title: Gauge Origami and BPS $qq$-characters

abstract: Gauge origami is a generalized supersymmetric quiver gauge theory where intersecting D-branes appear. The simplest setup is the gauge origami system in $\mathbb{C}^4$, where D2/D4/D6/D8 branes are present. Instantons are D0-branes bound to the D(2p)-branes, and the partition functions are the flavored Witten indices of the supersymmetric quantum mechanics of the D0-branes. We demonstrate that the contour integral formulas of the partition functions have free field interpretations, leading to the operator formalism of $qq$-characters, which we generally call BPS $qq$-characters. On one hand, the $qq$-characters of D2-D0 and D4-D0 bound states correspond to screening charges and generators of the affine quiver W-algebra, respectively. On the other hand, the $qq$-characters of D6-D0 and D8-D0 bound states give new types of $qq$-characters, where the monomial terms are characterized by plane partitions and solid partitions. The D4, D6, D8 $qq$-characters automatically reproduce the partition function of the spiked instanton, tetrahedron instanton, and the magnificent four, which eventually establish the BPS/CFT correspondence. We also discuss the relation with quantum toroidal algebras and generalizations to toric Calabi-Yau four-folds. This talk is based on arXiv: 2310.08545, 2404.17061, 2411.01987