Finite dimensional algebras arising from dimer models
and their derived equivalences
Yusuke Nakajima
Kavli IPMU, University of Tokyo

September 19th, 2018

Yusuke Nakajima (Kavli IPMU)

Algebras arising from dimer models

September 19th, 20181 / 15

Motivations
2-dimensional case (Reiten-Van den Bergh, Bockland-Schedler-Wemyss, etc.)

C[x, y ] ∗ G∏Morita∼2-CY algebra
gl.dim=2

degree 0 part
preprojective algebra

eCQ
rep. infinite
hereditary algebra
gl.dim=1

e extended Dynkin quiver of type ADE
where Q:G ⊂ SL(2, C): finite subgroupHigher dim. case (cf. Keller, Minamoto-Mori, Herchend-Iyama-Oppermann)degree 0 partAd-CY alg. of GP 1
gl.dim=dA0
d-preprojective algebra (d − 1)-rep. infinite alg.
gl.dim=d-1

Some examples are obtained by dimer models (cf. Amiot-Iyama-Reiten).
Yusuke Nakajima (Kavli IPMU)

Algebras arising from dimer models

September 19th, 20182 / 15

What is a dimer model ?
Definition
A dimer model (or brane tiling) is a finite bipartite graph inducing apolygonal cell decomposition of the real two-torus T := R2 /Z2 .
Therefore each node is colored either black or white so that each edge
connects a black node to a white node.

Yusuke Nakajima (Kavli IPMU)

Algebras arising from dimer models

September 19th, 20183 / 15

Quivers associated with dimer models
As the dual of a dimer model Γ, we define the quiver Q associated with Γ.
Dimer model Γfacesedges

←→
←→
←→

Quiver Q
vertices Q0arrows Q1
232

10

1

232

The orientation of arrows is determined so that the white node is on theright of the arrow.
(We also define a certain “potential WQ ” from a dimer model.)Yusuke Nakajima (Kavli IPMU)

Algebras arising from dimer models

September 19th, 20184 / 15

Jacobian algebras arising from dimer models
For the quiver Q associated with a dimer model, we consider the path
algebra CQ.
For each arrow α ∈ Q1 , ∃ two oppositely oriented cycles (αpα+ andαpα− ) containing α as a boundary.

pα−αpα+Define the Jacobian algebra of Q (or Γ) as
A := CQ/⟨pα+ − pα− | α ∈ Q1 ⟩.
Yusuke Nakajima (Kavli IPMU)

Algebras arising from dimer models

September 19th, 2018

5 / 15Perfect matchings of dimer models
Definition
A perfect matching of a dimer model Γ is a subset D of edges such thatfor any node n there is a unique edge in D containing n as the end point.
For example, perfect matchings of our dimer model are

D0

D1

D5
Yusuke Nakajima (Kavli IPMU)

D2

D6

D3

D7Algebras arising from dimer models

D4

D8
September 19th, 20186 / 15

Calabi-Yau properties on Jacobian algebras
We consider
Q : the quiver associated with a “consistent” dimer model Γ,
A : the Jacobian algebra of Q,
D : a perfect matching of Γ.
We define the degree dD on each arrow a ∈ Q1 of Q as{1 if a ∈ DdD (a) =0 otherwise
This makes the Jacobian algebra A a graded algebra.

Theorem (Broomhead, Amiot-Iyama-Reiten)
A is a bimodule 3-Calabi-Yau algebra of Gorenstein parameter 1,that is, A ∈ perAe and ∃P• : graded proj. resolution of A as Ae -mod. s.t.
P• ∼= P•∨ [3](−1)
where (−)∨ := HomAe (−, Ae ).
Yusuke Nakajima (Kavli IPMU)

Algebras arising from dimer models

September 19th, 20187 / 15

2-representation infinite algebras arising from dimer models
We consider
Q : the quiver associated with a consistent dimer model Γ,
A : the graded Jacobian algebra of Q whose grading induced by D(This is bimodule 3-CY algebra of GP 1),
AD : the degree zero part of A.

Theorem (Keller, Minamoto-Mori, Herchend-Iyama-Oppermann)
We assume that AD is finite dimensional.
Then AD is a 2-representation infinite algebra (or quasi 2-Fanoalgebra), that is,gl.dimAD ≤ 2andν2−i (AD ) ∈ modAD for all i ≥ 0

where ν2− := ν − ◦ [2] : Db (modAD ) → Db (modAD ).
On the other hand, the “ 3-preprojective algebra” of AD is A.
Yusuke Nakajima (Kavli IPMU)

Algebras arising from dimer models

September 19th, 20188 / 15

2-representation infinite algebras arising from dimer models
Question
When is the degree part AD finite dimensional ?

