Given one of an infinite class of supersymmetric quiver gauge theories, string theorists can associate a corresponding toric variety (which is a Calabi–Yau 3-fold) as well as an associated combinatorial model known as a brane tiling. In combinatorial language, a brane tiling is a bipartite graph on a torus and its perfect matchings are of interest to both combinatorialists and physicists alike. A cluster algebra may also be associated to such quivers and in this paper we study the generators of this algebra, known as cluster variables, for the quiver associated to the cone over the del Pezzo surface d P3. In particular, mutation sequences involving mutations exclusively at vertices with two in-coming arrows and two out-going arrows are referred to as toric cascades in the string theory literature. Such toric cascades give rise to interesting discrete integrable systems on the level of cluster variable dynamics. We provide an explicit algebraic formula for all cluster variables that are reachable by toric cascades as well as a combinatorial interpretation involving perfect matchings of subgraphs of the d P3 brane tiling for these formulas in most cases.
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Acknowledgements. The authors are grateful to Mihai Ciucu, Philippe Di Francesco, Richard Eager, Sebastian Franco, Michael Gehktman, Rinat Kedem, Richard Kenyon, Pasha Pylyavskyy, Michael Shapiro, David Speyer, and Dylan Thurston for a number of inspirational discussions. The second author was supported by NSF Grants DMS-#1148634 and DMS-#13692980. Much of this research was also aided by the open source mathematical software [Sage]. We are both appreciative of the hospitality of the Institute of Mathematics and its Applications (IMA) for providing support for this project. Much of this project was done during the time the first author worked at the IMA as a postdoctoral associate (2014–2016).
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