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arXiv · 1308.3926

Aztec Castles and the dP3 Quiver

Megan Leoni·Gregg Musiker·Seth Neel·Paxton Turner
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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 $τ-$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 $τ-$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 $τ$-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.

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Megan Leoni, Gregg Musiker, Seth Neel, Paxton Turner. 2014-08-25. Aztec Castles and the dP3 Quiver. https://arxiv.org/abs/1308.3926

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