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

1

Introduction and Preliminaries 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

1

Introduction and Preliminaries Quiver and cluster mutation The Del Pezzo 2 Quiver (dP2) and its brane tiling Toric mutations Two models of the dP2 quiver

2

Classification of Toric Mutation Sequences Adjacency between different models ρ-mutations

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 2 / 29

Overview

1

Introduction and Preliminaries Quiver and cluster mutation The Del Pezzo 2 Quiver (dP2) and its brane tiling Toric mutations Two models of the dP2 quiver

2

Classification of Toric Mutation Sequences Adjacency between different models ρ-mutations

3

Explicit Formula for Cluster Variables

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 2 / 29

Overview

1

Introduction and Preliminaries Quiver and cluster mutation The Del Pezzo 2 Quiver (dP2) and its brane tiling Toric mutations Two models of the dP2 quiver

2

Classification of Toric Mutation Sequences Adjacency between different models ρ-mutations

3

Explicit Formula for Cluster Variables

4

Subgraph of the Brane Tiling Weighting Scheme and Covering Monomial

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 2 / 29

Overview

1

Introduction and Preliminaries Quiver and cluster mutation The Del Pezzo 2 Quiver (dP2) and its brane tiling Toric mutations Two models of the dP2 quiver

2

Classification of Toric Mutation Sequences Adjacency between different models ρ-mutations

3

Explicit Formula for Cluster Variables

4

Subgraph of the Brane Tiling Weighting Scheme and Covering Monomial

5

Contour 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 2 3 4 5 μ1 1 2 3 4 5 Figure: Example of quiver mutation

Binomial Exchange Relation x′

1 = x2x5 + x3x4

x1 .

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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.

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 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

• ut-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

• ut-degree 2.

1 2 3 4 5

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 2 3 4 5 1 2 3 4 5

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 2 1 1 1 2 2 1 1

Figure: Adjacency between different models

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 7 / 29

ρ−mutation sequence

1 2 2 2 2 4 4 4 4 1 5 5

2 2 2 2 1 1 1 1 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 2 2 2 2 4 4 4 4 1 5 5

2 2 2 2 1 1 1 1 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)

ρ1 = µ1 ◦ (54321), ρ2 = µ5 ◦ (12345), ρ3 = µ2 ◦ µ4 ◦ (24), ρ4 = µ2 ◦ µ1 ◦ µ4 ◦ (531), ρ5 = µ4 ◦ µ5 ◦ µ2 ◦ (351), ρ6 = µ2 ◦ µ1 ◦ µ2 ◦ (531)(24), ρ7 = µ4 ◦ µ5 ◦ µ4 ◦ (135)(24).

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 8 / 29

ρ−mutation sequence

1 2 2 2 2 4 4 4 4 1 5 5

2 2 2 2 1 1 1 1 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)

ρ1 = µ1 ◦ (54321), ρ2 = µ5 ◦ (12345), ρ3 = µ2 ◦ µ4 ◦ (24), ρ4 = µ2 ◦ µ1 ◦ µ4 ◦ (531), ρ5 = µ4 ◦ µ5 ◦ µ2 ◦ (351), ρ6 = µ2 ◦ µ1 ◦ µ2 ◦ (531)(24), ρ7 = µ4 ◦ µ5 ◦ µ4 ◦ (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: ρ1 = µ1 ◦ (54321).

1 2 3 4 5 μ1 5 1 2 3 4

(54321)

1 2 3 4 5

(x1, x2, x3, x4, x5) − → (x2x5 + x3x4 x1 = x6, x2, x3, x4, x5) − → (x2, x3, x4, x5, x6)

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)

ρ4 = ρ2

1ρ3,

ρ5 = ρ2

2ρ3,

ρ6 = ρ2

1,

ρ7 = ρ2

2.

It suffices to consider ρ1, ρ2, ρ3.

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

ρ−mutations

Proposition (Relations between ρ−mutations)

ρ4 = ρ2

1ρ3,

ρ5 = ρ2

2ρ3,

ρ6 = ρ2

1,

ρ7 = ρ2

2.

It suffices to consider ρ1, ρ2, ρ3.

Proposition (Relations between ρ1, ρ2, ρ3)

ρ1ρ2 = ρ2ρ1 = ρ2

3 = 1.

