Review of instantons in noncommutative gauge theories across 4, 6, and 8 dimensions.
problem Understanding instantons in noncommutative gauge theories.
method Analysis of instantons in various dimensions, focusing on string theory and toric varieties.
result Geometric interpretations and applications of instantons in non-compact toric varieties.
Study G 2 \mathrm{G}_2 G 2 -instantons on 7-sphere from 4-sphere instantons.
problem Deformation theory of G 2 \mathrm{G}_2 G 2 -instantons on 7-sphere. method Analysis of pullback from 4-sphere instantons and identification of deformation spaces.
result Existence of a smooth, complete family of 15-dimensional G 2 \mathrm{G}_2 G 2 -instantons. Develops gluing theory for contact instantons and pseudoholomorphic curves.
problem Constructing contact instanton Floer cohomology and Fukaya-type category.
method Gluing theory of contact instantons and pseudoholomorphic curves in symplectization context.
result Construction of Legendrian contact instanton homology and moduli spaces of holomorphic buildings.
Study instantons on asymptotically conical Spin(7)-manifolds, identifying deformation spaces.
problem Deformation theory of instantons on specific Spin(7)-manifolds.
method Relating deformation complex to spinors, identifying kernel of twisted negative Dirac operator.
result Virtual dimension of moduli space calculated using index theorem and Dirac operator spectrum.
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.
Study deformations of G2-instantons on nearly G2 manifolds.
problem Deformations of G2-instantons on nearly G2 manifolds.
method Formulated in terms of spinors and Dirac operators, proved isomorphism of infinitesimal deformations to kernel of an elliptic operator.
result Proved abelian instantons are rigid and described the deformation space of the canonical connection on specific nearly G2 manifolds.
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…
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.
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.
Unified tau invariants in instanton and monopole Floer theories.
problem Unified tau invariants in instanton and monopole Floer theories.
method Identified τ_{G} with τ_{G}^{\sharp} via cobordism maps.
result Identified τ_{G} with τ_{G}^{\sharp} via cobordism maps.
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 of U ( 1 ) U(1) U ( 1 ) gauge theories on G 2 G_2 G 2 manifolds, including instantons and Chern-Simons.
problem Investigate U ( 1 ) U(1) U ( 1 ) gauge theories on G 2 G_2 G 2 manifolds. method Analyze U ( 1 ) U(1) U ( 1 ) -Yang-Mills and higher-order U ( 1 ) U(1) U ( 1 ) -Chern-Simons theories on G 2 G_2 G 2 manifolds. result Emergence of G 2 G_2 G 2 manifolds from anti-self-dual U ( 1 ) U(1) U ( 1 ) instantons and calculation of partition functions. 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.
Researchers construct explicit bundles for ALF metrics, revealing rational patching matrices for gravitational instantons.
problem Constructing explicit toric Ricci-flat metrics and their associated bundles.
method Explicit construction of patching matrices for ALF metrics and gravitational instantons.
result Rational form of patching matrices for gravitational instantons in the Chen--Teo family.
A quantum field theory for Spin(7)-instantons derived from moduli spaces.
problem Constructing a topological quantum field theory for Spin(7)-instantons.
method Using Mathai-Quillen formalism and AKSZ formalism, we derive the action and Batalin-Vilkovisky action.
result The Batalin-Vilkovisky action matches the Mathai-Quillen construction and provides a framework for classical observables.
Study of instantons on Sasakian 7-manifolds using gauge fields.
problem Understanding instantons on Sasakian manifolds.
method Fredholm theory, cohomological conditions, index of a transverse elliptic operator.
result Moduli space of selfdual contact instantons is Kähler.
Study uses instanton Floer theory to obstruct knot unknotting operations.
problem Obstructing knot unknotting operations and ribbon concordance.
method Equivariant singular instanton Floer theory with Chern--Simons filtration.
result For a large class of slice knots, any unknotting sequence must contain both signs.
