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Article

BKP and projective Hurwitz numbers

Letters in Mathematical Physics. 2017. Vol. 107. P. 1065-1109.

We consider $d$-fold branch covering of the real projective plane $RP^2$ 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 paticular we get the $RP^2$ analogue of the both $CP^1$ generating functions proposed by A.Okounkov and by Goulden-Jackson. Other examples are Hurwitz numbers weighted by Hall-Littlewood and by Macdonald polynomials. We also consider integrals of tau functions which generate projective Hurwitz numbers.