Double Quiver Gauge Theory and BPS/CFT Correspondence

We provide a formalism using the $q$-Cartan matrix to compute the instanton partition function of quiver gauge theory on various manifolds. Applying this formalism to eight dimensional setups, we introduce the notion of double quiver gauge theory characterized by a pair of quivers. We also explore the BPS/CFT correspondence in eight dimensions based on the $q$-Cartan matrix formalism.