Abstract relates G2 instantons to Seiberg-Witten monopoles.
problem Understanding the connection between G2 instantons and Seiberg-Witten monopoles.
method Relates Seiberg-Witten monopoles, Fueter sections, and G2 instantons.
result Describes a conjectural relation between G2 instantons and Seiberg-Witten monopoles.
In G2 manifolds, 3-dimensional associative submanifolds (instantons) play a role similar to J-holomorphic curves in symplectic geometry. In [21], instantons in G2 manifolds were constructed from regular J-holomorphic curves in coassociative submanifolds. In this exposition paper, after reviewing the background of G2 ge…
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 give a sufficient condition for an associative submanifold in a G2-manifold to appear as the bubbling locus of a sequence of G2-instantons, related to the existence of a Fueter section of a bundle of ASD instanton moduli spaces over said submanifold.
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.
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.
Method constructs G2-instantons over twisted connected sums.
problem Constructing G2-instantons over complex manifolds.
method Gluing G2-instantons from holomorphic bundles over building blocks.
result Explicit examples from semi-Fano 3-folds.
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.
Let M be a G2-manifold. We consider an almost CR-structure on the sphere bundle of unit tangent vectors on M, called the CR twistor space. This CR-structure is integrable if and only if M is a holonomy G2 manifold. We interpret G2-instanton bundles as CR-holomorphic bundles on its twistor space.
The heterotic G2 system is solved on Calabi-Yau 3-orbifolds.
problem Solving the heterotic G2 system on Calabi-Yau 3-orbifolds.
method Adjusting circle bundles and powers of α' to obtain cocalibrated G2 structures.
result Non-trivial solutions to the heterotic G2 system are found.
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.
Classifies G2-instantons on Aloff-Wallach spaces, distinguishing nearly parallel structures.
problem Classifying G2-instantons on Aloff-Wallach spaces. method Classification of invariant G2-instantons for homogeneous coclosed G2-structures. result Examples of irreducible G2-instantons merging into reducible and obstructed ones. 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. 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. 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. 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. 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…
We find G2-manifolds with specific asymptotic properties.
problem Existence and structure of G2-manifolds with ALC asymptotics.
method Robust Fredholm theory for ALC spaces, proving existence and rigidity results.
result Existence of a G2-analogue of the Atiyah-Hitchin metric and good moduli theory for ALC G2-holonomy metrics.
A concrete model for a 7-dimensional gauge theory under special holonomy is proposed, within the paradigm outlined by Donaldson and Thomas, over the asymptotically cylindrical G2-manifolds provided by Kovalev's noncompact version of the Calabi conjecture. One obtains a solution to the G2-instanton equation from the …
Study of U(1) gauge theories on G2 manifolds, including instantons and Chern-Simons.
problem Investigate U(1) gauge theories on G2 manifolds. method Analyze U(1)-Yang-Mills and higher-order U(1)-Chern-Simons theories on G2 manifolds. result Emergence of G2 manifolds from anti-self-dual U(1) instantons and calculation of partition functions. New equivalence found between Yang-Mills and heterotic G2 structures.
problem Understanding instanton connections and moduli spaces in string theory.
method Rephrasing heterotic G2 systems in terms of differential operators and instanton conditions.
result Strominger-Hull system equivalent to holomorphic Yang-Mills on restricted manifolds.
Study of instantons on Stiefel manifold with G2 and Sasakian structures.
problem Characterizing and classifying instantons on Stiefel manifold.
method Reductive decomposition, spinorial approach, Weitzenböck-type formula with torsion.
result Classification of invariant connections and rigidity results for G2-instantons. The paper examines properties of deformed Donaldson-Thomas connections on G2-manifolds.
problem Properties of deformed Donaldson-Thomas connections on G2-manifolds.
method Analyzes the existence, description, and flow of dDT connections.
result Observations in dDT connections mirror those in other geometric problems.
New solutions found for G2 system on squashed manifolds.
problem Finding solutions to the heterotic G2 system on squashed manifolds.
method Constructing families of solutions using squashed metrics on 7-sphere or Aloff-Wallach space.
result Obtained AdS3 solutions for most squashing parameters, excluding a specific case.
The paper defines a moduli space for heterotic G2 systems.
problem Deformations of heterotic string compactifications on G2 manifolds.
method Defines a covariant exterior derivative and shows it is equivalent to a heterotic G2 system.
result Determines the infinitesimal moduli space of heterotic G2 systems.
Study on L2 harmonic forms on special holonomy manifolds, proving vanishing results.
problem Analyzing L2 harmonic forms on complete special holonomy manifolds. method Examined L2 harmonic forms on G2 and Spin(7) manifolds with nonzero parallel forms. result Vanishing of L2 harmonic 2-forms on G2 and Spin(7) manifolds. Method constructs rigid associative submanifolds in twisted G2-manifolds.
problem Constructing rigid associative submanifolds in twisted G2-manifolds.
method Introducing a gluing theorem for asymptotically cylindrical associative submanifolds in ACyl G2-manifolds.
result Yields many new topological types of rigid associative submanifolds.
