Gluing Construction Research Articles

A Kummer construction for Chern–Ricci flat balanced manifolds

Given a non-Kähler Calabi–Yau compact orbifold with isolated singularities endowed with a Chern–Ricci flat balanced metric, we study, via a gluing construction, the existence of Chern–Ricci flat balanced metrics on its crepant resolutions, and discuss applications to the search of solutions for the Hull–Strominger system. We also describe the scenario of singular threefolds with ordinary double points, and see that similarly is possible to obtain balanced approximately Chern–Ricci flat metrics.

Exponential Decay Estimates and Smoothness of the Moduli Space of Pseudoholomorphic Curves

The compacified moduli space of bordered stable maps carries a Kuranishi structure with boundary. Smoothness of Kuranishi structure along the boundary requires smoothness of coordinate changes along the boundary. The proof of smoothness is written by the authors in [AMS/IP Stud. Adv. Math., 46.1, 2009, xii+396 pp.], [AMS/IP Stud. Adv. math., 46.2, 2009, pp. i–xii and 397–805], and [Surv. Differ. Geom., vol. 22, pp. 133–190, 2018] based on some uniform exponential decay estimates of the stable maps with respect to a parameter T T , the length of the gluing cylindrical neck, near the boundary of the moduli space. In this paper we establish this exponential decay estimates in a precise manner by carefully examining the dependence on the parameter T T of the gluing construction in the setting of bordered Riemann surfaces with boundary punctures. We also show that the smoothness of the collar follows from the aforementioned exponential estimates by taking s = 1 / T s = 1/T as the radial coordinate of the collar.

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We explain a construction of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_2$$\\end{document}-instantons on manifolds obtained by resolving G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_2$$\\end{document}-orbifolds. This includes the case of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_2$$\\end{document}-instantons on resolutions of T7/Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T^7/\\Gamma $$\\end{document} as a special case. The ingredients needed are a G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_2$$\\end{document}-instanton on the orbifold and a Fueter section over the singular set of the orbifold which are used in a gluing construction. In the general case, we make the very restrictive assumption that the Fueter section is pointwise rigid. In the special case of resolutions of T7/Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T^7/\\Gamma $$\\end{document}, improved control over the torsion-free G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_2$$\\end{document}-structure allows to remove this assumption. As an application, we construct a large number of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_2$$\\end{document}-instantons on the simplest example of a resolution of T7/Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T^7/\\Gamma $$\\end{document}. We also construct one new example of a G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_2$$\\end{document}-instanton on the resolution of (T3×K3)/Z22\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(T^3 \ imes \ ext {K3})/\\mathbb {Z}^2_2$$\\end{document}.

MHD equilibria with nonconstant pressure in nondegenerate toroidal domains

We prove the existence of piecewise smooth MHD equilibria in three-dimensional toroidal domains of \mathbb R^3 where the pressure is constant on the boundary but not in the interior. The pressure is piecewise constant and the plasma current exhibits an arbitrary number of current sheets. We also establish the existence of free boundary steady states surrounded by vacuum with an external surface current. The toroidal domains where these equilibria are shown to exist need not be small perturbations of an axisymmetric domain, and in fact they can have any knotted topology. The building blocks we use in our construction are analytic toroidal domains satisfying a certain nondegeneracy condition, which roughly states that there exists a force-free field that is ergodic on the surface of the domain. The proof involves three main ingredients: a gluing construction of piecewise smooth MHD equilibria, a Hamilton–Jacobi equation on the two-dimensional torus that can be understood as a nonlinear deformation of the cohomological equation (so the nondegeneracy assumption plays a major role in the corresponding analysis), and a new KAM theorem tailored for the study of divergence-free fields in three dimensions whose Poincaré map cannot be computed explicitly.

