Toric Mutations in the dP2 Quiver Yibo Gao, Zhaoqi Li, Thuy-Duong - PowerPoint PPT Presentation

Toric Mutations in the dP2 Quiver Yibo Gao, Zhaoqi Li, Thuy-Duong Vuong, Lisa Yang July 29, 2016 Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 1 / 29

Overview Introduction and Preliminaries 1 Quiver and cluster mutation The Del Pezzo 2 Quiver (dP2) and its brane tiling Toric mutations Two models of the dP2 quiver Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 2 / 29

Overview Introduction and Preliminaries 1 Quiver and cluster mutation The Del Pezzo 2 Quiver (dP2) and its brane tiling Toric mutations Two models of the dP2 quiver Classification of Toric Mutation Sequences 2 Adjacency between different models ρ -mutations Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 2 / 29

Overview Introduction and Preliminaries 1 Quiver and cluster mutation The Del Pezzo 2 Quiver (dP2) and its brane tiling Toric mutations Two models of the dP2 quiver Classification of Toric Mutation Sequences 2 Adjacency between different models ρ -mutations Explicit Formula for Cluster Variables 3 Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 2 / 29

Overview Introduction and Preliminaries 1 Quiver and cluster mutation The Del Pezzo 2 Quiver (dP2) and its brane tiling Toric mutations Two models of the dP2 quiver Classification of Toric Mutation Sequences 2 Adjacency between different models ρ -mutations Explicit Formula for Cluster Variables 3 Subgraph of the Brane Tiling 4 Weighting Scheme and Covering Monomial Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 2 / 29

Overview Introduction and Preliminaries 1 Quiver and cluster mutation The Del Pezzo 2 Quiver (dP2) and its brane tiling Toric mutations Two models of the dP2 quiver Classification of Toric Mutation Sequences 2 Adjacency between different models ρ -mutations Explicit Formula for Cluster Variables 3 Subgraph of the Brane Tiling 4 Weighting Scheme and Covering Monomial Contour 5 Fundamental Shape and Definitions Main Result Kuo’s Condensation Theorems Proof Sketch Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 2 / 29

Quiver and Cluster Mutation 1 1 5 2 5 2 μ 1 4 4 3 3 Figure: Example of quiver mutation 1 = x 2 x 5 + x 3 x 4 x ′ Binomial Exchange Relation . x 1 Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 3 / 29

The Del Pezzo 2 Quiver (dP2) and its Brane Tiling Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 4 / 29

The Del Pezzo 2 Quiver (dP2) and its Brane Tiling The second Del Pezzo Surface (dP2) is first introduced in the physics literature. 3 3 2 2 1 5 4 5 4 1 1 5 2 3 3 3 2 2 2 5 4 5 4 5 4 1 1 1 4 3 Figure: dP2 quiver and its corresponding brane tiling [HS12] Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 4 / 29

Toric Mutations Definition (Toric Mutations) A toric mutation is a cluster mutation at a vertex with in-degree 2 and out-degree 2. Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 5 / 29

Toric Mutations Definition (Toric Mutations) A toric mutation is a cluster mutation at a vertex with in-degree 2 and out-degree 2. 1 5 2 4 3 Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 5 / 29

Two Models of the dP2 Quiver Same Model: isomorphic or reverse isomorphic. Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 6 / 29

Two Models of the dP2 Quiver Same Model: isomorphic or reverse isomorphic. 1 1 5 2 5 2 4 3 4 3 Figure: Model 1 (left) and Model 2 (right) of the dP2 quiver [HS12] Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 6 / 29

Classification of Toric Mutation Sequences Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 7 / 29

Classification of Toric Mutation Sequences 2 1 1 1 1 2 2 2 1 Figure: Adjacency between different models Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 7 / 29

ρ − mutation sequence 1 1 1 4 2 4 2 1 5 2 2 2 2 4 2 4 2 1 1 5 1 1 1 Figure: All possible toric mutation sequences that start from model 1 and return to model 1 the first time. Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 8 / 29

ρ − mutation sequence 1 1 1 4 2 4 2 1 5 2 2 2 2 4 2 4 2 1 1 5 1 1 1 Figure: All possible toric mutation sequences that start from model 1 and return to model 1 the first time. Definition ( ρ -mutations) ρ 1 = µ 1 ◦ (54321) , ρ 2 = µ 5 ◦ (12345) , ρ 3 = µ 2 ◦ µ 4 ◦ (24) , ρ 4 = µ 2 ◦ µ 1 ◦ µ 4 ◦ (531) , ρ 5 = µ 4 ◦ µ 5 ◦ µ 2 ◦ (351) , ρ 6 = µ 2 ◦ µ 1 ◦ µ 2 ◦ (531)(24) , ρ 7 = µ 4 ◦ µ 5 ◦ µ 4 ◦ (135)(24) . Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 8 / 29

ρ − mutation sequence 1 1 1 4 2 4 2 1 5 2 2 2 2 4 2 4 2 1 1 5 1 1 1 Figure: All possible toric mutation sequences that start from model 1 and return to model 1 the first time. Definition ( ρ -mutations) ρ 1 = µ 1 ◦ (54321) , ρ 2 = µ 5 ◦ (12345) , ρ 3 = µ 2 ◦ µ 4 ◦ (24) , ρ 4 = µ 2 ◦ µ 1 ◦ µ 4 ◦ (531) , ρ 5 = µ 4 ◦ µ 5 ◦ µ 2 ◦ (351) , ρ 6 = µ 2 ◦ µ 1 ◦ µ 2 ◦ (531)(24) , ρ 7 = µ 4 ◦ µ 5 ◦ µ 4 ◦ (135)(24) . A ρ − mutation sequence is a sequence of ρ − mutations. Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 8 / 29

