2026/08/03 by Fridolin Melong, Raimar Wulkenhaar
Physics and Astronomy · Mathematics · #math-ph #math.AG #math.MP #math.RT
arxiv created 2026/08/03 · arxiv updated 2026/08/04
This work investigates the q-deformation of (r,s)-Airy structures and their realization via q-difference operators, providing a bridge between quantum spectral curves and integrable systems. We construct an all-order q-WKB solution for the matrix systems associated with the q-quantized curve Eq(x,y)=0. We demonstrate that the resulting non-perturbative connected q-amplitudes satisfy a set of shifted q-loop equations, which can be interpreted as the Ward identities of a q-deformed W(\mathfrakglr) algebra. Our main result provides a rigorous classification of admissible (r,s,q) pairs and q-Casimir configurations that satisfy the q-topological type property. This ensures that the semi-classical expansion is uniquely governed by the q-topological recursion, offering new insights into the q-quantization of mirror curves and their underlying algebraic structures.