Fermionic Approach to Weighted Hurwitz Numbers and Topological Recursion
Abstract
A fermionic representation is given for all the quantities entering in the generating function approach to weighted Hurwitz numbers and topological recursion. This includes: KP and 2 D Toda {τ} functions of hypergeometric type, which serve as generating functions for weighted single and double Hurwitz numbers; the Baker function, which is expanded in an adapted basis obtained by applying the same dressing transformation to all vacuum basis elements; the multipair correlators and the multicurrent correlators. Multiplicative recursion relations and a linear differential system are deduced for the adapted bases and their duals, and a ChristoffelDarboux type formula is derived for the pair correlator. The quantum and classical spectral curves linking this theory with the topological recursion program are derived, as well as the generalized cutandjoin equations. The results are detailed for four special cases: the simple single and double Hurwitz numbers, the weakly monotone case, corresponding to signed enumeration of coverings, the strongly monotone case, corresponding to Belyi curves and the simplest version of quantum weighted Hurwitz numbers.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 June 2018
 DOI:
 10.1007/s0022001730659
 arXiv:
 arXiv:1706.00958
 Bibcode:
 2018CMaPh.360..777A
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 57 pages. Details added to examples