arXiv subjects
Publications and source records attributed to Roger Casals.
We show that the higher Legendrian homology of any Legendrian link given by the rainbow closure of a positive braid must necessarily vanish. The result is proven in the most general setting: for any number of strands and crossings, integrally and with one basepoint per component of the Legendrian link.
The main result of this article shows that the quiver mutation class of a malleable real Morsification uniquely determines the integral monodromy module of a plane curve singularity. In particular, we show that the quiver mutation class of a malleable divide determines the complex topological type of an irreducible plane curve singularity. This establishes the algebraic-to-topological implication of a conjecture of S.~Fomin, P.~Pylyavskyy, E.~Shustin and D.~Thurston in the malleable irreducible case. The result is proven by developing representation-theoretic techniques based on an equivariant Euler pairing in the derived category of continuous finite-dimensional dg modules over a differential bigraded Ginzburg algebra. A key step uses these techniques to show that the quiver mutation class of a plabic fence uniquely recovers the torsion part of the Alexander module of the associated smooth link.
We establish the foundations of categorical weave calculus, developing the diagrammatic calculus of weaves and braid varieties within the study of Calabi-Yau triangulated categories and cluster tilting theory. This is achieved by associating a perverse sheaf of triangulated categories to each Demazure weave. A central contribution is the construction and study of the categorical Lusztig cycles and their duals, which we show form simple-minded and silting collections in the category of global sections of such a sheaf of categories. These categorical collections are built using the diagrammatics of weaves and we study their behavior under changes of weaves. For instance, we show that they undergo tilts under weave mutations. En route, we develop the study of categorical weighted braid words, as canonical rigid filtered dg modules over derived preprojective algebras, and the categorical incarnation of the tropical Lusztig rules, as a gluing mechanism for such filtered objects. Appendix A contains homological results, providing a novel construction of simple-minded and silting collections from full exceptional collections, and characterizing when these arise from a highest weight structure on an abelian category.
We relate Bruhat numbers in real Morse theory to cluster variables in braid varieties. This provides instances of wall-crossing coordinates in the study of Cerf diagrams.
Braid varieties parametrize linear configurations of flags with transversality conditions dictated by positive braids. They include and generalize reduced double Bruhat cells, positroid varieties, open Bott-Samelson varieties, and Richardson varieties, among others. Recently, two cluster algebra structures were independently constructed in the coordinate rings of braid varieties: one using weaves and the other using Deodhar geometry. The main result of the article is that these two cluster algebras coincide. More generally, our comparative study matches the different concepts and results from each approach to the other, both on the combinatorial and algebraic geometric aspects.
We study the boundedness of a mutation class for quivers with real weights. The main result is a characterization of bounded mutation classes for real quivers of rank 3.
We introduce and develop the theory of spectral networks in real contact and symplectic topology. First, we establish the existence and pseudoholomorphic characterization of spectral networks for Lagrangian fillings in the cotangent bundle of a smooth surface. These are proven via analytic results on the adiabatic degeneration of Floer trajectories and the explicit computation of continuation strips. Second, we construct a Family Floer functor for Lagrangian fillings endowed with a spectral network and prove its equivalence to the non-abelianization functor. In particular, this implies that both the framed 2d-4d BPS states and the Gaiotto-Moore-Neitzke non-abelianized parallel transport are realized as part of the $A_\infty$-operations of the associated 4d partially wrapped Fukaya categories. To conclude, we present a new construction relating spectral networks and Lagrangian fillings using Demazure weaves, and show the precise relation between spectral networks and augmentations of the Legendrian contact dg-algebra.
We establish a geometric criterion for local microlocal holonomies to be globally regular on the moduli space of Lagrangian fillings. This local-to-global regularity result holds for arbitrary Legendrian links and it is a key input for the study of cluster structures on such moduli spaces. Specifically, we construct regular functions on derived moduli stacks of sheaves with Legendrian microsupport by studying the Hochschild homology of the associated dg-categories via relative Lagrangian skeleta. In this construction, a key geometric result is that local microlocal merodromies along positive relative cycles in Lagrangian fillings yield global Hochschild 0-cycles for these dg-categories.
