$N^{3/2}$ Scaling from $3d$ $\mathcal{N}=2$ Dualities: an Alternative Approach to Chiral Quivers

Abstract: We investigate families of 3d $\mathcal{N}=2$ chiral quiver gauge theories conjectured to be dual to M2-branes probing toric SE$_7$ singularities. Geometrically, these families correspond to toric diagrams without internal points. At the field theory level, the models are constructed via an un-higgsing procedure applied to non-chiral quivers. While the moduli space of these theories was shown to match M-theory expectations, determining the $N^{3/2}$ scaling of the free energy remained an open problem for over a decade, with positive results emerging only very recently. In this work, we address this challenge by reformulating the three-sphere partition function as a hyperbolic hypergeometric integral. Using exact integral identities, we show that the free energy reduces precisely to that of non-chiral quivers with chiral flavors, for which the $N^{3/2}$ scaling is already established. Physically, this mathematical identity corresponds to the equivalence of three-sphere partition functions under a generalization of Giveon-Kutasov duality to chiral quivers. Our results thus provide a large $N$ duality between the chiral quivers and non-chiral quivers with chiral flavors, confirming the $N^{3/2}$ scaling for the chiral quivers under study.