Constellations and -functions for rationally weighted Hurwitz numbers

  • John Harnad

    Université de Montréal and Concordia University, Montreal, Canada
  • Boris Runov

    St. Petersburg State University, Russia
Constellations and $\tau$-functions for rationally weighted Hurwitz numbers cover
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Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the 2D Toda -functions of hypergeometric type serving as generating functions for weighted Hurwitz numbers in the case of polynomial weight generating functions. The product over all vertex and edge weights of a given weighted constellation, summed over all configurations, reproduces the -function. In the present work, this is generalized to constellations in which the weighting parameters are determined by a rational weight generating function. The associated -function may be expressed as a sum over the weights of doubly labelled weighted constellations, with two types of weighting parameters associated to each equivalence class of branched coverings. The double labelling of branch points, referred to as “colour” and “flavour” indices, is required by the fact that, in the Taylor expansion of the weight generating function, a particular colour from amongst the denominator parameters may appear multiply, and the flavour labels indicate this multiplicity.

Cite this article

John Harnad, Boris Runov, Constellations and -functions for rationally weighted Hurwitz numbers. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 8 (2021), no. 1, pp. 119–158

DOI 10.4171/AIHPD/104