Higher order constraints for the ($\beta$-deformed) Hermitian matrix models

TL;DR

This paper develops higher order derivative operators for ($eta$-deformed) Hermitian matrix models, deriving and reducing their constraints to Virasoro form, and proving the Itoyama-Matsuo conjecture, linking to $W$-representations.

Contribution

It introduces new higher order derivative operators for ($eta$-deformed) models and proves their reducibility to Virasoro constraints, confirming the Itoyama-Matsuo conjecture.

Findings

Higher order constraints are reducible to Virasoro constraints.

The Itoyama-Matsuo conjecture is proved.

Constraint operators relate to $W$-operators through variable rescaling.

Abstract

We construct the ($\beta$-deformed) higher order total derivative operators and analyze their remarkable properties. In terms of these operators, we derive the higher order constraints for the ($\beta$-deformed) Hermitian matrix models. We prove that these ($\beta$-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 ($\beta$-deformed) partition function hierarchies in the literature.