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. 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.
First non-trivial examples of deformed Spin(7)-instantons constructed.
problem Constructing deformed Spin(7)-instantons and connections.
method Constructing on cotangent bundles of CP2 and cones over 3-Sasakian 7-manifolds. result First non-trivial examples of deformed Spin(7)-instantons.
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.
Gluing instantons on 4-manifolds with group action.
problem Gluing invariant ASD connections on 4-manifolds with group action.
method Equivariant gluing theory for instanton moduli spaces.
result Construction of Γ-invariant ASD connections on connected sums. 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 on instantons over product manifolds with a codimension-4 form.
problem Characterize dimension reduction for moduli spaces of generalized ASD instantons.
method Integrability results for families of connections, topological criteria, and explicit descriptions of moduli spaces.
result Complete characterization of dimension reduction for moduli spaces of generalized ASD instantons.
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. 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 properties of instanton knot Floer homology and connected sum formula.
problem Properties of instanton knot Floer homology.
method General properties of sutured Floer theories, Heegaard Floer and monopole Floer settings.
result Connected sum formula for instanton knot Floer homology.
Instantons on ALF spaces constructed from bow data.
problem Constructing instantons on Asymptotically Locally Flat spaces.
method Using bow data to represent ALF spaces and their moduli spaces, constructing anti-self-dual connections.
result Anti-self-dual connections on ALF spaces are instantons with finite action.
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.
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.
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…
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. Study Spin(7) instantons and HYM connections on Stenzel metric.
problem Relationship between Spin(7) instantons and HYM connections.
method Reduced to ODEs, analyzed invariant solutions, constructed moduli space.
result Nonexistence of global invariant Spin(7) instantons, one parameter family of irreducible instantons.
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)).
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.
Study instantons on G2-manifolds, proving uniqueness.
problem Deforming G2-instantons on asymptotically conical manifolds. method Spinorial approach linking deformations to Dirac operators.
result Uniqueness of G2-instantons on R7. 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.
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…
We make some comments on noncommutative U(N)-instantons on Rθ4. We elaborate on the equations for the ASD-connection for free modules. Further we make some remarks on the computation of the topological index of ADHM instantons.
Using earlier work of Sá Earp and the author [SEW13] we construct an irreducible unobstructed G2-instanton on an SO(3)-bundle over a twisted connected sum recently discovered by Crowley-Nordström [CN14].
Study of 0-instantons on hyperbolic manifolds, proving invariants and energy formulas.
problem Understanding 0-instantons on hyperbolic manifolds and their properties.
method Analyzing asymptotic expansions and using Fefferman-Graham expansion for Poincaré-Einstein metrics.
result The 0-instanton obstruction tensor is a conformal invariant related to Weyl curvature, vanishing for smooth 0-instantons.
Essential obstruction found for gluing G2−instantons with singularities.
problem Obstructing the gluing of G2−instantons with 1-dimensional singularities. method Using Atiyah classes generated by curvature contraction and analyzing tangent connections.
result Gluing fails if tangent connection is not twisted Fubini-Study on P2. We define a twisted version of Manolescu and Woodward's Symplectic Instanton homology, prove that this invariant fits into the framework of Wehrheim and Woodward's Floer Field theory, and describe its behaviour for connected sum and Dehn surgery.
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…
We consider cones over manifolds admitting real Killing spinors and instanton equations on connections on vector bundles over these manifolds. Such cones are manifolds with special (reduced) holonomy. We generalize the scalar ansatz for a connection proposed by Harland and Nolle in such a way that instantons are parame…
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 $4π^2 \left( |κ| + 2
ight) + \varepsilon_κ$.
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.
We introduce a method to construct G2-instantons over compact G2-manifolds arising as the twisted connected sum of a matching pair of building blocks [Kov03,KL11,CHNP12]. Our construction is based on gluing G2-instantons obtained from holomorphic bundles over the building blocks via the first named author's wo…
We study the converse to the statement that instantons are minimizers of the Yang--Mills energy in four dimensions. We show that given an energy minimizing connection, A, the curvature of A takes values in a subbundle of the adjoint bundle which decomposes as a sum of instantons.
Constructs deformed G2-instantons on specific G2-manifolds.
problem Finding non-trivial deformed G2-instantons on G2-manifolds.
method Explicit construction of deformed G2-instantons on specified manifolds.
result First non-trivial examples of deformed G2-instantons on G2-manifolds.
Modified Laplacian connects to Yang-Mills instantons on manifolds.
problem Understanding instantons on 4D manifolds.
method Infinite dimensional Lévy Laplacian defined on manifolds, parameterized by curves in orthogonal rotations.
result Instantons on 4D manifolds are related to the modified Lévy Laplacian under specific curve conditions.
The article finds non-Abelian DT-instantons on non-Kähler manifolds.
problem Finding solutions to DT-instanton equations on non-Kähler manifolds.
method Constructing examples of DT-instantons for homogeneous almost Hermitian structures on the manifold of full flags in C^3.
result Explicit classification and phenomena of reducibility and disappearance of DT-instantons.
We propose a method to construct G_2-instantons over a compact twisted connected sum G_2-manifold, applying a gluing result of Sá Earp and Walpuski to instantons over a pair of 7-manifolds with a tubular end (see arXiv:1310.7933). In our example, the moduli spaces of the ingredient instantons are non-trivial, and their…
Study on Yang-Mills fields blow-up in 4D, proving certain configurations impossible.
problem Prohibiting specific configurations of Yang-Mills fields in 4D.
method Expanding connection forms on long cylinders, proving equations relating bubble and limit connections.
result Proves certain configurations of Yang-Mills fields in 4D are impossible.
Deformed holomorphic Chern-Simons theory yields new instantons.
problem Deforming classical holomorphic Chern-Simons theory on Calabi-Yau manifolds.
method Deformation of complex structure by a parameter \( h \) leading to new instanton solutions.
result Existence of instanton solutions invariant under re-scalings of \( h \) and their connection to \( G_2 \)-instantons.
Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi-Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi-Yau four-orbifold divided by an anti-holomorphi…
Explicit construction of the basic SU(2) anti-instantons over the multi-Taub--NUT geometry via the classical conformal rescaling method is exhibited. These anti-instantons satisfiy the so-called weak holonomy condition at infinity with respect to the trivial flat connection and decay rapidly. The resulting unital energ…
The paper introduces new knot invariants using singular instanton gauge theory.
problem Developing new knot invariants using singular instanton gauge theory.
method Using SU(2) singular instanton gauge theory, the paper constructs invariants and Morse chain complexes. result The constructions lead to a triad of groups and several concordance invariants.
This is the first nontrivial construction to date of instantons over a compact manifold with holonomy exactly G2. The HYM connections on asymptotically stable bundles over Kovalev's noncompact Calabi-Yau 3-folds, obtained in the first article, are glued compatibly with a twisted connected sum, to produce a G2−ins…
I describe a relation (mostly conjectural) between the Seiberg-Witten monopoles, Fueter sections, and G2 instantons. In the last part of this article I gathered some open questions connected with this relation.
We construct finite energy instanton connection on R4 which are periodic in two directions via an analogue of the Nahm transform for certain singular solutions of Hitchin's equations defined over a 2-torus.
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.
Symplectic instanton homology is an invariant for closed oriented three-manifolds, defined by Manolescu and Woodward, which conjecturally corresponds to a symplectic version of a variant of Floer's instanton homology. In this thesis we study the behaviour of this invariant under connected sum, Dehn surgery, and four-di…
We investigate Yang--Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these spaces introduced by Hausel--Hunsicker--Mazzeo. First referring to the codimension 2 …