ρ − mutations An example: ρ 1 = µ 1 ◦ (54321) . 1 1 5 5 2 5 2 4 1 μ 1 (54321) 4 3 4 3 3 2 → ( x 2 x 5 + x 3 x 4 ( x 1 , x 2 , x 3 , x 4 , x 5 ) − = x 6 , x 2 , x 3 , x 4 , x 5 ) − → ( x 2 , x 3 , x 4 , x 5 , x 6 ) x 1 Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 9 / 29

ρ − mutations Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 10 / 29

ρ − mutations Proposition (Relations between ρ − mutations) ρ 4 = ρ 2 ρ 5 = ρ 2 ρ 6 = ρ 2 ρ 7 = ρ 2 1 ρ 3 , 2 ρ 3 , 1 , 2 . It suffices to consider ρ 1 , ρ 2 , ρ 3 . Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 10 / 29

ρ − mutations Proposition (Relations between ρ − mutations) ρ 4 = ρ 2 ρ 5 = ρ 2 ρ 6 = ρ 2 ρ 7 = ρ 2 1 ρ 3 , 2 ρ 3 , 1 , 2 . It suffices to consider ρ 1 , ρ 2 , ρ 3 . Proposition (Relations between ρ 1 , ρ 2 , ρ 3 ) ρ 1 ρ 2 = ρ 2 ρ 1 = ρ 2 3 = 1 . ρ 2 1 ρ 3 = ρ 3 ρ 2 ρ 2 2 ρ 3 = ρ 3 ρ 2 1 , 2 , ρ 1 ρ 3 ρ 2 = ρ 2 ρ 3 ρ 1 . Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 10 / 29

ρ − mutation sequence: a visualization Proposition (Relations between ρ 1 , ρ 2 , ρ 3 ) ρ 1 ρ 2 = ρ 2 ρ 1 = ρ 2 3 = 1 . ρ 2 1 ρ 3 = ρ 3 ρ 2 ρ 2 2 ρ 3 = ρ 3 ρ 2 1 , 2 , ρ 1 ρ 3 ρ 2 = ρ 2 ρ 3 ρ 1 . Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 11 / 29

ρ − mutation sequence: a visualization Proposition (Relations between ρ 1 , ρ 2 , ρ 3 ) ρ 1 ρ 2 = ρ 2 ρ 1 = ρ 2 3 = 1 . ρ 2 1 ρ 3 = ρ 3 ρ 2 ρ 2 2 ρ 3 = ρ 3 ρ 2 1 , 2 , ρ 1 ρ 3 ρ 2 = ρ 2 ρ 3 ρ 1 . Figure: A visualization for ρ − mutation sequence. ρ 1 : → , ρ 2 : ← , ρ 3 : ↑ / ↓ . Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 11 / 29

ρ − mutation sequence Theorem Every toric mutation sequence that starts at Q (the original dP2 quiver) and ends in model 1 can be written as either ρ k 1 ( ρ 3 ρ 1 ) m ρ k 1 ( ρ 3 ρ 1 ) m ρ 3 , or where k ∈ Z , m ∈ Z ≥ 0 and ρ − 1 = ρ 2 . 1 Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 12 / 29

Explicit Formula for Cluster Variables Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 13 / 29

Explicit Formula for Cluster Variables Definition (Laurent Polynomial for Somos-5 Sequence) Let x 1 , x 2 , x 3 , x 4 , x 5 be our initial variables. Define x n for each n ∈ Z by x n x n − 5 = x n − 1 x n − 4 + x n − 2 x n − 3 . Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 13 / 29

Explicit Formula for Cluster Variables Definition (Laurent Polynomial for Somos-5 Sequence) Let x 1 , x 2 , x 3 , x 4 , x 5 be our initial variables. Define x n for each n ∈ Z by x n x n − 5 = x n − 1 x n − 4 + x n − 2 x n − 3 . Notice that { x n } n ≥ 1 is the somos-5 sequence if x 1 = x 2 = x 3 = x 4 = x 5 = 1. Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 13 / 29

Explicit Formula for Cluster Variables Definition (Laurent Polynomial for Somos-5 Sequence) Let x 1 , x 2 , x 3 , x 4 , x 5 be our initial variables. Define x n for each n ∈ Z by x n x n − 5 = x n − 1 x n − 4 + x n − 2 x n − 3 . Notice that { x n } n ≥ 1 is the somos-5 sequence if x 1 = x 2 = x 3 = x 4 = x 5 = 1. Definition (Some Constants) A := x 1 x 5 + x 2 B := x 2 x 6 + x 2 = x 1 x 2 4 + x 2 x 3 x 4 + x 2 2 x 5 � � 3 4 , . x 2 x 4 x 3 x 5 x 1 x 3 x 5 Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 13 / 29

Explicit Formula for Cluster Variables Theorem � s � s − 1 � � s � � s +1 � � Define g ( s , k ) := if k is even and g ( s , k ) := if k is 2 2 2 2 odd. Then we have, for k ∈ Z and s ∈ Z ≥ 0 , ρ k 1 ( ρ 3 ρ 1 ) s { x 1 , x 2 , x 3 , x 4 , x 5 } = { A g ( s +1 , k ) B g ( s +1 , k +1) x k + s +1 , A g ( s , k ) B g ( s , k +1) x k + s +2 , A g ( s +1 , k ) B g ( s +1 , k +1) x k + s +3 , A g ( s , k ) B g ( s , k +1) x k + s +4 , A g ( s +1 , k ) B g ( s +1 , k +1) x k + s +5 } . Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 14 / 29