The Geometry of Conformal Quivers

We study the geometry of the gauged quiver quantum mechanics realizing $D(2,1;0)$ superconformal symmetry. These models arise as effective descriptions of multi-centered D-brane systems in type II Calabi-Yau compactifications, in the $AdS_2$ scaling limit, and they describe the near-horizon microscopics of multi-centered BPS black holes in the four dimensional $\mathcal{N}=2$ supergravity. We work in the gauged sigma model formulation with off-shell $(4,4,0)$ multiplets, which allows us to introduce a coordinate system adapted to the complex structure of the gauged sigma model target space, that assigns a radial-angular frame to each quiver node in the physical configuration space. It provides an explicit decomposition of the sigma model quiver metric into distinct sectors governed by couplings between each node. We compute the full target space metric for 2-node and 3-node quiver configurations, and generalize to symmetric $N$-node crystal quivers. Finally we implement Gaiotto-Simons-Strominger-Yin's $D(2,1;0)$ index for this system and show that the corresponding gauged superconformal index -- at least in a rich family of physical examples -- provides a new tool for resolving the conical singularity complications generically present in the ungauged superconformal mechanics.