Related papers: Gluing Localized Mirror Functors

We introduce an algorithm to piecewise dualise linear quivers into their mirror dual. The algorithm uses two basic duality moves and the properties of the $S$-wall which can all be derived by iterative applications of Seiberg-like…

In this note we discuss the geometry of Riemannian surfaces having a discrete set of singular points. We assume the conformal structure extends through the singularities and the curvature is integrable. Such points are called \emph{simple…

We present a novel approach to the concept of gluing in mathematics by introducing the notions of a gluing data category and a gluing data functor. Our work provides a formal categorical characterization of the notion of gluing in algebraic…

We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2875 lines on…

In this paper, we study Hamiltonian stationary Lagrangian surfaces in complex space forms. We first show that when the mean curvature is a non-zero constant, the second fundamental form is parallel. We then consider the case in which the…

We construct a Lagrangian in the cotangent bundle of a 3-torus whose projection to the fiber is a neighborhood of a tropical curve with a single 4-valent vertex. This Lagrangian has an isolated conical singular point, and its smooth locus…

The Doran-Harder-Thompson "gluing/splitting" conjecture unifies mirror symmetry conjectures for Calabi-Yau and Fano varieties, relating fibration structures on Calabi-Yau varieties to the existence of certain types of degenerations on their…

The special geometry of calibrated cycles, closely related to mirror symmetry among Calabi--Yau 3-folds, is itself a real form of a new subject, which we call slightly deformed algebraic geometry. On the other hand, both of these geometries…

We describe a general procedure, based on Gerstenhaber-Schack complexes, for extending to quantized twistor spaces the Donaldson-Friedman gluing of twistor spaces via deformation theory of singular spaces. We consider in particular various…

We use Lagrangian torus fibrations on the mirror $X$ of a toric Calabi-Yau threefold $\check X$ to construct Lagrangian sections and various Lagrangian spheres on $X$. We then propose an explicit correspondence between the sections and line…

This paper lays the groundwork for the theory of categorical diagonalization. Given a diagonalizable operator, tools in linear algebra (such as Lagrange interpolation) allow one to construct a collection of idempotents which project to each…

We propose a procedure which allows one to construct local symmetry generators of general quadratic Lagrangian theory. Manifest recurrence relations for generators in terms of so-called structure matrices of the Dirac formalism are…

Given a smooth toric variety X and an ample line bundle O(1), we construct a sequence of Lagrangian submanifolds of (C^*)^n with boundary on a level set of the Landau-Ginzburg mirror of X. The corresponding Floer homology groups form a…

This paper is motivated by a relatively recent work by Joyce in special Lagrangian geometry, but the basic idea of the present paper goes back to an earlier pioneering work of Donaldson in Yang--Mills gauge theory; Donaldson discovered a…

We compute the Lagrangian cobordism group of the standard symplectic 2-torus and prove that it is isomorphic to the Grothendieck group of its derived Fukaya category. The proofs use homological mirror symmetry for the 2-torus.

This paper introduces differential-geometric methods to study $n$-dimensional locally conformally flat spaces as submanifolds in $\mathbb{R}^{n+2}$. We derive explicit formulas relating intrinsic and ambient differential-geometric objects,…

Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of PDEs. We explain these conjectures and verify them in some cases.

We construct a four-parameter family of affine Yangian algebras by gluing two copies of the affine Yangian of $\mathfrak{gl}_1$. Our construction allows for gluing operators with arbitrary (integer or half integer) conformal dimension and…

Let (M,w) be a compact symplectic manifold, and L a compact, embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono construct Lagrangian Floer cohomology for such M,L, yielding groups HF^*(L,b;\Lambda) for one Lagrangian or…

We find two natural spherical functors associated to the Kummer surface and analyse how their induced twists fit with Bridgeland's conjecture on the derived autoequivalence group of a complex algebraic K3 surface.