On constraints of the (β-deformed) partition functions with W-representations

We investigate the constraints of the (β-deformed) partition functions with W-representations in the literature, which can be realized by the interpolating two-matrix models (two β-ensembles). For the non-deformed partition functions, we construct their Virasoro and higher order constraints using the total derivative operators. For the β-deformed partition functions, we derive their Virasoro constraints in terms of the constructed Virasoro operators. Meanwhile, we obtain a new realization of the Virasoro algebra in terms of the Dunkl operators and the conserved quantities in the rational Calogero model.