Study rho-invariants for torus knots, computing instanton chain complex.
problem Computing rho-invariants for specific involutions and torus knots.
method Equivariant rho-invariants, Floer gradings, singular instanton chain complex.
result Computed rho-invariants and Floer gradings for specific cases.
This paper upgrades instanton TQFT to infinity-categories for better simplification.
problem Developing a more structured approach to instanton TQFT.
method Constructing an infinity-cobordism category and a derived category for instantons.
result A simplified construction of the hypercube of chain complexes for link spectral sequences.
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. 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.
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.
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.
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…
Infinitely many M2-instantons affect M-theory on G2-manifolds.
problem Understanding non-perturbative effects in M-theory on G2-manifolds.
method Analyzing superpotential contributions from M2-instantons on TCS G2-manifolds. result Infinitely many M2-instanton corrections to M-theory on G2-manifolds. We define an elementary relatively Z/4 graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram obtained by taking traceless SU(2) character varieties of 2-tangle decompositions of knots. Calculations for torus knots…
Study tangle invariants using pillowcase Fukaya category and character varieties.
problem Tangle invariants and their relationships with Khovanov cohomology.
method Define a twisted complex of compact Lagrangians in the pillowcase's Fukaya category, show functoriality, and relate to Khovanov cohomology.
result Hom set with a specific Lagrangian is naturally isomorphic to reduced Khovanov chain complex.
Monads link instantons and bow solutions in complex geometry.
problem Linking instantons and bow solutions via monads.
method Generalized ADHM-Nahm transform and monads.
result Established one-to-one correspondence between instantons and bow solutions.
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.
Paper constructs Thom-Smale complex using instantons from Morse functions.
problem Constructing Thom-Smale complex for Morse functions.
method Analytic instanton construction using eigenspaces of mapping cone Laplacian.
result Instanton complex is cochain isomorphic to Thom-Smale complex.
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.
Gravitational instantons collapse to a punctured plane with a special Kahler metric.
problem The collapse of gravitational instantons from a complex structure limit.
method Analysis of a sequence of ALH*-gravitational instantons and their collapse to a punctured plane.
result The moduli space of pointed ALH*-gravitational instantons collapses to a punctured plane with a special Kahler metric.
The paper constructs instanton complexes on stratified pseudomanifolds.
problem Analyzing functions with non-isolated critical points on singular spaces.
method Constructing Witten instanton complexes and Hilbert complexes.
result Proves Morse inequalities for stratified pseudomanifolds.
We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line C, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to TxP1.…
This paper extends Witten's holomorphic Morse inequalities to singular spaces.
problem Applying Witten's holomorphic Morse inequalities to singular spaces.
method Constructing Witten instanton complexes for Kähler Hamiltonian Morse functions on stratified pseudomanifolds.
result Extends Witten's holomorphic Morse inequalities to singular spaces.
There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphi…
We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special…
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.
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…
Researchers smooth out moduli space of Spin(7)-instantons.
problem Regularizing the moduli space of Spin(7)-instantons.
method Constructing and perturbing the moduli space to achieve smoothness and expected dimension.
result Smooth and of expected dimension over the irreducible locus.
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.
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.
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.
Charge measurements for instantons and gravitational perturbations.
problem Evaluating charges in Hermitian non-Kähler Einstein 4-manifolds and their perturbations.
method Evaluation of charges via Killing spinors and perturbation analysis of gravitational instantons.
result Generic gravitational perturbations admit a closed 2-form measuring the charge change.
Study contact instantons on Sasakian 5-manifolds with Calabi-Yau structures.
problem Anti-self-dual contact instantons on Sasakian 5-manifolds with transverse Calabi-Yau structures.
method Singularity data of leaf spaces and computation of moduli spaces.
result Explicit computation of complex dimensions of moduli spaces for 95 orbifold K3 surfaces.
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.
We revisit the problem of constructing instantons on ADE orbifolds R^4/Γand point out some subtle relations with the complex structure on the orbifold. We consider generalized instanton equations on R^4/Γwhich are BPS equations for the Yang-Mills equations with an external current. The relation between level sets of th…
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. Proves complex homology three-spheres can be bounded by many handles.
problem Complex homology three-spheres and their bounding four-manifolds.
method Uses homology cobordism invariant Γ from instanton Floer homology.
result Any bounding four-manifold must be built out of many handles.
