CONJUGATE FILLINGS AND LEGENDRIAN WEAVES A COMPARATIVE STUDY ON LAGRANGIAN FILLINGS ROGER CASALS AND WENYUAN LI

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CONJUGATE FILLINGS AND LEGENDRIAN WEAVES

– A COMPARATIVE STUDY ON LAGRANGIAN FILLINGS –

ROGER CASALS AND WENYUAN LI

Abstract. First, we show that conjugate Lagrangian ﬁllings, associated to plabic graphs,

and Lagrangian ﬁllings obtained as Reeb pinching sequences are both Hamiltonian isotopic

to Lagrangian projections of Legendrian weaves. In general, we establish a series of new

Reidemeister moves for hybrid Lagrangian surfaces. These allow for explicit combinatorial

isotopies between the diﬀerent types of Lagrangian ﬁllings and we use them to show that

Legendrian weaves indeed generalize these previously known combinatorial methods to

construct Lagrangian ﬁllings. This generalization is strict, as weaves are typically able to

produce inﬁnitely many distinct Hamiltonian isotopy classes of Lagrangian ﬁllings, whereas

conjugate surfaces and Reeb pinching sequences produce ﬁnitely many ﬁllings.

Second, we compare the sheaf quantizations associated to each such types of Lagrangian

ﬁllings and show that the cluster structures in the corresponding moduli of pseudo-perfect

objects coincide. In particular, this shows that the cluster variables in Bott-Samelson

cells, given as generalized minors, are geometric microlocal holonomies associated to sheaf

quantizations. Similar results are presented for the Fock-Goncharov cluster variables in

the moduli spaces of framed local systems. In the course of the article and its appendices,

we also establish several technical results needed for a rigorous comparison between the

diﬀerent Lagrangian ﬁllings and their microlocal sheaf invariants.

Contents

1. Introduction 1

2. Constructions of Lagrangian ﬁllings 7

3. Hamiltonian Isotopies for Hybrid Lagrangians 26

4. Conjugate Fillings and Reeb Pinchings as Legendrian weaves 37

5. Sheaf Quantization of Conjugate Fillings and Legendrian weaves 53

6. Comparison of Cluster Coordinates and Related Applications 65

Appendix A. DG-Categories in the microlocal theory of sheaves 80

Appendix B. Results on the microlocal theory of constructible sheaves 83

References 98

1. Introduction

The object of this article will be to show that the Lagrangian ﬁllings obtained via the

combinatorics of plabic graphs are Hamiltonian isotopic to the Lagrangian ﬁllings obtained

from Legendrian weaves. The main geometric contribution of the article is the comparison

of such Hamiltonian isotopy classes through the study of hybrid Lagrangian surfaces, in-

cluding new results and applications from a series of Reidemeister moves that we establish

for such Lagrangian surfaces. The main algebraic contribution is the comparison of Legen-

drian invariants from the microlocal theory of sheaves where we show that, under the above

Hamiltonian isotopies, the independently deﬁned cluster coordinates, through conjugate

arXiv:2210.02039v1 [math.SG] 5 Oct 2022

2 ROGER CASALS AND WENYUAN LI

surfaces and through weaves, match. These cluster coordinates are associated to the ring

of functions of the moduli derived stack of pseudo-perfect objects in the dg-subcategory

of compact objects within the dg-category of constructibles sheaves with singular support

along the corresponding Legendrians.

Plabic combinatorics

Alternating strand diagrams [57],

Goncharov-Kenyon conjugate surfaces [31,65],

n-triangulations [15] and ideal webs [30],

plabic fences in R-morsiﬁcations [5,17],

generalized minors and Pl¨ucker coordinates [4,19].

Legendrian weaves

Legendrian (-1)-closures of positive braids [9,51],

Exact Lagrangian spectral curves [11,20,70],

Legendrian weaves for triangulations [7,8,11],

Lagrangian A’Campo-Gusein-Zade skeleta [5],

weaves for GP-graphs and cluster coordinates [10,26].

This

paper

Table 1. Schematics of contributions in the present manuscript, connect-

ing the study of plabic combinatorics (left) with the symplectic topology of

Legendrian weaves (right). The former was initiated by A. Postnikov [57]

and is rooted in the study of cluster algebras [4,17,19] and their topological

incarnations by A. Goncharov and V. Fock [15,16,31]. The latter has been

a central ingredient in many of the proofs of recent results in the study of

Lagrangian ﬁllings, e.g. [1,2,6–8,10,37,38].

