Abstract
Brane tilings are infinite, bipartite, periodic, planar graphs that are dual to quivers. In this paper, we study the del Pezzo 2 (dP$_2$) quiver and its associated brane tiling which arise in theoretical physics. Specifically, we prove explicit formulas for all cluster variables generated by toric mutation sequences of the dP$_2$ quiver. Moreover, we associate a subgraph of the dP$_2$ brane tiling to each toric cluster variable such that the sum of weighted perfect matchings of the subgraph equals the Laurent polynomial of the cluster variable.
Highlights
Cluster algebras are a class of commutative rings generated by cluster variables, which are partitioned into sets called clusters
Combinatorial interpretations of the cluster variables have been obtained by associating a subgraph of the brane tiling to each cluster variable such that the Laurent polynomial of the cluster variable is recoverable from a weighting scheme applied to the subgraph ([13], [14], [12])
We prove that the weighted perfect matchings of a subgraph are terms in the Laurent polynomial of the cluster variable
Summary
Cluster algebras are a class of commutative rings generated by cluster variables, which are partitioned into sets called clusters. The notion of brane tilings was first introduced in theoretical physics [3] They are important in physics since perfect matchings of brane tilings carry information on the geometry of certain toric varieties which are Calabi-Yau 3-folds. For such quivers, combinatorial interpretations of the cluster variables have been obtained by associating a subgraph of the brane tiling to each cluster variable such that the Laurent polynomial of the cluster variable is recoverable from a weighting scheme applied to the subgraph ([13], [14], [12]). We classify all cluster variables generated by toric mutations and give combinatorial interpretations for their Laurent polynomials.
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