ρ2

1ρ3 = ρ3ρ2 1,

ρ2

2ρ3 = ρ3ρ2 2,

ρ1ρ3ρ2 = ρ2ρ3ρ1.

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

ρ−mutation sequence: a visualization

Proposition (Relations between ρ1, ρ2, ρ3)

ρ1ρ2 = ρ2ρ1 = ρ2

3 = 1.

ρ2

1ρ3 = ρ3ρ2 1,

ρ2

2ρ3 = ρ3ρ2 2,

ρ1ρ3ρ2 = ρ2ρ3ρ1.

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 11 / 29

ρ−mutation sequence: a visualization

Proposition (Relations between ρ1, ρ2, ρ3)

ρ1ρ2 = ρ2ρ1 = ρ2

3 = 1.

ρ2

1ρ3 = ρ3ρ2 1,

ρ2

2ρ3 = ρ3ρ2 2,

ρ1ρ3ρ2 = ρ2ρ3ρ1.

Figure: A visualization for ρ−mutation sequence.

ρ1 :→, ρ2 :←, ρ3 :↑ / ↓ .

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 ρk

1(ρ3ρ1)m

• r

ρk

1(ρ3ρ1)mρ3,

where k ∈ Z, m ∈ Z≥0 and ρ−1

1

= ρ2.

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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 x1, x2, x3, x4, x5 be our initial variables. Define xn for each n ∈ Z by xnxn−5 = xn−1xn−4 + xn−2xn−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 x1, x2, x3, x4, x5 be our initial variables. Define xn for each n ∈ Z by xnxn−5 = xn−1xn−4 + xn−2xn−3. Notice that {xn}n≥1 is the somos-5 sequence if x1 = x2 = x3 = x4 = x5 = 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 x1, x2, x3, x4, x5 be our initial variables. Define xn for each n ∈ Z by xnxn−5 = xn−1xn−4 + xn−2xn−3. Notice that {xn}n≥1 is the somos-5 sequence if x1 = x2 = x3 = x4 = x5 = 1.

Definition (Some Constants)

A := x1x5 + x2

3

x2x4 , B := x2x6 + x2

4

x3x5

• = x1x2

4 + x2x3x4 + x2 2x5

x1x3x5

• .

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

Explicit Formula for Cluster Variables

Theorem

Define g(s, k) := s

2

s+1

2

• if k is even and g(s, k) :=

s−1

2

s

2

• if k is
• dd. Then we have, for k ∈ Z and s ∈ Z≥0,

ρk

1(ρ3ρ1)s{x1, x2, x3, x4, x5} = {Ag(s+1,k)Bg(s+1,k+1)xk+s+1,

Ag(s,k)Bg(s,k+1)xk+s+2, Ag(s+1,k)Bg(s+1,k+1)xk+s+3, Ag(s,k)Bg(s,k+1)xk+s+4, Ag(s+1,k)Bg(s+1,k+1)xk+s+5}.

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 14 / 29

Explicit Formula for Cluster Variables

Theorem

Define g(s, k) := s

2

s+1

2

• if k is even and g(s, k) :=

s−1

2

s

2

• if k is
• dd. Then we have, for k ∈ Z and s ∈ Z≥0,

ρk

1(ρ3ρ1)s{x1, x2, x3, x4, x5} = {Ag(s+1,k)Bg(s+1,k+1)xk+s+1,

Ag(s,k)Bg(s,k+1)xk+s+2, Ag(s+1,k)Bg(s+1,k+1)xk+s+3, Ag(s,k)Bg(s,k+1)xk+s+4, Ag(s+1,k)Bg(s+1,k+1)xk+s+5}.

Corollary

All cluster variables generated by toric mutations can be written as either An2Bn(n−1)x2m

• r

An(n−1)Bn2x2m−1 for some m, n ∈ Z.

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 14 / 29

Subgraph of the Brane Tiling

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 15 / 29

Subgraph of the Brane Tiling

Definition (Weighting Scheme)

Associate a weight w(e) :=

1 xixj to each edge bordering blocks labeled i

and j. Let M(G) be the collection of perfect matchings of G. For each M ∈ M(G), define its weight w(M) =

e∈M w(e).

Define the weight of the graph G as w(G) :=

• M∈M(G)

w(M).

