KP integrability of triple Hodge integrals, I. From Givental group to hierarchy symmetries

In this paper, we investigate a relation between the Givental group of rank one and the Heisenberg–Virasoro symmetry group of the KP hierarchy. We prove, that only a two-parameter family of the Givental operators can be identified with elements of the Heisenberg–Virasoro symmetry group. This family describes triple Hodge integrals satisfying the Calabi–Yau condition. Using the identification of the elements of two groups we prove that the generating function of triple Hodge integrals satisfying the Calabi–Yau condition and its $\Theta$-version are tau-functions of the KP hierarchy. This generalizes the result of Kazarian on KP integrability in the case of linear Hodge integrals.

2010 Mathematics Subject Classification

14N10, 14N35, 37K10, 81R10

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This work was supported by the Institute for Basic Science (IBS-R003-D1) and by RFBR grant 18-01-00926.

Received 3 November 2020

Accepted 12 May 2021

Published 15 July 2021