Monotone orbifold Hurwitz numbers

Norman Do, Maksim Karev

Research output: Contribution to journalArticleResearch


In general, Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. In this paper, we initiate the study of monotone orbifold Hurwitz numbers. These are simultaneously variations of the orbifold case and generalisations of the monotone case, both of which have been previously studied in the literature. We derive a cut-and-join recursion for monotone orbifold Hurwitz numbers, determine a quantum curve governing their wave function, and state an explicit conjecture relating them to topological recursion.
Original languageEnglish
Pages (from-to)40-69
Number of pages30
JournalZapiski Nauchnykh Seminarov POMI
Publication statusPublished - 2016


  • Hurwitz numbers
  • monotone Hurwitz numbers
  • monodromy groups
  • topological recursion
  • quantum curve

Cite this