Aztec castles and the dP3 quiver

Megan Leoni, Gregg Musiker, Seth Neel, Paxton Turner

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

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 (dP3) 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 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 Zhang (2012 Cluster Variables and Perfect Matchings of Subgraphs of the dP3 Lattice http://www.math.umn.edu/∼/REU/Zhang2012.pdf), and forms part of the emerging story in combinatorics and theoretical high energy physics relating cluster variables to subgraphs of the associated brane tiling. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Cluster algebras in mathematical physics'.

Original languageEnglish (US)
Article number474011
JournalJournal of Physics A: Mathematical and Theoretical
Volume47
Issue number47
DOIs
StatePublished - Nov 28 2014

Keywords

  • brane tiling
  • del Pezzo 3 lattice
  • dimer model

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