Brane Tilings and Cluster Algebras

Transcription

1 Brane Tilings and Cluster Algebras Gregg Musiker (U. Minnesota), In-Jee Jeong (Brown Univ.), and Sicong Zhang (Columbia Univ.) Algebra, Geometry and Combinatorics Day Loyola University Chicago November 9, 0 Thank you to NSF Grants DMS-08, DMS-009, and DMS-8; and the Columbia University Rabi Scholars Program. Work done during 0-0 Research Experiences for Undergraduates in Combinatorics at University of Minnesota, Twin Cities Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

2 Outline. Introduction to Cluster Algebras. What is a Brane Tiling The Del Pezzo Quiver and Lattice Gale-Robinson Sequences Further Examples: Aztec Castles (work of Leoni-Neel-Turner) http//math.umn.edu/ musiker/brane.pdf Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

3 Introduction to Cluster Algebras In the late 990 s: Fomin and Zelevinsky were studying total positivity and canonical bases of algebraic groups. They noticed recurring combinatorial and algebraic structures. Let them to define cluster algebras, which have now been linked to quiver representations, Poisson geometry Teichmüller theory, tilting theory, mathematical physics, discrete integrable systems, string theory, and many other topics. Cluster algebras are a certain class of commutative rings which have a distinguished set of generators that are grouped into overlapping subsets, called clusters, each having the same cardinality. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

4 What is a Cluster Algebra? Definition (Sergey Fomin and Andrei Zelevinsky 00) A cluster algebra A (of geometric type) is a subalgebra of k(x,..., x n, x n+,..., x n+m ) constructed cluster by cluster by certain exchange relations. Generators: Specify an initial finite set of them, a Cluster, {x, x,..., x n+m }. Construct the rest via Binomial Exchange Relations: x α x α = x d+ i γ i + x d i γ i. The set of all such generators are known as Cluster Variables, and the initial pattern of exchange relations (described as a valued quiver, i.e. a directed graph) determines the Seed. Relations: Induced by the Binomial Exchange Relations. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

5 Example: Rank Cluster Algebras [ ] 0 b Let B =, b, c Z c 0 >0. ({x, x }, B) is a seed for a cluster algebra A(b, c) of rank. (E.g. when b = c, B = B(Q) where Q is a -vertex quiver with b arrows from v v.) µ (B) = µ (B) = B and x x = x c +, x x = + x b. Thus the cluster variables in this case are { xn b + if n is odd {x n : n Z} satisfying x n x n = xn c + if n is even. Example (b = c = ): (Finite Type, of Type A ) x = x + x = x = x + x. x = x + x = x + x + x = x + x + x x. x +x + x x + = x (x + x + + x x ) = x +. x = x. (x + )/x x x (x + ) x Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

6 Example: Rank Cluster Algebras Example (b = c = ): (Affine Type, of Type Ã) x = x +. x = x + = x + x + + x x x x x. x = x + = x + x + x + + x + x + x x x x x,... If we let x = x =, we obtain {x, x, x, x } = {,,, }. The next number in the sequence is x = + = = 89, an integer! Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

7 What is a Brane Tiling (in Physics & Algebraic Geometry) In physics, Brane Tilings are combinatorial models that are used to Decribe the world volume of both D and M branes, and describe certain ( + )-dimensional superconformal field theories arising in string theory (Type II B). In Algebraic Geometry, they are used to Probe certain toric Calabi-Yau singularities, and relate to non-commutative crepant resolutions and the -dimensional McKay correspondence. Certain examples of path algebras with relations (Jacobian Algebras) can be constructed by a quiver and potential coming from a brane tiling. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

8 What is a Brane Tiling (Combinatorially) However, this is a combinatorics talk, not a physics talk, so I will henceforth focus on combinatorial motivation instead. Most simply stated, a Brane Tiling is a Bipartite graph on a torus. We view such a tiling as a doubly-periodic tiling of its universal cover, the Euclidean plane. Examples:, Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 8 /

9 Brane Tilings from a Quiver Q with Potential W A Brane Tiling can be associated to a pair (Q, W ), where Q is a quiver and W is a potential (called a superpotential in the physics literature). A quiver Q is a directed graph where each edge is referred to as an arrow, and multiple edges are allowed. A potential W is a linear combination of cyclic paths in Q (possibly an infinite linear combination). For combinatorial purposes, we assume other conditions on (Q, W ), such as Each arrow of Q appears in one term of W with a positive sign, and one term with a negative sign. The number of terms of W with a positive sign equals the number with a negative sign. All coefficients in W are ±. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 9 /

