Around spin Hurwitz numbers

• Published in: Letters in mathematical physics Vol. 111; no. 5
• Main Authors: Mironov, A. D., Morozov, A., Natanzon, S. M., Orlov, A. Yu
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-10-2021; Springer Nature B.V
• Subjects: Boris Dubrovin Memorial Issue; Complex Systems; Fermions; Geometry; Group Theory and Generalizations; Hierarchies; Mathematical and Computational Physics; Numbers; Physics; Physics and Astronomy; Solitary waves; Theoretical; Thermodynamics; 05E05; Spin Hurwitz numbers; Q Schur functions; 05A15; 37K10; Integrable hierarchies
• ISSN: 0377-9017; 1573-0530
• DOI: 10.1007/s11005-021-01457-3

We present a review of the spin Hurwitz numbers, which count the ramified coverings with spin structures. It is known that they are related to the characters of the Sergeev group and to the
Q
Schur functions. This allows one to put the whole story into the context of matrix models and integrable hierarchies. The generating functions of the spin Hurwitz numbers
τ
±
are hypergeometric
τ
-functions of the BKP integrable hierarchy; we present their fermionic realization. The cut-and-join equation in the form of a heat equation is written down. We explain, how a special
d
-soliton
τ
-functions of KdV and Veselov–Novikov hierarchies generate the spin Hurwitz numbers
H
±
Γ
d
b
and
H
±
Γ
d
b
,
Δ
. We present the well-known Kontsevich matrix integral as the BKP
τ
-function in the form of special neutral fermion vacuum expectation values (few different ones). We also explain how to rewrite certain BKP
τ
-functions (including the Kontsevich one) as the hypergeometric BKP
τ
-functions using certain relations between the projective Schur functions.