Exotic Lagrangian tori in Grassmannians

We describe an iterative construction of Lagrangian tori in the complex Grassmannian $\operatorname{Gr}(k,n)$, based on the cluster algebra structure of the coordinate ring of a mirror Landau-Ginzburg model proposed by Marsh-Rietsch. Each torus comes with a Laurent polynomial, and local systems controlled by the $k$-variables Schur polynomials at the $n$-th roots of unity. We use this data to give examples of monotone Lagrangian tori that are neither displaceable nor Hamiltonian isotopic to each other, and that support nonzero objects in different summands of the spectral decomposition of the Fukaya category over $\mathbb{C}$.

1. Introduction 1.1. Lagrangian tori. The construction and classification of Lagrangian submanifolds is a driving question in symplectic topology, with Lagrangian tori having a prominent role. One reason for this is the origin of the field in the Hamiltonian formulation of classical mechanics. In this context, the Arnold-Liouville theorem constrains the level sets of a completely integrable system to be Lagrangian tori; see e.g. Duistermaat [12]. A more recent motivation is the geometric description of mirror symmetry, where Lagrangian tori arise as generic fibers of Strominger-Yau-Zaslow fibrations [44]. Lagrangian tori are also of interest in low-dimensional topology: the Luttinger surgery [27] operation was used by Auroux-Donaldson-Katzarkov [4] to study symplectic isotopy classes of plane curves; Vidussi [48] and Fintushel-Stern [13] found connections between Seiberg-Witten invariants and Lagrangian tori. In general dimension, Lagrangian tori in the standard symplectic R 2n have been the subject of much investigation: Viterbo [49] and Buhovsky [5] constrained their Maslov class; Chekanov [9] classified those of product type [9]; Chekanov-Schlenk [8] and Auroux [3] constructed examples that are not products.

Disk potentials.
A unifying way to think about these results is to consider Lagrangian tori L Ă R 2n " C n as boundary conditions for maps u : D 2 Ñ C n satisfying the nonlinear Cauchy-Riemann type equation B J puq " 0, where J is an almost-complex structure on the target that may vary from point to point and be non-integrable. One can try to understand how J-holomorphic disks change as L is deformed through Lagrangian embeddings; many known results focus on deformations by Hamiltonian isotopies. This line of thought generalizes to the global case, when L Ă X is not in a Darboux chart of the symplectic manifold X; however, J-holomorphic disks are not easy to describe for an arbitrary target X. Since the work of Floer [14] and Oh [34], the monotone case has been the focus of much investigation. A symplectic manifold pX 2N , ωq is monotone if rωs and the first Chern class c 1 pXq are positively proportional in H 2 pX; Rq; a Lagrangian L Ă X is monotone if the area ωpβq of disk classes β P H 2 pX, L; Rq is positively proportional to their Maslov index µpβq. In this setting, for Partially supported by NSF grant DMS 1711070. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. arXiv:1910.10888v2 [math.SG] 12 Jun 2021 generic J the moduli space M J pL, βq of unparametrized J-holomorphic disks with boundary on L, homology class β and a boundary marked point ‚ is a compact manifold of dimension µpβq`dimpLq´2. One can encode counts of J-holomorphic disks in a finite generating function called disk potential, and try to establish general properties of the function that may imply something about its coefficients. The disk potential of a monotone Lagrangian torus L N Ă X 2N is defined as W L " ÿ βPH 2 pX,L;Zq c β pLqx Bβ P Crx1 , . . . , xN s ; here the degree c β pLq " deg pev ‚ : M J pL, βq Ñ Lq P Z of the evaluation map ev ‚ at the marked point ‚ P BD 2 is independent of J, and rigid disks have µpβq " 2; monotonicity implies that c β pLq ‰ 0 for finitely many classes β. When writing the disk potential, we implicitly assume the choice of a basis of cycles γ 1 , . . . , γ N P H 1 pL; Zq -Z N , so that J-holomorphic disks with boundary of class Bβ " k 1 γ 1`¨¨¨`kN γ N contribute to the monomial x Bβ " x k 1¨¨¨x k N . A Hamiltonian isotopy φ t gives an isomorphism pφ t q˚: H 1 pL; Zq Ñ H 1 pφ t pLq; Zq, and W φ t pLq " W L in the induced basis of cycles. It is known that the critical points of W L obstruct Hamiltonian displaceability; see Cho-Oh [10,Proposition 7.2] for toric moment fibers, Auroux [2, Proposition 6.9] and Sheridan [43,Proposition 4.2] for a general discussion. Disk potentials have been used by Vianna [46,47] to distinguish infinitely many monotone Lagrangian tori in complex surfaces X of Fano type; see also Pascaleff-Tonkonog [36].

A cluster construction.
In this article, we construct Lagrangian tori in a class of Fano manifolds of arbitrarily large dimension: the Grassmannians Grpk, nq of complex kdimensional linear subspaces in C n . Construction 1.1. Given integers 1 ď k ă n, for any Plücker sequence s of type pk, nq there is a corresponding Lagrangian torus L s Ă Grpk, nq, equipped with a canonical basis of cycles γ d P H 1 pL s ; Zq labeled by Young diagrams d Ď kˆpn´kq. The torus comes with a rational function W s of formal variables x d .
The Plücker sequences s are based on the notion of quiver mutation from representation theory; see Section 2 for more details and Example 1.3 below. The Lagrangian tori L s are obtained from algebraic degenerations to (singular) toric varieties Grpk, nq XpΣ s q, using a general technique for constructing completely integrable systems on complex projective manifolds studied by Harada-Kaveh [21]; the notion of toric degeneration is explained in Section 3. Such degenerations of Grassmannians have been studied by Rietsch-Williams [40] in connection with the theory of Okounkov bodies [35,25,23]. All Plücker sequences start from a single initial seed, and the rational functions W s are obtained by explicit rational changes of variable from a single initial Laurent polynomial W 0 , whose variables are labeled by those Young diagrams d Ď kˆpn´kq that are rectangles. In [7], it was proved that W 0 is in fact the disk potential of the monotone Lagrangian torus fiber of the Gelfand-Cetlin integrable system introduced by Guillemin-Sternberg [20]. The formulation of the construction as iterative procedure is particularly convenient for computational purposes. To illustrate this point, we created a random walk that generates Plücker sequences s of arbitrary length, and computes the corresponding Laurent polynomials W s explicitly; the code is available for inspection and experiments [6]. 1.4. Topology of Laurent/positivity phenomena. By computing a few examples of W s , one quickly notices the following two phenomena, which are not a direct consequence of the construction: (1) the rational function W s is a Laurent polynomial ; (2) the coefficients of each Laurent polynomial W s are natural numbers . Property (1) is related to the Laurent phenomenon of cluster algebras, a notion developed by Fomin-Zelevinsky [15]. Think each x d as a Plücker coordinate on the dual Grassmannian Gr _ pk, nq " Grpn´k, nq Ă P p n k q´1 in its Plücker embedding; the definition of Plücker coordinate is recalled in Definition 2.1. Each Plücker sequence s singles out an open algebraic torus chart T s Ă U k,n " Gr _ pk, nqzD _ F Z in the complement of a particular divisor D _ F Z , and the global regular functions A k,n " OpU k,n q form a cluster algebra; see Scott [41]. The space U k,n is a smooth affine variety known as open positroid stratum, and its properties have been the focus of several works in representation theory, combinatorics, topology and mirror symmetry [26,37,24,42,40]. By a result of Marsh-Rietsch [28], one can think of each W s as restriction W s " W |Ts of a single global regular function W P A k,n called Landau-Ginzburg potential. Property (2) is related to positivity of cluster algebras, which has been proved by Gross-Hacking-Keel-Kontsevich [19]. Their proof consists in interpreting the coefficients of certain elements of a cluster algebra as generating functions of tropical curves called broken lines. In mirror symmetry, broken lines are expected to correspond to the J-holomorphic disks of symplectic topology, and this heuristic leads us to the following.  Example 1.3. Let k " 2 and n " 5. Figure 5 represents a Plücker sequence s : pQ 0 , W 0 q Ñ pQ 1 , W 1 q Ñ pQ 2 , W 2 q Ñ pQ 3 , W 3 q Ñ pQ 4 , W 4 q Ñ pQ 5 , W 5 q of type p2, 5q and length five, where the final step and the initial one coincide. At each step pQ i , W i q, the graph Q i is a quiver whose nodes are labeled by Plücker coordinates x d for some collection of Young diagrams d Ď 2ˆ3, and W i is a Laurent polynomial of the variables x d . A step pQ i , W i q Ñ pQ i`1 , W i`1 q in the sequence consists in performing a quiver mutation at a mutable node v of Q i as described in Section 2. This procedure changes the label lpvq of the node v in Q i to a new label l 1 pvq of the same node in Q i`1 . The two labels are related by the following exchange relation: lpvqlpv 1 q is a sum of two terms, obtained by taking the product of labels lpwq from incoming/outgoing nodes w adjacent to v respectively. The rational function W i`1 is obtained from W i by using the previous relation to replace the label lpvq with lpv 1 q, and becomes Laurent modulo Plücker relations, i.e. when interpreted as element of the function field FracpA 2,5 q " CpU 2,5 q " CpGr _ p2, 5qq of the dual Grassmannian Gr _ p2, 5q " Grp3, 5q. The intermediate steps of Construction 1.1 produce Lagrangian tori L 0 , L 1 , L 2 , L 3 , L 4 Ă Grp2, 5q. In this case, all the tori are monotone and the Laurent polynomials W i for 0 ď i ď 4 match their disk potentials W L i : The equality W s " W Ls is an application of a general result of Nishinou-Nohara-Ueda [30] on the behavior of disk potentials under toric degeneration, which also implies monotonicity of L s ; see Proposition 3.12. This result gives a sufficient condition for the equality W s " W Ls , which is the existence of a small toric resolution for the singular toric variety XpΣ s q; see Definition 3.11. Due to the combinatorial nature of toric varieties, for any given Plücker sequence s one can check this condition in finitely many steps. In Section 4 we use this to describe a sample application in the smallest example not accessible by previous techniques. We call these tori exotic, because only one monotone torus was previously known: the Gelfand-Cetlin torus. The new examples are of the form L s for some Plücker sequence s, and are distinguished by a combination of two invariants: the number of critical points of their disk potential W Ls and the f -vector of its Newton polytope. This strategy applies without modification to arbitrary Grassmannians. If Conjecture 3.8 holds, the same arguments of Theorem 4.16 imply that the tori L s Ă Grpk, nq are always nondisplaceable, and generally not Hamiltonian isotopic. Note that Conjecture 3.8 may still hold when the toric variety XpΣ s q has no small toric resolution, and the result of Nishinou-Nohara-Ueda [30] does not apply. In this case, positivity of the coefficients of W s suggests an enumerative interpretation in terms of counts of J-holomorphic disks with boundary on L s . We plan to explore this in a separate work, simply pointing out here a possible interpretation in terms of low-area Floer theory in the sense of Tonkonog-Vianna [45]. For k " 1 one has projective spaces Grp1, nq " P n´1 , and there is only one Plücker sequence s of lenght 0; in this case W s is the disk potential of the Clifford torus. In particular, Construction 1.1 does not imply the existence of exotic tori in P 2 established by Vianna [46,47]. For k " 2 Construction 1.1 recovers a different one studied by Nohara-Ueda [32], who introduced a collection of Lagrangian tori in Grp2, nq corresponding to triangulations of an n-gon; the relation is explained in Lemma 4.4, and it implies that Conjecture 3.8 holds when k " 2.
1.5. Probing the spectral decomposition. Since for k " 2 all tori L s Ă Grp2, nq are monotone, it is natural to think of them as objects of the monotone Fukaya category. As described by Sheridan [43], the Fukaya category of a monotone symplectic manifold X has a spectral decomposition FpXq " à λ F λ pXq .
The summands are A 8 -categories indexed by the eigenvalues λ of the operator c 1 ‹ of multiplication by the first Chern class acting on the small quantum cohomology. The objects of the λ-summand are monotone Lagrangians L ξ equipped with a rank one local system ξ such that m 0 pL ξ q " ÿ β c β pLq hol ξ pBβq " λ .
Definition 3.7 introduces some natural local systems supported on the Lagrangian tori L s Ă Grpk, nq, that are controlled by the values of k-variables Schur polynomials at certain roots of unity. These local systems generalize the ones studied in [7] for the Gelfand-Cetlin torus, that were controlled by Schur polynomials corresponding rectangular Young diagrams. When k " 2, we show that the corresponding objects split-generate the derived Fukaya category DFpGrp2, nqq in some cases, notably including examples where the Gelfand-Cetlin torus alone fails to do so.
Theorem 1.5. (see Theorem 4.8) If n " 2 t`1 for some t P N`, the derived Fukaya category DFpGrp2, 2 t`1 qq is split-generated by objects supported on a single Plücker torus.
The Lagrangian torus in the statement is associated to a special special triangulation of the n-gon, that we call dyadic. In fact, Section 4 contains a criterion to prove split-generation of DFpGrp2, nqq by objects supported on any number of tori L s Ă Grp2, nq, whenever n is odd. The criterion is based on a construction of triangulations of the n-gon whose sides lengths avoid the prime numbers appearing in the factorization of n. Theorem 1.6. (see Theorem 4.11) Let n ą 2 be odd, and consider its prime factorization n " p e 1 1¨¨¨p e l l . If for all 1 ď i ď l there exists a triangulation Γ i of rns that is p i -avoiding, then DFpGrp2, nqq is split generated by objects supported on l Plücker tori.
Remark 1.7. A standard consequence of split-generation is that any monotone Lagrangian supporting nonzero objects of the Fukaya category must intersect the generator.
These results seem to suggest that objects supported on the tori L s could split-generate DFpGrpk, nqq in general, although split-generation over C has a subtle relation with the location of the critical points of W P A k,n relative to the torus charts T s Ă U k,n . For example, split-generation over C fails for Grp2, 4q, where W has two critical points in a complex codimension 2 locus of U k,n which is not covered by cluster charts. We plan to investigate in a separate work how the situation changes when considering bulk-deformations in the sense of Fukaya-Oh-Ohta-Ono [17].
1.6. Mirror symmetry and abundance of Lagrangian tori. This article can be thought of as part of a broader program aimed at investigating the abundance of Lagrangian tori in Fano manifolds X with an anti-canonical divisor D Ă X whose complement U " XzD is a cluster variety. This class includes many homogeneous varieties X " G{P with P Ă G parabolic subgroup of a complex linear algebraic group. The cluster variety U comes with a Langlands dual cluster variety U _ , and Gross-Hacking-Keel-Kontsevich [19] proposed that pX, Dq has a Landau-Ginzburg model pU _ , W q in the sense of homological mirror symmetry. Here W P OpU _ q is a regular function intrinsically defined by the cluster structure and given as a sum of theta functions, which are generating functions of discrete objects called broken lines in a scattering diagram. We expect that the cluster charts of U _ will correspond to certain Lagrangian tori in L Ă G{P , and that the restriction of W to different cluster charts will fully determine their disk potential W L in some cases, and in general suffice to distinguish many of their Hamiltonian isotopy classes in the spirit of Conjecture 1.2.

1.7.
Algebraic and topological wall-crossing. When W Ls " W s , the Lagrangian tori L s Ă Grpk, nq constructed in this article have disk potentials related by algebraic wall-crossing formulas by construction. It is natural to ask if these formulas correspond to a topological wall-crossing, i.e. if the tori L s are connected by families of Lagrangian immersions that bound Maslov 0 J-holomorphic disks at some intermediate time. We do not investigate this question here, but only point out that it would be interesting to see if there is a relation between our examples and the model of Lagrangian mutation studied by Pascaleff-Tonkonog [36].
Acknowledgements I thank my PhD advisor Chris Woodward for his constant encouragement and useful conversations. I also thank Mohammed Abouzaid for useful conversations about mirror symmetry, Lev Borisov for useful remarks on toric resolutions of singularities, and Lauren Williams for pointing out that only quiver mutations at 4-valent nodes are allowed as transition between plabic cluster charts, which was crucial at some point of the project.

The iterative construction
Throughout this article, k and n are integers with 1 ď k ă n. The symbol d denotes a Young diagram in the kˆpn´kq grid, obtained by placing d i consecutive boxes in the i-th row for all 1 ď i ď k, starting from the left in each row and with d 1 ě d 2 ě¨¨¨ě d k . Chosen 0 ď i ď k and 0 ď j ď pn´kq, one has a rectangular Young diagram iˆj, with iˆj " H empty diagram if i " 0 or j " 0. A full rank nˆpn´kq matrix M determines an pn´kq-dimensional linear subspace of C n by taking its column-span. If rM s is the equivalence class of M modulo column operations, write rM s P Gr _ pk, nq " Grpn´k, nq and think of it as a point of the complex Grassmannian. Each Young diagram d has a profile path, which connects the top-right corner of the kˆpn´kq grid to the bottom-left one. Labeling the steps of the path by rns " t1, . . . , nu, the vertical steps of d determine a set d | Ă rns with |d | | " k, while the horizontal steps determine a set d´Ă rns with |d´| " n´k; see Figure 2 for an example with k " 3 and n " 8. The Plücker coordinates define a projective embedding of Gr _ pk, nq in P p n k q´1 . If I k,n Ă Crx d : d Ď kˆpn´kqs is the corresponding homogeneous ideal, each x d is an element of the algebra A k,n " Crx d : d Ď kˆpn´kqs{I k,n of regular functions of the affine cone over Gr _ pk, nq.

Initial seed.
Definition 2.2. A quiver with potential of type pk, nq is a pair pQ, W q, where: (1) Q is an oriented connected graph, with no edge connecting a node to itself and no oriented loops with two edges, whose nodes are labeled by Plücker coordinates x d P A k,n ; (2) W is a Laurent polynomial in the labels of the nodes of Q .
As part of the data, the nodes of Q are partitioned in two groups, called frozen and mutable. Remark 2.3. To avoid confusion, we point out that Definition 2.2 is not a special case of the notion of quiver with potential in representation theory: although Q is a quiver in the classical sense, the potential W is an element of the commutative algebra A k,n as opposed to the non-commutative path algebra of Q.
The iterative construction we describe in this section begins with a specific quiver with potential.
Definition 2.4. The initial seed of type pk, nq is the quiver with potential pQ 0 , W 0 q, where: (1) Q 0 is the oriented labeled graph in Figure 3 ; A node of Q 0 is frozen if its label is x iˆj with iˆj " H, i " k or j " n´k; the remaining nodes are mutable.  Observe that the labels on the nodes of Q 0 are precisely the kpn´kq`1 variables x d where d is a rectangular Young diagram, and n of the nodes are frozen.

Mutation step.
Given a quiver with potential pQ, W q as in Definition 2.2, and fixed a mutable node v of Q, one can form a new labeled quiver Q 1 as follows: (1) start with Q 1 " Q, and for all length 2 paths a Ñ v Ñ b with at least one mutable node among a and b, add to Q 1 a new edge a Ñ b ; (2) modify Q 1 by reversing all the edges incident to v ; (3) remove all oriented 2-cycles formed in Q 1 , by deleting their arrows . Calling lpwq the label of a node w in Q, define new labels l 1 pwq in Q 1 by declaring l 1 pwq " lpwq if w ‰ v, and l 1 pvq " ś wÑv lpwq`ś vÑw lpwq lpvq .
Since Q and Q 1 have the same nodes, the nodes of Q 1 inherit the property of being frozen or mutable from Q.
Definition 2.5. The mutation of pQ, W q along v is the pair pQ 1 , W 1 q with Q 1 constructed as above, and W 1 obtained from W by substitution lpvq " p ś wÑv l 1 pwq`ś vÑw l 1 pwqql 1 pvq´1. A priori, mutations of quivers with potentials as in Definition 2.2 are not necessarily quivers with potentials, since l 1 pvq and W 1 are only rational functions of the Plücker coordinates x d . The following guarantees that certain iterated mutations of the initial seed of Definition 2.4 remain quivers with potentials. (1) if pQ, W q " pQ 0 , W 0 q is the initial seed of Definition 2.4, then W 1 is a Laurent polynomial in the labels of Q 1 ; (2) if in addition each mutation of the sequence is based at some node with two incoming and two outgoing edges, then the labels of Q 1 are Plücker coordinates x d .
Proof. For the reader's convenience, we explain how the statements follow from the cited results. It suffices to prove them when the sequence of mutations consists of a single mutation, as the general case follows by applying repeatedly the same argument.
(1) Marsh-Rietsch [28, Section 6.3] (see also Rietsch-Williams [40, Proposition 9.5]) showed that the potential W 0 of the initial seed is the restriction W 0 " W |T 0 of a regular function W P A k,n to an algebraic torus T 0 Ă Gr _ pk, nq defined by T 0 " trM s P Gr _ pk, nq : lpM q ‰ 0 @l label of Q 0 u .

By Scott [41, Theorem 3]
A k,n is a cluster algebra, and the rational functions labeling the nodes of Q 1 are cluster variables. Just as with the labels of Q 0 , one can use the labels of Q 1 to define an algebraic torus T 1 Ă Gr _ pk, nq via this torus is called toric chart in [41,Section 6]. By Definition 2.5, W 1 is obtained from W 0 by substitution lpvq " p ś wÑv l 1 pwq`ś vÑw l 1 pwqql 1 pvq´1. This means that W 1 is the pull-back of W 0 along the birational map from T 1 to T 0 defined by the substitution formula. It is part of the statement that A k,n is a cluster algebra that the substitution formula gives a relation so that W 1 " W |T 1 is a restriction of W as well. In particular, W 1 is a regular function on the algebraic torus T 1 , and hence a Laurent polynomial.
(2) If the mutation from pQ, W q to pQ 1 , W 1 q is based at some node v with two incoming and two outgoing edges, denote tv1 , v2 u and tv1 , v2 u the corresponding nodes of Q adjacent to v. The substitution formula of Definition 2.5 simplifies to lpvq " pl 1 pv1 ql 1 pv2 q`l 1 pv1 ql 1 pv2 qql 1 pvq´1. By definition of mutation l 1 pvì q " lpvì q and l 1 pví q " lpví q for i " 1, 2. Moreover, by assumption the l labels are Plücker coordinates, meaning that lpvì q " x dì and lpví q " x dí for i " 1, 2 and lpvq " x d for some Young diagrams d, d1 , d2 , d1 , d2 Ď kˆpn´kq. Scott [41, proof of Theorem 3] proves that this implies l 1 pvq " x d 1 for some Young diagram d 1 Ď kˆpn´kq too, using combinatorial objects called wiring arrangements. The same phenomenon is discussed in Rietsch-Williams [40,Lemma 5.6] in the combinatorial framework of plabic graphs; see also the proof of Proposition 3.2 for a comparison between plabic graphs and quivers.
Definition 2.7. A length l Plücker sequence of mutations of type pk, nq, denoted s, is a finite sequence of pairs pQ i , W i q with 0 ď i ď l such that: (1) pQ 0 , W 0 q is the initial seed of type pk, nq of Definition 2.4 ; (2) pQ i`1 , W i`1 q is obtained from pQ i , W i q by mutation along a mutable node with two incoming and outgoing edges, as in Definition 2.5 . If we want to suppress the length l, we denote pQ l , W l q " pQ s , W s q and call it the final quiver with potential of s.

