Quantum Airy Structures and Matrix Models: a supercurrent approach

Abstract

We develop a super-current formulation of $\mathcal{N}=1$ super-Virasoro constraints for external-source models. By assigning bosonic and fermionic monodromies independently, we unify the NS-NS, NS-R, R-NS, and R-R sectors. For each, a super-Miwa transformation represents the projected super energy-momentum tensor as a differential operator in spectral variables. Dilaton shifts yield Super Quantum Airy Structures in the NS-NS and R-NS sectors, whereas the R-R and NS-R sectors require an additional odd coordinate. Our principal result is an exact similarity transformation relating the NS-NS and R-R differential constraints. The intertwining prefactor combines a bosonic determinant, a cubic Airy weight, and a Grassmann exponential kernel defined by the difference of Neveu--Schwarz and Ramond fermionic propagators. These results establish a concrete candidate system of Ward identities for a supersymmetric extension of the Kontsevich model, delineating the precise algebraic and analytic properties of the measure and fermionic kernel required for its potential matrix-model realization.