Study calculates deformations of instantons on a specific S p i n ( 7 ) Spin(7) S p in ( 7 ) -manifold.
problem Analyzing instantons on a specific S p i n ( 7 ) Spin(7) S p in ( 7 ) -manifold. method Deformation theory developed by the author.
result Virtual dimensions of moduli spaces of instantons calculated.
New cobordism maps for instanton knot homology help compute surgeries on knots.
problem Computing surgeries on knots using instanton knot homology.
method Construct cobordism maps for instanton knot homology.
result Compute the framed instanton Floer homology of certain knots.
We show that the integral of the first Pontrjagin class is given by an integer and it is identified with instanton number of the U(n) gauge theory on noncommutative R 4 {\bf R^4} R 4 . Here the dimension of the vector space V V V that appear in the ADHM construction is called Instanton number. The calculation is done in operato…
New instanton homology theories for 3-manifolds and bundles.
problem Defining instanton homology for a class of 3-manifolds and bundles.
method Functorial construction for 4-manifold cobordisms, algebraic construction of homology theories for dg-modules.
result Isomorphic to Floer's instanton homology for admissible bundles, calculates I ∞ I^\infty I ∞ in all cases it is defined. Instanton homology detects 2-torsion in fibered knots.
problem Detecting 2-torsion in instanton homology for fibered knots.
method Using sutured instanton theory to derive a formula for I ♯ ( Y , K ; C ) I^\sharp(Y,K;\mathbb{C}) I ♯ ( Y , K ; C ) and comparing dimensions. result Proves the presence of 2-torsion in instanton homology for null-homologous fibered knots.
This work extends knot homology theory to links, proving exact triangles and categorifying link signatures.
problem Extending knot homology theory to links and proving exact triangles.
method Equivariant singular instanton Floer theory, circle-equivariant Morse-Floer theory, cobordism constructions.
result Established unoriented skein exact triangles and categorified link signatures.
Study on deformations of LC Spin(7) instantons simplifies the problem.
problem Deformation theory of instantons on locally conformal Spin(7) manifolds.
method Reformulated linearized deformation equations using a t-parameter family of Dirac operators, demonstrating cancellation of torsion terms.
result The deformation space H^1 is governed by Levi-Civita geometry, reducing the problem to a torsion-free setting.
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 ) U(1) U ( 1 ) -bundles and computes instantons explicitly. result Unique anti-self-dual Yang-Mills instantons exist and are described explicitly.
Expanding on supersymmetric Yang-Mills theory, this work focuses on cohomological field theory aspects.
problem Exploring the mathematical aspects of supersymmetric Yang-Mills theory on 4D manifolds.
method Constructing a cohomological field theory using the Atiyah-Jeffrey construction and demonstrating the transversally elliptic complex.
result The deformation theory of flipping instantons is controlled by a transversally elliptic complex, crucial for non-degeneracy and calculability.
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 \Gamma Γ -invariant ASD connections on connected sums. In four-dimensional gauge theory there exists a well-known correspondence between instantons and holomorphic curves, and a similar correspondence exists between certain octonionic instantons and triholomorphic curves. We prove that this latter correspondence stems from the dynamics of various dimensional reductions of …
The paper introduces new knot invariants using singular instanton gauge theory.
problem Developing new knot invariants using singular instanton gauge theory.
method Using S U ( 2 ) SU(2) S U ( 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.
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 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)).
We examine the anomalies arising in instanton calculus as detailed by Damiano Anslemi in 1994. Whereas Anselmi uses BRST theory, we use the ADHM construction to arrive at the same conclusions from a differential-geometric way. We observe that Anselmi's TQFT is similar to Donaldson Theory applied to charge 1 instantons …
New one-parameter families of S U ( 2 ) 2 SU(2)^2 S U ( 2 ) 2 -invariant instantons found on Calabi-Yau 3-folds.
problem Behavior of Calabi-Yau instantons and monopoles with S U ( 2 ) 2 SU(2)^2 S U ( 2 ) 2 -symmetry. method Gauge theory on asymptotically conical Calabi-Yau 3-folds with S U ( 2 ) 2 SU(2)^2 S U ( 2 ) 2 co-homogeneity one action. result New one-parameter families of invariant instantons found.