The manuscript also shows that Lagrangian ﬁllings obtained via Reeb pinching sequences are

compactly supported Hamiltonian isotopic to Lagrangian projections of Legendrian weaves.

In general, a recurring theme of our results is that Legendrian weaves strictly generalize all

standard constructions of Lagrangian ﬁllings. Explicit diagrammatic methods to transform

such ﬁllings into Legendrian weaves are also provided. Schematics of a few implications from

this paper are depicted in Table 1. Finally, in addition to the above results, the article and

its appendices establish rigorous comparisons between the diﬀerent objects and techniques

employed in diﬀerent sources in the literature.

1.1. Scientiﬁc context. The combinatorics of plabic graphs [57] have found several ap-

plications to the study of cluster algebras [18,22,23,61–63] and the birational geometry of

moduli of local systems [15,20,21,29–31]. Independently, recent advances [8,10,26] have

been able to establish new connections between contact and symplectic topology and the

study of cluster algebras. For instance, Legendrian weaves [7,11] have been used to con-

struct cluster algebras on braid varieties, in particular resolving Leclerc’s conjecture [46],

and, conversely, cluster algebras have been used to provide the ﬁrst instances of Legendrian

links with inﬁnitely many ﬁllings [6,26], up to compactly supported Hamiltonian isotopy.

In Type A, the common denominator of these two areas of research is anchored in the study

of Lagrangian ﬁllings of Legendrian knots, in the language of the latter, and conjugate sur-

faces bounding alternating strand diagrams, in the language of the former. There are two

layers of study: the geometry of the surfaces and their boundaries, as smooth or Lagrangian

surfaces, and the algebraic invariants that can be associated to them. The ﬁrst layer is en-

tirely geometric, with a focus on the isotopy classes of the surfaces and links at hand, may

they be smooth or Hamiltonian isotopies. The second layer leads to the construction of

cluster algebras, cluster seeds and categoriﬁcations thereof. This manuscript is structured

in the same manner, with Sections 2,3and 4focusing on the geometry, and Sections 5and

6studying the algebraic invariants.

From the perspective of low-dimensional symplectic topology, our results serve to both

add and connect recent developments in the study of Lagrangian ﬁllings, including the

CONJUGATE FILLINGS AND LEGENDRIAN WEAVES 3

obstruction methods in [6,9,11,13,27,54,65] and the constructions in [1,2,6,9,11,13,27,38,

65,71]. In brief, there are currently three combinatorial methods to construct Lagrangian

ﬁllings for Legendrian (−1)-closures of positive braids:

(1) conjugate Lagrangian ﬁllings [65].

(2) pinching sequences of Reeb chords [13],

(3) free Legendrian weaves [11].

In a nutshell, this article will establish that Legendrian weaves, Method (3), generalizes the

prior two methods whenever they can be compared, i.e. conjugate Lagrangian ﬁllings and

Lagrangian ﬁllings obtained via pinching sequences are Lagrangian projections of Legen-

drian weaves. In addition, both Methods (1) and (2) can only yield ﬁnitely many Lagrangian

ﬁllings, whereas Method (3) is known to produce inﬁnitely many Lagrangian ﬁllings [6,11]

in many cases.

From the perspective of cluster algebras, our results show that the cluster variables

constructed from plabic graphs, in the form of Pl¨ucker coordinates and their generalizations,

actually coincide with the microlocal holonomies studied in [10]. Note that the former,

in the shape of (factors of) generalized minors, are key in the constructions of [22,23],

whereas weaves and microlocal holonomies (both monodromies and merodromies) are a core

ingredient in [8,10]. These algebraic comparisons are guided by the Hamiltonian isotopies

that we construct between conjugate Lagrangian surfaces and Lagrangian projections of

Legendrian weaves.

1.2. Main results. Let G⊆Σ be a plabic graph in a smooth surface Σ. In [31, Section

1.1], an embedded smooth surface C(G)⊆T∗Σ is constructed; it is called the conjugate sur-

face. The alternating strand diagram of G[57, Section 14] is a cooriented immersed curve in

Σ and thus naturally lifts to a Legendrian link Λ(G)⊆(T∗,∞Σ,ker λst) in the ideal contact

boundary of the standard cotangent bundle (T∗Σ, λst). In [65, Section 4.2] it is shown that

there exists an embedded exact Lagrangian L(G)⊆(T∗Σ, λst) which is smoothly isotopic

to C(G). We show in Proposition 2.4 that the Hamiltonian isotopy class of L(G) is unique,

in that it is independent of the choices used in its construction, and it is thus well-deﬁned to

speak of the conjugate Lagrangian ﬁlling of Λ(G) given by L(G), up to Hamiltonian isotopy.

