Geodesic
Toric mutations in the \(\mathrm{dp}_2\) quiver and subgraphs of the \(\mathrm{dp}_2\) brane tiling
Yibo Gao1 ; Zhaoqi Li2 ; Thuy-Duong Vuong ; Lisa Yang1
1Massachusetts Institute of Technology
2Macalester College
The electronic journal of combinatorics, Tome 26 (2019) no. 2
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Brane tilings are infinite, bipartite, periodic, planar graphs that are dual to quivers. In this paper, we study the del Pezzo 2 (dP$_2$) quiver and its associated brane tiling which arise in theoretical physics. Specifically, we prove explicit formulas for all cluster variables generated by toric mutation sequences of the dP$_2$ quiver. Moreover, we associate a subgraph of the dP$_2$ brane tiling to each toric cluster variable such that the sum of weighted perfect matchings of the subgraph equals the Laurent polynomial of the cluster variable.
DOI : 10.37236/6825
Classification : 13F60, 05B45, 05C20, 16G20

Yibo Gao  1   ; Zhaoqi Li  2   ; Thuy-Duong Vuong    ; Lisa Yang  1

1 Massachusetts Institute of Technology
2 Macalester College 
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     author = {Yibo Gao and Zhaoqi Li and Thuy-Duong Vuong and Lisa Yang},
     title = {Toric mutations in the \(\mathrm{dp}_2\) quiver and subgraphs of the \(\mathrm{dp}_2\) brane tiling},
     journal = {The electronic journal of combinatorics},
     year = {2019},
     volume = {26},
     number = {2},
     doi = {10.37236/6825},
     zbl = {1442.13068},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/6825/}
}
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Yibo Gao; Zhaoqi Li; Thuy-Duong Vuong; Lisa Yang. Toric mutations in the \(\mathrm{dp}_2\) quiver and subgraphs of the \(\mathrm{dp}_2\) brane tiling. The electronic journal of combinatorics, Tome 26 (2019) no. 2. doi: 10.37236/6825

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