Extended Heun Hierarchy in Quantum Seiberg-Witten Geometry

We investigate the quantum geometry of the Seiberg-Witten curve for $\mathcal{N}=2$, $\mathrm{SU(2)}^n$ linear quiver gauge theories. By applying the Weyl quantization prescription to the algebraic curve, we derive the corresponding second-order differential equation and demonstrate that it is isomorphic to the Extended Heun Equation with $n+3$ regular singular points. The physical parameters of the gauge theory are linked to the canonical coefficients of the Heun equation via a polynomial representation of the Seiberg-Witten curve. This framework provides the necessary mathematical foundation to apply non-perturbative gauge-theoretic techniques, such as instanton counting, to spectral problems in gravitational physics, most notably for higher-dimensional black holes.