Let us consider the set C(Σ) of plabic graphs on Σ associated to either of the following

combinatorial objects: an n-triangulation, a grid plabic fence and the reduced plabic graphs

for Gr(2, m). These are three of the most common general constructions of plabic graphs.

For the ﬁrst type, Σ is an arbitrary (marked) smooth surface, whereas Σ = D2is the 2-disk,

possibly with marked points on the boundary, for the second and third types. Now, the

plabic graphs for each of these objects were respectively introduced in [15] (see also ideal

webs [30]) in the study of higher Teichm¨uller theory and moduli spaces of local systems, in

[17] in the study of real Morsiﬁcations of isolated plane curve singularities (see also [10]),

and in [19,62] in the study of cluster algebras of ﬁnite type Am−3, which corresponds to the

coordinate ring of functions on Gr(2, m), m≥3. Note that the latter type can be under-

stood as a special case of an ideal triangulation of a disk with boundary marked points; it

is emphasized as a separate case because all cluster seeds are actually described by plabic

graphs. The Legendrian weaves in (J1Σ,ker(dz −λst)) associated to each of these com-

binatorial objects were respectively introduced in [11, Section 3.1] for n-triangulations, in

[10, Section 3] for (grid) plabic graphs, and the weave associated to a reduced plabic graph

for Gr(2, m) is deﬁned to be the 2-weave dual to the triangulation, see [11,71]. We refer to

them as standard weaves and denote them by w(G); we denote by w(G) both the planar

weave itself and the Legendrian surface associate to it, as this distinction is always clear by

4 ROGER CASALS AND WENYUAN LI

context.

First, the main symplectic geometric result shows that conjugate Lagrangian surfaces

are Hamiltonian isotopic to Legendrian weaves:

Theorem 1.1. Let Σbe a smooth surface, G∈ C(Σ) and L(G)⊆(T∗Σ, λst)its conjugate

Lagrangian surface. Then there exists an embedded Lagrangian w(G)⊆(T∗Σ, λst)Hamil-

tonian isotopic to w(G)which is the Lagrangian projection of the standard weave w(G).

The Hamiltonian isotopy in Theorem 1.1 is not a compactly supported isotopy. In fact, a

non-trivial Legendrian isotopy needs to be applied to even compare the Legendrian links of

∂w(G) and the alternating strand diagram of G. Theorem 1.1 is proven by ﬁrst develop-

ing a series of new Reidemeister moves for hybrid Lagrangian surfaces, which allow us to

interpolate between conjugate surfaces and weaves. These moves are shown in Table 1and

proven in Theorem 3.1. These moves are of independent interest as well. In addition, they

are likely to be also necessary when comparing the two recent resolutions [8] and [22,23] of

Leclerc’s conjecture on cluster algebras for Richardson varieties, as the former uses weaves

and the latter use conjugate surfaces.

Figure 1. Reidemeister-type moves used in order to transition from a con-

jugate surface towards a weave. These are proven in Section 3. In Section 4

these moves are used to prove Theorem 1.1.

In the comparison in Theorem 1.1 several subtleties appear. These include the behaviour

at inﬁnity of conjugate Lagrangian surfaces versus that of weaves, and the uniqueness of

the Hamiltonian isotopy class of the Lagrangian conjugate surface L(G). The necessary

technical results to address these diﬀerences are established in Section 2. In addition, both

Section 2and Section 4discuss Lagrangian ﬁllings obtained via Reeb pinching sequences. In

particular, it is established in Section 4.4 that such Lagrangian ﬁllings are also Lagrangian

projections of Legendrian weaves.

Second, Section 5studies the sheaf quantizations of these Lagrangian ﬁllings. In partic-

ular, we review the alternating sheaf quantization of [65] and set up the sheaf quantization

for Legendrian weaves following [11]. The technical details needed for the comparison of

these sheaf quantizations are provided (see also Appendices Aand B). Sheaf quantizations

at hand, we can then compare the a priori diﬀerent cluster algebra structures on the moduli

derived stack of pseudo-perfect objects of the dg-category of wrapped sheaves. Namely,

CONJUGATE FILLINGS AND LEGENDRIAN WEAVES 5

the combinatorial cluster structures arising from the theory of plabic graphs (with Pl¨ucker

coordinates and generalized minors, e.g. see [26]) and that recently constructed using mi-

crolocal sheaf theory [10]. Note that the latter enjoys Hamiltonian functoriality, whereas

the former is only known to be invariant under plabic graph moves. Section 6shows that

the cluster structures coincide, as follows.