2.3.
Relation with polytope mutations. Later on in this article, we will be interested in how the Newton polytope P s " NewtpW s q changes throughout a sequence of mutations in the sense of Definition 2.5.
Proposition 2.10 proves that each P s is a Fano polytope. The Fano condition has the following interpretation in terms of toric geometry; see [18,11] for some general background on toric varieties. Lemma 2.9. If P is a Fano polytope, then the polyhedral fan Σ " Σ f P consisting of the cones spanned by its faces is such that the associated toric variety XpΣq is Fano, meaning that: (1) the anti-canonical toric Weil divisor D Σ is Q-Cartier ; (2) D Σ is ample .
Proof. The anti-canonical toric Weil divisor is defined to be where the sum is over the one-dimensional cones ρ P Σp1q and D ρ is the prime divisor corresponding to ρ. By [11,Theorem 4 where ΣpN q is the set of maximal cones of Σ, σp1q is the set of one-dimensional cones in σ, and u ρ P Z N is the primitive generator of ρ; if m σ exists then it is unique, and when all m σ P Z N one recorvers the stronger Cartier condition. By assumption (3) in Definition 2.8 and the fact that Σ " Σ f P , one has u ρ P V pP q for all ρ P Σp1q. Now consider the polar dual polytope P˝" t v P R N : xv, uy ě´1 @u P P u ; since pP˝q˝" P and V pP q Ă Z N , the polytope P˝is reflexive. One equivalent formulation of reflexivity is to say that each vertex of the polar dual polytope u ρ P V ppP˝q˝q " V pP q defines a facet (or codimension one face) We claim that for each σ P ΣpN q one has where m σ P Q N and it satisfies the Q-Cartier condition. Polar duality exchanges the face fan Σ f and the normal fan Σ n , so that σ P Σ " Σ f P " Σ n P˝; from this point of view the vector u ρ can be thought of as inward-pointing normal to the facet F uρ Ă P˝. By [11, Proposition 2.3.8] the intersection above describes the unique vertex m σ P V pP˝q corresponding to the N -dimensional cone σ of the normal fan. Although in general m σ R Z N , one always has m σ P Q N , because P˝is the polar dual of a polytope P with V pP q Ă Z N ; see [11, Exercise 2.2.1(a)]. Finally, xm σ , u ρ y "´1 for all ρ P σp1q because m σ P F uρ by construction. This proves part (1), for part (2) proceed as follows. The Q-Cartier divisor D Σ has a support function φ D Σ : R N Ñ R, which is piecewise-linear on Σ and such that φ D Σ pu ρ q "´1 for all ρ P Σp1q. By [11, Theorem 6.1.7, Lemma 6.1.13] D Σ is ample if and only if the points t m σ : σ P ΣpN q u are the vertices of the polytope and moreover m σ ‰ m σ 1 for σ ‰ σ 1 . This is true because P D Σ " P˝and the fact that the correspondence between faces of P˝and cones of its normal fan is a bijection.
Akhtar-Coates-Galkin-Kasprzyk [1] proposed a general notion of polytope mutation that should describe how the Newton polytope of a Laurent polynomial changes under the action of special birational maps of a torus. More precisely, consider a birational map of the form for i " 1, 2 are automorphisms of the torus, specified by invertible integer matrices. If f is a Laurent polynomial, one can think of it as a polynomial in the xN variables with coefficients C h which are Laurent polynomials with B x N C h " 0 and write f " Then the rational function is again a Laurent polynomial whenever A´h|C t for all´h min ď h ă 0, and so is φ˚f " g because φ M 1 , φ M 2 are automorphisms. The Newton polytopes P " Newtpf q and P 1 " Newtpgq are convex hulls in R N of the exponent vectors of the monomials in f and g respectively. The special form of φ A singles out the x N variable, and a width vector w " p0, . . . , 0, 1q P Z N corresponding to this choice. For heights´h min ď h ď h one can form lattice polytopes w h pP q Ď P by taking the convex hull of lattice points in hyperplane sections orthogonal to the w-direction: w h pP q " ConvpP X tx¨, wy " hu X Z N q . In fact, h max´hmin P N can be thought as the width of the polytope P with respect to the w-direction, and h as a height coordinate. Calling the polytope F Ă R N has codimension at least one and lies at height h " 0. Denoting V pP q the vertices of P one has V pP q X tx¨, wy " hu Ď G h`p´h qF Ď w h pP q @´h min ď h ă 0 .
The notation P 1 " mut w pP, F q expresses the fact that P 1 is a polytope mutation of P in direction w and with factor F , and the quantity h max´hmin is called width of the mutation. We now explain how mutation in the sense of Definition 2.5 is related to polytope mutations.
Proposition 2.10. If pQ 1 , W 1 q is obtained from pQ, W q by mutation along a node v, then the Newton polytopes P " NewtpW q and P " NewtpW 1 q are related by polytope mutation. In particular, P s and XpΣ f P s q are Fano for any Plücker sequence s.
Proof. Define the complex tori T " SpecpCrxd : d label of Qsq , T 1 " SpecpCrxd : d label of Q 1 sq , and think of W and W 1 as regular functions on them. Quiver mutation along v changes the label x lpvq of Q into a new label x l 1 pvq in Q 1 , and the two are related by Up to automorphisms of tori, one can arrange the coordinates in such a way that x lpvq and x l 1 pvq go last, and the common ones appear in the same order. With this choice, there is a birational transition map between the tori The Laurent polynomial A " ś wÑv x lpwq`śvÑw x lpwq satisfies B x lpvq A " 0, and using the notation introduced in this section φ " φ A˝φM 1 , where φ M 1 is the automorphism of T that inverts the last coordinate. By direct inspection, one sees that the polytope P 0 " NewtpW 0 q corresponding to the initial potential W 0 given in Definition 2.4 is Fano in the sense of Definition 2.8. It follows from [1, Proposition 2] that P s is Fano for every Plücker sequence s, and thus XpΣ f P s q is too, thanks to Lemma 2.9.

Plücker Lagrangians
In this section, Σ denotes a complete fan in R kpn´kq , and XpΣq its associated proper toric variety; see for example [18,11] for background material on toric geometry. The reader familiar with symplectic manifolds and Hamiltonian torus actions can think of Σ as the normal fan Σ " Σ n ∆ of a moment polytope ∆, with the important caveat that XpΣq is typically singular, and not even an orbifold; in this case ∆ should be thought as the closure of the open convex region obtained from the moment map of the maximal torus orbit.
We will assume that the primitive generators of the rays of Σ in the lattice Z kpn´kq Ă R kpn´kq are the vertices of a convex polytope P , and alternatively think of Σ as its face fan Σ " Σ f P . This condition is equivalent to XpΣq being Fano, and P is sometimes called a Fano polytope. The reader should not confuse the polytopes ∆ and P : the second is always a lattice polytope, whereas the first may not be. The two polytopes are related by polar duality ∆ " P˝.