23223210

110123223

2

AD0 : finite dimensional

AD1 : not finite dimensional⇒ To understand this problem, consider the perfect matching polygon.
Yusuke Nakajima (Kavli IPMU)

Algebras arising from dimer models

September 19th, 20189 / 15

The perfect matching polygon
Give the orientation to each edge of Γ:
Fix a perfect matching D ′ of Γ.
For each perf. match. D, the diﬀerence D−D ′ will be a 1-cycle on T.
Consider D−D ′ as the element in [D−D ′ ] ∈ H1 (T) ∼= Z2 .

D0

D1

D2

D3

D4
Remark that[Di − D0 ]=(0, 0)
if i = 0, 5, · · ·, 8

[D1 −D0 ]= (1, 0)[D2 −D0 ]= (0, 1)

Yusuke Nakajima (Kavli IPMU)[D3 −D0 ]= (−1, 0)
Algebras arising from dimer models[D4 −D0 ]= (0, −1)
September 19th, 201810 / 15

The perfect matching polygon
The perfect matching polygon of Γ:∆Γ := conv{[D − D ′ ] ∈ Z2 | D is a perfect matching of Γ}For our example, the PM polygon of our dimer model is
D2
D3

D0 , D5 , D6 , D7 , D8
D1

D4This ∆Γ is determined uniquely up to translations.
D is called an internal perfect matching if it corresponds to aninterior lattice point of ∆Γ .
Yusuke Nakajima (Kavli IPMU)

Algebras arising from dimer models

September 19th, 201811 / 15

2-representation infinite algebras arising from dimer models
Theorem A (N., Bocklandt etc.)
Let Q be the quiver associated with a consistent dimer model Γ.
Then, the following conditions are equivalent.
(1) D is an internal perfect matching.
(2) Q − D is an acyclic quiver.
(3) AD is a finite dimensional algebra.
When this is the case, AD is a 2-representation infinite algebra.
We recall that
2

3

2

2

3

2

10

1

1

0

1

2

3

2

2

3

2

D2
D3

AD0 : fin. dim.

AD1 : not fin. dim.

D0 , D5 , D6 , D7 , D8D1D4∆Γ : PM polygon

Is there a relationship between internal perfect matchings ?
Yusuke Nakajima (Kavli IPMU)

Algebras arising from dimer models

September 19th, 201812 / 15

Mutations of perfect matchings and derived equivalences
The mutation of perfect matchings:
µ+
ke.g.
kk

µ−
k

In particular, if k is a source (resp. sink) of Q − D,−µ+k (D) (resp. µk (D)) is a perfect matching.
−k is a sink of Q − µ+k (D) (resp. a souce of Q − µk (D) ).
++ −µ−k (µk (D)) = D (resp. µk (µk (D)) = D).

Theorem B (N.)
Let Γ be a consistent dimer model and Q be the associated quiver. LetD, D ′ be internal perfect matchings. Then, the followings are equivalent.
(1) D and D ′ are “mutation equivalent”.
(2) D and D ′ correspond to the same interior lattice point of ∆Γ .
Yusuke Nakajima (Kavli IPMU)

Algebras arising from dimer models

September 19th, 201813 / 15

Mutations of perfect matchings and derived equivalences
2121230322112231230
3222

1

122322

12

Yusuke Nakajima (Kavli IPMU)

02

10233033032

1202

12

Algebras arising from dimer models

September 19th, 201814 / 15

Mutations of perfect matchings and derived equivalences
Corollary (Theorem B + Iyama-Oppermann)
Let Γ be a consistent dimer model and Q be the associated quiver. If
D, D ′ are internal perfect matchings corresponding to the same interior
lattice point of ∆Γ , then we have thatDb (modAD ) ∼= Db (modAD ′ ).
Note that
Iyama-Oppermann showed that if k is a source of Q − D and′∼µ+k (D) = D , then ∃tilting AD -module Tk s.t. EndAD (Tk ) = AD ′ .
The converse of this corollary is not true, that is, even if
Db (modAD ) ∼= Db (modAD ′ ), D and D ′ might not correspond to
the same interior lattice point.
Yusuke Nakajima (Kavli IPMU)

Algebras arising from dimer models

September 19th, 201815 / 15