Let β∈Br+be a positive braid word, Λβ⊆T∗,∞R2its associated Legendrian cylin-

drical closure and Λ≺

β⊆(R3, ξst) its rainbow closure. Denote by Gβ∈ C(D2) the plabic

fence associated to β, see [10, Section 2.5] or [17,65]. Section 6compares diﬀerent mod-

uli spaces: the moduli spaces Mfr

1(D2,Λβ∆2)0, resp. Mfr

1(D2,Λ≺

β)0, of microlocal rank-one

framed sheaves on Λβ∆2, resp. Λ≺

β, as introduced in Appendix B.4, and the ﬂag conﬁg-

uration space Confe

β(C), as introduced in [26, Deﬁnition 4.3], where e= id is the empty

braid. Following the geometry in Theorem 1.1, and once is established in Section 6that

these moduli spaces are all isomorphic, it is natural to study the two known cluster algebras:

(i) The generalized minor cluster variables on ﬂag conﬁguration space C[Confe

β(C)] de-

ﬁned in [26, Section 4]. This is a cluster algebra whose cluster variables in the initial

seed are given by generalized minors dictated by the plabic fence Gβ. In particular,

they generalize the initial seeds associated to double Bruhat cells and reduced plabic

graphs for the open positroid cells.

(ii) The microlocal cluster variables on C[Mfr

1(D2,Λ≺

β)0] deﬁned in [10, Section 4].

This is a cluster structure whose cluster variables in the initial seed are given by

microlocal merodromies of a sheaf quantization of the Legendrian weave w(G).1

Since the microlocal cluster A-variables are equally deﬁned for Mfr

1(D2,Λβ∆2)0and

Mfr

1(D2,Λ≺

β)0, and these moduli are isomorphic, we focus on the latter.

Note that both structures above give cluster algebras [18,19], not just upper cluster algebras

[4], or merely cluster X-structures [15] (or the partial structures deﬁned in [65]). Namely,

the comparison of cluster A-variables for cluster algebras is the strongest setting possible.

In the second part of the article, we show that these cluster algebras coincide:

Theorem 1.2. Let βbe a positive braid and G(β)its associated plabic fence. Then the

coordinate ring C[Confe

β(C)], endowed with the minor cluster algebra structure, and the co-

ordinate ring C[Mfr

1(D2,Λ≺

β)0]endowed with the microlocal cluster algebra structure, are

isomorphic cluster algebras.

Furthermore, there exists an isomorphism that sends the initial seed in C[Confe

β(C)],

given by the toric chart associated to the conjugate Lagrangian surface L(Gβ), to the initial

seed in C[Mfr

1(D2,Λ≺

β)0]given by the toric chart associated to w(Gβ).

It follows from Theorem 1.2 and the Hamiltonian functoriality in [10] that the minor clus-

ter variables can, a posteriori, be deﬁned intrinsically in terms of symplectic topological

techniques and are invariant under Legendrian Reidemeister moves applied to the relative

Lagrangian skeleton. Note also that the generalized minor cluster algebra is used crucially

in the cluster algebra construction for braid varieties [8] and the resolution of Leclerc’s

1The article [10] constructs cluster structures in the more general setting of grid plabic graphs, which

include the cases of plabic fences. Nevertheless, we must restrict to plabic fences to compare to [26] as the

latter is only deﬁned in that setting.

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CONJUGATEFILLINGSANDLEGENDRIANWEAVES{ACOMPARATIVESTUDYONLAGRANGIANFILLINGS{ROGERCASALSANDWENYUANLIAbstract.First,weshowthatconjugateLagrangianllings,associatedtoplabicgraphs,andLagrangianllingsobtainedasReebpinchingsequencesarebothHamiltonianisotopictoLagrangianprojectionsofLegendrianweaves.Ingene...

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CONJUGATE FILLINGS AND LEGENDRIAN WEAVES A COMPARATIVE STUDY ON LAGRANGIAN FILLINGS ROGER CASALS AND WENYUAN LI

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