KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions

Abstract

In this paper, we continue our investigation of the triple Hodge integrals satisfying the Calabi-Yau condition. For the tau-functions, which generate these integrals, we derive the complete families of the Heisenberg-Virasoro constraints. We also construct several equivalent versions of the cut-and-join operators. These operators describe the algebraic version of topological recursion. For the specific values of parameters associated with the KdV reduction, we prove that these tau-functions are equal to the generating functions of intersection numbers of ψ and κ classes. We interpret this relation as a symplectic invariance of the Chekhov-Eynard-Orantin topological recursion and prove this recursion for the general Θ-case.

GUE via Frobenius Manifolds. I. From Matrix Gravity to Topological Gravity and Back

“…But, the recent studies [26,28,30,68] all together show that one can start with Witten's topological gravity and come back to the matrix gravity without a deformation theory, again at least in the higher genera. More precisely, we first go to the special cubic Hodge partition function by a space/time duality [18,67,68] (see also [2,3]) (in [68] this is revealed by the Hodge-BGW correspondence), and then go to the so-called modified even GUE partition function by the Hodge-GUE correspondence [26,28], and finally back to the even GUE partition function via a product formula [30], again at least to the higher genera in jets. (We note that the genus zero parts for the above-mentioned models are relatively easy, so for us the non-trivial things are in higher genera.)…”

On the Hodge-BGW correspondence

“…Recently, they play important roles in the Hodge-GUE correspondence [15,16,17], which implies the ELSV-like (cf. [22]) formula for even GUE correlators [7,24]; see [2,3,51] for interesting connections to the KP hierarchy and 2D Toda lattice. Denote by…”

The ordered exponential representation of GKM using the W1+∞ operator

“…Examples about the conjugation of Heisenberg-Virasoro operators on constraints can be found in other articles such as [12] and [5].…”