TY - JOUR

T1 - Beyond Aztec Castles

T2 - Toric Cascades in the dP 3 Quiver

AU - Lai, Tri

AU - Musiker, Gregg

PY - 2017/12/1

Y1 - 2017/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85030105038&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85030105038&partnerID=8YFLogxK

U2 - 10.1007/s00220-017-2993-8

DO - 10.1007/s00220-017-2993-8

M3 - Article

AN - SCOPUS:85030105038

VL - 356

SP - 823

EP - 881

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -