Constructs instantons on a specific G_2-manifold limit.
problem Finding instantons on a particular G_2-manifold limit.
method Dynamical systems approach involving perturbations of abelian and cone solutions.
result Obtains a 1-parameter family of SU(2) instantons with bounded curvature.
New one-parameter families of SU(2)2-invariant instantons found on Calabi-Yau 3-folds.
problem Behavior of Calabi-Yau instantons and monopoles with SU(2)2-symmetry. method Gauge theory on asymptotically conical Calabi-Yau 3-folds with SU(2)2 co-homogeneity one action. result New one-parameter families of invariant instantons found.
Study conically singular instantons over SU(3)-manifolds, proving existence and dimension formulas.
problem Existence and dimension of moduli spaces of conically singular instantons.
method Develop Fredholm deformation theory, investigate cokernel of instanton operator.
result Formula for virtual dimension of moduli space of conically singular instantons with structure group P(U(n)).
Study G2-instantons with SU(2)2-symmetry on R4imesS3.
problem Investigate G2-instantons with specific symmetry. method Develop foundational theory, existence, non-existence, and classification results.
result Provide existence and non-existence results for G2-instantons with SU(2)2-symmetry. Study of SU(N)-instanton Floer homology and its behavior under Dehn surgery.
problem Behavior of SU(N)-instanton Floer homology under Dehn surgery. method Construction of surgery exact triangles and tetragons/pentagons for SU(3) and SU(4) instanton Floer homologies, construction of asymptotically cylindrical metrics on ALE spaces. result SU(N)-instanton Floer homology admits a surgery exact (N+1)-gon. SU(3) instanton homology counts Tait colorings for webs and foams.
problem Counting Tait colorings for webs and foams.
method SU(3) gauge theory with structure group, eigenspace decomposition, skein exact triangles.
result SU(3) homology counts Tait colorings, Euler characteristic interpretable as polynomial invariant value.
Study G2-instantons on R4imesS3 with invariant properties.
problem Existence of G2-instantons on R4imesS3. method Explicit construction and study of invariant properties.
result Existence of 1-parameter families of G2-instantons. 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 …
Study knots with bounded SU(2)-cyclic slopes and unique limit point.
problem Infinitely many SU(2)-cyclic surgeries on knots. method Holonomy perturbation techniques applied to instanton knot homology.
result Knots have bounded SU(2)-cyclic slopes with a unique limit point. New invariants show distinct contact structures have unique instanton homology.
problem Distinguishing contact structures on 3-manifolds using instanton Floer homology.
method Defined and used sutured instanton homology to prove contact invariants are linearly independent.
result Contact structures on 3-manifolds with distinct Chern classes have linearly independent contact invariants.
New instantons found on Euclidean Schwarzschild manifold, challenging existing theories.
problem Challenging the understanding of instantons on the Euclidean Schwarzschild manifold.
method Exploring a correspondence between planar Abelian vortices and spherically symmetric instantons.
result Complete description of a connected component of the moduli space of unit energy SU(2) instantons.
New G2-instantons found on 3-sphere's spinor bundle.
problem Classifying and constructing G2-instantons. method Used SU(2)3-symmetries and asymptotically conical, co-homogeneity one G2-metric. result Found new examples of G2-instantons with obstructed deformations. New insights into knot surgeries via instanton 2-torsion.
problem Understanding rational surgeries on knots and their implications.
method Extending earlier results on integral surgeries to rational surgeries using framed instanton homology.
result If framed instanton homology is 2-torsion-free, the knot is an instanton L-space knot and the surgery parameter is greater than 2g(K)-1.
We present a general procedure to construct 6-dimensional manifolds with SU(3)-structure from SU(2)-structure 5-manifolds. We thereby obtain half-flat cylinders and sine-cones over 5-manifolds with Sasaki-Einstein SU(2)-structure. They are nearly Kahler in the special case of sine-cones over Sasaki-Einstein 5-manifolds…
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 $4π^2 \left( |κ| + 2
ight) + \varepsilon_κ$.
Study finds unique instanton for small α values in SU(2) Hopf fibration.
problem Identifying Yang-Mills connections in the context of α-energy.
method Approximations with Yang-Mills α-energy, focusing on SU(2) Hopf fibration.
result SO(4) invariant ADHM instanton is the unique α-critical point for small α values.
