Classifies gravitational instantons with quadratic volume growth.
problem Classifying gravitational instantons with specific growth properties.
method Proves a classification theorem for ALG∗ gravitational instantons, determines topology, and proves existence of uniform coordinates. result Proves a relationship between ALG gravitational instantons of different orders.
Given two hyperkähler manifolds M and N and a quaternionic instanton on their product, a hyperkähler Nahm transform can be defined, which maps quaternionic instantons on M to quaternionic instantons on N. This construction includes the case of Nahm transform for periodic instantons on $\bR^4$, the Fourier-Mukai…
Study calculates ring structure in instanton homology for a surface with points.
problem Calculating the ring structure in instanton homology for a surface with points.
method Used singular instanton Floer homology and excision formula.
result Proved an excision formula for instanton Floer homology when n=1.
Survey of recent progress in gravitational instantons
problem Classification of hyperkähler and Hermitian gravitational instantons
method Survey of recent progress in gravitational instantons
result Survey of recent progress in gravitational instantons
Study shows instantons and monopoles decompose into U(1) components.
problem Understanding the asymptotic structure of instantons and monopoles.
method Analysis of multi-centered Taub-NUT manifolds, calorons, and monopoles on R^3.
result Instantons and monopoles asymptotically decompose into U(1) components.
We construct a spectral sequence from the reduced odd Khovanov homology of a link converging to the framed instanton homology of the double cover branched over the link, with orientation reversed. Framed instanton homology counts certain instantons on the cylinder of a 3-manifold connect-summed with a 3-torus. En route…
Sp(n)-instantons linked to complex Lagrangian graphs via Fourier-Mukai transform.
problem Understanding Sp(n)-instantons on hyperkahler manifolds with conical singularities.
method Relating Sp(n)-instantons to deformed instantons and studying their properties on hyperkahler manifolds.
result Sp(n)-instantons on hyperkahler manifolds correspond to tri-contact instantons on the 3-Sasakian link.
Study on Yang-Mills equation near instanton-anti-instanton configurations with energy constraints.
problem Understanding Yang-Mills connections near instanton-anti-instanton configurations.
method Analyzing the Uhlenbeck limit and bubble configurations, determining obstructions and proving solutions.
result Instantons are the only solutions with energy less than $4π^2 \left( |κ| + 2
ight) + \varepsilon_κ$.
Proves exact triangle linking knot instanton Floer homology to surgeries.
problem Relating knot instanton Floer homology to surgeries.
method Proves an exact triangle using integer coefficients.
result Poincaré Homology Sphere is not an instanton L-space with integer coefficients.
The paper proves unique characterization of gravitational instantons with specific volume growth.
problem Characterizing gravitational instantons with quadratic volume growth.
method Defining a period mapping and proving its surjectivity and openness.
result The periods uniquely characterize ALG∗ and ALG gravitational instantons up to diffeomorphism. We study the deformation theory of G2-instantons on the 7-sphere, specifically those obtained from instantons on the 4-sphere via the quaternionic Hopf fibration. We find that the pullback of the standard ASD instanton lies in a smooth, complete, 15-dimensional family of G2-instantons. In genera…
Proves unique ALE instanton with toric Hermitian structure.
problem Classify Ricci flat ALE instantons with toric Hermitian non-Kähler structure.
method Direct global analysis of Tod form in Weyl-Papapetrou coordinates, avoiding toric Kähler geometry.
result Eguchi-Hanson instanton is the only smooth, Ricci flat, ALE instanton with toric Hermitian non-Kähler structure.
Classifies instantons on a specific gravitational instanton and computes partition functions.
problem Classifying finite energy harmonic 2-forms and anti-self-dual Yang-Mills instantons.
method Analyzes U(1)-bundles and computes instantons explicitly. result Unique anti-self-dual Yang-Mills instantons exist and are described explicitly.
We investigate instantons on manifolds with Killing spinors and their cones. Examples of manifolds with Killing spinors include nearly Kaehler 6-manifolds, nearly parallel G_2-manifolds in dimension 7, Sasaki-Einstein manifolds, and 3-Sasakian manifolds. We construct a connection on the tangent bundle over these manifo…
Period maps surjective for certain gravitational instantons.
problem Understanding the uniformization of gravitational instantons.
method Uniformization theorems by Chen--Chen, Chen--Viaclovsky, Collins--Jacob--Lin, and Hein--Sun--Viaclovsky--Zhang.
result Period maps surjective for ALH*, ALG, and ALG* gravitational instantons.
