BKP and projective Hurwitz numbers |
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Authors: | Sergey M. Natanzon Aleksandr Yu. Orlov |
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Affiliation: | 1.National Research University Higher School of Economics,Moscow,Russia;2.Institute for Theoretical and Experimental Physics,Moscow,Russia;3.Institute of Oceanology,Moscow,Russia;4.International Laboratory of Representation Theory and Mathematical Physics,National Research University Higher School of Economics,Moscow,Russia |
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Abstract: | We consider d-fold branched coverings of the projective plane (mathbb {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 (mathbb {RP}^2) analogues of the (mathbb {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 (textsc {e}), in particular projective Hurwitz numbers (textsc {e}=1). |
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