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…
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48 results for “G2 holonomy”
In 1981, covariantly constant spinors were introduced into Kaluza-Klein theory as a way of counting the number of supersymmetries surviving compactification. These are related to the holonomy group of the compactifying manifold. The first non-trivial example was provided in 1982 by D=11 supergravity on the squashed S7,…
The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. Riemannian manifolds with these holonomy groups are Ricci-flat. This is a survey paper on exceptional holonomy, in two parts. Part I introduces the exceptional holonomy groups, and explains constructions for compact 7- and 8-manifolds …
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.
Bryant and Salamon gave a construction of metrics of G2 holonomy on the total space of the bundle of anti-self-dual (ASD) 2-forms over a 4-dimensional self-dual Einstein manifold. We generalise it by considering the total space of an SO(3) bundle (with fibers R^3) over a 4-dimensional base, with a connection on this bu…
Non-formal G2 manifold found with holonomy.
problem Existence of non-formal G2 manifolds.
method Construction method of compact torsion-free G2 manifolds.
result Found a compact, simply connected G2 manifold that is non-formal.
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…
The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. Riemannian manifolds with these holonomy groups are Ricci-flat. This is a survey paper on constructions for compact 7- and 8-manifolds with holonomy G2 and Spin(7). The simplest such constructions work by using techniques from complex …
We construct propose an anzatz for Spin(7) metrics as an R-bundle over closed G2 structures. These G2 structures are R3 bundles over 4-dimensional compact quaternion Kahler spaces. The inspiration for the anzatz metric comes from the Bryant-Salamon construction of G2 holonomy metrics and from the fact that the twistor …
We show how certain diffeomorphism-invariant functionals on differential forms in dimensions 6,7 and 8 generate in a natural way special geometrical structures in these dimensions: metrics of holonomy G2 and Spin(7), metrics with weak holonomy SU(3) and G2, and a new and unexplored example in dimension 8. The general f…
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.
We study the duality between M-theory on compact holonomy G2-manifolds and the heterotic string on Calabi-Yau three-folds. The duality is studied for K3-fibered G2-manifolds, called twisted connected sums, which lend themselves to an application of fiber-wise M-theory/Heterotic Duality. For a large class of such G2-man…
Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew-symmetric. We show that on a compact manifold with holonomy G2 or Spin7 any Killing form has to be parallel. The main tool is a universal Weitzenboeck formula. We show how such a formula can be obtained for any given…
This is a survey paper. We explain the known constructions for two geometrically different classes of examples of compact Riemannian 7-manifolds with holonomy G2. One method uses resolutions of singularities of appropriately chosen 7-dimensional orbifolds, with the help of asymptotically locally Euclidean spaces. Anoth…
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…
There is a rich theory of so-called (strict) nearly Kaehler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kaehler 6-manifolds play a distinguished role both in the general structure theory and also because of their con…
The article confirms Joyce's examples of G2-holonomy are formal spaces.
problem Formality of special holonomy manifolds and nearly Kähler manifolds.
method Analyzing rational homotopy theory and cohomology algebra.
result Joyce's examples of G2-holonomy are formal spaces.
A knot space in a manifold M is a space of oriented immersions from a circle S^1 to M up to Diff(S^1). Brylinski has shown that a knot space of a Riemannian threefold is formally Kahler. We prove that a space of knots in a holonomy G2 manifold is formally Kahler.
Study on Dolbeault complexes for special holonomy manifolds.
problem Characterizing Dolbeault complexes on manifolds with exceptional holonomy.
method Analysis of Dolbeault complexes based on holonomy group representations.
result All Dolbeault complexes on compact manifolds with exceptional holonomy are identified.
We consider immersions of a Riemann surface into a manifold with -holonomy and give criteria for them to be conformal and harmonic, in terms of an associated Gauss map.
Promotes Poisson deformations to hyperkähler structures.
problem Deforming hyperkähler cone metrics.
method Twistor methods for universal Poisson deformations.
result Produces incomplete hyperkähler metrics with applications.
In this expository article we discuss the relations between Sasakian geometry, reduced holonomy and supersymmetry. It is well known that the Riemannian manifolds other than the round spheres that admit real Killing spinors are precisely Sasaki-Einstein manifolds, 7-manifolds with a nearly parallel G2 structure, and nea…
We develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M over B satisfying…
We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an …
Study of special geometric structures on Lie groups.
problem Investigating left-invariant -structures with specific holonomy properties.
method Classification of indecomposable holonomy algebras, determination of infinitesimal holonomy algebras.
result Only abelian subalgebras of dimension 2 or 3 arise as holonomy algebras.
A homotopy classification of -structures with applications to exceptional Riemannian holonomymath.DG
Classifies Spin(7) structures on compact 8-manifolds with abelian fundamental group.
problem Classifying Spin(7) structures on compact 8-manifolds.
method Obstruction theory applied to Spin(7) structures on compact 8-manifolds with abelian fundamental group.
result Compact Riemannian 8-manifolds with holonomy Spin(7) have exactly two Spin(7) structures extending the induced G2 structure on the boundary.
