G2 Manifolds Research Articles

Towards generalized mirror symmetry for twisted connected sum G2 manifolds

We revisit our construction of mirror symmetries for compactifications of Type II superstrings on twisted connected sum G2 manifolds. For a given G2 manifold, we discuss evidence for the existence of mirror symmetries of two kinds: one is an autoequivalence for a given Type II superstring on a mirror pair of G2 manifolds, the other is a duality between Type II strings with different chiralities for another pair of mirror manifolds. We clarify the role of the B-field in the construction, and check that the corresponding massless spectra are respected by the generalized mirror maps. We discuss hints towards a homological version based on BPS spectroscopy. We provide several novel examples of smooth, as well as singular, mirror G2 backgrounds via pairs of dual projecting tops. We test our conjectures against a Joyce orbifold example, where we reproduce, using our geometrical methods, the known mirror maps that arise from the SCFT worldsheet perspective. Along the way, we discuss non-Abelian gauge symmetries, and argue for the generation of the Affleck-Harvey-Witten superpotential in the pure SYM case.

Tops as building blocks for G2 manifolds

A large number of examples of compact G2 manifolds, relevant to supersymmetric compactifications of M-Theory to four dimensions, can be constructed by forming a twisted connected sum of two building blocks times a circle. These building blocks, which are appropriate K3-fibred threefolds, are shown to have a natural and elegant construction in terms of tops, which parallels the construction of Calabi-Yau manifolds via reflexive polytopes. In particular, this enables us to prove combinatorial formulas for the Hodge numbers and other relevant topological data.

Mirror symmetry for G2-manifolds: twisted connected sums and dual tops

Recently, at least 50 million of novel examples of compact G2 holonomy manifolds have been constructed as twisted connected sums of asymptotically cylindrical Calabi-Yau threefolds. The purpose of this paper is to study mirror symmetry for compactifications of Type II superstrings in this context. We focus on G2 manifolds obtained from building blocks constructed from dual pairs of tops, which are the closest to toric CY hypersurfaces, and formulate the analogue of the Batyrev mirror map for this class of G2 holonomy manifolds, thus obtaining several millions of novel dual superstring backgrounds. In particular, this leads us to conjecture a plethora of novel exact dualities among the corresponding 2d mathcal{N} = 1 sigma models.

T4 fibrations over Calabi–Yau two-folds and non-Kähler manifolds in string theory

We construct a geometric model of eight-dimensional manifolds and realize them in the context of type II string theory. These eight-manifolds are constructed by non-trivial T4 fibrations over Calabi–Yau two-folds. These give rise to eight-dimensional non-Kähler Hermitian manifolds with SU(4) structure. The eight-manifold is also a circle fibration over a seven-dimensional G2 manifold with skew torsion. The eight-manifolds of this type appear as internal manifolds with SU(4) structure in type IIB string theory with F3 and F7 fluxes. These manifolds have generalized calibrated cycles in the presence of fluxes.

Supersymmetric backgrounds and generalised special holonomy

We define intrinsic torsion in generalised geometry and use it to introduce a new notion of generalised special holonomy. We then consider generic warped supersymmetric flux compactifications of M theory and Type II of the form . Using the language of generalised geometry, we show that, for , preserving minimal supersymmetry is equivalent to the manifold M having generalised special holonomy and list the relevant holonomy groups. We conjecture that this result extends to backgrounds preserving any number of supersymmetries. As a prime example, we consider in D = 4. The corresponding generalised special holonomy group is , giving the natural M theory extension to the notion of a G2 manifold, and, for Type II backgrounds, reformulating the pure spinor conditions as an integrable structure.

SO(10) Grand Unification from M theory on a G2 manifold

We consider Grand Unified Theories based on SO(10) which originate from M theory on G2 manifolds. In this framework we are naturally led to a novel solution of the doublet- triplet splitting problem involving an extra vector-like pair by considering discrete symmetries of the extra dimensions and preserving unification. Since Wilson line breaking preserves the rank of the gauge group, the necessary U(1) gauge breaking is generated from extra multiplets. The main prediction of the approach is the existence of light states with the quantum numbers of a vector-like pair which could show up in future LHC searches.

Algebraic topology of [formula omitted] manifolds

In this paper we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G2. Some of these results are new. We give self-contained proofs here. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. For the sake of completeness we decided to collect them here in a self-contained way to be easily accessible for future usage in calibrated geometry. As an application we deduce existence of certain special 3 and 4 dimensional submanifolds of G2 manifolds with special properties, which appear in the first named author’s work with S. Salur about G2dualities.

