurwitz…
The construction of hypergeometric $2D$ Toda $\tau$-functions as generating functions for weighted Hurwitz numbers is extended to multispecies families. Both the enumerative geometrical significance of multispecies weighted Hurwitz numbers,…
A multiparametric family of 2D Toda $\tau$-functions of hypergeometric type is shown to provide generating functions for composite, signed Hurwitz numbers that enumerate certain classes of branched coverings of the Riemann sphere and paths…
We derive an explicit formula for the connected $(n,m)$-point functions associated to an arbitrary diagonal tau-function $\tau_f(\boldsymbol{t}^+,\boldsymbol{t}^-)$ of the 2d Toda lattice hierarchy using fermionic computations and the…
We present the multi-matrix models that are the generating functions for 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,…
Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the $2$D Toda $\tau$-functions of hypergeometric type serving as…
The quantum spectral curve equation associated to KP $\tau$-functions of hypergeometric type serving as generating functions for rationally weighted Hurwitz numbers is solved by generalized hypergeometric series. The basis elements spanning…
We consider weighted double Hurwitz numbers, with the weight given by arbitrary rational function times an exponent of the completed cycles. Both special singularities are arbitrary, with the lengths of cycles controlled by formal…
We consider $d$-fold branched coverings of the projective plane $\mathbb{RP}^2$ and show that the hypergeometric tau function of the BKP hierarchy of Kac and van de Leur is the generating function for weighted sums of the related Hurwitz…
We present a study of real Hurwitz numbers enumerating a special kind of real meromorphic functions, which we call simple framed purely real functions. We deduce partial differential equations of cut-and-join type for generating functions…
The basis elements spanning the Sato Grassmannian element corresponding to the KP $\tau$-function that serves as generating function for rationally weighted Hurwitz numbers are shown to be Meijer $G$-functions. Using their Mellin-Barnes…
In this paper, we discuss the properties of the generating functions of spin Hurwitz numbers. In particular, for spin Hurwitz numbers with arbitrary ramification profiles, we construct the weighed sums which are given by Orlov's…
Hurwitz theory provides a large variety of enumerative problems related to algebraic geometry, mathematical physics, and combinatorics. We give a general framework to approach the large genus asymptotics of Hurwitz theory using only…