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

Abstract We construct the ( $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -deformed) higher order total derivative operators and analyze their remarkable properties. In terms of these operators, we derive the higher order constraints for the ( $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -deformed) Hermitian matrix models. We prove that these ( $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -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 ( $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -deformed) partition function hierarchies in the literature.