Two $$\beta $$-ensemble realization of $$\beta $$-deformed WLZZ models
А. Миронов, A. Oreshina, A. Popolitov
Abstract
Abstract We consider a two $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -ensemble realization of the series of $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -deformed WLZZ matrix models. We demonstrate that such a realization involves $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -deformed Harish-Chandra–Itzykson–Zuber integrals, one of them providing a coupling to the external field. We also construct Ward identities in the corresponding two $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -ensemble model, which requires a set of identities for partition function of the one $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -ensemble in the external field, and a set of identities for the $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -deformed Itzykson-Zuber integral. These both sets of identities are formulated in terms of the Dunkl operators.