2015/01/06 by Natanzon, Sergei, Orlov, Alexander
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1501.01283
We consider d-fold branched coverings of the projective plane \mathbbRP2 and show that the hypergeometric tau function of the BKP hierarchy of Kac and van de Leur is the generating function for weighted sums of the related Hurwitz numbers. In particular we get the \mathbbRP2 analogues of the \mathbbCP1 generating functions proposed by Okounkov and by Goulden and Jackson. Other examples are Hurwitz numbers weighted by the Hall-Littlewood and by the Macdonald polynomials. We also consider integrals of tau functions which generate projective Hurwitz numbers and Hurwitz numbers related to different Euler characteristics of the base Klein surfaces.