In this paper, we prove a conjecture of Mironov and Morozov which expresses Kontsevich-Witten tau-function as a linear combination of Schur Q-polynomials. We also prove Alexandrov's conjecture that Kontsevich-Witten tau-function is a hypergeometric tau-function of the BKP hierarchy after re-scaling.
In this paper, we prove a conjecture of Alexandrov that the generalized Brezin-Gross-Witten tau-functions are hypergeometric tau functions of BKP hierarchy after re-scaling. In particular, this shows that the original BGW tau-function, which has enumerative geometric interpretations, can be represented as a linear combination of Schur Q-polynomials with simple coefficients. * Research was partially supported by NSFC grants 11890662 and 11890660.
Preliminaries
Brézin-Gross-Witten tau-function and its generalizationsThe BGW model was originally proposed in [BG] and [GW] as a unitary matrix model. As it was proved in [N] that the partition function of this model, i.e. τ BGW , has the following geometric interpretation.Let M g,n be the moduli space of stable genus g curves C with n distinct smooth marked points x 1 , . . . , x n ∈ C. For each i ∈ {1, • • • , n}, there is a tautological class ψ i ∈ H 2 (M g,n ), which is the first Chern class of the line bundle on M g,n whose fiber at a point (C;is the cotangent space of C at the marked point x i . The Kontsevich-Witten tau function is the generating function of intersection numbers of these ψ-classes (c.f. [W] and [K]). To recover the BGW tau function, Norbury constructed a new family of classes Θ g,n ∈ H 4g−4+2n (M g,n ), which are well behaved with respect to pullbacks of gluing and forgetful maps among moduli spaces of stable curves. He proved that τ BGW is the generating function of intersection numbers of Θ g,n and ψ-classes.Following notations in [A20], define intersection numbers
Given a tau-function τ (t) of the BKP hierarchy satisfying τ (0) = 1, we discuss how to recover its BKP-affine coordinates on the isotropic Sato Grassmannian from BKP-wave function. Using this result, we formulate a type of Kac-Schwarz operators for τ (t) in terms of BKP-affine coordinates. As an example, we compute the affine coordinates of the BKP tau-function for spin single Hurwitz numbers with completed cycles, and find a pair of Kac-Schwarz operators (P, Q) satisfying [P, Q] = 1. By doing this, we obtain the quantum spectral curve for spin single Hurwitz numbers.
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