Colored BPS pyramid partition functions, quivers and cluster transformations

Abstract

We investigate the connections between flavored quivers, dimer models, and BPS pyramids for generic toric Calabi-Yau threefolds from various perspectives. We introduce a purely field theoretic definition of both finite and infinite pyramids in terms of quivers with flavors. These pyramids are associated to the counting of BPS invariants for generic toric Calabi-Yau threefolds. We discuss how cluster transformations provide an efficient recursive method for computing pyramid partition functions and show that the recursion is equivalent to the multidimensional octahedron recurrence. Transitions between different pyramids are related to Seiberg dualities, and we offer complimentary characterizations of these transitions in terms of the motion of zonotopes and duality webs. Our methods apply to completely general geometries including those with vanishing 4-cycles, which are associated to chiral quivers, thus overcoming one of the main limitations in the existing literature. We illustrate our ideas with explicit results for the infinite family of La,b,c geometries, dP2, pseudo-dP2, and dP3. The counting of pyramid partitions for dP1 gives rise to the Somos-4 sequence, while dP2 and pseudo-dP2 generate the Somos-5 sequence. Our results for dP3 reproduce and extend those previously obtained for this theory, which were originally obtained from dimer shuffling.