arXiv research

New transitions found in Spin(7) holonomy metrics related to a dynamical system.

In this paper we complete the classification of spin manifolds admitting parallel spinors, in terms of the Riemannian holonomy groups. More precisely, we show that on a given n-dimensional Riemannian manifold, spin structures with parallel spinors are in one to one correspondence with lifts to Spin_n of the Riemannian …

We prove that Riemannian $Spin(7)$ holonomy manifolds carry octonionic-Kähler structure.

Study on harmonic flow of Spin(7)-structures in 8D manifolds.

Classifies Spin(7) structures on compact 8-manifolds with abelian fundamental group.

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…

We give a necessary and suffcient condition for almost-flat manifolds with cyclic holonomy to admit a Spin structure. Using this condition we find all 4-dimensional orientable almost- flat manifolds with cyclic holonomy that do not admit a Spin structure.

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 …

No Hantzsche-Wendt manifolds over 3D admit spin^c structures.

This note fills a hole in the author's previous paper ``Ricci-Flat Holonomy: a Classification'', by dealing with irreducible holonomy algebras that are subalgebras or real forms of $\mbb{C} \oplus \mf{spin}(10,\mbb{C})$. These all turn out to be of Ricci-type.

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…

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 study holonomy algebras generated by an algebraic element of the Clifford algebra, or equivalently, the holonomy algebras of certain spin connections in flat space. We provide series of examples in arbitrary dimensions and establish general properties of the holonomy algebras under some mild conditions on the genera…

Study on Dolbeault complexes for special holonomy manifolds.

Study finds holonomy algebras for Lorentzian Weyl spin manifolds with specific spinors.

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

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…

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…

An explicit expression of the canonical 8-form on a Riemannian manifold with a Spin(9)-structure, in terms of the nine local symmetric involutions involved, is given. The list of explicit expressions of all the canonical forms related to Berger's list of holonomy groups is thus completed. Moreover, some results on Spin…

We describe the possible holonomy groups of simply connected irreducible non-locally symmetric pseudo-Riemannian spin manifolds which admit parallel spinors.

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 $G_2$ holonomy. It is also …

The real form Spin(6,H) in End(R^{32}) of Spin(12,C) in End(C^{32}) is absolutely irreducible and thus satisfies the algebraic identities (40) and (41). Therefore, it also occurs as an exotic holonomy and the associated supermanifold M_g admits a SUSY-invariant polynomial. This real form has been erroneously omitted in…

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 completely explore the system of ODE's which is equivalent to the existence of a parallel $Spin(7)$-structure on the cone over a 7-dimensional 3-Sasakian manifold. The one-dimensional family of solutions of this system is constructed. The solutions of this family correspond to metrics with holonomy SU(4) which gener…

The paper explores warped-like product metrics with exceptional holonomy groups.

All candidates to the weakly-irreducible not irreducible holonomy algebras of Lorentzian manifolds are known. In the present paper metrics that realize all these candidates as holonomy algebras are given. This completes the classification of the Lorentzian holonomy algebras. Also new examples of metrics with the holono…

We consider noncompact complete manifolds with Spin(9) holonomy and proved an one end result and a splitting type theorem under different conditions on the bottom of the spectrum. We proved that any harmonic functions with finite Dirichlet integral must be Cayley-harmonic, which allowed us to conclude an one end result…

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 …

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

It is shown that on a compact spin symmetric space with a Kähler or Quaternion-Kähler structure, the first eigenvalue of the Dirac operator is linked to a ''{lowest}'' action of the holonomy, given by the fiberwise action on spinors of the canonical forms characterized by this holonomy. The result is also verified for …

We show that the $G_2$-manifolds and certain ${\rm Spin}(7)$-manifolds are endowed with natural Riemannian twistorial structures. Along the way, the exceptional holonomy representations are reviewed and other related facts are considered.

For each integer $d$ at least two, we construct non-spin closed oriented flat manifolds with holonomy group $\mathbb Z_2^d$ and with the property that all of their finite proper covers have a spin structure. Moreover, all such covers have trivial Stiefel-Whitney classes.

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…

New criterion for Ricci-flat manifolds with non-vanishing Rosenberg index.

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

The article confirms Joyce's examples of G2-holonomy are formal spaces.

We describe, by their holonomy groups, all complete simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalize the Riemannian SpinC case and the pseudo-Riemannian Spin one.

We consider relations between two families of flat manifolds with holonomy group (Z_2)^k of diagonal type. The family ${\cal RBM}$ of real Bott manifolds and the family ${\cal GHW}$ of generalized Hantzsche-Wendt manifolds. In particular, we prove that the intersection ${\cal GHW}\cap {\cal RBM}$ is not empty. We also …

Let G be one of the Ricci-flat holonomy groups SU(n), Sp(n), Spin(7) or G_2, and M a compact manifold of dimension 2n, 4n, 8 or 7, respectively. We prove that the natural map from the moduli space of torsion-free G-structures on M to the moduli space of Ricci-flat metrics is open, and that the image is a smooth manifol…

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group cyclic of odd prime order p. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p=n=3. We use the expressions ob…

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.

We resolve Spin(7)-orbifolds using algebraic and symplectic techniques.

Paper proves rigidity for spin bands with specific conditions.

We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups $G_2$ and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum $G_2$ manifolds, mirrors of such Spin(7) manifolds can be fou…

In this paper we examine the structure of Riemannian manifolds with a special kind of Codazzi tensors. We use them to construct globally hyperbolic Lorentzian manifolds with complete Cauchy hypersurfaces for any weakly irreducible holonomy representation with parallel spinors, i.e. with a holonomy group which is a semi…

We characterise simply-connected biquotients which potentially admit metrics of holonomy G_2. We prove that there are at most three real homotopy types of rationally elliptic such manifolds---all of them being formal. In the course of this examination we classify rationally elliptic homotopy types and characterise 7-di…

We construct left invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven dimensional…

Study linear perturbations of Spin(7) metrics, finding only rank one nilpotent matrices.