Lagrangian tori from degenerations.
Definition 3.1. If X Ă P M is a smooth subvariety of complex dimension N , an embedded toric degeneration X XpΣq is a closed subscheme X Ă P MˆC such that the map p : X Ñ C obtained by restriction of the projection satisfies the following properties: ‚ p´1pCˆq -XˆCˆas schemes over Cˆ; ‚ p´1p0q Ă P M is an orbit closure for some linear torus action pCˆq N P M ; ‚ p´1p0q is a toric variety with fan Σ . Proof. For the reader's convenience, we provide details on how to specialize the result of Rietsch-Williams [40, Theorem 1.1] to recover this statement. Each step pQ i , W i q of the Plücker sequence s corresponds to a reduced plabic graph G i of type π k,n [40, Section 3], which is a combinatorial object encoding the quiver Q i and the Laurent polynomial W i simultaneously. Nodes in Q i correspond to faces in G i , and each arrow of Q i is dual to an edge of G i , with black/white nodes of the plabic graph respectively to the right/left of the arrow. The frozen nodes of Q i correspond to boundary faces of G i , and the mutable nodes to interior faces. Mutations at some mutable node with two incoming and outgoing arrows in Q i correspond to a square move on the plabic graph G i . The Plücker variables on nodes of Q i are labeled by the Young diagrams appearing on the faces of G i , which are induced by trips as in [40,Definition 3.5]. The Laurent polynomial W i is a generating function counting matchings on the plabic graph G i [40, Theorem 18.2]; see also Marsh-Scott [29] for a proof. The initial seed pQ 0 , W 0 q corresponds to a particular plabic graph G 0 " G rec k,n , called the rectangle plabic graph in [40,Section 4]. Consider the divisor D i Ă Grpk, nq cut out by the equation x d i " 0, with d i Ď kˆpn´kq one of the n frozen Young diagrams, and call D " r 1 D 1`¨¨¨`rn D n a general effective divisor with the same support. One can associate to the pair pD, G s q a convex polytope ∆ Gs pDq known as Okounkov body [40, Section 1.2]. From now on set r 1 " . . . " r n " 1, and call D F Z " D 1`¨¨¨`Dn the corresponding divisor. There exists a scaling factor r s P Q`such that r s ∆ Gs pD F Z q is a normal lattice polytope [11, Definition 2.2.9]; normality is referred to as integer decomposition property in [40,Definition 17.7], and from [40,Proposition 19.4] one sees that the scaling factor mentioned there is related to ours by r s " r Gs n . From [40, Section 17] one gets a degeneration of Grpk, nq to the toric variety associated with the polytope r s ∆ Gs pD F Z q, and this is an embedded toric degeneration in the sense of Definition 3.1 with fan Σ s " Σ n r s ∆ Gs pD F Z q " Σ n ∆ Gs pD F Z q, where we used that the normal fan of a polytope doesn't change under scaling. In [40, Theorem 1.1] and [40, Definition 10.14], an interpretation of ∆ Gs pr 1 D 1`¨¨¨`rn D n q is given in terms of the tropicalization of W s . Setting r 1 " . . . " r n " 1, one finds in particular that for D F Z " D 1`¨¨¨`Dn in fact ∆ Gs pD F Z q " tv P R kpn´kq : xv, uy ě´1 for every vertex u P P s u ; here P s denotes the Newton polytope of the Laurent polynomial W s , i.e. the convex hull of its exponents. To see this, observe that from [40, Theorem 1.1] and [40, Definitions 10.7, 10.14] one has v P ∆ Gs pD F Z q ðñ TroppW i|Ts qpvq ě´1 for i " 1, . . . , n ; here each W i is a special term of a rational function W " W 1`. . .`W n on Gr _ pk, nq defined in [40, Definition 10.1], and T s " t rM s P Gr _ pk, nq : lpM q ‰ 0 @l label of Q s u is a complex torus chart such that W |Ts " W s ; compare (1) of Proposition 2.6. The symbol Tropp¨q denotes tropicalization of Laurent polynomials, which produces a piece-wise linear function defined as Trop˜ÿ u c u x u¸p vq " min u xv, uy .
Summarizing, v P ∆ Gs pD F Z q is equivalent to xv, uy ě´1 for every u exponent of a monomial in W s . By convexity, the latter condition is equivalent to asking xv, uy ě´1 only for those u that are vertices of the Newton polytope P s of W s . We have thus recovered the polar dual polytope, i.e. ∆ Gs pD F Z q " Ps . Since the normal fan of a polytope equals the face fan of its polar dual and polar duality is an involution, we find that Σ s " Σ n ∆ Gs pD F Z q " Σ f P s as in the statement.
Remark 3.3. The toric variety XpΣ s q depends only the final step of s in the following sense. Suppose that Q s and Q s 1 are the final quivers of two different Plücker sequences. Consider the charts T s " t rM s P Gr _ pk, nq : lpM q ‰ 0 @l label of Q s u and T s 1 " t rM s P Gr _ pk, nq : l 1 pM q ‰ 0 @l 1 label of Q s 1 u ; these were already considered in (1) of Proposition 2.6, were it was observed that W s " W |Ts and W s 1 " W |Ts . If the quivers Q s and Q 1 s have equal sets of labels, then T s " T s 1 and therefore the final Laurent polynomials of the two Plücker sequences are equal: W s " W s 1 . Since Σ s and Σ s 1 are the face fans of their Newton polytopes P s " P s 1 , it follows that Σ s " Σ s 1 and thus XpΣ s q " XpΣ s 1 q.
In what follows, we endow the Grassmannian Grpk, nq with the symplectic structure obtained by restriction of the Fubini-Study form on the target projective space of the Plücker embedding. open set U s Ă Grpk, nq and a smooth submersion µ s : U s Ñ R kpn´kq whose image is the interior of the polytope, and whose fibers are Lagrangian tori. Call L s " µ´1 s p0q. If p P L s , the tangent space to the fiber at p is T p pL s q " kerpd p µ s q. Therefore, the standard basis of R kpn´kq lifts under d p µ s to a basis of T p pU s q{T p pL s q. Since the symplectic structure vanishes on L s , the lift defines a symplectic-dual basis of T p pL s q. Since L s is a torus, the vectors of this basis are tangent to natural closed loops in L s , and their homology classes give a basis of H 1 pL s ; Zq which is independent of the point p P L s . Remark 3.6. Note that x H is the label of a frozen node in the initial quiver Q 0 . Since frozen labels do not change under mutation, x H is in fact a the label of a forzen node in any quiver Q s .

3.2.
Local systems from Schur polynomials. Given a Young diagram d Ď kˆpn´kq, the corresponding k-variables Schur polynomial is defined as where the sum runs over semi-standard tableaux T d on d. The tableaux T d are obtained by labeling d with integers in t1, . . . ku, in such a way that rows are weakly increasing and columns are strictly increasing. The exponent t i is the number of times that the integer i appears in the tableaux T d . If I is any of the`n k˘s ets of roots of z n " p´1q k`1 with |I| " k, it makes sense to evaluate S d pIq P C without specifying an order on the elements of I because Schur polynomials are symmetric functions.
Definition 3.7. For each Plücker torus L s Ă Grpk, nq and set I, denote ξ I the rank one local system whose holonomy hol ξ I : H 1 pL s ; Zq Ñ Cˆaround the canonical cycles of Definition 3.5 is given by the formula hol ξ I pγ d q " S d pIq P Cˆ; if S d pIq " 0 for some d appearing in a label x d of the final quiver of s, then ξ I is not defined.

3.3.
A conjecture and some evidence. If s is a Plücker sequence of type pk, nq, after setting x H " 1 the Laurent polynomial W s can be thought of as a regular function on the algebraic torus H 1 pL s ; Zq b Cˆ-pCˆq kpn´kq . Setting x H " 1 corresponds to thinking A k,n " OpU k,n q as algebra of regular functions on U k,n " Gr _ pk, nqzD _ F Z rather than on its affine cone. The canonical cycles γ d P H 1 pL s ; Zq of Definition 3.5 give an isomorphism of schemes H 1 pL s ; Zq b Cˆ-pCˆq kpn´kq , where one thinks the latter torus as having coordinates x d labeled by Young diagrams d Ď kˆpn´kq such that d ‰ H and d appears in some label of the quiver Q s . Under the identification described by Scott [41,Theorem 4], one can think that H 1 pL s ; Zq b Cˆ-T s Ă Gr _ pk, nq where T s " trM s P Gr _ pk, nq : x d pM q ‰ 0 @d label of Q s u .
Conjecture 3.8. If s and s 1 are two Plücker sequences of type pk, nq, and φ is a Hamiltonian isotopy of Grpk, nq such that φpL s q " L s 1 , then the induced map φ˚: H 1 pL s ; Zq Ñ H 1 pL s 1 ; Zq is such that W s " W s 1˝pφ˚b id Cˆq , where " denotes equality up to automorphisms of T s . Remark 3.9. The reason for " in the conjecture above is the following. Suppose that W s is the disk potential of L s , i.e. W s " W Ls . By Hamiltonian invariance of the disk potential, if L s 1 " φpL s q then W L s 1 " W Ls " W s as long as we express the disk potential of L s 1 in the basis of cycles induced by φ˚: H 1 pL s ; Zq Ñ H 1 pL s 1 ; Zq. Instead, the Laurent polynomial W s 1 expresses the disk potential of L s 1 in the canonical basis of cycles of Definition 3.5, which is a priori different from the one induced by φ˚.
Under some assumptions on the singularities of the toric varieties XpΣ s q appearing as limits of the degenerations Grpk, nq XpΣ s q, the conjecture above can be verified. We describe below how, and give some sample applications in Section 4. Σq with a toric morphism r : Xp r Σq Ñ XpΣq which is a birational equivalence.
Any toric variety XpΣq has a toric resolution; see for example [11,Chapter 11]. Toric resolutions can be constructed by taking refinements r Σ of the fan Σ, which have natural associated morphisms r. The refined fan r Σ has in general more rays than Σ, and these correspond to torus invariant divisors in the exceptional locus r´1pSing XpΣqq.
Definition 3.11. A toric resolution r : Xp r Σq Ñ XpΣq is small if r Σ and Σ have the same rays. Being small is equivalent to codim C pr´1pSing XpΣqqq ě 2; see for example [11,Proposition 11.1.10].
Proposition 3.12. If s is a Plücker sequence of type pk, nq, and the toric variety XpΣ s q admits a small resolution, then L s Ă Grpk, nq is monotone and has disk potential W s with respect to the basis of canonical cycles for H 1 pL s ; Zq.
Proof. Recall from Proposition 3.4 that there is a smooth submersion µ s : U s Ñ R kpn´kq with Lagrangian torus fibers, defined on some open set U s Ă Grpk, nq. If P s is the Newton polytope of the Laurent polynomial W s , the image of this map is the interior of the polytope described in Proposition 3.2: r s ∆ Gs pD F Z q " tv P R kpn´kq : xv, uy ě´r s for every vertex u P P s u .
Call v a point in the interior, and L s pvq " µ´1 s pvq the corresponding Lagrangian torus fiber. Observe that XpΣ s q is Fano, thanks to Proposition 2.10. The assumption that XpΣ s q has a small toric resolution allows to invoke a theorem of Nishinou-Nohara-Ueda [30,Theorem 10.1], and conclude that the disk potential of L s pvq Ă Grpk, nq has one monomial for each facet xv, uy "´r s of r s ∆ Gs pD F Z q, with exponent u P H 1 pL s pvq; Zq. The subgroup of disk classes in H 2 pGrpk, nq, L s pvq; Zq is generated by Maslov index 2 classes. This holds because U s Ă Grpk, nq is the domain of a symplectomporhism ψ : U s Ñ XpΣ s qzD Σs with the maximal torus orbit of the toric variety XpΣ s q obtained by removing the standard torus invariant divisor D Σs . Harada-Kaveh [21, Theorem A (1)] shows that ψ extends to a continuous map ψ : Grpk, nq Ñ XpΣ s q. As explained in Nishinou-Nohara-Ueda [30, Lemma 9.2, Corollary 9.3], the assumption that XpΣ s q has a small resolution Xp Ă Σ s q allows to use the map ψ to identify disk classes in H 2 pGrpk, nq, L s pvq; Zq with classes of disks in Xp Ă Σ s q with boundary on the toric moment fiber over v, and these are generated by Maslov index 2 classes; see Cho-Oh [10, Theorem 5.1] for a general formula computing the Maslov index of disks with boundary in a toric moment fiber. In conclusion, L s pvq is monotone if and only if all Maslov index 2 classes have the same symplectic area. The symplectic area of a Maslov 2 disk with boundary u is 2πpxv, uy`r s q; see Cho-Oh [10, Theorem 8.1] for a proof. The choice v " 0 guarantees that all the areas are equal, and thus L s " L s p0q is monotone.