Proves SU(2) representations for certain 3-spheres with embedded tori.
problem Characterizing SU(2) representations for specific 3-manifolds.
method Instanton Floer homology, surgery exact triangle, holonomy perturbations, non-vanishing results, and cable surgery results.
result Fundamental groups of certain 3-manifolds admit irreducible SU(2) representations.
New knot classification based on SU(2) representations and instanton homology.
problem Classifying knots based on SU(2) representations and Alexander polynomials.
method Large surgery formula connecting instanton knot homology and framed instanton homology.
result Non-SU(2)-abundant knots are prime with restricted Alexander polynomial coefficients. New proof shows certain 3D shapes can't be instanton L-spaces.
problem Proving certain 3D shapes cannot be instanton L-spaces.
method Proving by gluing complements of knots and analyzing their fundamental groups.
result Proven that the resulting 3-manifold cannot be an instanton L-space.
The study connects knot representations to Seifert hypersurfaces and instanton Floer homology.
problem Existence of Seifert hypersurfaces and irreducible representations for 2-knots.
method Quantitative instanton Floer homology and related techniques.
result Existence of at least four irreducible SU(2) representations for certain knots.
We present a construction of self-dual Yang-Mills connections on the Taub-NUT space. We illustrate it by finding explicit expressions for all SU(2) instantons of instanton number one and generic monodromy at infinity.
For each partial flag manifold of SU(N), we define a Floer homology theory for knots in 3-manifolds, using instantons with codimension-2 singularities.
We present a classification of SU(2) instantons on T2×R2 according to their asymptotic behaviour. We then study the existence of such instantons for different values of the asymptotic parameters, describing explicitly the moduli space for unit charge.
Proves irreducible SU(2) representations for 3-surgery knots.
problem Irreducible SU(2) representations for 3-surgery knots.
method Instanton Floer homology and symplectic Floer homology.
result Proves irreducible SU(2) representations for 3-surgery knots.
Study examines corrections to heterotic geometry on SU(3) manifolds.
problem Analyzing α′2 corrections to heterotic supersymmetry algebra. method Derives integrability condition and pure gauge correction from graviton equation of motion.
result Curvature of tangent bundle connection acquires a (0,2) component, disrupting semi-classical intuition.
New insights on ACYT 6-manifolds reveal instanton conditions for curvature.
problem Understanding instanton conditions on ACYT 6-manifolds.
method Analyzing curvature and torsion properties of ACYT 6-manifolds.
result Curvature is an SU(3)-instanton if and only if torsion is parallel. Study of gauge theories and instantons on special spheres.
problem Understanding instantons on special geometric spaces.
method Equivariant quiver gauge theories, Hermitian Yang-Mills instantons, and moduli spaces.
result Moduli spaces of instantons described for specific geometric cones.
The main result is a computation of the Nahm transform of a SU(2)-instanton over RxT^3, called spatially-periodic instanton. It is a singular monopole over T^3, a solution to the Bogomolny equation, whose rank is computed and behavior at the singular points is described.
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. Null-homotopic knots in certain 3-manifolds are uniquely identified by their complements.
problem Identifying null-homotopic knots in specific 3-manifolds.
method Instanton Floer homology and SU(2)-representation varieties.
result Null-homotopic knots are determined by their complements in certain 3-manifolds.
Study instanton moduli spaces on class VII surfaces, proving complex structure extension.
problem Extend complex structure on Donaldson-Uhlenbeck compactification of instanton moduli spaces.
method Analyzing moduli spaces of SU(2)-instantons on class VII surfaces.
result Prove complex structure extension for moduli spaces of instantons with c2=1.
New proof shows L-space knots are fibered and have specific properties.
problem Proving L-space knots are fibered and have strong quasipositive properties.
method Conceptually simpler proof using Heegaard Floer homology, translated to instanton Floer homology.
result Generalization of L-space knot properties to other Floer homologies.
Study of SU(2) calorons with rotation map symmetry.
problem Understanding the symmetry groups of SU(2) calorons. method Utilizing a modified ADHM construction for calorons, derived from Charbonneau and Hurtubise's work.
result Construction of new calorons through fixed points of symmetry groups.