10 Brane Tilings from a Quiver Q with Potential W Example (The dp Quiver): Q dp = Q =, W = A A A A A A + A A A + A A A A A A A A A A A A A A A. We now unfold Q onto the plane, letting the three positive (resp. negative) terms in W depict clockwise (resp. counter-clockwise) cycles on Q. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 0 /

11 Brane Tilings from a Quiver Q with Potential W Example (continued): F C A D B F E F C D A B F Q = unfolds to Q = W = A A A A A A (A) + A A A (B) + A A A (C) A A A A (D) A A A A (E) A A A A (F ). Locally, the configurations around vertices of Q and Q are identical. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

12 Brane Tilings from a Quiver Q with Potential W Taking the planar dual yields a bipartite graph on a torus (Brane Tiling): F A D B A C F E F F B D C Q T Q = F A D B A C F F F B D E B E C B D C Negative Term in W Counter-Clockwise cycle in Q in T Q Positive Term in W Clockwise cycle in Q in T Q (To obtain Q from T Q, we dualize edges so that white is on the right.) Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

13 Brane Tilings from a Quiver Q with Potential W Summarizing the dp Example: Q T Q Negative Term in W Counter-Clockwise cycle in Q in T Q Positive Term in W Clockwise cycle in Q in T Q (To obtain Q from T Q, we dualize edges so that white is on the right.) Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

14 Brane Tilings in Physics Face U(N) Gauge Group Edge Bifundamental Chiral Fields (Representations) Vertex Gauge-invariant operator (Term in the Superpotential) Together, this data yields a quiver gauge theory. One can apply Seiberg duality to get a different quiver gauge theory. Combinatorial connection: Seiberg duality corresponds to mutation in cluster algebra theory. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

15 Description of Seiberg Duality (from physics) From Brane Dimers and Quiver Gauges Theories (00) by Franco, Hanany, Kennaway, Wegh, and Wecht: After picking a node to dualize at: Reverse the direction of all arrows entering or exiting the dualized node. This is because Seiberg duality requires that the dual quarks transform in the conjugate flavor representations to the originals.... Next, draw in... bifundamentals which correspond to composite (mesonic) operators.... the Seiberg mesons are promoted to the fields in the bifundamental representation of the gauge group.... It is possible that this will make some fields massive, in which case the appropriate fields should then be integrated out. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

16 Description of Seiberg Duality (rephrased combinatorially) Pick a vertex j of the quiver Q (equiv. face of the brane tiling T Q ) at which to mutate. Then, reverse the direction of all arrows incident to j, i.e. A ij A ji. Next, for every two-path i j k, meson, in Q draw in a new arrow i k, the Seiberg mesons are promoted to the fields. Let Q denote this new quiver. We similarly alter the superpotential W to get W. For every -path i j k in Q, we replace any appearance of the product A ij A jk in W with the singleton A ik, and add or subtract a new degree -term, A ik A kj A ji. It is possible, that this will make some of the terms of W of degree two, massive, in which case there should be an associated -cycle in the mutated quiver Q that can be deleted, the appropriate fields should then be integrated out. This is in fact Mutation of Quivers with potential from cluster algebras (as defined by Derksen-Weyman-Zelevinsky)! Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

17 Description of Seiberg Duality (on the Brane Tiling) In the special case, that we are mutating at a vertex with two arrows in and out, a toric vertex, this corresponds to a Urban Renewal of a square face in the brane tiling. Example (Q (,) ): with potential W = A A A + A A A A + A A A + A A A + A A A A A A A A A A A A A A A A A A A A A. Consider the corresponding Brane Tiling T (,) and mutation of (Q, W ) at the toric vertex labeled. (Associated to Gale-Robinson Sequence) Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

18 Description of Seiberg Duality (on the Brane Tiling) Example (Q (,) ): with potential W = A A A + A A A (V ) A + A (H) A A + A A A + A A A A A A A A A A A A A (H) A A A A (V ) A A A A. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 8 /

19 Description of Seiberg Duality (on the Brane Tiling) Example (Q (,) ): Rotate potential terms containing W = A A A + A A A A (V ) + A(H) A A + A A A + A A A A A A A A A A A A A A (H) A A A (V ) A A A A. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 9 /

