Quantum spectral curve for $(q,t)$-matrix model and XXZ spin chain

In the large $N$ limit the properties of matrix models are generally governed by their spectral curves. For some models there is a natural deformation of this setup to the more general Nekrasov–Shatashvili limit. This deformation corresponds to quantization of the spectral curve, which can be interpreted as an operator acting on the wave function of a finite dimensional integrable system in separated variables. We consider the realisation of these ideas in the case of the $(q,t)$-matrix model (also known as the $q$-deformed beta-ensemble), where the quantum spectral curve turns out to be certain difference operator and the corresponding integrable system is the closed inhomogeneous XXZ spin chain. We also discuss the relation between our results, AGT duality and spectral duality in integrable systems, CFT and gauge theories.