Toric Mutations in the dP$_2$ Quiver and Subgraphs of the dP$_2$ Brane Tiling

TL;DR

This paper explores the dP2 quiver and its brane tiling, providing explicit formulas for cluster variables generated by toric mutations and associating subgraphs of the tiling to these variables.

Contribution

It offers explicit formulas for all cluster variables from toric mutations and links each to a subgraph of the brane tiling, advancing understanding of dP2 quiver mutations.

Findings

Explicit formulas for cluster variables from toric mutations.

Association of subgraphs of the brane tiling with cluster variables.

Enhanced understanding of the dP2 quiver and brane tiling structure.

Abstract

Brane tilings are infinite, bipartite, periodic, planar graphs that are dual to quivers. In this paper, we examine the del Pezzo 2 (dP$_2$) quiver and its brane tiling, which arise from the physics literature, in terms of toric mutations on its corresponding cluster. Specifically, we give explicit formulas for all cluster variables generated by toric mutation sequences. Moreover, for each such variable, we associate a subgraph of the dP$_2$ brane tiling to it such that its weight matches the variable.