20 Description of Seiberg Duality (on the Brane Tiling) Example (Q (,) ): W = A A + A A A (V ) + A(H) Mutating at yields AA + AAA + AAAA A (D) AA AAA A(H) A(H) AAA(V ) + A A (D) A + A A (H) A A A A A A A. A AAA Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 0 /

21 Description of Seiberg Duality (on the Brane Tiling) Example (Q (,) ): W = A A + A A A (V ) + A(H) Highlighting Massive terms AA + AAA + AAAA A (D) AA AAA A(H) A(H) AAA(V ) + A A (D) A + A A (H) A A A A A A A. A AAA Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

22 Description of Seiberg Duality (on the Brane Tiling) Example (Q (,) ): W = A A + A A A (V ) + A(H) Highlighting complementary terms AA + AAA + AAAA A (D) AA AAA A(H) A(H) AAA(V ) + A A (D) A + A (H) A A A A A A A A. A AAA Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

23 Description of Seiberg Duality (on the Brane Tiling) Example (Q (,) ): W = A A A (V ) Reduces the potential to ) + AAA + AAAA A(D) AA AAA(V A A A A + A A (D) A A A A A A A A + A A A A. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

24 Description of Seiberg Duality (on the Brane Tiling) Example (Q (,) ): W = A A A (V ) If we cyclically permute vertices ) + AAA + AAAA A(D) AA AAA(V A A A A + A A (D) A A A A A A A A + A A A A. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

25 Description of Seiberg Duality (on the Brane Tiling) Example (Q (,) ): The cyclic permutation yields the original Brane Tiling and (Q, W )! W = A A A (V ) ) + AAA + AAAA A(D) AA AAA(V A A A A + A A (D) A A A A A A A A + A A A A W = A A A + A A A (V ) A + A(H) AA + AAA + AAAA A A A A A A A A A (H) ) A AAA(V A AAA. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

26 Enter Combinatorics The quiver Q dp has a similar periodicity property. µ µ = If we mutate Q dp by,,,,,,,,..., after the first two mutations, we obtain same quiver back up to cyclically permuting the vertex labels. Point: Mutating once in the Q (r,s) N case, or twice in the Q dp case, yields a quiver with potential that is equivalent up to cyclic rotation. Such quivers are called periodic in the Fordy-Marsh sense. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

27 Cluster Variable Mutation In addition to the mutation of quivers, there is also a complementary cluster mutation that can be defined. Cluster mutation yields a sequence of Laurent polynomials in Q(x, x,..., x n ) known as cluster variables. Given a quiver Q (the potential is irrelevant here) and an initial cluster {x,..., x N }, then mutating at vertex yields a new cluster variable x N+ defined by x N+ = x i + / x. i Q i Q Example (Q (r,s) ): In Q, r +, N r + and s +, N s +. N x i For r =, s =, N =, we get x 8 = (x x + x x ) /x. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

28 The Gale-Robinson Sequence Example (Q (r,s) N ): (e.g. r =, s =, N = ) = Mutating at,,,..., N,,,... yields the same quiver, up to cyclic permutation, at each step, hence we obtain the infinite sequence of x N+, x N+,... satsifying x n = (x n r x n N+r + x n s x n N+s ) /x n N for n > N. Known as the Gale-Robinson Sequence of Laurent polynomials. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 8 /

29 The Gale-Robinson Sequence (with coefficients) Example (Q (r,s) ): (e.g. r =, s =, N = ) N We add N frozen vertices to Q (r,s) N with incoming arrows. Let y i denote the cluster variable corresponding to vertex i. Mutating again at,,,..., N,,,... (never at frozen vertices) yields a infinite sequence of cluster variables with a more complicated recurrence: x nx n N = x n r x n N+r + ny i= y d(n n i,s,n s) i x n sx n N+s for n > N. where d(m, s, s ) = # ways to write M as A s + B s with A, B Z 0 Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 9 /

30 Gale-Robinson Sequence Example For Q (,), x 8 = x x y +x x x, x 9 = x x y +x x x, x 0 = x x x y y +x x y +x x x x x, x = x x x x y y y +x x x x y y +x x x y +x x x x x x x,... x 8, x 9, x 0, x, x,... Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 0 /