We consider the Hermitian Yang-Mills (instanton) equations for connections on vector bundles over a 2n-dimensional Kähler manifold X which is a product Y x Z of p- and q-dimensional Riemannian manifold Y and Z with p+q=2n. We show that in the adiabatic limit, when the metric in the Z direction is scaled down, the gauge…
Gravitational instantons are non-Kähler, providing a counterexample to Euclidean Black Hole Uniqueness.
problem Classical Euclidean Black Hole Uniqueness conjecture
method Analyzing the Chen-Teo gravitational instanton
result The Chen-Teo instanton is Hermitian and non-Kähler
New homological action on sutured instanton homology defined.
problem Detecting link splitting in knot homology.
method Defining a homological action on sutured instanton Floer homology.
result Instanton knot homology detects link splitting for two-component links.
Smooth family of G2-instantons over a Kummer construction.
problem Constructing a smooth family of G2-instantons over a Kummer construction.
method Extending gluing construction for G2-instantons to Kummer constructions, using a Z2-action to overcome obstructions.
result Proves the existence of a smooth 1-parameter family of G2-instantons, showing their infinitesimal rigidity and non-flatness.
Proves uniqueness of certain S1-symmetric gravitational instantons.
problem Proving uniqueness of S1-symmetric gravitational instantons. method Using a divergence identity and results from the G-signature theorem. result Proof of the S1-symmetric Euclidean Black Hole Uniqueness conjecture. First non-trivial examples of deformed G_2-instantons, distinguishing nearly parallel G_2-structures.
problem Distinguishing between nearly parallel G_2-structures and isometric G_2-structures.
method Provided first non-trivial examples of deformed G_2-instantons and studied their deformation theory.
result Found non-trivial deformed G_2-instantons with obstructed deformation theory and moduli spaces of different dimensions.
Instantons on multi-Taub-NUT spaces are mapped to bow representations.
problem Mapping instantons to bow representations for multi-Taub-NUT spaces.
method Proving gauge equivalence classes correspondence and isometry of moduli spaces.
result Instantons on multi-Taub-NUT spaces are isometric to bow representations.
In this paper we explicitly calculate the analogue of the 't Hooft SU(2) Yang--Mills instantons on Gibbons--Hawking multi-centered gravitational instantons which come in two parallel families: the multi-Eguchi--Hanson, or A_k ALE gravitational instantons and the multi-Taub--NUT, or A_k ALF gravitational instantons. We …
Explains a 1978 construction for Yang-Mills instantons.
problem None explicitly stated; focuses on explaining a construction.
method Atiyah-Drinfeld-Hitchin-Manin construction
result Explains the ADHM construction of Yang-Mills instantons.
Existence proved for specific types of gravitational instantons.
problem Existence of toric ALE and ALF gravitational instantons.
method Established existence and uniqueness results for ALE and ALF gravitational instantons.
result Existence of a unique, Ricci-flat, toric ALE and ALF gravitational instanton for every admissible rod structure.
New Spin(7)-instantons constructed on Joyce's manifold.
problem Constructing Spin(7)-instantons on Joyce's compact manifold. method Gluing non-flat connections on local model spaces to a flat connection on the Spin(7)-orbifold. result More than 20,000 new four-parameter families of Spin(7)-instantons. We formulate the deformation theory for instantons on nearly Kähler six-manifolds using spinors and Dirac operators. Using this framework we identify the space of deformations of an irreducible instanton with semisimple structure group with the kernel of an elliptic operator, and prove that abelian instantons are rigid…
Study calculates instanton Floer homology for surgeries on L-space knots.
problem Computing the framed instanton Floer homology of surgeries on L-space knots.
method Used Baldwin-Sivek contact invariant and proved homogeneity of the invariant.
result Framed instanton Floer homology agrees with Heegaard Floer homology for L-space knots.
Study on instantons in G2 and Spin(7) manifolds.
problem Investigate instanton properties in G2 and Spin(7) manifolds. method Analyze the characteristic and torsion connections on G2 and Spin(7) manifolds, focusing on parallel torsion and Lee forms. result Conditions for the curvature of the characteristic connection to be a G2 instanton and for the torsion connection to be a Spin(7) instanton are established. Authors prove Torelli theorem for a specific type of gravitational instantons.
problem Identifying diffeomorphism types of gravitational instantons.
method Using mirror special Lagrangian fibrations and recent work on rational elliptic surfaces.
result Identify 10 diffeomorphism types of ALHb∗ gravitational instantons. Sutured instanton homology uses Heegaard diagrams to bound homology dimensions.
problem Bounding the dimension of sutured instanton homology.
method Using admissible Heegaard diagrams, proving a bound on the number of generators of the associated sutured Heegaard Floer complex.
result Strong L-spaces are instanton L-spaces.
