Lagrangian Fibrations on Grassmannians
and Cluster Transformations
Yuichi Nohara
Joint work with Kazushi Ueda
Toric Topology 2019 in Okayama
Okayama University of Science
November 21, 20191

Introduction
For each triangulation of a convex n-gon, one can associate(1) a completely integrable system on Gr(2, n) = Gr(2, Cn)(N.-Ueda)(2) a toric degeneration of Gr(2, n)(Speyer-Sturmfels)(3) a cluster chart on the Landau-Ginzburg mirror of Gr(2, n)(Fomin-Zelevinsky)“Theorem”. (1) and (3) are SYZ mirror.
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Triangulations
For a triangulation of a convex n-gon( ←→a trivalent tree with n leaves), set12dual graph8Γdiag:= the set of diagonals (i, j),Γside:= {(i, i + 1) | i = 1, . . . , n},
Γ:= Γside ∪ Γdiag,3

7
4

where (n, n + 1) := (1, n).

65Remark. #Γside = n, #Γdiag = n − 3, and#Γ = (2n − 4) + 1 = dimC Gr(2, n) + 1.

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Lagrangian torus fibrations on the Grassmannian Gr(2, n)Theorem (N.-Ueda). For each triangulation Γ of a convex n-gon, one
can associate a completely integrable systemΦΓ = (φij )(i,j)∈Γ\{(1,n)} : Gr(2, n) → RN ,where N = dimC Gr(2, n) = 2n − 4. The image ∆Γ = ΦΓ(Gr(2, n)) isan N -dimensional convex polytope.
Example (n = 4). ΦΓ : Gr(2, 4) → R4
coincides with the Gelfand-Cetlin system for each Γ (up to U (4)-action). ∆Γ∼ [0, λ] on which the fibers
has an edge =of ΦΓ are non-torus Lagrangians:∼ U(2) =∼ S1 × S3
Lt =(0 < t < λ).

0λ/2λ4

Construction of ΦΓ : Gr(2, n) → RNIdentify Gr(2, n) with the adjoint orbit Oλ of √diag(λ, λ, 0, . . . , 0) in −1u(n):√ iOλ = {x ∈ −1u(n) | eigenval. = λ, λ, 0, . . . , 0}. For each (i, j) ∈ Γ \ {(1, n)}, definei



j









 j


φij (x) = max. eigenvalue of (xkl )i≤k,l<j .

Remark. If the triangulation Γ is “caterpillar”,ΦΓ is the Gelfand-Cetlin system (GuilleminSternberg).

5Toric degenerations of Gr(2, n)Theorem (Speyer-Sturmfels).
{toric degenerations of Gr(2, n)}/ ∼1-1←→ {trivalent trees with n leaves Γ}.
Theorem (N.-Ueda). For each triangulation Γ of a convex n-gon, thecentral fiber XΓ of the corresponding toric degenerationX Γ ⊃ XΓ↓↓C ∋ 0
has moment polytope ∆Γ. Furthermore, ΦΓ : Gr(2, n) → ∆Γ can bedeformed into the toric moment map XΓ → ∆Γ.

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Mirror Symmetry for Fano Manifolds∨ , W ) : Landau-Ginzburg model:( N -dim. Fano←→(X)()∨c1(X) > 0
X : a non-cpt cpx mfd.
⇔ Ric > 0
W : X ∨ → C (or Λ)
Symp. (resp. cpx.) geom. ←→ Cpx. (resp. symp.) geom.

X“Classical” Mirror Symmetry∼ Jac(W )“ = ”Λ[y ±1, . . . , y ±1]
QH ∗(X; Λ) =1N{∑∞ a T λiwhere Λ =i=1 i/(}∂W
∂W,...,∂y1∂yN),ai ∈ C, λi ∈ R, λi → ∞ is the Novikov field.
()
Q∨
×λFS ) ←→ X = C , W (y) = y + y , Q = T .
/()±2
Jac(W ) = Λ[y ] 1 − Q/y = Λ[y]/(y 2 − Q) ∼ QH ∗(P1; Λ)

Example. (P1, λωmirror=∼ Λ[y]/(y 2 = 0).
cf. H ∗(P1; Λ) =7

Strominger-Yau-Zaslow Conjecture
X and X ∨ admit dual (special) Lagrangian torus fibrations
∨,Xzz

XAAA

AAAAA

Φ

Bzz
zzz
∨
|zz

Φ

i.e., (Φ∨)−1(u) = moduli of flat U(1)-connections on L(u) = Φ−1(u).
X ∨ = moduli space of “Lagrangian branes” (L(u), ∇).
The superpotential W is given by the disk potential
PO(L, ∇)=∑

n(β)zβ (L, ∇),β∈π2 (X,L)
zβ (L, ∇)= T ω(β) hol∇(∂β),

wherehol.