31 Gale-Robinson Sequence Example (continued) With Minimal Matchings Highlighted: For Q (,), x 8 = x x y +x x x, x 9 = x x y +x x x, x 0 = x x x y y +x x y +x x x x x, x = x x x x y y y +x x x x y y +x x x y +x x x x x x x,... x 8, x 9, x 0, x, x,... Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

32 Main Theorem (Jeong-M-Zhang) For certain periodic quivers Q, which include the Gale-Robison quiver family, the dp quiver, and some other -periodic quivers, we can use the Brane Tiling T Q to obtain combinatorial formulas for an infinite sequence of cluster variables in A Q. For n > N, x n = cm(g n ) M= perfect matching of G n x(m)y(m), where {G n : n > N} s are a collection of subgraphs of T Q, x(m) = edge e M x i x j (for edge e straddling faces i and j), y(m) = height of M (recording what faces need to be twisted to obtain matching M starting from the minimal matching, and cm(g n ) = the covering monomial of the graph G n (which records what face labels are contained in G n and along its boundary). Remark: This weighting scheme is a reformulation of schemes appearing in works of Speyer ( Octahedron Recurrence ) and Goncharov-Kenyon. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

33 Gale-Robinson Example (Q (,), Mutating,,...,,... ) Q = T Q = x 8, x 9, x 0, x, x, x, x, x, x,... Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

34 Gale-Robinson Example (Q (,), Mutating,,...,,... ) Obtain pinecone graphs from Bousquet-Mélou, Propp, and West in terms of Brane Tilings Terminology. Furthermore, to get cluster variable formulas with coefficients, need only use weights (Goncharov-Kenyon, Speyer) and heights (Kenyon-Propp-...) x 8, x 9, x 0, x, x, x, x, x, x,... Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

35 Gale-Robinson Example (Q (,), Mutating,,...,,... ) Similar connections (without principal coefficients) also observed in Brane tilings and non-commutative geometry by Richard Eager. Eager uses physics terminology where he looks at Y p,q and L a,b,c quiver gauge theories, and their periodic Seiberg duality (i.e. quiver mutations). x 8, x 9, x 0, x, x, x, x, x, x,... Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

36 dp Example (Mutating,,,,,,,,... ) Q T Q : D D D D D D s0 s s s s s Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

37 dp Example (Mutating,,,,,,,,... ) These subgraphs appear in work by Cottrell-Young and a subsequence of them appear in M. Ciucu s work Perfect matchings and perfect powers, where they are called Aztec Dragons. More on Aztec Dragons and the dp lattice shortly. D D D D D D s0 s s s s s Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

38 Hirzebruch Quiver F 0 : Aztec Diamonds and Fortresses The quiver Q below is -periodic, as illustrated by mutating in order,,,,,,... Q =... Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 8 /

39 Hirzebruch Quiver F 0 : Aztec Diamonds and Fortresses The quiver Q below is -periodic, as illustrated by mutating in order,,,,,,... Q =... Fortresses from M. Ciucu s work Perfect matchings and perfect powers. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 9 /

40 Non-periodic mutation sequences in F 0 Q =... If we instead mutate by,,,,,... or,,,,,..., we obtain quivers where the number of arrows grows without bound. Nonetheless, a combinatorial interpretation for the cluster variables is or Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 0 /

41 Non-periodic mutation sequences in the dp Lattice Mutating at a vertex of dp and its antipode commute so a mutation such as,,, can be reordered to,,,, and hence is the identity. Letting τ = µ µ, τ = µ µ, τ = µ µ, the above can be written as τ = τ = τ =. As discussed with Pylyavskyy, we see the further relations (τ τ ) = (τ τ ) = (τ τ ) =, and it can be shown that there are no other relations. Thus τ, τ, τ = Ã. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

42 τ-mutation-sequences We call a mutation sequence built out of a concatenation of τ, τ, and τ a τ-mutation sequence. As an example, the periodic sequences,,,,,,,,... yielding the Aztec Dragons, were examples of τ-mutation sequences given by τ, τ, τ, τ,.... Up to the à relations and relabeling vertices, other τ-mutation sequences do not necessarily give new cluster variables. First non-trivial τ-mutation sequence: τ, τ, τ, τ =,,,,,,, yields the following as the last cluster variable: (x x x x + x x x x + x x x x x + x x x x x x + x x x x x + x x x x + x x x x x x + x x x x x x + x x x x + x x x x x + x x x x x x + x x x x x + x x x x + x x x x )/x x x x x What is a combinatorial interpretation of this Laurent polynomial? Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

