Schur \(\mathrm{Q}\)-polynomials and Kontsevich-Witten tau function (Q6615574)

Kontsevich-Witten tau function is the generating function for intersection numbers of certain tautological classes on moduli spaces of stable curves. It is also the partition function in the two dimensional pure topological quantum gravity. It was conjectured by Witten and proved by Kontsevich that this function is a tau function of the Korteweg-de Vries (KdV) hierarchy (cf. [\textit{M. Kontsevich}, Commun. Math. Phys. 147, No. 1, 1--23 (1992; Zbl 0756.35081); \textit{E. Witten}, in: Surveys in differential geometry. Vol. I: Proceedings of the conference on geometry and topology, held at Harvard University, Cambridge, MA, USA, April 27-29, 1990. Providence, RI: American Mathematical Society; Bethlehem, PA: Lehigh University. 243--310 (1991; Zbl 0757.53049)]). Using matrix model, Mironov and Morozov recently gave a formula which represents Kontsevich-Witten tau function as a linear expansion of Schur \(Q\)-polynomials (also known as \(Q\)-functions). Such functions provide polynomial solutions to BKP hierarchy. The aim of this paper is to show directly that the \(Q\)-polynomial expansion in this formula satisfies the Virasoro constraints, only using properties of \(Q\)-polynomials. The advantage of this proof using Virasoro constraints is that it does not need matrix model. They also give a proof for Alexandrov's conjecture that Kontsevich-Witten tau function is a hypergeometric tau function of the BKP hierarchy after re-scaling. The paper is supported by some appendices about elementary identities which are needed in the proof of combinatorial identities.