TL;DR
This paper extends the theory of weighted Hurwitz numbers to multispecies cases, connecting enumerative geometry with combinatorics and exploring quantum weighted variants.
Contribution
It introduces multispecies weighted Hurwitz numbers and links them to hypergeometric 2D Toda tau-functions, enriching both geometric and combinatorial frameworks.
Findings
Extended hypergeometric 2D Toda tau-functions to multispecies cases
Established geometric and combinatorial interpretations of multispecies weighted Hurwitz numbers
Analyzed quantum weighted Hurwitz numbers in detail
Abstract
The construction of hypergeometric Toda -functions as generating functions for weighted Hurwitz numbers is extended to multispecies families. Both the enumerative geometrical significance of multispecies weighted Hurwitz numbers, as weighted enumerations of branched coverings of the Riemann sphere, and their combinatorial significance in terms of weighted paths in the Cayley graph of are derived. The particular case of multispecies quantum weighted Hurwitz numbers is studied in detail.
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