Topological Recursion for Orlov–Scherbin Tau Functions, and Constellations with Internal Faces

We study the correlators W g , n arising from Orlov–Scherbin 2-Toda tau functions with rational content-weight G ( z ), at arbitrary values of the two sets of time parameters. Combinatorially, they correspond to generating functions of weighted Hurwitz numbers and ( m ,  r )-factorisations of permut...

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Vydáno v:Communications in mathematical physics Ročník 405; číslo 8; s. 189
Hlavní autoři: Bonzom, Valentin, Chapuy, Guillaume, Charbonnier, Séverin, Garcia-Failde, Elba
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2024
Springer Nature B.V
Springer Verlag
Témata:
Boundaries
Classical and Quantum Gravitation
Combinatorial analysis
Combinatorics
Complex Systems
Geometry
Mathematical and Computational Physics
Mathematical Physics
Parameters
Permutations
Physics
Physics and Astronomy
Polynomials
Quantum Physics
Rational functions
Relativity Theory
Theoretical
Topology
Weighting functions
ISSN:0010-3616, 1432-0916
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Popis
Shrnutí:We study the correlators W g , n arising from Orlov–Scherbin 2-Toda tau functions with rational content-weight G ( z ), at arbitrary values of the two sets of time parameters. Combinatorially, they correspond to generating functions of weighted Hurwitz numbers and ( m ,  r )-factorisations of permutations. When the weight function is polynomial, they are generating functions of constellations on surfaces in which two full sets of degrees (black/white) are entirely controlled, and in which internal faces are allowed in addition to boundaries. We give the spectral curve (the “disk” function W 0 , 1 , and the “cylinder” function W 0 , 2 ) for this model, generalising Eynard’s solution of the 2-matrix model which corresponds to G ( z ) = 1 + z , by the addition of arbitrarily many free parameters. Our method relies both on the Albenque–Bouttier combinatorial proof of Eynard’s result by slice decompositions, which is strong enough to handle the polynomial case, and on algebraic arguments. Building on this, we establish the topological recursion (TR) for the model. Our proof relies on the fact that TR is already known at time zero (or, combinatorially, when the underlying graphs have only boundaries, and no internal faces) by work of Bychkov–Dunin-Barkowski–Kazarian–Shadrin (or Alexandrov–Chapuy–Eynard–Harnad for the polynomial case), and on the general idea of deformation of spectral curves due to Eynard and Orantin, which we make explicit in this case. As a result of TR, we obtain strong structure results for all fixed-genus generating functions. Our techniques also cover the case where G ( z ) is a rational function times an exponential (containing in particular the case of classical Hurwitz numbers).
Bibliografie:ObjectType-Article-1
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content type line 14
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-024-05048-w