n(β) = “#”{v : (D2, ∂D2) → (X, L) | [v] = β}hol∇(∂β) = holonomy of ∇ along v(∂D2) ⊂ L.
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Example: SYZ mirror of P1
For a Lagrangian fiber L(u) ⊂ P1 of
the moment map
Φ : (P1, λωFS) → ∆ = [0, λ]and a flat U (1)-connection√∼ L(u),∇ = d + −1xdθ, θ ∈ R/Z =PO(L(u), ∇) =2∑T ω(Di) hol∇(∂Di)

i=1 √√u−1xλ−u−=T e+Te −1xQ=y+ ,y√uwhere y = T e −1x, Q = T λ.
9Potential function for Lagrangian torus fibers L(u)Theorem (Cho-Oh, Fukaya-Oh-Ohta-Ono). Let X be a toric Fano
manifold, and write its moment polytope as∆ = {u ∈ RN | ℓj (u) = ⟨v j , u⟩ − τj ≥ 0, j = 1, . . . , m}for primitive v i ∈ ZN . Then for a Lagrangian torus fibers L(u) and a√∑flat U (1)-connection ∇ = d + −1 i xidθi on L(u),PO(u, x) = PO(L(u), ∇) =√m∑zβj (u, x),j=1zβj (u, x) = T ℓj (u)e −1⟨v j ,x⟩ = T −τj y v j .
Furthermore, PO coincides with the superpotentialof the LG mirror of X.
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Landau-Ginzburg mirror of Grassmannians
Marsh-Rietsch: The LG mirror of Gr(k, n) = Gr(k, Cn) is given by(X ∨ = Gr(n − k, (Cn)∗) \ D, W ),where D is an anti-canonical divisor of Gr(n − k, (Cn)∗).
Remark. X ∨ is a partial compactification of (C×)N , N = dimC Gr(k, n).
∼ Gr(2, Cn) = Gr(2, n).
k = 2 case: Identify Gr(n − 2, (Cn)∗) =D=∪{pi,i+1 = 0},i∈Z/nZn−2∑ pi,i+2

p1,n−1p2,nW=+,+Qpp
pn−1,n1,ni=1 i,i+1∧
where [pij ]1≤i<j≤n are the Plücker coordinates on Gr(2, n) ⊂ P( 2 Cn).
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The disk potential for ΦΓ : Gr(2, n) → ∆Γ
Theorem (N.-Ueda). For each triangulation Γ, the disk potential POΓ
for the Lagrangian torus fibers of ΦΓ is given by the same formula as
in the toric Fano case:POΓ(y ) =m∑

T −τj y v j ,∆Γ = {u | ⟨v j , u⟩ − τj ≥ 0, j = 1, . . . , m}.

i=1

which gives a rational functionPOΓ : (C×)N = (C×)Γ\{(1,n)} −→ C.
Moreover, there exists an embedding ιΓ : (C×)N ,→ X ∨ such thatι∗ΓW = POΓ.
Idea of the proof. Deform ΦΓ into a toric moment map, and countholomorphic disks in a (singular) toric variety.
12Cluster charts on Gr(2, n)
A cluster algebra is a commutative ring generated by a “system” ofcluster variables. Two sets of cluster variables x = {x1, . . . , xN } andx′ = (x \ {xk }) ∪ {x′k } are related by an cluster mutation
xk x′k =∏i→kxi +∏xii←kCluster algebra structure on C[Gr(2, n)]cluster var. {pij }(i,j)∈Γ = {(pij )(i,j)∈Γdiag , (pij )(i,j)∈Γside },|{zmutable} |{zfrozen}

cluster mutation = Plücker rel. pik pjl = pij pkl + pil pjk .
The “SYZ mirror” of ΦΓ Gr(2, n) → ∆Γ gives a cluster chartUΓ := ιΓ((C×)N ) = {[pij ] ∈ Gr(2, n) | pij ̸= 0, (i, j) ∈ Γ}.
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Wall-crossing formula for disk counting
If a Lagrangian torus fibration Φ : X → B
has a singular fiber, the weighted countzβ (L, ∇) of pseudo-holo. disks for Lag.
fibers L = Φ−1(u) changes when u crosses
a wall{u ∈ B | L bounds hol. disk of µL = 0}.

X

B
wall

Theorem (Auroux, Pascaleﬀ-Tonkonog). Assume that each fiber L(u0)on the wall bounds a unique simple pseudo-holo. disk α of Maslov
index 0. Then zβ ’s on chambers separated by the wall are related by(zβ (L, ∇) 7→ zβ (L, ∇) 1 +∑ak zα(L, ∇)k)[∂β]·[∂α].

k≥1{z}|disk bubbles
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Wall-crossing formula and Plücker relations
Theorem (N.-Ueda). Suppose that Γ and Γ′ are related by a flip.
Then there exists a 1-parameter family Φt (0 ≤ t ≤ 1) of Lagrangianfibrations on Gr(2, n) such thatΦ0 = ΦΓ,Φ1 = Φ Γ ′ .

For 0 < t < 1, the wall-crossing formula zβ ′ = zβ (1 + zα) for Φt coincides with the Plücker relation(pjl pi,i+1pil pjk
pkl pi,i+1=··1+pijpilpikpilpij pkl)

disk bubbles

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Remarks

1. In general,∪ΓUΓ⊊ X ∨, and X ∨ \∪UΓ contains critical points ofΓ

the superpotential W . The complement should corresponds to• non-torus Lagrangian fibers of ΦΓ (N.-Ueda).
• singular Lagrangian fibers of Φt (Hong-Kim-Lau).
2. [Scott] For k ≥ 3, C[Gr(k, n)] is a cluster algebra such that{Plücker coordinates} ⊊ {cluster variables}.
• ∃Lagrangian fibrations corresponding to cluster charts given
by Plücker coord. (Castronovo).
• wall-crossing on Gr(3, 6) (In progress).
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