43 Aztec Castles UMN 0 REU Students Megan Leoni, Seth Neel, and Paxton Turner investigated these non-periodic mutation sequences in the dp Lattice. They developed a two-parameter family of graphs, Aztec Castles, to encode cluster variables arising from τ-mutation sequences. NW Example: x τ,τ,τ,τ corresponds to (Up to a shift in indexing on faces by one.)! N 0!! NE 0!! 0!!!!!! W SE SW. Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

44 Aztec Castles (cont.) For a general τ-mutation sequence, due to à symmetry, the last cluster variable can be associated to an alcove in the à Coxeter Lattice, i.e. indexed by a pair (k, l). More precisely, they cut the lattice into twelve cones and by symmetry it suffices to describe the families of graphs for two adjacent cones. j = j = j =0 j = j = (, ) D0 σ j = j = j =8 j = (, ) D (, ) D (, ) D (, ) (, 0) (, ) (, ) D σ σ (, ) σ (, ) σ (, ) (, 0) (, ) (, ) D σ σ (0, ) (0, ) (0, ) (0, ) D σ σ (, ) σ σ σ (, ) D σ D8 σ σ σ σ σ (, ) (, ) (, ) (, ) (, 0) (, ) (, ) (, ) (, ) (, 0) σ σ σ σ σ (, ) (, ) (, ) (, 0) (, ) (, ) (, ) (, ) (, ) (, 0) (, ) (, ) σ σ σ σ σ σ (, ) (, ) (, 0) (, ) (, ) (, ) (, ) (, ) (, 0) (, ) (, ) σ σ σ σ (, ) (, 0) (, ) (, ) (, ) (, ) (, 0) (, ) (, ) σ σ σ σ (, 0) (, ) (, ) (, ) Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

45 Aztec Castles (cont.) For a general τ-mutation sequence, due to à symmetry, the last cluster variable can be associated to an alcove in the à Coxeter Lattice, i.e. indexed by a pair (k, l). More precisely, they cut the lattice into twelve cones and by symmetry it suffices to describe the families of graphs for two adjacent cones. 0!!!!!! 0!! 0!!!!!!! 0!! 0!! 0!!!!!!!!!!!!!! 0!!! 0!!! 0!! 0!!!!!!!!!!!!!!!!!!!!!! 0!!! 0!! 0!!!! 0!!! 0!!! 0!!!!!!!!!!!!!!!!!!! 0!! 0!! 0!! 0!! 0!! 0!!!!!!!!!!!!!!!!!! 0!!! 0!!!!!! Example (k =, l = ): 0!!! Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

46 Open Problems and Work in Progress τ-mutation sequences and the periodic mutation sequences discussed are examples of toric mutation sequences. Mutation only performed at vertices with at most two incoming and two outgoing arrows. Colored BPS Pyramid Partition Functions, Quivers, and Cluster Transformations by Richard Eager and Sebastian Franco gives a recipe for getting combinatorial interpretations for toric mutation sequences but there is a gap for examples like τ τ τ τ. Current work in progress with S. Franco to resolve this issue using their formulation involving Jacobian algebras to find a general working recipe. Extend definition of heights to other cases to obtain cluster variables with principal coefficients. Obtain more combinatorial interpretations for non-toric mutation sequences (like the F 0 case). Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /

47 Thank You For Listening In-Jee Jeong, Bipartite Graphs, Quivers, and Cluster Variables, REU Report, reiner/reu/jeong0.pdf Sicong Zhang, Cluster Variables and Perfect Matchings of Subgraphs of the dp Lattice, REU Report, reiner/reu/zhang0.pdf I. Jeong, G. Musiker, and S. Zhang. Gale-Robinson Sequences and Brane Tilings. Discrete Mathematics and Theoretical Computer Science Proc. AS (0), -8. (Longer version in preparation.) Megan Leoni, Seth Neel, and Paxton Turner, Aztec Castles and the dp Quiver, arxiv:08.9. Slides Available at http//math.umn.edu/ musiker/brane.pdf Jeong-Musiker-Zhang (AlGeCom 9) Brane Tilings and Cluster Algebras November 9, 0 /