Study on instantons on nearly Kähler manifolds, focusing on deformations and rigidity.
problem Understanding instantons on nearly Kähler manifolds and their deformations.
method Formulated deformation theory using spinors and Dirac operators, identified kernel of elliptic operator.
result Abelian instantons are rigid, while canonical connections on specific manifolds admit large spaces of deformations.
In this paper we show how hypercomplex function theoretical objects can be used to construct explicitly self-dual SU(2)-Yang-Mills instanton solutions on certain classes of conformally flat 4-manifolds. We use a hypercomplex argument principle to establish a natural link between the fundamental solutions of DΔf=0 …
The information metric arises in statistics as a natural inner product on a space of probability distributions. In general this inner product is positive semi-definite but is potentially degenerate. By associating to an instanton its energy density, we can examine the information metric {\bf g} on the moduli spaces $\M…
Necessary and sufficient conditions to the existence of a hermitian connection with totally skew-symmetric torsion and holonomy contained in SU(3) are given. Non-compact solution to the supergravity-type I equations of motion with non-zero flux and non-constant dilaton is found in dimensions 6. Non-conformally flat non…
Study shows moduli space of Spin(7)-instantons is orientable under certain conditions.
problem Determining orientability of moduli space of Spin(7)-instantons.
method Analyzing the homology group and gauge group properties.
result Moduli space of Spin(7)-instantons is orientable when specific homology condition holds.
The study reveals the moduli of instantons on product manifolds.
problem Understanding instantons on product manifolds.
method Using a broken gauge transformation and Hermitian Yang-Mills connections.
result Moduli spaces of G2 and Spin(7)−instantons on product manifolds are equivalent to Hermitian Yang-Mills connections. We present ADHM-Nahm data for instantons on the Taub-NUT space and encode these data in terms of Bow Diagrams. We study the moduli spaces of the instantons and present these spaces as finite hyperkahler quotients. As an example, we find an explicit expression for the metric on the moduli space of one SU(2) instanton. W…
We prove an energy identity for anti-self-dual connections on the product C\timesΣof the complex plane and a Riemann surface. The energy is a multiple of a basic constant that is determined from the values of a corresponding Chern-Simons functional on flat connections and its ambiguity under gauge transformations. For …
Instantons prove knots are fibered and strongly quasipositive.
problem Proving properties of knots using instantons and L-space surgeries.
method New cobordism maps in framed instanton Floer homology.
result Proves fibered and strongly quasipositive knots have specific homomorphisms.
In this paper we exhibit a one-parameter family of new Taub-NUT instantons parameterized by a half-line. The endpoint of the half-line will be the reducible Yang-Mills instanton corresponding to the Eguchi-Hanson-Gibbons L^2 harmonic 2-form, while at an inner point we recover the Pope-Yuille instanton constructed as a …
We consider SU(3)-equivariant dimensional reduction of Yang-Mills theory over certain cyclic orbifolds of the 5-sphere which are Sasaki-Einstein manifolds. We obtain new quiver gauge theories extending those induced via reduction over the leaf spaces of the characteristic foliation of the Sasaki-Einstein structure, whi…
Khovanov homology can identify trefoil knots.
problem Detecting trefoil knots using Khovanov homology.
method Floer homology, contact geometry, open books, instanton Floer setting, bypass exact triangle, spectral sequence.
result Khovanov homology detects trefoils.
Study irreducible SU(2) representations for knots in 3D.
problem Existence of irreducible SU(2) representations for cyclic branched covers of knots.
method Combining covering space techniques, SICUP matrices, and instanton Floer homology.
result Irreducible SU(2) representations exist for certain knots and branched covers.
We study and construct non-abelian hermitian Yang-Mills (HYM) instantons on Calabi-Yau cones. By means of a particular isometry preserving ansatz, the HYM equations are reduced to a novel Higgs-Yang-Mills flow on the Einstein-Kahler base. For any 2d-dimensional Calabi-Yau cone, we find explicit solutions of the flow eq…
Researchers solve field equations for special gravitational instantons.
problem Solving field equations for conformally Kähler Riemannian four-manifolds.
method Developed a framework to solve the field equations for generalised gravitational instantons using conformal self-duality and cosmological Einstein-Maxwell.
result Found conformally self-dual and Einstein-Maxwell generalisations of specific geometries.