Factorized Quantum Curves and Minuscule Vertices in 3D Duality Cascades with FI Parameters

In the study of duality cascades in three-dimensional gauge theories without FI parameters, an important role is played by a fundamental domain whose vertices correspond to brane configurations with vanishing relative ranks. Through the Fermi gas formalism, such brane configurations are known to be represented by factorized quantum curves. In this paper, we show that this factorized description extends naturally to quantum curves associated with del Pezzo geometries possessing exceptional Weyl-group symmetries in the presence of FI parameters. We find that the vertices of the fundamental domain corresponding to the weights of minuscule representations are realized as factorized quantum curves built from canonical operators interpreted as 5-branes dressed with FI parameters. This reveals an unexpected relation between factorized quantum curves and minuscule representations, providing a physical realization of these vertices in terms of ``extremal’’ brane configurations.