Higher order constraints for the (${\hat{I}}^2$-deformed) Hermitian matrix models

Abstract

We construct the (${\hat{I}}^2$-deformed) higher order total derivative operators and analyze their remarkable properties. In terms of these operators, we derive the higher order constraints for the (${\hat{I}}^2$-deformed) Hermitian matrix models. We prove that these (${\hat{I}}^2$-deformed) higher order constraints are reducible to the Virasoro constraints. Meanwhile, the Itoyama-Matsuo conjecture for the constraints of the Hermitian matrix model is proved. We also find that through rescaling variable transformations, two sets of the constraint operators become the $W$-operators of $W$-representations for the (${\hat{I}}^2$-deformed) partition function hierarchies in the literature.

16 pages