Weighted Hurwitz numbers and topological recursion: An overview

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Title
Weighted Hurwitz numbers and topological recursion: An overview
Author(s)
Alexandrov A.; Chapuy G.; Eynard B.; Harnad J.
Publication Date
2018-08
Journal
Journal of Mathematical Physics, v.59, no.8, pp.081102 -
Publisher
American Institute of Physics
Abstract
Multiparametric families of hypergeometric τ-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 of the weighting is given in terms of constellations mapped onto the covering surface. The theory is placed within the framework of topological recursion, with the Baker function at t = 0 shown to satisfy the quantum spectral curve equation, whose classical limit is rational. A basis for the space of formal power series in the spectral variable is generated that is adapted to the Grassmannian element associated with the τ-function. Multicurrent correlators are defined in terms of the τ-function and shown to provide an alternative generating function for weighted Hurwitz numbers. Fermionic vacuum state expectation value representations are provided for the adapted bases, pair correlators, and multicurrent correlators. Choosing the weight generating function as a polynomial and restricting the number of nonzero second KP flow parameters in the Toda τ-function to be finite implies a finite rank covariant derivative equation with rational coefficients satisfied by a finite window of adapted basis elements. The pair correlator is shown to provide a Christoffel-Darboux type finite rank integrable kernel, and the WKB series coefficients of the associated adjoint system are computed recursively, leading to topological recursion relations for the generators of the weighted Hurwitz numbers. © 2018 Author(s)
URI
https://pr.ibs.re.kr/handle/8788114/4932
ISSN
0022-2488
Appears in Collections:
Center for Geometry and Physics(기하학 수리물리 연구단) > Journal Papers (저널논문)
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