Decoupling Limit of Quiver Theories and the Angular Spectra of Extreme C-metrics

We investigate the angular eigenvalue problem of the extreme charged C-metric.

In the extreme limit ($Q \to M$), the governing differential equation degenerates from a Fuchsian equation with five regular singular points into a Confluent Extended Heun Equation.

To evaluate the angular spectrum analytically, we formulate a decoupling limit within the dual four-dimensional $\mathcal{N}=2$, $\mathrm{SU(2)}\times \mathrm{SU(2)}$ linear quiver gauge theory.

Within this framework, we derive the parameter dictionary and renormalized Matone relations, which absorb the macroscopic residue shifts induced by the singularity fusion.

Based on the regular boundary conditions of the angular equation, we utilize the instanton counting method to establish an algebraic quantization condition, yielding angular eigenvalues consistent with numerical results.