Generating functions for weighted Hurwitz numbers
| Autor: | John Harnad, Mathieu Guay-Paquet |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2014 |
| Předmět: |
High Energy Physics - Theory
FOS: Physical sciences Riemann sphere 01 natural sciences Classical limit Combinatorics symbols.namesake 0103 physical sciences FOS: Mathematics Mathematics - Combinatorics 0101 mathematics Mathematical Physics Mathematics Cayley graph Nonlinear Sciences - Exactly Solvable and Integrable Systems Riemann surface Probability (math.PR) 010102 general mathematics Generating function Statistical and Nonlinear Physics Graph theory Mathematical Physics (math-ph) High Energy Physics - Theory (hep-th) symbols Combinatorics (math.CO) 010307 mathematical physics Exactly Solvable and Integrable Systems (nlin.SI) Toda lattice Branch point Mathematics - Probability |
| Popis: | Double Hurwitz numbers enumerating weighted $n$-sheeted branched coverings of the Riemann sphere or, equivalently, weighted paths in the Cayley graph of $S_n$ generated by transpositions are determined by an associated weight generating function. A uniquely determined $1$-parameter family of 2D Toda $\tau$-functions of hypergeometric type is shown to consist of generating functions for such weighted Hurwitz numbers. Four classical cases are detailed, in which the weighting is uniform: Okounkov's double Hurwitz numbers, for which the ramification is simple at all but two specified branch points; the case of Belyi curves, with three branch points, two with specified profiles; the general case, with a specified number of branch points, two with fixed profiles, the rest constrained only by the genus; and the signed enumeration case, with sign determined by the parity of the number of branch points. Using the exponentiated quantum dilogarithm function as weight generator, three new types of weighted enumerations are introduced. These determine {\em quantum} Hurwitz numbers depending on a deformation parameter $q$. By suitable interpretation of $q$, the statistical mechanics of quantum weighted branched covers may be related to that of Bosonic gases. The standard double Hurwitz numbers are recovered in the classical limit. Comment: Final version, appearing in J. Math. Phys. 58, 083503 (2017) |
| Databáze: | OpenAIRE |
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