Microlocal Theory of Legendrian Links and Cluster Algebras

TL;DR

The paper builds a bridge between symplectic topology and cluster algebras by constructing full quasi-cluster A-structures on moduli stacks 𝔐(Λ,T) of decorated microlocal rank-one sheaves with singular support in Legendrian lifts of grid plabic graphs. It introduces microlocal merodromies along relative cycles, realized as regular global functions, and shows that Lagrangian surgeries along L-compressing disks implement cluster mutations, yielding an explicit initial seed and mutations for all seeds. It further proves cluster duality for the associated cluster ensembles and constructs Donaldson-Thomas transformations in the shuffle-graph setting via Legendrian isotopy and a half-twist contactomorphism, grounding the X-structure in symplectic geometry. The work provides a comprehensive framework tying Legendrian weaves, sugar-free hulls, and relative homology to cluster algebra data, enabling geometric computation of A-variables and mutation rules. Overall, it delivers a canonical, Hamiltonian-invariant realization of cluster structures on highly nontrivial moduli spaces with strong applications to 3-dimensional contact/topology and representation-theoretic categories.

Abstract

We show the existence of quasi-cluster $\mathcal{A}$-structures and cluster Poisson structures on moduli stacks of sheaves with singular support in the alternating strand diagram of grid plabic graphs by studying the microlocal parallel transport of sheaf quantizations of Lagrangian fillings of Legendrian links. The construction is in terms of contact and symplectic topology, showing that there exists an initial seed associated to a canonical relative Lagrangian skeleton. In particular, mutable cluster $\mathcal{A}$-variables are intrinsically characterized via the symplectic topology of Lagrangian fillings in terms of dually $\mathbb{L}$-compressible cycles. New ingredients are introduced throughout this work, including the initial weave associated to a grid plabic graph, cluster mutation along a non-square face of a plabic graph, the concept of the sugar-free hull, and the notion of microlocal merodromy. Finally, a contact geometric realization of the DT-transformation is constructed for shuffle graphs, proving cluster duality for the cluster ensembles.

Figures (56)

• Figure 1: The quasi-cluster $\mathrm{K}_2$-structure we construct for this grid plabic graph is on the coordinate ring of the moduli of decorated sheaves on $\mathbb{R}^2$ with singular support in a max-tb Legendrian representative of the $m(9_6)$ knot.
• Figure 2: The three types of elementary columns in a GP-graph.
• Figure 3: The four corners depicted on the left, in yellow, are allowed in a sugar-free region. The four corners depicted on the right, in orange, are not allowed in a sugar-free region, they have sugar content.
• Figure 4: Four types of staircase building blocks for the boundary $\partial R$ of a sugar-free region $R\subset\mathbb{G}$. In each instance, the letter $R$ marks the location of the region in the plane. The dashed lines indicate that $\partial R$ can continue in either of the two branches
• Figure 5: The local models for an alternating strand diagram associated to a GP-graph $\mathbb{G}$. The small hairs indicate the co-orienting direction, which is needed to specify a Legendrian lift.

Theorems & Definitions (140)

• Theorem 1.1: Main Result
• Corollary 1.2
• Corollary 1.3
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Corollary 2.5
• Definition 2.6