Defines a new field theory in 1+1+1 dimensions.
problem Developing a new field theory in 1+1+1 dimensions.
method Defines an extended field theory as a quasi 2-functor with values in a completed 2-category, H a m ^ \widehat{\mathcal{H}am} H am . result Extends existing theories and provides a real analog of a construction by Moore and Tachikawa.
Study instantons on G 2 G_2 G 2 -manifolds, proving uniqueness.
problem Deforming G 2 G_2 G 2 -instantons on asymptotically conical manifolds. method Spinorial approach linking deformations to Dirac operators.
result Uniqueness of G 2 G_2 G 2 -instantons on R 7 R^7 R 7 . This paper is one step toward infinite energy gauge theory and the geometry of infinite dimensional moduli spaces. We generalize a gluing construction in the usual Yang-Mills gauge theory to an ``infinite energy'' situation. We show that we can glue an infinite number of instantons, and that the resulting instantons ha…
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.
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 initiate the systematic study of G 2 G_2 G 2 -instantons with S U ( 2 ) 2 SU(2)^2 S U ( 2 ) 2 -symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on R 4 × S 3 \mathbb{R}^4\times S^3 R 4 × S 3 with its two explicitly known distinct holonomy G 2 G_2 G 2 metrics, whic…
New invariants help study satellite knots and their concordance.
problem Understanding the concordance of satellite knots.
method Filtered instanton homology and classical knot theory techniques.
result Criteria for satellite knots to be linearly independent.
Classifies instantons on ALF multi-Taub-NUT spaces and ties them to bow solutions.
problem Classifying instantons on specific spaces.
method Uses Nahm's equations and bow solutions to construct instantons.
result Constructs holomorphic instanton bundles on multi-Taub-NUT spaces.
Adjusting conventional Chern-Simons theory to G 2 {\rm G}_2 G 2 -manifolds, one describes G 2 {\rm G}_2 G 2 -instantons on bundles over a certain class of 7 7 7 -dimensional flat tori which fiber non-trivially over T 4 T^4 T 4 , by a pullback argument. Moreover, if c 2 ≠ 0 c_2\neq0 c 2 = 0 , any (generic) deformation of the G 2 {\rm G}_2 G 2 -structure away from s…
The paper constructs knot homologies for monopole and instanton theories.
problem Constructing knot homologies in monopole and instanton theories.
method Developed techniques to compute sutured monopole and instanton Floer homologies.
result Proved properties and formulas for knot homologies in non-null-homologous cases.
We study quiver gauge theories on the round and squashed seven-spheres, and orbifolds thereof. They arise by imposing G G G -equivariance on the homogeneous space G / H = S U ( 4 ) / S U ( 3 ) G/H=\mathrm{SU}(4)/\mathrm{SU}(3) G / H = SU ( 4 ) / SU ( 3 ) endowed with its Sasaki-Einstein structure, and G / H = S p ( 2 ) / S p ( 1 ) G/H=\mathrm{Sp}(2)/\mathrm{Sp}(1) G / H = Sp ( 2 ) / Sp ( 1 ) as a 3-Sasakian manifold. In both cases …
This paper extends knot invariants using instantons to study torus knot groups.
problem Understanding the topology of knots and their representations.
method Generalization of equivariant singular instanton Floer theory.
result Irreducible singular instanton homology of torus knots for rational holonomy parameters are Z / 4 \mathbb{Z}/4 Z /4 -graded abelian groups. New instanton Floer homology for sutured manifolds with tangles defined.
problem Defining instanton Floer homology for complex geometric structures.
method Extending excision theorems to handle singularities and tangles.
result Detection of unlink and trivial braid closures in annular Khovanov homology.
The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the sin…
Paper generalizes sutured Floer homologies with new algorithms and polytopes.
problem Computing and understanding sutured Floer homologies for complex manifolds.
method Grading shifting property, algorithms, polytopes, canonical decompositions.
result Established new bounds and detection results for sutured Floer homologies.