BKP and projective Hurwitz numbers

We consider d -fold branched coverings of the 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 particular, we get the RP 2 analogues of the CP 1 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 Hurwitz numbers related to base surfaces with arbitrary Euler characteristics E , in particular projective Hurwitz numbers E = 1 .