Calorons and Constituent Monopoles

We study anti-self-dual Yang–Mills instantons on {mathbb {R}}^{3}times S^{1}, also known as calorons, and their behaviour under collapse of the circle factor. In this limit, we make explicit the decomposition of calorons in terms of constituent pieces which are essentially charge 1 monopoles. We give a gluing construction of calorons in terms of the constituents and use it to compute the dimension of the moduli space. The construction works uniformly for structure group an arbitrary compact semi-simple Lie group.

Free boundary minimal surfaces with connected boundary in the $3$-ball by tripling the equatorial disc

In the Euclidean unit three-ball, we construct compact, embedded, two-sided free boundary minimal surfaces with connected boundary and prescribed high genus, by a gluing construction tripling the equatorial disc. Aside from the equatorial disc itself, these are the first examples in the three-ball of compact free boundary minimal surfaces with connected boundary.

Helicoids and vortices

We point out an interesting connection between fluid dynamics and minimal surface theory: When gluing helicoids into a minimal surface, the limit positions of the helicoids correspond to a ‘vortex crystal’, an equilibrium of point vortices in two-dimensional fluid that move together as a rigid body. While vortex crystals have been studied for almost 150 years, the gluing construction of minimal surfaces is relatively new. As a consequence of the connection, we obtain many new minimal surfaces and some new vortex crystals by simply comparing notes.

On the gluing construction of translating solitons to Lagrangian mean curvature flow

We construct Lagrangian translating solitons by desingularizing the intersection points between Lagrangian Grim Reaper cylinders with the same phase using special Lagrangian Lawlor necks. The resulting Lagrangian translating solitons could have arbitrarily many ends and non-contractible loops.

A self-dual complete resolution

In this paper, we construct a self-dual complete resolution of a module defined by a pair of embedded complete intersection ideals in a local ring. Our construction is based on a gluing construction of Herzog and Martsinkovsky and exploits the structure of Koszul homology in the embedded complete intersection case. As a consequence of our construction, we produce an isomorphism between certain stable homology and cohomology modules.

A gluing construction of projective K3 surfaces

We construct a non-Kummer projective K3 surface $X$ which admits compact Levi-flats by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective plane at nine general points.

Homological mirror symmetry for invertible polynomials in two variables

In this paper, we give a proof of homological mirror symmetry for two variable invertible polynomials, where the symmetry group on the B-side is taken to be maximal. The proof involves an explicit gluing construction of the Milnor fibres, and, as an application, we prove derived equivalences between certain nodal stacky curves, some of whose irreducible components have non-trivial generic stabiliser.

Higher Order Automatic Differentiation of Higher Order Functions

We present semantic correctness proofs of automatic differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Throughout, we show how the analysis extends to AD methods for computing higher order derivatives using a Taylor approximation.

Growing random graphs with a preferential attachment structure

The aim of this paper is to develop a method for proving almost sure convergence in Gromov-Hausodorff-Prokhorov topology for a class of models of growing random graphs that generalises Remy's algorithm for binary trees. We describe the obtained limits using some iterative gluing construction that generalises the famous line-breaking construction of Aldous' Brownian tree. In order to do that, we develop a framework in which a metric space is constructed by gluing smaller metric spaces, called \emph{blocks}, along the structure of a (possibly infinite) discrete tree. Our growing random graphs seen as metric spaces can be understood in this framework, that is, as evolving blocks glued along a growing discrete tree structure. Their scaling limit convergence can then be obtained by separately proving the almost sure convergence of every block and verifying some relative compactness property for the whole structure. For the particular models that we study, the discrete tree structure behind the construction has the distribution of an affine preferential attachment tree or a weighted recursive tree. We strongly rely on results concerning those two models of random trees and their connection, obtained in a companion paper.

Ginzburg algebras of triangulated surfaces and perverse schobers

AbstractGinzburg algebras associated to triangulated surfaces provide a means to categorify the cluster algebras of these surfaces. As shown by Ivan Smith, the finite derived category of such a Ginzburg algebra can be embedded into the Fukaya category of the total space of a Lefschetz fibration over the surface. Inspired by this perspective, we provide a description of the unbounded derived category in terms of a perverse schober. The main novelty is a gluing formalism describing the Ginzburg algebra as a colimit of certain local Ginzburg algebras associated to discs. As a first application, we give a new construction of derived equivalences between these Ginzburg algebras associated to flips of an edge of the triangulation. Finally, we note that the perverse schober as well as the resulting gluing construction can also be defined over the sphere spectrum.

