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 L a,b,c geometries, dP 2 , pseudo-dP 2 , and dP 3 . The counting of pyramid partitions for dP 1 gives rise to the Somos-4 sequence, while dP 2 and pseudo-dP 2 generate the Somos-5 sequence. Our results for dP 3 reproduce and extend those previously obtained for this theory, which were originally obtained from dimer shuffling.

Integrability on the master space

“…The study of the physical implication of the integrability of the bipartite graph in field theory started in [9][10][11]. In [9] the connection between the brane tiling, the quantum Teichmuller theory and the Poisson structure was investigated.…”

“…Then in [10] the Poisson manifold was studied in the language of quiver gauge theory and interesting connections with five dimensional N = 1 theories were made. Then in [11] a connection with cluster algebra and flavored quiver models was proposed.…”

A comprehensive survey of brane tilings

“…In parallel, in mathematics, the dialogue between gauge theory and the geometry and combinatorics of toric CY 3-folds also engendered numerous developments, including: new directions in Calabi-Yau algebras and quiver representations [20][21][22][23][24][25][26], non-commutative crepant resolutions of toric singularities [27][28][29][30][31], connections with Grothendieck's dessins d'enfants and certain isogenies of elliptic curves [32][33][34][35] and a geometric perspective on cluster algebras [36][37][38][39].…”

Introduction to Cluster Algebras

“…The topic of cluster algebras quickly grew into its own as a subject deserving independent study mainly fueled by its emergent close relationship to many areas of mathematics. Here is a partial list of related topics: combinatorics [39], hyperbolic geometry [11,12,40], Lie theory [19], Poisson geometry [24], integrable systems [9,26], representations of associative algebras [6,7,5,46,45,47], mathematical physics [10,1], and quantum groups [33,20,34,3].…”