arXiv research

Study calculates deformations of instantons on a specific $Spin(7)$-manifold.

The paper studies curvature identities and solitons on Spin(7)-manifolds.

Study instantons on asymptotically conical Spin(7)-manifolds, identifying deformation spaces.

New $Spin(7)$-instantons constructed on Joyce's manifold.

Study $\mathrm{Spin}(7)$-manifolds with a 4-torus action using a symmetric matrix ansatz.

Study on instantons in $G_2$ and $Spin(7)$ manifolds.

Study shows only two topological configurations for Spin(7)-manifold fibrations, ruling out smooth Cayley fibrations.

New Spin(7) manifolds created from Calabi-Yau bundles.

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…

Constructs Spin(7) manifolds from self-dual Einstein 4-orbifolds.

Study classifies certain 7-manifolds with specific cohomology and proves positive Ricci curvature.

Study Cayley fibrations on Bryant-Salamon $Spin(7)$ manifolds.

Desingularizes conically singular Cayley submanifolds.

We present a classification theorem for closed smooth spin 2-connected 7-manifolds M. This builds on the almost-smooth classification from the first author's thesis. The main additional ingredient is an extension of the Eells-Kuiper invariant for any closed spin 7-manifold, regardless of whether the spin characteristic…

We characterize compact locally conformal parallel $G_2$ (respectively, $Spin(7)$) manifolds as fiber bundles over $S^1$ with compact nearly Kähler (respectively, compact nearly parallel $G_2$) fiber. A more specific characterization is provided when the local parallel structures are flat.

Any Spin(7)-manifold admits a metric connection \nabla^c with totally skew-symmetric torsion T^c preserving the underlying structure. We classify those with \nabla^c-parallel T^c\neq0 and non-Abelian isotropy algebra iso(T^c)<spin(7). These are isometric to either Riemannian products or homogeneous naturally reductive …

We show that on every Spin(7) manifold there always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor and the Spin(7) structure. We express its torsion and the Riemannian scalar curvature in terms of the fundamental 4-form. We present an explicit formula for the Rieman…

Twistor theory applied to special holonomy manifolds.

This paper explores 4-planes in Spin(7) manifolds with symplectic structures.

Deformations of singular Cayley submanifolds studied.

M-theory on compact eight-manifolds with $\mathrm{Spin}(7)$-holonomy is a framework for geometric engineering of 3d $\mathcal{N}=1$ gauge theories coupled to gravity. We propose a new construction of such $\mathrm{Spin}(7)$-manifolds, based on a generalized connected sum, where the building blocks are a Calabi-Yau four…

We find new examples of compact Spin(7)-manifolds using a construction of Joyce. The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces …

The paper studies quotient spaces of Spin(7) manifolds by S^1 actions and their G2 structures.

Study classifies 7-manifolds with specific homology and finds nonconnected moduli spaces of positive Ricci curvature metrics.

In an earlier paper we showed that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with the direct sum of the space of smooth functions and closed 2-forms on HL. In that paper, we also introduced a new class of Lagrangian-type 4-dimensional su…

The main purpose of this paper is to give a mathematical definition of ``mirror symmetry'' for Calabi-Yau and G_2 manifolds. More specifically, we explain how to assign a G_2 manifold (M,φ,Λ), with the calibration 3-form φand an oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2 and 3-forms of M. …

Alternative definition of dDT connections for Spin(7) manifolds.

We prove an existence theorem for Spin(7)-instantons, which are highly concentrated near a Cayley submanifold; thus giving a partial converse to Tian's foundational compactness theorem. As an application, we show how to construct Spin(7)-instantons on Spin(7)-manifolds with suitable local K3 Cayley fibrations. This rec…

This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson-Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces. They correspond to Yukawa couplings and correlation functions in M-theory. We ex…

We give a differential-geometric construction of compact manifolds with holonomy $\mathrm{Spin}(7)$ which is based on Joyce's second construction of compact $\mathrm{Spin}(7)$-manifolds in \cite{Joyce00} and Kovalev's gluing construction of $G_2$-manifolds in \cite{Kovalev03}. We also give some examples of compact $\ma…

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…

A nearly parallel $G_{2}$-manifold $Y$ is a Riemannian 7-manifold whose cone $C(Y) = \mathbb{R}_{>0} \times Y$ has the holonomy group contained in ${\rm Spin(7)}$. In other words, it is a spin 7-manifold with a real Killing spinor. We have a special class of calibrated submanifolds called Cayley submanifolds in $C(Y)$.…

We define a measure of spectral asymmetry for G_2 and Spin(7) manifolds. We show that this invariant can be computed in terms of characteristic classes and the covariant constant form defining the G_2 or Spin(7) structure.

Spin(7) geometry linked to multisymplectic geometry.

Study mirror symmetry on manifolds with exceptional holonomy groups.

The abstract discusses instantons on flat spaces and provides explicit constructions.

In this paper we show that a parallel differential form $Ψ$ of even degree on a Riemannian manifold allows to define a natural differential both on $Ω^\ast(M)$ and $Ω^\ast(M, TM)$, defined via the Frölicher-Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbe…

Local SU(3)-structures on an oriented submanifold of Spin(7)-manifold are determined and their types are characterized in terms of the shape operator and the type of the Spin(7)-structure. An application to Bryant \cite{MR89b:53084} and Calabi \cite{MR24 #A558} examples is given. It is shown that the product of a Cayle…

Article finds a geometric connection between structures on manifolds.

Explains calibrated submanifolds in special holonomy manifolds.

Metrics of exceptional holonomy are vacuum solutions to the Einstein equation. In this paper we describe manifolds with holonomy contained in Spin(7) preserved by a three-torus symmetry in terms of tri-symplectic geometry of four-manifolds. These complement examples that have appeared in the context of domain wall prob…

Cocalibrated G_2-structures are structures naturally induced on hypersurfaces in Spin(7)-manifolds. Conversely, one may start with a seven-dimensional manifold M endowed with a cocalibrated G_2-structure and construct via the Hitchin flow a Spin(7)-manifold which contains M as a hypersurface. In this article, we consid…

Study free circle actions on specific 7-manifolds with positive Ricci curvature.

Study on deformations of LC Spin(7) instantons simplifies the problem.

We construct examples of asymptotically cylindrical Riemannian 8-manifolds with holonomy group Spin(7). To our knowledge, these are the first such examples. The construction uses an extension to asymptotically cylindrical setting of Joyce's existence result for torsion-free Spin(7)-structures. One source of examples ar…

Study ${ m Spin}(7)$-dDT connections on manifolds with ${ m Spin}(7)$-structures.

We extend the characterization of the integrability of an almost complex structure $J$ on differentiable manifolds via the vanishing of the Frölicher-Nijenhuis bracket $[J, J] ^{FN}$ to an analogous characterization of torsion-free $G_2$-structures and torsion-free Spin(7)-structures. We also explain the Fernández-Gray…

First non-trivial examples of deformed Spin(7)-instantons constructed.