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 15 / 29

Subgraph of the Brane Tiling

Definition (Weighting Scheme)

Associate a weight w(e) :=

1 xixj to each edge bordering blocks labeled i

and j. Let M(G) be the collection of perfect matchings of G. For each M ∈ M(G), define its weight w(M) =

e∈M w(e).

Define the weight of the graph G as w(G) :=

• M∈M(G)

w(M).

3 2 3 5 1 4 5 3 2 3 5 1 4 5 3 2 3 5 1 4 5

Figure: w(G) =

1 x1x5x2

2 x2 3 +

1 x2

1 x2 5 x2 2 +

1 x2

1 x2x3x4x5

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 15 / 29

Subgraph of the Brane Tiling

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 16 / 29

Subgraph of the Brane Tiling

Definition (Covering Monomial)

Given a subgraph G, let aj be the number of blocks labeled j in G. Let bj be the number of blocks labeled j adjacent to G. Let c3 be the number of blocks labeled 3 adjacent to G with 4 edges inside G. The covering monomial m(G) is the product xa1+b1

1

xa2+b2

2

x2a3+b3+c3

3

xa4+b4

4

xa5+b5

5

.

3 2 3 5 1 4 5 Figure: Example of Covering Monomial: x1x2x2

3x4x2 5

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 16 / 29

Subgraph of the Brane Tiling

Definition (Covering Monomial)

Given a subgraph G, let aj be the number of blocks labeled j in G. Let bj be the number of blocks labeled j adjacent to G. Let c3 be the number of blocks labeled 3 adjacent to G with 4 edges inside G. The covering monomial m(G) is the product xa1+b1

1

xa2+b2

2

x2a3+b3+c3

3

xa4+b4

4

xa5+b5

5

. For any graph G, denote the product of its weight and its cover monomial as c(G) := w(G)m(G).

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 17 / 29

Contour: Fundamental Shape

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Contour: Fundamental Shape

e a b c d

Figure: 5-sided fundamental shape.

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 18 / 29

Contour: Length

Similar to [LM15], we define the length of our contour.

Definition (Length of Contour)

∀i ∈ {a, b, c, d, e}, len(i) =

• |i|,

if same direction as the associated side −|i|,

• therwise

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Contour: Length

e a b c d

Figure: 5-sided fundamental shape.

a=6 b=-4 c=2 d=2 e=-4

Figure: Length of Contour.

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From Contour to Subgraph

Definition (Rules to Get Subgraph)

positive length → keep black points; negative length → keep white points. b ≡ d (mod 2), keep special point; b ≡ d (mod 2), remove special point.

a=6 b=-4 c=2 d=2 e=-4

Figure: Length of Contour.

2 1 2 3 4 2 1 2 3 4 1 2 3 4 5 2 1 2 3 4 1 2 4 5 1 2 3 4 1 2 3 5 1 1 1

Figure: Example of Subgraph.

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Main Result

Theorem (Formula of Contours)

Define the contours as follows: An2Bn2−nx2k =

• k − 2 + n, −

k − 4 + 5n 2

• , 2n − 1,

k − 3n 2

• , 1 + n − k
• An2+nBn2x2k−1 =
• k − 2 + n, −

k − 2 + 5n 2

• , 2n,

k − 2 − 3n 2

• , 2 + n − k
• For any such cluster variable, if G is the subgraph of its corresponding contour,

then c(G) is the Laurent polynomial of the cluster variable.

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Kuo’s Condensation Theorems

Kuo’s Condensation Theorems [Kuo04] tell us how to write the weight of a large graph in terms of smaller ones.

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 23 / 29

Kuo’s Condensation Theorems

Kuo’s Condensation Theorems [Kuo04] tell us how to write the weight of a large graph in terms of smaller ones. Let G = (V1, V2, E) be a weighted planar bipartite graph. Let p1, p2, p3, p4 be four vertices in a cyclic order on the boundary of G.

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 23 / 29

Kuo’s Condensation Theorems

Kuo’s Condensation Theorems [Kuo04] tell us how to write the weight of a large graph in terms of smaller ones. Let G = (V1, V2, E) be a weighted planar bipartite graph. Let p1, p2, p3, p4 be four vertices in a cyclic order on the boundary of G.