In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and use these correspondences to obtain interesting stratifications of braid varieties and their quotients. It is shown that the maximal charts of these stratifications are exponential Darboux charts for the holomorphic symplectic structures, and we relate these strata to exact Lagrangian fillings of Legendrian links.
We show the existence of cluster $\mathcal{A}$-structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig's coordinates. Several explicit seeds are provided and the quiver and cluster variables are readily computable. We prove that these upper cluster algebras equal their cluster algebras, show local acyclicity, and explicitly determine their DT-transformations as the twist automorphisms of braid varieties. The main result also resolves the conjecture of B. Leclerc on the existence of cluster algebra structures on the coordinate rings of open Richardson varieties.
We show that each cluster seed in the augmentation variety is inhabited by an embedded exact Lagrangian filling. This resolves the matter of surjectivity of the map from Lagrangian fillings to cluster seeds. The main new technique to produce these Lagrangian fillings is the construction and study of a quiver with potential associated to curve configurations. We prove that its deformation space is trivial and show how to use it to manipulate Lagrangian fillings with $\mathbb{L}$-compressing systems via Lagrangian disk surgeries.
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.
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 associated cluster structures, and prove that the conjugate surface of the graph is Hamiltonian isotopic to the Lagrangian filling associated to the weave. The T-duality map for plabic graphs has a surprising key role in the construction of these weaves. Second, we use the above established bridge between weaves and reduced plabic graphs to show that the Muller-Speyer twist map on positroid varieties is the Donaldson-Thomas transformation. This latter statement implies that the Muller-Speyer twist is a quasi-cluster automorphism. An additional corollary of our results is that target labeled seeds and the source labeled seeds are related by a quasi-cluster transformation.
This note explores the use of Newton polytopes in the study of Lagrangian fillings of Legendrian submanifolds. In particular, we show that Newton polytopes associated to augmented values of Reeb chords can distinguish infinitely many distinct Lagrangian fillings, both for Legendrian links and higher-dimensional Legendrian spheres. The computations we perform work in finite characteristic, which significantly simplifies arguments and also allows us to show that there exist Legendrian links with infinitely many non-orientable exact Lagrangian fillings.
We construct a binary mutation invariant for skew-symmetric integer matrices. The invariant is not an integer congruence invariant for matrices of odd size: we provide examples of congruent such matrices with different values for the invariant.
This article introduces a new method to construct volume-filling symplectic embeddings of 4-dimensional ellipsoids by employing polytope mutations in toric and almost-toric varieties. The construction uniformly recovers the full sequences for the Fibonacci Staircase of McDuff-Schlenk, the Pell Staircase of Frenkel-Muller and the Cristofaro-Gardiner-Kleinman's Staircase, and adds new infinite sequences of ellipsoid embeddings. In addition, we initiate the study of symplectic tropical curves for almost-toric fibrations and emphasize the connection to quiver combinatorics.
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.
We present the first examples of elements in the fundamental group of the space of Legendrian links in the standard contact 3-sphere whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer-theoretic techniques. These families include the first known Legendrian links with infinitely many fillings that are not rainbow closures of positive braids, and the smallest Legendrian link with infinitely many fillings known to date. We discuss how to use our examples to construct other links with infinitely many fillings, in particular giving the first Floer-theoretic proof that Legendrian (n,m) torus links have infinitely many Lagrangian fillings, if n is greater than 3 and m greater than 6, or (n,m)=(4,4),(4,5). In addition, for any given higher genus, we construct a Weinstein 4-manifold homotopic to the 2-sphere whose wrapped Fukaya category can distinguish infinitely many exact closed Lagrangian surfaces of that genus. A key technical ingredient behind our results is a new combinatorial formula for decomposable cobordism maps between Legendrian contact DGAs with integer (group ring) coefficients.