Sample applications
We describe some sample applications of what seen so far to the symplectic topology of Grassmannians. These results are by no means optimal; they are meant to illustrate new phenomena, and we give some indications on how one can prove analogous statements using the same techniques.

4.1.
Generating the Fukaya category of Grp2, nq. In this subsection we focus on Grassmannians of planes Grp2, nq. For this class of Grassmannians, Nohara-Ueda [31,32] have used symplectic reduction techniques to construct a Catalan number C n´2 of integrable systems on Gr 2 pnq, each labeled by a triangulation Γ of the n-gon; the generic fibers of these systems are Lagrangian tori L Γ Ă Gr 2 pnq, and the images of their Hamiltonians are lattice polytopes ∆ Γ . Explicit formulas for the disk potentials W L Γ were given as sums over edges of the triangulation Γ. We compare this construction with the case k " 2 of our Construction 1.1, and explore some consequences for the Fukaya category of Grp2, nq. When k " 2, the Plücker coordinates appearing as labels of a quiver Q s have a simple combinatorial description.
Definition 4.1. Let n ą 2, and consider n points on S 1 , labeled counter-clockwise from 1 to n. A triangulation Γ of rns is a collection of subsets ti, ju Ă rns with i ‰ j, such that connecting i and j with an arc in D 2 for all ti, ju P Γ one gets a triangulation of the n-gon with vertices at rns.  t1, 2u , t2, 3u , . . . , tn´1, nu , t1, nu ; these correspond to the edges of the n-gon; the other sets in Γ correspond to interior edges of the triangulation. Note that the n sets above are also the vertical steps d | of those Young diagrams d Ď 2ˆpn´2q that label the frozen nodes in Definition 2.4 (specialized to k " 2).  Proof. In view of Proposition 3.12, it suffices to prove that for any Plücker sequence s of type p2, nq the toric variety XpΣ s q has a small toric resolution. From Lemma 4.3, the labels of Q s define a triangulation Γ s of rns. Nohara-Ueda [32, Theorem 1.1] describe an open embedding ι Γs : pCˆq 2pn´2q Ñ Gr _ pk, nq such that ιΓ s W " W Γs , where W P A k,n is the Landau-Ginzburg potential defined by Marsh-Rietsch [28] and W Γs is a Laurent polynomial associated to the triangulation Γ s . It was shown earlier by Nohara-Ueda [31,Proposition 7.4] that the polar dual of the Newton polytope of W Γs is a lattice polytope ∆ Γs " Newt˝pW Γs q (as opposed to just a rational polytope) and that the associated toric variety XpΣ n ∆ Γs q has a small toric resolution [31, Theorem 1.5]. Since polar duality exchanges normal and face fans XpΣ n ∆ Γs q " XpΣ f NewtpW Γs qq. The image of the embedding ι Γs is the cluster chart T s and W s " W |Ts , so W Γs and W s are Laurent polynomials related by an automorphism of the torus, therefore their Newton polytopes NewtpW Γs q and P s are equivalent under the action of GLp2pn´2q, Zq. We conclude that XpΣ f NewtpW Γs qq -XpΣ f P s q " XpΣ s q and therefore XpΣ s q has a small toric resolution too.
As described by Sheridan [43], the Fukaya category of a monotone symplectic manifold like the Grassmannian has a spectral decomposition The summands are A 8 -categories indexed by the eigenvalues λ of the operator c 1 ‹ of multiplication by the first Chern class acting on the small quantum cohomology. The objects of the λ-summand are monotone Lagrangians with rank one local systems L ξ as described in Section 1. The following proposition holds for general Grassmannians.
Proposition 4.5. For any 1 ď k ă n and any Plücker sequence s of type pk, nq: (1) if F λ pGrpk, nqq ‰ 0 then λ " npζ 1`. . .`ζ k q for some tζ 1 , . . . , ζ k u " I Ă t ζ P C : ζ n " p´1q k`1 u with |I| " k ; (2) if XpΣ s q has a small toric resolution, then pL s q ξ I is a defined and nonzero in F λ pGrpk, nqq if and only if S d pIq ‰ 0 for all Young diagrams d appearing as labels on the nodes of Q s , and moreover λ " npζ 1`¨¨¨`ζk q . Proof.
(1) By [7, Proposition 1.3], λ P C is an eigenvalue of the operator c 1 ‹ of multiplication by the first Chern class acting on the small quantum cohomology if and only if λ " npζ 1`. . .`ζ k q for some tζ 1 , . . . , ζ k u " I Ă t ζ P C : ζ n " p´1q k`1 u with |I| " k. By Auroux [3, Proposition 6.8], any monotone Lagrangian L with a rank one local system ξ having Floer cohomology HF pL ξ , L ξ q ‰ 0 must have m 0 pL ξ q which is an eigenvalue of c 1 ‹.
(2) Think of the holonomy of a local system ξ on L s as a point on a complex torus hol ξ P HompH 1 pL s ; Zqq -pCˆq kpn´kq .
The identification depends on the choice of a basis for H 1 pL s ; Zq, and we use the canonical γ d P H 1 pL s ; Zq of Definition 3.5. By Proposition 3.12 and the assumption of small resolution, the Lagrangian torus L s is monotone and has disk potential W Ls " W s . By Auroux [3, Proposition 6.9] and Sheridan [43,Proposition 4.2], one has Floer cohomology HF ppL s q ξ , pL s q ξ q ‰ 0 if and only if hol ξ is a critical point of the disk potential. By Definition 3.7, the local system ξ " ξ I has hol ξ I pγ d q " S d pIq. Rietsch [38,Lemma 4.4] proves that S d pIq " S d T pI _ q, where d T Ď pn´kqˆk is the transpose Young diagram of d and I _ " tζ 1 , . . . ζ n´k u is the set of n´k distinct roots of ζ n " p´1q n´k`1 obtained by looking at the roots I c of ζ n " p´1q k`1 that are not in I and declaring I _ " e πi I c . Consider the points rM I _ s P Gr _ pk, nq defined as these are known to be the critical points of the Landau-Ginzburg potential W P A k,n defined by Marsh-Rietsch [28]; see [22,Theorem 1.1 and Corollary 3.12]. Observe that S d T pI _ q " x d pM I _ q{x H pM I _ q; this follows from the expression of Schur polynomials as determinants [7,Proposition 2.3 (1)]. After setting x H " 1, one can think of W s as a regular function on the cluster chart T s Ă Gr _ pk, nq such that W s " W |Ts , as explained in (1) of Proposition 2.6. This means that the critical points of W s are precisely those critical points rM I _ s P Gr _ pk, nq of W such that rM I _ s P T s . By definition of T s the latter condition is equivalent to x d pM I _ q ‰ 0 for all Young diagrams d appearing as labels of Q s , and thus ξ I is a well-defined local system on L s such that HF ppL s q ξ , pL s q ξ q ‰ 0 if and only if S d pIq ‰ 0 for all d appearing as labels of Q s . Lemma 4.6. If n is odd, all the eigenvalues of c 1 ‹ acting on QHpGrp2, nqq have algebraic multiplicity one.