Atiyah classes and the essential obstructions in deforming a singular $$G_{2}$$-instanton

When the rank of the bundle is \(\ge 2\), in a certain sense, we found an essential obstruction for the gluing construction of \(G_{2}\)-instantons with 1-dimensional singularities. It involves the Atiyah classes generated by contracting a vector in \(\mathbb {C}^{3}\) with the curvature. Intuitively speaking, the gluing does not work if the tangent connection at a component of the 1-dimensional singular locus is not the twisted Fubini-Study connection on a twisted tangent bundle of \(\mathbb {P}^{2}\). Particularly, it fails if the rank of the bundle is \(\ge 3\).

Model Higgs bundles in exceptional components of the Sp(4, ℝ)-character variety

We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the [Formula: see text]-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the [Formula: see text] exceptional components of the maximal [Formula: see text]-Higgs bundle moduli space, which correspond to components solely consisting of Zariski dense representations. This also allows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by Guichard and Wienhard.

Gluing Non-commutative Twistor Spaces

AbstractWe 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 possible quantizations of twistor spaces that leave the underlying spacetime manifold classical, including the geometric quantization of twistor spaces originally constructed by the second author, as well as some variants based on non-commutative geometry. We discuss specific aspects of the gluing construction for these different quantization procedures.

Derived gluing construction of chiral algebras

We discuss the gluing construction of class $\mathcal{S}$ chiral algebras in derived setting. The gluing construction in non-derived setting was introduced by Arakawa to construct a family of vertex algebras of which the associated varieties give genus zero Moore-Tachikawa symplectic varieties. Motivated by the higher genus case, we introduce a dg vertex algebra version $\mathsf{MT}_{\mathrm{ch}}$ of the category of Moore-Tachikawa symplectic varieties, where a morphism is given by a dg vertex algebra equipped with action of the universal affine vertex algebra, and composition of morphisms is given by the BRST reduction. We also show that the procedure taking the associated scheme of gives a functor from $\mathsf{MT}_{\mathrm{ch}}$ to the category $\mathsf{MT}$ of derived Moore-Tachikawa varieties, which would imply compatibility of gluing constructions in both categories.

Spectra of perfect state transfer Hamiltonians on fractal-like graphs

In this paper we study the spectral features, on fractal-like graphs, of Hamiltonians which exhibit the special property of perfect quantum state transfer (PQST): the transmission of quantum states without dissipation. The essential goal is to develop the theoretical framework for understanding the interplay between PQST, spectral properties, and the geometry of the underlying graph, in order to design novel protocols for applications in quantum information science. We present a new lifting and gluing construction, and use this to prove results concerning an inductive spectral structure, applicable to a wide variety of fractal-like graphs. We illustrate this construction with explicit examples for several classes of diamond graphs.

Dk Gravitational Instantons as Superpositions of Atiyah–Hitchin and Taub–NUT Geometries

Abstract We obtain Dk ALF gravitational instantons by a gluing construction which captures, in a precise and explicit fashion, their interpretation as nonlinear superpositions of the moduli space of centred SU(2) monopoles, equipped with the Atiyah–Hitchin metric, and k copies of the Taub–NUT manifold. The construction proceeds from a finite set of points in euclidean space, reflection symmetric about the origin, and depends on an adiabatic parameter which is incorporated into the geometry as a fifth dimension. Using a formulation in terms of hyperKähler triples on manifolds with boundaries, we show that the constituent Atiyah–Hitchin and Taub–NUT geometries arise as boundary components of the five-dimensional geometry as the adiabatic parameter is taken to zero.