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b -Hurwitz numbers from refined topological recursion.

Nitin Kumar Chidambaram1,2, Maciej Dołęga3, Kento Osuga4,5

  • 1School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Rd, EH9 3FD Edinburgh, U.K.

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This study shows that G-weighted b-Hurwitz numbers are calculated using refined topological recursion on a rational spectral curve. This finding applies to various enumeration problems, including maps and beta-ensembles.

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Area of Science:

  • Combinatorics
  • Mathematical Physics
  • Algebraic Geometry

Background:

  • Hurwitz numbers enumerate branched coverings of surfaces.
  • Topological recursion is a powerful tool for studying random matrix theory and enumerative geometry.
  • Rational spectral curves provide a framework for analyzing certain combinatorial quantities.

Purpose of the Study:

  • To establish a connection between G-weighted b-Hurwitz numbers and refined topological recursion.
  • To demonstrate that the b-Hurwitz generating function analytically continues to a rational curve.
  • To apply these findings to enumerate maps and analyze beta-ensembles.

Main Methods:

  • Utilizing refined topological recursion on a rational spectral curve.
  • Developing a framework for G-weighted b-Hurwitz numbers.
  • Analytical continuation of generating functions.

Main Results:

  • Proved that single G-weighted b-Hurwitz numbers with internal faces are computed by refined topological recursion.
  • Showed that the b-Hurwitz generating function analytically continues to a rational curve.
  • Demonstrated that this framework encompasses b-monotone Hurwitz numbers, map enumeration, and beta-ensembles.

Conclusions:

  • Refined topological recursion provides a computational framework for G-weighted b-Hurwitz numbers.
  • The study unifies diverse combinatorial objects under a single theoretical umbrella.
  • The results have direct applications in random matrix theory and enumerative geometry.