Proof. It was explained in part (2) of Proposition 4.5 that the eigenvalues of c 1 ‹ acting on QHpGrpk, nqq correspond to critical values of the Landau-Ginzburg potential W on Gr _ pk, nq defined by Marsh-Rietsch [28], and that the corresponding critical points can be explicitly described. In particular, there are`n k˘c ritical points, and thus at most the same number of critical values. Therefore the statement is equivalent to proving that there are preciselỳ n 2˘d istinct eigenvalues. From part (1) of Proposition 4.5, each eigenvalue is of the form λ " npζ 1`ζ2 q, with ζ 1 and ζ 2 distinct roots of ζ n "´1. Write ζ 1 " e πi n a and ζ 1 " e πi n b with 0 ă a ă b ă 2n odd integers. The norm of one such eigenvalue is The function cospxq is decreasing for 0 ď x ď π and cosp2π´xq " cospxq; in our case 0 ď π n pb´aq ď π whenever 0 ď b´a ď n. Since n is odd by assumption, by varying a and b among all odd integers with 0 ă a ă b ă 2n one finds l " pn´1q{2 eigenvalues with 0 ă |λ 1 | ă¨¨¨ă |λ l |, corresponding to b´a attaining all the even integer values in the interval r2, n´1s. Moreover, fixed any 1 ď t ď l, the n complex numbers λ t , pe 2π n i qλ t , . . . , pe 2π n i q n´1 λ t are eigenvalues of c 1 ‹ too, and they have the same norm as λ t ; see also [7,Proposition 1.12] for more on the symmetries of the spectrum of c 1 ‹. Overall, we found npn´1q{2 "`n 2˘d istinct eigenvalues.
Lemma 4.7. Let d Ď 2ˆpn´2q be a Young diagram, and denote by d | " ts, tu with 1 ď s ă t ď n its vertical steps. Writing an arbitrary set I Ă tζ P C : ζ n "´1u with |I| " 2 as I a,b " te π n ia , e π n ib u with 0 ă a ă b ă 2n odd integers, then Proof. Consider the full rank nˆ2 matrix rM I a,b s " " 1 e π n ia pe π n ia q 2 . . . pe π n ia q n´1 1 e π n ib pe π n ib q 2 . . . pe π n ib q n´1  T As pointed out in part (2) of Proposition 4.5, one has S d pI a,b q " 0 if and only if x d T pM I a,b q " 0. Observe that the the horizontal steps of the transpose diagram are pd T q´" tn`1´t, n`1´su, so that x d T pM I a,b q " e π n iapn´tq e π n ibpn´sq´e π n iapn´sq e π n ibpn´tq , from which x d T pM I a,b q " 0 if and only if e π n ipas`bt´at´bsq " 1. The last condition is verified precisely when 2n pb´aqpt´sq, and since b´a is the difference of two odd integers this can be rewritten as in the statement. Theorem 4.8. If n " 2 t`1 for some t P N`, the derived Fukaya category DFpGrp2, 2 t`1 qq is split-generated by objects supported on a single Plücker torus.
Proof. Up to replacing 2 with n´2, we can think of the critical points rM I a,b s of the Landau-Ginzburg potential W on Gr _ p2, nq defined by Marsh-Rietsch [28] as being parametrized by sets I a,b " te π n ia , e π n ib u with 0 ă a ă b ă 2n odd integers; compare with part (2) of Proposition 4.5. We claim that there exists a Plücker sequence s of type p2, nq such that the corresponding cluster chart T s Ă Gr _ p2, nq contains all critical points rM I a,b s. If this is true, then these will be also critical points of the Laurent polynomial W s " W |Ts , which is the disk potential of the monotone Lagrangian torus L s Ă Grp2, nq by Proposition 3.12 and Lemma 4.4. By Sheridan [43,Corollary 2.19], if the generalized eigenspace QH λ pXq of the operator c 1 ‹ is one-dimensional, any monotone Lagrangian brane L ξ with HF pL ξ , L ξ q ‰ 0 split-generates DF λ pXq. Since n " 2 t`1 is odd, by Lemma 4.6 we can apply this to X " Grp2, nq, L " L s and any ξ " ξ I a,b for all 0 ă a ă b ă 2n odd integers, thus concluding that the objects pL s q I a,b split-generate every summand of DFpGrp2, 2 t`1 qq. The construction of the Plücker sequence s mentioned in the claim above proceeds as follows.  Consider the following incremental construction of a set Γ (an example with t " 3 is given in Figure 4): (1) start with a segment partitioned in n´1 " 2 t intervals, which are added to Γ as new edges t1, 2u, t2, 3u, . . . , t2 t , 2 t`1 u ; (2) partition the segment into pn´1q{2 " 2 t´1 pairs of consecutive intervals, and add a new arc connecting the left end of the left interval to the right end of the right interval in each pair, thus adding new edges t1, 3u, t3, 5u, . . . ,t2 t´1 , 2 t`1 u to Γ ; (3) partition the segment in pn´1q{2 2 " 2 t´2 tuples of 2 2 consecutive intervals, and add a new arc connecting the left end of the leftmost interval to the right end of the rightmost interval in each tuple, thus adding new edges t1, 5u, t5, 9u, . . . ,t2 t`1´22 , 2 t`1 u to Γ ; (4) proceed as above until the initial segment is partitioned in two tuples of 2 t´1 consecutive intervals, and add the edge t1, nu " t1, 2 t`1 u to Γ, so that it becomes a triangulation of rns in the sense of Definition 4.1 . Let pQ 0 , W 0 q be the initial seed of Definition 2.4, and call Γ 0 the triangulation of rns corresponding to Young diagrams labeling the nodes of Q 0 as in Lemma 4.3. The triangulation Γ 0 is connected to Γ constructed above by a sequence of flips, which correspond to mutations of the quiver Q 0 at nodes with to incoming and two outgoing arrows. From Proposition 2.6, this gives a Plücker sequence of mutations of type p2, nq in the sense of Definition 2.7, which ends at pQ s , W s q and such that the labels of Q s correspond to the triangulation Γ s " Γ, again by Lemma 4.3. It remains to show that rM I a,b s P T s for all odd integers a and b such that 0 ă a ă b ă 2n. Suppose not, then there exist some a, b and some Young diagram d Ď 2ˆpn´2q such that x d pM I a,b q " 0. By Lemma 4.7, this implies that n b´a 2 pt´sq, where d | " ts, tu are the vertical steps of d. By construction, for any d | P Γ s , if d | " ts, tu then t´s is a power of 2, and since n " 2 t`1 is odd by assumption we must have n b´a 2 . This is impossible, because b´a 2 ă n. Example 4.9. DFpGrp2, 9qq is generated by a single Plücker torus. Note that instead the Gelfand-Cetlin torus mentioned in Section 1 does not support enough nonzero objects to generate; compare [7, Figure 2(C)].
The arguments above can be generalized to prove that certain collections of Plücker tori split generate DFpGrp2, nqq. Theorem 4.11. Let n ą 2 be odd, and consider its prime factorization n " p e 1 1¨¨¨p e l l . Assume that for all 1 ď i ď l there exists a triangulation Γ i of rns that is p i -avoiding, then DFpGrp2, nqq is split generated by objects supported on l Plücker tori.