The 7-dimensional link K of a weighted homogeneous hypersurface on the round 9-sphere in C5 has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed G2-structure φ induced by the Calabi-Yau 3-orbifold basic geometry. We disti…
Witten's approach to Khovanov homology of knots is based on the five-dimensional system of partial differential equations, which we call Haydys-Witten equations. We argue for a one-to-one correspondence between its solutions and solutions of the seven-dimensional system of equations. The latter can be formulated on any…
We describe the infinitesimal moduli space of pairs (Y,V) where Y is a manifold with G2 holonomy, and V is a vector bundle on Y with an instanton connection. These structures arise in connection to the moduli space of heterotic string compactifications on compact and non-compact seven dimensional spaces, e.…
G2-manifolds with a cohomogeneity-one action of a compact Lie group G are studied. For G simple, all solutions with holonomy G2 and weak holonomy G2 are classified. The holonomy G2 solutions are necessarily Ricci-flat and there is a one-parameter family with SU(3)-symmetry. The weak holonomy G2 solutions are Einstein o…
Investigates counting associative 3-folds in G2-manifolds, analogous to holomorphic curves in Calabi-Yau.
problem Counting associative 3-folds in G2-manifolds and defining invariants that are independent of the taming 4-form.
method Generalizes Gromov-Witten theory to tamed almost G2-manifolds, focusing on associative 3-folds.
result Conjectures a superpotential counting associative 3-spheres, invariant under certain reparametrizations.
The article classifies G2-structures with conformally flat metrics.
problem Identifying G2-structures with specific geometric properties.
method Classifying closed G2-structures with conformally flat metrics.
result Any closed G2-structure with conformally flat metric is locally equivalent to one of three explicit examples.
The paper explores Einstein warped G2 and Spin(7) manifolds.
problem Understanding Einstein warped G2 and Spin(7) manifolds.
method Warped G2 and Spin(7) structures with Einstein fiber are studied.
result Explicit descriptions of torsion forms for these structures.
We overview the properties of non-infinitesimal deformations of G2-structures on seven-manifolds, and in particular, focus on deformations that lie in the seven-dimensional representation of G2 and are thus defined by a vector. We then consider deformations from G2-structures with the torsion class having one-dimension…
The study shows that certain nearly G2 and nearly Kähler conifolds cannot be resolved by gluing asymptotically conical G2 and Calabi-Yau manifolds.
problem The inability to resolve nearly G2 and nearly Kähler conifolds by gluing asymptotically conical G2 and Calabi-Yau manifolds.
method Topological analysis of asymptotically conical G2 and Calabi-Yau manifolds to show conditions under which resolutions are impossible.
result For certain rates of the metric, the G2 4-form and Kähler form cannot be simultaneously exact, leading to non-existence of resolutions.
Study on G2-structures on solvmanifolds, focusing on Laplacian solitons and Ricci pinching.
problem Existence and interplay of Laplacian solitons and Ricci pinched G2-structures on solvmanifolds.
method Exploration of left-invariant G2-structures on solvable Lie groups, analysis of Ricci pinching properties.
result Obtained Ricci pinching properties and extremal values for G2-structures on solvmanifolds.
We give an answer to a question posed recently by R.Bryant, namely we show that a compact 7-dimensional manifold equipped with a G2-structure with closed fundamental form is Einstein if and only if the Riemannian holonomy of the induced metric is contained in G2. This could be considered to be a G2 analogue of the Gold…
This paper is a review of current developments in the study of moduli spaces of G2 manifolds. G2 manifolds are 7-dimensional manifolds with the exceptional holonomy group G2. Although they are odd-dimensional, in many ways they can be considered as an analogue of Calabi-Yau manifolds in 7 dimensions. They play an impor…
New perspective on G2-structures flow from DeTurck Laplacian.
problem Understanding G2-structures and their flows.
method Introducing a new flow (DeTurck Laplacian flow) for G2-structures.
result DeTurck Laplacian flow is a flow of G2-structures.
Found a new compact G2-structure on a 7-manifold.
problem Identifying compact G2-structures on manifolds.
method Lattice in solvable Lie group, Laplacian coflow analysis.
result Obtained a 4-parameter subfamily of expanding solitons.
Article explores G2-structures with quadratic conditions, finding new ERP and complete solitons.
problem Investigating closed G2-structures satisfying quadratic conditions.
method Analyzing a second-order PDE system involving parameter λ, producing new examples of ERP and complete solitons.
result First examples of ERP G2-structures, including complete inhomogeneous ERP G2-structure.
Study G2-structures with non-closed four-forms under local conformal conditions.
problem Characterizing G2-structures with non-closed four-forms under local conformal conditions.
method Investigated G2-structures with non-closed four-forms and non-zero Lee form, focusing on Lie algebra properties.
result Found solutions for G2-structures in preposition1 and theorem1.
New G2-structures found on Lie groups with strong structural conditions.
problem Existence and structure of extremally Ricci pinched G2-structures on Lie groups.
method Strong structural conditions on Lie algebra, deformation and rigidity studies.
result Three new examples of extremally Ricci pinched G2-structures, all steady Laplacian solitons.
Study curvature properties of G2 connections with skew-symmetric torsion.
problem Investigate curvature identities and solitons on G2 manifolds. method Analyzes curvature identities and properties of G2 connections with skew-symmetric torsion. result Characterizes conditions for curvature to be symmetric and Ricci flat.
Study dualities between G2-manifolds and string theories.
problem Determine dualities between M-theory and heterotic string theories on G2-manifolds.
method Analyze K3-fibered G2-manifolds and their dual heterotic and F-theory realizations.
result Identify dual heterotic and F-theory realizations for a large class of G2-manifolds.
Study G2-structures in N=1 AdS4 solutions of M-theory.
problem Characterize G2-structures in N=1 AdS4 solutions of M-theory.
method Reformulate Killing spinor equations using octonion bundle structure; study G2-structures and their torsion.
result Define single complexified G2-structure or two real G2-structures on M.
Classification of G2-structures on Lie groups with Ricci pinched conditions.
problem Classifying G2-structures on Lie groups under specific geometric conditions.
method Complete classification of left-invariant closed G2-structures on Lie groups, extremally Ricci pinched, up to equivalence and scaling.
result Five distinct G2-structures on five different completely solvable Lie groups, with one unimodular case being exact.