Computes upper bounds for instanton knot homology.
problem Computing the exact dimension of instanton knot homology.
method Developed an algorithm using knot diagrams and Heegaard diagrams.
result Algorithm provides sharp bounds for knots up to seven crossings.
Yang-Mills instantons on ALE gravitational instantons were constructed by Kronheimer and Nakajima in terms of matrices satisfying algebraic equations. These were conveniently organized into a quiver. We construct generic Yang-Mills instantons on ALF gravitational instantons. Our data are formulated in terms of matrix-v…
Defines cobordism maps connecting Khovanov and instanton homologies.
problem No direct correspondence between Khovanov and instanton homology cobordism maps.
method Defines a cobordism map on the instanton cube complex as a filtered chain map.
result Proves the cobordism map recovers both Khovanov and instanton homology cobordism maps.
Constructs non-abelian G2-instantons on ALC members of B7 family.
problem Constructing non-abelian G2-instantons on ALC members of B7 family.
method Using co-homogeneity one symmetries, classify and describe the solutions as perturbations of abelian instantons.
result Find a one-parameter family of instantons with polynomial decay.
The abstract discusses instantons on flat spaces and provides explicit constructions.
problem Understanding instantons on flat spaces.
method Explains and provides explicit constructions of instantons on R4, R7, R8, and Hn. result Natural generalizations of instantons on flat R4 to R7 and R8. New instantons show Einstein-Maxwell fields are more complex.
problem Disprove Euclidean Einstein-Maxwell black hole uniqueness.
method Explicit construction of new three-parameter family of asymptotically flat instantons.
result Demonstrate subtle properties of coupled gravitational and electromagnetic fields.
Study instanton metrics via Taub-NUT deformations.
problem Understanding deformations of instanton metrics.
method Using generalized Legendre transform on bow varieties.
result Found Kähler potential on instanton moduli spaces.
This is our third paper in a series on the gravitational instantons. In this paper, we classify ALG and ALH gravitational instantons. In ALG case, we extend Hein's construction slightly and show that it's the only ALG gravitational instanton. In ALH case, we prove a Torelli-type theorem.
Proves infinitesimal rigidity of Hermitian gravitational instantons.
problem Understanding the moduli space of Hermitian gravitational instantons.
method Proof of infinitesimal rigidity and integrability using boundary conditions and conformal Kähler properties.
result Completes the understanding of Hermitian gravitational instantons, both compact and non-compact.
Study G2-instantons on specific Lie groups, finding conditions and structures.
problem Characterize G2-instantons on 2-step nilpotent Lie groups.
method Analyze connections arising from characteristic connections, use Lie group structure and torsion.
result Establish necessary and sufficient conditions for G2-instantons, define naturally reductive structures.
Solve Painleve VI to relate instanton bundles.
problem Relate instanton bundles to Painleve VI solutions.
method Generalize Hitchin's logarithmic connection to vector bundles with SL2 action.
result Identify Okamoto transformations as creation operators.
New instanton invariants for rational homology spheres defined and shown to be functorial.
problem Defining and proving invariance of instanton homology groups for rational homology spheres.
method Novel suspended flow category technique to handle obstructed cobordisms and prove wall-crossing formula.
result Instanton invariant λI(Y) conjecturally equals Casson-Walker invariant for rational homology spheres. Researchers classify and construct various instantons and connections on a specific 8-manifold.
problem Classifying and constructing instantons and connections on a specific Calabi manifold.
method Invariant Hermitian Yang-Mills connections, Sp(2)-instantons, Spin(7)-instantons, and gauge group analysis.
result Classification of SU(3)-invariant primitive Hermitian Yang-Mills connections and Sp(2)-instantons over T∗CP2. New technique connects instanton Floer homology to Heegaard diagrams for knots and 3-manifolds.
problem Accessing structural properties of instanton Floer homology.
method Developed a new technique using Heegaard diagrams and sutures.
result Established a relation between instanton knot homology and Heegaard diagrams for (1,1)-knots. New invariants from framed instanton homology for knot concordance.
problem Concordance of knots and their properties.
method Framed instanton homology to define invariants.
result Computations and bounds on knot concordance invariants.
The first irreducible solution of the SU(2) self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on R4 were discovered. While soon after, in 1978, the ADHM construction gave a complete descrip…
Taubes proved that all compact oriented four-manifolds admit non-flat instantons. We show that there exists a non-compact oriented four-manifold having no non-flat instanton.