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Affine cluster varieties are covered up to codimension 2 by open algebraic tori. We put forth a general conjecture (based on earlier conversation between Vivek Shende and the last author) characterizing their deep locus, i.e. the complement…
The purpose of this article is to initiate a study of a class of Lorentz invariant, yet tractable, Lagrangian Field Theories which may be viewed as an extension of the Klein-Gordon Lagrangian to many scalar fields in a novel manner. These…
We introduce the notion of Wall-Crossing Structure and discuss it in several examples including complex integrable systems, Donaldson-Thomas invariants and Mirror Symmetry. For a big class of non-compact Calabi-Yau 3-folds we construct…
This work has two goals. The first is to provide a conceptual introduction to Heegaard Floer homology, the second is to survey the current state of the field, without aiming for completeness. After reviewing the structure of Heegaard Floer…
We show how the change from Eulerian to Lagrangian coordinates for the two-component Camassa-Holm system can be understood in terms of certain reparametrizations of the underlying isospectral problem. The respective coordinates correspond…
Let $M$ be a fibered 3-manifold with multiple boundary components. We show that the fiber structure of $M$ transforms to closely related transversely oriented taut foliations realizing all rational multislopes in some open neighborhood of…
The goal of this paper is to formalize the notion of The Compositional Integral in The Complex Plane. We prove a convergence theorem guaranteeing its existence. We prove an analogue of Cauchy's Integral Theorem--and suggest an approach at…
We exhibit the one-loop multi-gluon effective Lagrangian in any dimension for a field theory with a quasilocal background, using the background-field formalism. Specific results, including counter terms (up to 12 spacetime dimensions), have…
We show that Floer cohomology of an immersed Lagrangian brane is invariant under smoothing of a self-intersection point if the quantum valuation of the weakly bounding cochain vanishes and the Lagrangian has dimension at least two. The…
The purpose of this paper is to give an application of the gluing theorem for special Lagrangian submanifolds of a Calabi-Yau 3-fold. We proved a gluing theorem before to smooth a codimension-two singularity of a particular special…
The width of a Lagrangian is the largest capacity of a ball that can be symplectically embedded into the ambient manifold such that the ball intersects the Lagrangian exactly along the real part of the ball. Due to Dimitroglou Rizell,…
We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then…
The SYZ conjecture suggests a folklore that "Lagrangian multi-sections are mirror to holomorphic vector bundles". In this paper, we prove this folklore for Lagrangian multi-sections inside the cotangent bundle of a vector space, which are…
We introduce the orthogonal Grassmannian as a novel kinematic space for describing correlators of massless spinning fields in de Sitter space. By automatically encoding the constraints of conformal symmetry and current conservation, the…
Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known "elementary" building blocks for Lagrangian cobordisms that…
We find a relation between Lagrangian Floer pairing of a symplectic manifold and Kapustin-Li pairing of the mirror Landau-Ginzburg model under localized mirror functor. They are conformally equivalent with an interesting conformal factor…
The structure of integrable field theories in the presence of jump defects is discussed in terms of boundary functions under the Lagrangian formalism. Explicit examples of bosonic and fermionic theories are considered. In particular, the…
We introduce the notion of (graded) anchored Lagrangian submanifolds and use it to study the filtration of Floer' s chain complex. We then obtain an anchored version of Lagrangian Floer homology and its (higher) product structures. They are…
We show that the interior of the convex core of a quasifuchsian punctured-torus group admits an ideal decomposition (usually an infinite triangulation) which is canonical in two different senses: in a combinatorial sense via the pleating…
We give a construction of the Floer homology of the pair of {\it non-compact} Lagrangian submanifolds, which satisfies natural continuity property under the Hamiltonian isotopy which moves the infinity but leaves the intersection set of the…