Spin(7) duality for N $$ \mathcal{N} $$ = 1 CS-matter theories

In this paper we propose a duality for non-holomorphic $$ \mathcal{N} $$ = 1 CS-matter theories living on M2 branes probing Spin(7) cones. We call this duality Spin(7) duality. Two theories are named Spin(7) dual if they have the same moduli space: a real Spin(7) cone with base a weak G2 manifold, and they are hence holographic dual to the same AdS4 × G2 M theory solution. We provide a systematic way to generate these dualities, derived by combining toric duality for $$ \mathcal{N} $$ = 2 CS-matter theories and generalized non-holomorphic orientifold projections to $$ \mathcal{N} $$ = 1. Brane construction, AdS/CFT correspondence, and the computation of the moduli space support our proposal at the classical level and provide some arguments at the quantum strong coupling regime. The relation with Seiberg-like duality is also analyzed.

Generalized Killing spinors on Einstein manifolds

We study generalized Killing spinors on compact Einstein manifolds with positive scalar curvature. This problem is related to the existence of compact Einstein hypersurfaces in manifolds with parallel spinors, or equivalently, in Riemannian products of flat spaces, Calabi–Yau, hyperkähler, G2 and Spin(7) manifolds.

EINSTEIN MANIFOLDS WITH SKEW TORSION

This paper is devoted to the first systematic investigation of manifolds that are Einstein for a connection ∇ with skew-symmetric torsion. We derive the Einstein equation from a variational principle and prove that, for parallel torsion, any Einstein manifold with skew torsion has constant scalar curvature; and if it is complete of positive scalar ∇-curvature, it is necessarily compact and it has finite first fundamental group π1. The longest part of the paper is devoted to the systematic construction of large families of examples. We discuss when a Riemannian Einstein manifold can be Einstein with skew torsion. We give examples of almost Hermitian, almost metric contact, and G2 manifolds that are Einstein with skew torsion. For example, we prove that any Einstein–Sasaki manifold and any seven-dimensional 3-Sasakian manifolds admit deformations into an Einstein metric with parallel skew torsion.

REVIEW AND UPDATE OF THE COMPACTIFIED M/STRING THEORY PREDICTION OF THE HIGGS BOSON MASS AND PROPERTIES

The August 2011 Higgs mass prediction was based on an ongoing six-year project studying M-theory compactified on a manifold of G2 holonomy, with significant contributions from Jing Shao, Eric Kuflik and others and particularly co-led by Bobby Acharya and Piyush Kumar. The M-theory results include stabilization of all moduli in a de Sitter vacuum, gauge coupling unification, derivation of TeV scale physics (solving the hierarchy problem), the derivation that generically scalar masses are equal to the gravitino mass which is larger than about 30 TeV, derivation of the Higgs mechanism via radiative electroweak symmetry breaking, absence of the flavor and CP problems, and the accommodation of string axions. The tan β and the μ parameter are part of the theory and are approximately calculated; as a result, the little hierarchy problem is greatly reduced. The heavy scalars imply that decoupling rare decays such as Bs →μ+ μ- should not deviate from their Standard Model values. This paper summarizes the results relevant to the Higgs mass prediction. A recent review [Int. J. Mod. Phys. A27, 1230012 (2012)] describes the program more broadly. Some of the results such as the scalar masses being equal to the gravitino mass and larger than about 30 TeV, derived early in the program, hold generically for compactified string theories as well as for compactified M-theory, while some other results may or may not. If the world is described by M-theory compactified on a G2 manifold and has a Higgs mechanism (so it could be our world) then Mh was predicted to be 126±2 GeV before the measurement. The derivation has some assumptions not related to the Higgs mass, but involves no free parameters.

On scalar curvature of toric selfdual four-manifolds

In this paper, we prove the nonexistence of constant scalar curvature of four dimensional selfdual Joyce manifolds. Since the manifold admits a torus symmetry, our work is greatly reduced to the case of two dimensional upper-half planes.

A formally Kähler structure on a knot space of a G 2-manifold

A knot space in a manifold M is a space of oriented immersions \({S^{1} \hookrightarrow M}\) up to Diff(S1). J.-L. 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.

