Quaternionic Manifolds Research Articles

GEOMETRY OF TYPE II SUPERSTRINGS AND THE MODULI OF SUPERCONFORMAL FIELD THEORIES

We study general properties of the low-energy effective theory for 4D type II superstrings obtained by the compactification on an abstract (2,2) superconformal system. This is the basic step towards the construction of their moduli space. We give an explicit and general algorithm to convert the effective Lagrangian for the type IIA into that of type IIB superstring defined by the same (2,2) superconformal system (and vice versa). This map converts Kahler manifolds into quaternionic ones (and quaternionic into Kahlerian ones) and has a deep geometrical meaning. The relationship with the theory of normal quaternionic manifolds (and algebras), as well as with Jordan algebras, is outlined. It turns out that only a restricted class of quarternionic geometries is allowed in the string case. We derive a general and explicit formula for the (fully nonlinear) couplings of the vector-multiplets (IIA case) in terms of the basic three-point functions of the underlying superconformal theory. A number of illustrative examples is also presented.

The manifolds of scalar background fields in Z N orbifolds

We determine the Kähler and quaternionic manifolds of the untwisted multiplets of superstrings compactified on Z N ⊂SU(3) symmetric orbifolds. In particular we show that, when moduli and family replicas are considered, there is a nontrivial interference between the massless modes related (1, 1) and (2, 1) harmonic forms: the manifolds naturally depend on the net chiral family content and on the family-antifamily paired multiplets.

Yang-Mills fields on quaternionic spaces

A study is made of solutions of the Yang-Mills equations over a quaternionic Kahler manifold. The corresponding notion of self-duality is interpreted in terms of holomorphic geometry on a twistor space. Self-dual connections are constructed on various vector bundles over quaternionic projective spaces.

The axiom of spheres in quaternionic geometry

We show that a quaternion manifold satisfying the axiom of quaternionic spheres is a quaternion space-form.

A new class of quaternionic manifolds

Riemannian manifolds with structure group Sp(n)·Sp(1) are studied. Moreover for the Riemannian manifolds with structure group Sp(n) a new class is defined, and example of compact nilmanifolds in this class are constructed.

The effective interactions of chiral families in four-dimensional superstrings

The low-energy effective interactions in four-dimensional superstrings with N = 2 and N = 1 space-time supersymmetry and massless twisted (family) sector are obtained. Our results rely on some general symmetry properties of superstring particle states and on tensor-calculus techniques for supergravity couplings. The novel feature is that the N = 2 quaternionic manifold and N = 1 Kähler space of the scalar superpartners of family multiplets are non-symmetric spaces whose structure can be obtained by “integrating out” the massive superstring modes.

Effective lagrangians for four-dimensional superstrings

We derive effective supergravity lagrangians for the massless sector of new four-dimensional superstrings. The untwisted sector is obtained by suitable Z 2 or Z 2 ⊗ Z 2 ′ truncations of N = 4, D = 4 lagrangians. For the N = 1 “minimal” models, related to the N = 4, SO(44) superstrings, we derive the Kähler potential, the superpotential and the ƒ gauge kinetic function. The underlying nonlinear sigma model is based on the Kähler manifold [SU(1,1)/U(1)] ⊗ Π i = 1 3 {SO(2, n i )/[SO(2) ⊗ SO( n i )]}. For models without massless twisted sector, this result provides the full superstring low-energy effective action. For models with massless twisted sector, the N = 1 theory may be obtained once the quaternionic manifold of the corresponding N = 2 hypermultiplets is specified.

A generalization of the momentum mapping construction for quaternionic K�hler manifolds

We present a method of reduction of any quaternionic Kahler manifold with isometries to another quaternionic Kahler manifold in which the isometries are divided out. Our method is a generalization of the Marsden-Weinstein construction for symplectic manifolds to the non-symplectic geometry of the quaternionic Kahler case. We compare our results with the known construction for Kahler and hyperKahler manifolds. We also discuss the relevance of our results to the physics of supersymmetric non-linear σ-models and some applications of the method. In particular, we show that the Wolf spaces can be obtained as theU(1) andSU(2) quotients of quaternionic projective spaceHP(n). We also construct an interesting example of compact riemannianV-manifolds(orbifolds) whose metrics are quaternionic Kahler and not symmetric.

