Daping Weng
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View article: Microlocal theory of Legendrian links and cluster algebras
Microlocal theory of Legendrian links and cluster algebras Open
We show the existence of quasicluster Ꮽ-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 sh…
View article: Intersections of Dual $SL_3$-Webs
Intersections of Dual $SL_3$-Webs Open
We introduce a topological intersection number for an ordered pair of $\operatorname{SL}_3$-webs on a decorated surface. Using this intersection pairing between reduced $(\operatorname{SL}_3,\mathcal{A})$-webs and a collection of $(\operat…
View article: Augmentations, Fillings, and Clusters for 2-Bridge Links
Augmentations, Fillings, and Clusters for 2-Bridge Links Open
We produce the first examples relating non-orientable exact Lagrangian fillings of Legendrian links to cluster theory, showing that the ungraded augmentation variety of certain max-tb representatives of Legendrian $2$-bridge links is isomo…
View article: Demazure weaves for reduced plabic graphs (with a proof that Muller-Speyer twist is Donaldson-Thomas)
Demazure weaves for reduced plabic graphs (with a proof that Muller-Speyer twist is Donaldson-Thomas) Open
First, this article develops the theory of weaves and their cluster structures for the affine cones of positroid varieties. In particular, we explain how to construct a weave from a reduced plabic graph, show it is Demazure, compare their …
View article: F-Polynomials of Donaldson-Thomas Transformations
F-Polynomials of Donaldson-Thomas Transformations Open
$F$-polynomials are integer coefficient polynomials encoding the mutations of cluster variables inside a cluster algebra. In this article, we study the $F$-polynomials associated with the action of Donaldson-Thomas transformations on clust…
View article: Microlocal Theory of Legendrian Links and Cluster Algebras
Microlocal Theory of Legendrian Links and Cluster Algebras Open
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 tr…
View article: Cluster Structures on Double Bott–Samelson Cells
Cluster Structures on Double Bott–Samelson Cells Open
Let $\mathsf {C}$ be a symmetrisable generalised Cartan matrix. We introduce four different versions of double Bott–Samelson cells for every pair of positive braids in the generalised braid group associated to $\mathsf {C}$ . We prove that…
View article: Positive Braid Links with Infinitely Many Fillings
Positive Braid Links with Infinitely Many Fillings Open
We prove that any positive braid Legendrian link not isotopic to a standard finite type link admits infinitely many exact Lagrangian fillings.
View article: Augmentations, Fillings, and Clusters
Augmentations, Fillings, and Clusters Open
We investigate positive braid Legendrian links via a Floer-theoretic approach and prove that their augmentation varieties are cluster K2 (aka. A-) varieties. Using the exact Lagrangian cobordisms of Legendrian links in [EHK16], we prove th…
View article: Cyclic Sieving and Cluster Duality of Grassmannian
Cyclic Sieving and Cluster Duality of Grassmannian Open
We introduce a decorated configuration space $\\mathscr{C}\\!{\\rm\nonf}_n^\\times(a)$ with a potential function $\\mathcal{W}$. We prove the cluster\nduality conjecture of Fock-Goncharov for Grassmannians, that is, the\ntropicalization of…
View article: Cluster Structures on Double Bott-Samelson Cells
Cluster Structures on Double Bott-Samelson Cells Open
Let $C$ be a symmetrizable generalized Cartan matrix. We introduce four different versions of double Bott-Samelson cells for every pair of positive braids in the generalized braid group associated to $C$. We prove that the decorated double…
View article: Donaldson-Thomas Transformation of Double Bruhat Cells in Semisimple Lie Groups
Donaldson-Thomas Transformation of Double Bruhat Cells in Semisimple Lie Groups Open
Double Bruhat cells $G^{u,v}$ were studied by Fomin and Zelevinsky. They provide important examples of cluster algebras and cluster Poisson varieties. Cluster varieties produce examples of 3d Calabi-Yau categories with stability conditions…
View article: Donaldson-Thomas Transformation of Double Bruhat Cells in General Linear Groups
Donaldson-Thomas Transformation of Double Bruhat Cells in General Linear Groups Open
Kontsevich and Soibelman defined the Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety can produce an example of such a category, whose corresponding Donaldson-Thomas invariants are enc…
View article: Donaldson-Thomas Transformation of Grassmannian
Donaldson-Thomas Transformation of Grassmannian Open
Kontsevich and Soibelman defined the notion of Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. A family of examples of such categories can be constructed from an arbitrary cluster variety. The correspond…