Theorem (Balanced Kuo Condensation)

Assume |V1| = |V2|, p1, p3 ∈ V1 and p2, p4 ∈ V2. Then w(G)w(G − {p1, p2, p3, p4}) =w(G − {p1, p2})w(G − {p3, p4}) + w(G − {p1, p4})w(G − {p2, p3}).

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 23 / 29

Kuo’s Condensation Theorems

Kuo’s Condensation Theorems [Kuo04] tell us how to write the weight of a large graph in terms of smaller ones. Let G = (V1, V2, E) be a weighted planar bipartite graph. Let p1, p2, p3, p4 be four vertices in a cyclic order on the boundary of G.

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 24 / 29

Kuo’s Condensation Theorems

Kuo’s Condensation Theorems [Kuo04] tell us how to write the weight of a large graph in terms of smaller ones. Let G = (V1, V2, E) be a weighted planar bipartite graph. Let p1, p2, p3, p4 be four vertices in a cyclic order on the boundary of G.

Theorem (Unbalanced Kuo Condensation)

Assume |V1| = |V2| + 1, p1, p2, p3 ∈ V1 and p4 ∈ V2. Then w(G − {p2})w(G − {p1, p3, p4}) =w(G − {p1})w(G − {p2, p3, p4}) + w(G − {p3})w(G − {p1, p2, p4}).

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 24 / 29

Kuo’s Condensation Theorems

Kuo’s Condensation Theorems [Kuo04] tell us how to write the weight of a large graph in terms of smaller ones. Let G = (V1, V2, E) be a weighted planar bipartite graph. Let p1, p2, p3, p4 be four vertices in a cyclic order on the boundary of G.

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 25 / 29

Kuo’s Condensation Theorems

Kuo’s Condensation Theorems [Kuo04] tell us how to write the weight of a large graph in terms of smaller ones. Let G = (V1, V2, E) be a weighted planar bipartite graph. Let p1, p2, p3, p4 be four vertices in a cyclic order on the boundary of G.

Theorem (Non-alternating Kuo Condensation)

Assume |V1| = |V2|, p1, p2 ∈ V1 and p3, p4 ∈ V2. Then w(G − {p1, p4})w(G − {p2, p3}) =w(G)w(G − {p1, p2, p3, p4}) + w(G − {p1, p3})w(G − {p2, p4}).

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 25 / 29

Proof Sketch

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Proof Sketch

We use induction on n. Base case: xk → Somos-5 Sequence. Inductive Step: (A(n+1)2Bn2+nx2k)(An2Bn2−nx2k+2) =(An2+nBn2x2k+3)(An2+nBn2x2k−1) + (An2+nBn2x2k+1)2 w(G − {p1, p2})w(G − {p3, p4}) =w(G)w(G − {p1, p2, p3, p4}) + w(G − {p1, p3})w(G − {p2, p4}).

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 26 / 29

Proof Sketch

2 1 4 2 5 1 4 5

R K p1

2 3 2 3 2 4 1 4 5 1 4 5 1 4 5 5

p2 p

3 4 2 5 1 4 2 5 1 4 2 5 1 4 5 2 3 2 1 4 1 4 5 5

p

4

K R

Figure: Position of p1 through p4 when (a, b, c, d, e) = (+, −, +, +, −) − R

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References

Amihay Hanany and R-K Seong. Brane tilings and reflexive polygons. Fortschritte der Physik, 60(6):695–803, 2012. Eric H Kuo. Applications of graphical condensation for enumerating matchings and tilings. Theoretical Computer Science, 319(1):29–57, 2004. Tri Lai and Gregg Musiker. Beyond aztec castles: Toric cascades in the dp3 quiver. arXiv preprint arXiv:1512.00507, 2015.

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• Acknowledgements. This research was carried out as part of the 2016

REU program at the University of Minnesota, Twin Cities, and was supported by NSF RTG grant DMS-1148634 and by NSF grant DMS-1351590. The authors would like to thank Victor Reiner, Sunita Chepuri and Elise delMas for their advice and comments. The authors are especially grateful to Gregg Musiker for his mentorship, support, and valuable advice.

Thank you!!

Gao, Li, Vuong, Yang dP2 Cluster Variables July 29, 2016 29 / 29