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SUMMARY:Generating functions for weighted Hurwitz numbers and topological
recursion
DTSTART;VALUE=DATE-TIME:20170523T123000Z
DTEND;VALUE=DATE-TIME:20170523T133000Z
DTSTAMP;VALUE=DATE-TIME:20210623T122724Z
UID:indico-event-2349@indico.math.cnrs.fr
DESCRIPTION:\nA brief survey will be given of the use of KP and 2D-Toda ta
u functions of special “hypergeometric type” as generating functions
for weighted Hurwitz numbers (i.e. weighted enumerations of N-sheeted bra
nched coverings of the Riemann sphere\, or equivalently\, weighted pat
hs in the Cayley graph of the symmetric group S_N generated by transposi
tions). The weights depend on parametric families of auxiliary parameters\
, and consist of evaluations of basis elements of the algebra of symmetric
functions of the latter. An alternative generating function is provid
ed by certain correlation functions W_{n\,g}(x_1\,. …\, x_n) depending
on a pair of integers that play a role analogous to the multidifferenti
als in the Topological Recursion approach to intersection theory on moduli
spaced of marked Riemann surfaces. As in that case\, an associated inva
riant classical and quantum “spectral curve” is derived and a set of r
ecursion relations that determine the general term quadratically in terms
of finite sums over preceding ones. \n\n \n\n\nExamples include: 1) the
“simple” (double or single) Hurwitz numbers studied originally by Okou
nkov and Pandharipande\, 2) The case of "Belyi curves”\, having just thr
ee branch points\, one of them weighted\, and the related “dessins d’e
nfants”\; 3) The “weakly monotonic” paths in the Cayley graph\, for
which the generating tau function is the Itzykson-Zuber-Harish-Chandra int
egral and (if time permits) 4) The case of simple "quantum Hurwitz numbers
"\, in which the weighting is shown to coincide with that of a quantum Bos
e-Einstein gas. (Partly based on joint work with M. Guay-Paquet\, A. Orl
ov\, B. Eynard\, A. Alexandrov and G. Chapuy)\n\nhttps://indico.math.cnrs.
fr/event/2349/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/2349/
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