Classifies 7D manifolds with specific geometric properties.
problem Classifying 7D manifolds with parallel skew-symmetric torsion and holonomy.
method Extending Friedrich's work, using classification techniques for naturally reductive spaces and nearly parallel -structures.
result Complete classification of 7D manifolds with the specified properties.
Constructs harmonic 1-forms on K3-fibred Calabi-Yau 3-folds.
problem Analyzing harmonic 1-forms on K3-fibred Calabi-Yau 3-folds.
method Analytic construction of nowhere-vanishing harmonic 1-forms.
result Produces examples of compact 7-manifolds with holonomy G2.
Researchers find new -conifolds in -theory with potential field theory duals.
problem Exploring the field theory interpretation of -theory -conifolds.
method Constructing -holonomy orbifolds from circle bundles over Calabi-Yau cones.
result Many UV perturbative gauge theories have an infrared dual described by smooth -holonomy backgrounds in -theory.
Nearly Kähler manifolds are the Riemannian 6-manifolds admitting real Killing spinors. Equivalently, the Riemannian cone over a nearly Kähler manifold has holonomy contained in G2. In this paper we study the deformation theory of nearly Kähler manifolds, showing that it is obstructed in general. More precisely, we show…
The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. In a previous paper (Invent. math. 123 (1996), 507-552) the author constructed the first examples of compact 8-manifolds with holonomy Spin(7), by resolving orbifolds T^8/G, where T^8 is the 8-torus and G a finite group of automorphism…
Researchers found G2-structures with zero torsion on specific Lie groups.
problem Finding G2-structures with divergence-free torsion.
method Isometric flow to evolve G2-structures with divergence-free torsion as critical points.
result Three families of G2-structures on solvable Lie groups have zero torsion.
New compact manifolds with closed G2 structures found.
problem Existence of torsion-free G2 structures on compact manifolds.
method Equivariant resolution procedure for G2 orbifolds with Z2 isotropy.
result Construction of new compact manifolds with closed G2 structures.
New geometric approach realizes 5D bulk theories with 4D edge modes.
problem Realizing novel higher-dimensional junctions of theories coupled to localized edge modes.
method M-theory on singular, asymptotically conical G2-holonomy orbifolds.
result Geometric approach shows how bulk generalized symmetries are inherited in the boundary system.
The paper explores warped-like product metrics with exceptional holonomy groups.
problem Exploring new metrics with exceptional holonomy groups.
method Presented a general ansatz of warped-like product metric and studied specific examples.
result Explicit examples of and warped-like product manifolds with and holonomy, respectively.
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 -instanton equation from the …
The paper connects -manifolds to Coulomb and Higgs phases of gauge theories.
problem Exploring the physical interpretation of special singularities in -holonomy manifolds.
method Analyzing desingularizations of orbifold singularities and relating them to gauge theories.
result Shows an isomorphism between moduli spaces of Ricci flat metrics and flat ADE-connections.
IAs is well known, when D6 branes wrap a special lagrangian cycle on a non compact CY 3-fold in such a way that the internal string frame metric is Kahler there exists a dual description, which is given in terms of a purely geometrical eleven dimensional background with an internal metric of holonomy. It is also …
The paper proves Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.
problem Proving Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.
method Defining cohomology spaces analogous to Bott-Chern cohomology and relating them to harmonic forms on the manifolds.
result Geometric interpretation of cohomology classes in terms of submanifolds and gerbes for G2 manifolds.
This is the first nontrivial construction to date of instantons over a compact manifold with holonomy exactly . 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 ins…
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 study the special algebraic properties of alternating 3-forms in 6 and 7 dimensions and introduce a diffeomorphism-invariant functional on the space of differential 3-forms on a closed manifold M in these dimensions. Restricting the functional to closed forms in a fixed cohomology class, we find that a critical poin…
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.
The paper studies deformations of calibrated subbundles in special holonomy manifolds.
problem Deforming calibrated subbundles in noncompact manifolds of special holonomy.
method Twisting calibrated subbundles by special sections and deriving conditions for deformations to remain calibrated.
result Twisting conormal bundles of Lagrangian submanifolds in by 1-forms does not provide new examples.
This article classifies closed G2-structures such that the induced metric is conformally flat. It is shown that any closed G2-structure with conformally flat metric is locally equivalent to one of three explicit examples. In particular, it follows from the classification that any closed G2-structure inducing a metric t…
In this paper most of the classes of G2-structures with Einstein induced metric of negative, null or positive scalar curvature are realized. This is carried out by means of warped G2-structures with fiber an Einstein SU(3) manifold. The torsion forms of any warped G2-structure are explicitly described in terms of the t…
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.