Aztec Castles and the dP3 Quiver

TL;DR

This paper investigates algebraic and geometric properties of the del Pezzo 3 quiver and its brane tiling, introducing Aztec castles and a factorization formula for cluster variables generated by specific mutation sequences.

Contribution

It introduces Aztec castles, a new family of subgraphs, and provides explicit formulas and a lattice walk interpretation for cluster variables in the dP3 quiver.

Findings

Proves a factorization formula for cluster variables from $ au$-mutation sequences.

Establishes a correspondence between mutation sequences and walks in a triangular lattice.

Expresses cluster variables as sums over weighted perfect matchings of Aztec castles.

Abstract

Bipartite, periodic, planar graphs known as brane tilings can be associated to a large class of quivers. This paper will explore new algebraic properties of the well-studied del Pezzo 3 quiver and geometric properties of its corresponding brane tiling. In particular, a factorization formula for the cluster variables arising from a large class of mutation sequences (called $\tau-$mutation sequences) is proven; this factorization also gives a recursion on the cluster variables produced by such sequences. We can realize these sequences as walks in a triangular lattice using a correspondence between the generators of the affine symmetric group $\tilde{A_2}$ and the mutations which generate $\tau-$mutation sequences. Using this bijection, we obtain explicit formulae for the cluster that corresponds to a specific alcove in the lattice. With this lattice visualization in mind, we then express each cluster variable produced in a $\tau$-mutation sequence as the sum of weighted perfect matchings of a new family of subgraphs of the dP3 brane tiling, which we call Aztec castles. Our main result generalizes previous work on a certain mutation sequence on the dP3 quiver in [Zha12], and forms part of the emerging story in combinatorics and theoretical high energy physics relating cluster variables to subgraphs of the associated brane tiling.