Topology of ϕ-convex domains in calibrated manifolds

In [5], Harvey and Lawson showed that for any calibration ϕ there is an integer bound for the homotopy dimension of a strictly ϕ-convex domain and constructed a method to get these domains by using ϕ-free submanifolds. Here, we show how to get examples of ϕ-free submanifolds with different homotopy types for the quaternion calibration in ℍn, associative calibration, and coassociative calibration in G2 manifolds. Hence we give examples of strictly ϕ-convex domains with different homotopy types allowed by Morse Theory.

Betti Numbers of a Class of Barely G 2 Manifolds

We calculate explicitly the Betti numbers of a class of barely G2 manifolds - that is, G2 manifolds that are realised as a product of a Calabi-Yau manifold and a circle, modulo an involution. The particular class which we consider are those spaces where the Calabi-Yau manifolds are complete intersections of hypersurfaces in products of complex projective spaces and the involutions are free acting.

A note on mirror symmetry for manifolds with spin(7) holonomy

Starting from the extended worldsheet superconformal algebras associated with G2 manifolds, we extend the algebra to Joyce's orbifolds with holonomy. We show how the mirror symmetry in manifolds with holonomy arises as the automorphism in the extended sperconformal algebra. In one class of Joyce's orbifolds, the automorphism of the superconformal algebra can be realized as 14 kinds of T-dualities along the supersymmetric T4 toroidal fibrations in the manifolds with holonomy. In this class of examples, Joyce's orbifolds are pairwise identified under the mirror symmetry. We then discuss some interesting features of the mirror symmetry on the manifolds with exceptional holonomy and how it is different from the Calabi–Yau mirror symmetry.

Calabi-Yau manifolds from pairs of non-compact Calabi-Yau manifolds

Most of Calabi-Yau manifolds that have been considered by physicists are complete intersection Calabi-Yau manifolds of toric varieties or some quotients of product types. Purpose of this paper is to introduce a different and rather new kind of construction method of Calabi-Yau manifolds by pasting two non-compact Calabi-Yau manifolds. We will also in some details explain a curious and mysterious similarity with construction of some $G_2$-manifolds (also called Joyce manifolds), which are base spaces for M-theory.

A supermembrane with central charges on a G2 manifold

We construct an 11D supermembrane with topological central charges induced through an irreducible winding on a G2 manifold realized from the T7/Z32 orbifold construction. The Hamiltonian H of the theory on a T7 target has a discrete spectrum. Within the discrete symmetries of H associated with large diffeomorphisms, the Z2 × Z2 × Z2 group of automorphisms of the quaternionic subspaces preserving the octonionic structure is relevant. By performing the corresponding identification on the target space, the supermembrane may be formulated on a G2 manifold, preserving the discreteness of its supersymmetric spectrum. The corresponding 4D low energy effective field theory has N = 1 supersymmetry.

Consistent supersymmetric Kaluza-Klein truncations with massive modes

We construct consistent Kaluza-Klein reductions of D = 11 supergravity to four dimensions using an arbitrary seven-dimensional Sasaki-Einstein manifold. At the level of bosonic fields, we extend the known reduction, which leads to minimal N = 2 gauged supergravity, to also include a multiplet of massive fields, containing the breathing mode of the Sasaki-Einstein space, and still consistent with N = 2 supersymmetry. In the context of flux compactifications, the Sasaki-Einstein reductions are generalizations of type IIA SU(3)-structure reductions which include both metric and form-field flux and lead to a massive universal tensor multiplet. We carry out a similar analysis for an arbitrary weak G2 manifold leading to an N = 1 supergravity with massive fields. The straightforward extension of our results to the case of the seven-sphere would imply that there is a four-dimensional Lagrangian with N = 8 supersymmetry containing both massless and massive spin two fields. We use our results to construct solutions of M-theory with non-relativistic conformal symmetry.

Semi-flatland

We study perturbative compactifications of Type II string theory that rely on a fibration structure of the extra dimensions a la SYZ. Non-geometric spaces are obtained by using T-dualities as monodromies. These vacua generically preserve N=1 supersymmetry in four dimensions, and are U-dual to M-theory on G2 manifolds. Several examples are discussed, some of which admit an asymmetric orbifold description. The massless spectrum is matched to that of the dual M-theory compactification on a Joyce manifold when a comparison is possible. We explore the possibility of twisted reductions where left-moving spacetime fermion number Wilson lines are turned on in the fiber. We also give an explanation from this semiflat viewpoint for the Hanany-Witten brane-creation effect and for the equivalence of the Type IIA orientifold on T5/Z2 and Type IIB on S1 x K3.