Constructs BPS complexes and Chern--Simons theories from G-structures.
problem Infinitesimal moduli space computation and supersymmetric systems.
method Universal algebraic construction of BPS complexes and associated linearised BV Chern--Simons theories.
result Reproduces classic examples in gauge theory and constructs heterotic superpotential functionals.
Given two compact hyperkähler surfaces X and Y and a holomorphic vector bundle Q on X×Y, which is a generalized instanton, one can define a Fourier-Mukai transform, which, under suitable assumptions, maps vector bundles on X to vector bundles on Y. If X and Y are dual complex tori, this transform …
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.
The resolved conifold geometry is linked to a special Kähler manifold and an instanton-corrected hyperkähler manifold.
problem Understanding the geometry of the resolved conifold and its associated structures.
method Explicit description of ASK and instanton-corrected HK manifolds, relating them to twistor coordinates and solving Riemann-Hilbert problems.
result The instanton-corrected hyperkähler manifold realizes a smoothing of the semi-flat HK metric associated with the ASK geometry.
Analytic realization of Thom-Smale complex for G-manifolds.
problem Realizing Thom-Smale complex for G-manifolds with Lie group action.
method Using G-invariant Witten instanton complex associated with a Morse-Bott function.
result Generalized Thom-Smale complex for G-manifolds including horizontal direction influence.
Study on resurgence in Chern-Simons theory, recovering flat connection contributions.
problem Understanding the resurgence properties of Chern-Simons theory partition functions.
method Analyzing Borel transforms and asymptotic expansions of partition functions.
result Recovery of irreducible flat connection contributions from abelian flat connections.
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. Paper extends cobordism maps in Khovanov and instanton homologies.
problem Define and prove compatibility of cobordism maps in Khovanov and instanton homologies.
method Extend embedded cobordism map to immersed cobordisms and prove compatibility.
result Induced map on Khovanov homology is injective for certain concordances.
In earlier work of the authors, the Khovanov complex of a knot or link appeared as the first page in a spectral sequence abutting to the instanton homology. The quantum and (co)homological gradings on Khovanov homology do not survive as gradings, but we show that they survive as filtrations.
Continuous process closes cusps in complex algebraic surfaces.
problem Developing cusps in Kähler-Einstein metrics on algebraic surfaces.
method Continuous cusp closing process via gluing construction.
result Cusps form in Kähler-Einstein metrics near isolated singularities.
In this paper, we study gravitational instantons (i.e., complete hyperkäler 4-manifolds with faster than quadratic curvature decay). We prove three main theorems: 1.Any gravitational instanton must have known end----ALE, ALF, ALG or ALH. 2.In ALG and ALH-non-splitting cases, it must be biholomorphic to a compact comple…
We show that the moduli space M of holomorphic vector bundles on CP3 that are trivial along a line is isomorphic (as a complex manifold) to a subvariety in the moduli of rational curves of the twistor space of the moduli space of framed instantons on R4, called the space of twistor sections. We then use this c…
This paper explores adiabatic solutions of Haydys-Witten equations for knot homology.
problem Investigating instanton Floer homology and its relation to Khovanov homology.
method Analyzes decoupled Haydys-Witten equations and their equivalence to EBE solutions.
result Proposes an equivalence between adiabatic solutions of decoupled Haydys-Witten equations and non-vertical paths in EBE moduli space.
Let Sigma be a smooth complex curve, and let S be the product ruled surface Sigma \times CP^1. We prove a correspondence conjectured by Donaldson between finite energy U(2)-instantons over the cylinder Sigma \times S^1 \times R, and rank 2 holomorphic bundles over S whose restrictions to the divisors at infinity are st…
The paper studies traceless SU(2) representations and instanton gradings for specific knot types.
problem Analyzing traceless SU(2) representations and instanton gradings for two-bridge and (3,n)-torus knots. method Representation-theoretic data assembly and verification; explicit computation of character varieties and spectral-flow gradings.
result Explicit characterization of traceless representations and instanton gradings for specific knot types.
A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has constant rank 2n, n or 0, and degenerate forms in Ω belong to a non-degenerate quadric hypersurface. We show that a trisymplectic manifold is equipped with a holomorphic 3…