Proof. Recall that up to replacing 2 with n´2, we can think of the critical points rM I a,b s of the Landau-Ginzburg potential W on Gr _ p2, nq defined by Marsh-Rietsch [28] as being parametrized by sets I a,b " te π n ia , e π n ib u with 0 ă a ă b ă 2n odd integers; compare with part (2) of Proposition 4.5. Denote C the set of all critical points of W , and for 1 ď i ď l define Observe that C " C p 1 Y¨¨¨Y C p l . Indeed, if p e i i pb´aq{2 for all 1 ď i ď l then p e 1 1¨¨¨p e l l " n pb´aq{2, against the fact that pb´aq{2 ă n. By assumption, for each 1 ď i ď l there exist a triangulation Γ i of rns that is p i -avoiding, and arguing as in Theorem 4.8 one finds a Plücker sequence s i of type p2, nq that starts with the initial seed pQ 0 , W 0 q and ends with pQ s i , W s i q, and such that the labels of Q s i correspond to the triangulation Γ s i " Γ i as in Lemma 4.3. Each of the l Plücker tori L s i Ă Grp2, nq has an associated cluster chart T s i Ă Gr _ p2, nq, and we claim that C p i Ă T s i . Suppose not, then there exists some rM I a,b s P C p i such that rM I a,b s R T s i . This means that p e i i b´a 2 and there exists some Young diagram d Ď 2ˆpn´2q such that x d pM I a,b q " 0, and denoting d | " ts, tu its vertical steps ts, tu P Γ s i . By Lemma 4.7, this implies that n b´a 2 pt´sq, and so in particular p e i i b´a 2 pt´sq. Since Γ s i is p i -avoiding, this means that p e i i b´a 2 , against the fact that rM I a,b s P C p i . As in Theorem 4.8, the assumption n odd and Lemma 4.6 guarantee, by Sheridan [43,Corollary 2.19], that any nonzero object of the Fukaya category supported on one of the l monotone Plücker tori L s 1 , . . . , L s l Ă Grp2, nq split-generates the summand DF λ pGrp2, nqq of the derived Fukaya category containing it. The objects supported on L s i are obtained by endowing it with local systems ξ I a,b as in Definition 3.7 corresponding to critical points rM I a,b s P T s i ; these are such that HF ppL s i q ξ I a,b , pL s i q ξ I a,b q ‰ 0 because the disk potential of L s i is W s i " W |Ts i .  Example 4.12. DFpGrp2, 15qq is generated by two Plücker tori, whose corresponding triangulations are shown in Figure 5. To get the two triangulations, one starts by constructing a partial triangulation of the 15-gon with dyadic arcs as in Theorem 4.8 (solid arcs in Figure 5). The partial triangulation is p-avoiding for every prime p ą 2 by construction. Since 15 " 3¨5, by Theorem 4.11 one needs to find completions of the partial triangulation to full triangulations that are 3-avoid and 5-avoiding respectively. In Figure 5, the remaining arcs ti, ju with 3 pj´iq are coarsely dashed, while the one with 5 pj´iq is finely dashed; triangulation (A) is obtained by adding two shaded arcs and is 3-avoiding, while triangulation (B) is obtained by adding two different shaded arcs and is 5-avoiding.
Definition 4.13. If L s Ă Grpk, nq is a Plücker Lagrangian of type pk, nq, define its f -vector to be f pL s q " pf 1 , . . . , f kpn´kq q P N kpn´kq , where f i is the number of pi´1q-dimensional faces in the Newton polytope P s of the potential W s . Definition 4.14. If L s Ă Grpk, nq is a Plücker Lagrangian of type pk, nq, define its weight wtpL s q P N to be the number of sets I Ă tζ P C : ζ n " p´1q k`1 u such that |I| " k and S d pIq ‰ 0 for all Young diagrams d appearing as labels of the quiver Q s . Lemma 4.15. Assume s, s 1 are Plücker sequences of type pk, nq satisfying Conjecture 3.8. If f pL s q ‰ f pL s 1 q or wtpL s q ‰ wtpL s 1 q, then the Lagrangian tori L s , L s 1 Ă Grpk, nq are not Hamiltonian isotopic.
Proof. Suppose that there exists a Hamiltonian isotopy φ such that φpL s q " L s 1 . Then by assumption the induced map φ˚: H 1 pL s ; Zq Ñ H 1 pL s 1 ; Zq is such that where " denotes equality up to automorphisms of T s . This means that the Newton polytopes P s and P s 1 of the Laurent polynomials W s and W s 1 are related by a transformation of GLpkpnḱ q, Zq, and hence have the same f -vector, because the number of faces of any given dimension of a polytope is a unimodular invariant; this proves that f pL s q " f pL s 1 q. Moreover, the Laurent polynomials W s and W s 1 can be thought of as regular functions on a torus pCˆq kpn´kq , which agree up to an automorphism. Since the number of critical points of a function is invariant under automorphisms of its domain, it follows from part (2) of Proposition 4.5 that wtpL s q " wtpL s 1 q.
k " 3, n " 6 L s Labels of Q s f pL s q wtpL s q  1 123, 124, 125, 126, 156, 234, 245, 256, 345, 456 (14,83,276,571,766,670,372,122,20) appear as labels on the nodes of the quiver Q s . Calling Σ s " Σ f P s the face fan of the Newton polytope, by Proposition 3.12 the Lagrangian torus L s Ă Grpk, nq is monotone and has disk potential W s whenever the toric variety XpΣ s q has a small toric resolution in the sense of Definition 3.11. This condition can be checked algorithmically at each step, since every fan has finitely many simplicial refinements with the same rays, and every smooth refinement is in particular simplicial. For the 34 steps in the table, the code [6] finds small resolutions in 32 cases; the remaining 2 cases are marked gray in the table (we did not actually check all possible simplicial refinements in these cases, so small toric resolutions for them may still exist). From Lemma 4.15, we conclude that Grp3, 6q contains at least 6 monotone Lagrangian tori that are pairwise not Hamiltonian isotopic: rows 1, 3,7,8,15,29. Regarding nondisplaceability, it suffices to show that the 32 tori L s Ă Grp3, 6q have Floer cohomology HF pL ξ , L ξ q ‰ 0 for some local system ξ. By Auroux [3, Proposition 6.9] and Sheridan [43,Proposition 4.2], the Floer cohomology of a monotone Lagrangian torus brane L ξ is nonzero if and only if the holonomy hol ξ of its local system ξ is a critical point of the disk potential disk potential W s . Therefore, it suffices to show that each of the 32 Laurent polynomials W s has at least one critical point. Thinking W s as restriction W s " W |Ts of the Landau-Ginzburg potential W on Gr _ p3, 6q defined by Marsh-Rietsch [28] to the cluster chart T s Ă Gr _ p3, 6q, it suffices to show that each of the charts contains at least one critical point of W . In fact, something stronger is true: there is a critical point of W that is contained in T s for all s. As proved by Rietsch [39] (see also Karp [22]), for any 1 ď k ă n there is a (unique) critical point of W in the totally positive part Gr _ pk, nq ą0 Ă Gr _ pk, nq, i.e. the locus where all Plücker coordinates are real and positive. Following the notation of part (2) in Proposition 4.5, this point is rM I 0 s P Gr _ pk, nq with I 0 the set of k roots of ζ n " p´1q k`1 closest to 1. Applying this to pk, nq " p3, 6q, and recalling that rM I 0 s P T s if and only if x d pM I 0 q for all Young diagrams d Ď 3ˆ3 appearing as labels on the nodes of Q s , we conclude that the the total positivity of rM I 0 s implies that it belongs to every cluster chart T s , and this proves that all L s are nondisplaceable.
Remark 4.17. We emphasize that the arguments of Theorem 4.16 prove that any L s Ă Grpk, nq is nondisplaceable as long as W s " W Ls . This is due to the fact that the tori L s correspond to cluster charts T s Ă Gr _ pk, nq by construction, and that W has a critical point in the intersection of all such charts.
Remark 4.18. It was shown in [7,Theorem 4.8] that the dihedral group D n " x r, s | r n " s 2 " 1, rs " sr´1 y acts on the set of critical points of W P A k,n , and that the cluster chart T s is invariant under the action of the subgroup xry. Since wtpL s q is the number of critical points in T s , the fact that Z{nZ acts on it puts some arithmetic constraints on this number.