Connections for vector bundles over quaternionic Kähler manifolds

On definit des proprietes des connexions pour des fibres vectoriels sur des varietes de Kahler quaternioniques

σ-Models on quaternionic manifolds and anomalies

We define σ-models on quaternionic manifolds and discuss aspects of their geometry and quantization, in particular the question of anomalies.

Einstein metrics on quaternionic line bundles

The authors investigate the Einstein equation for a class of (4n+4)-dimensional SU(2)-invariant metrics on S4 and R4 fibre bundles over quaternionic Kahler base manifolds. Using numerical techniques, they establish the existence of complete compact inhomogeneous Einstein spaces of this form with positive Ricci curvature, and complete non-compact inhomogeneous Einstein spaces with zero or negative Ricci curvature.

Note on the existence of almost omplex structures on compact manifolds

In the paper we define a multiplicative genus of a compact orientable manifold. We use this genus for the study of the existence of almost complex structures on manifolds. A few applications are given, namely, we prove the nonexistence of an almost complex structure on quaternionic flag manifolds and give a theorem on the existence of an almost complex structure on the product of manifolds.

Differential geometry of quaternionic manifolds

On developpe l'algebre necessaire pour comprendre les proprietes d'une variete quaternionique. On caracterise la structure quaternionique par l'existence de certains complexes d'operateurs differentiels

N = 2 Maxwell-matter-Einstein supergravities in d = 5,4 aand 3

We consider the coupling of the N = 2 Maxwell and matter multiplets to N = 2 supergravity in d = 5. It is shown that the scalar manifold factorizes into a quaternionic manifold for the matter scalar and a cubic hypersurfrace for the Maxwell scalar fields. This factorization is carried out to d = 4 and 3 through ordinary dimensional reduction. We study also the N = 2 theories obtained from the truncation of the N = 8 and N = 4 theories in those dimensions. In d = 3 we find a remarkable N = 4 theory described by the coset space ( E 8(-24)/( E 7× SU(2)))×( E 8(-24)×/( E 7× SU(2))) which would yield a Kac-Moody algebra of the type ( E 8 (1)× E 8 1 when dimensionally reduced to d = 2.

Non-Parallelizability of Grassmann Manifolds

AbstractThe fact that no real Grassmann manifolds Gk(Rn) are parallelizable (or even stably parallelizable) except for the obvious cases G1R2≅S1, G1(R4)≅G3(R4) ≅ RP3, and G1(R8)≅ G7(R8) ≅ RP7 was first noted by Hiller and Stong. Their work in turn depends on induction and the work of Oproiu, who examined detailed calculations of Stiefel-Whitney classes for k = 2, 3. In this note we give a short proof of this result, using elementary results from K-theory, that also covers the complex and quaternionic Grassmann manifolds.

Almost contact metric 3‐submersions

An almost contact metric 3‐submersion is a Riemannian submersion, π from an almost contact metric manifold onto an almost quaternionic manifold which commutes with the structure tensors of type (1, 1);i.e., π*φi = Jiπ*, for i = 1, 2, 3. For various restrictions on ∇φi, (e.g., M is 3‐Sasakian), we show corresponding limitations on the second fundamental form of the fibres and on the complete integrability of the horizontal distribution. Concommitantly, relations are derived between the Betti numbers of a compact total space and the base space. For instance, if M is 3‐quasi‐Saskian (dΦ = 0), then b1(N) ≤ b1(M). The respective φi‐holomorphic sectional and bisectional curvature tensors are studied and several unexpected results are obtained. As an example, if X and Y are orthogonal horizontal vector fields on the 3‐contact (a relatively weak structure) total space of such a submersion, then the respective holomorphic bisectional curvatures satisfy: . Applications to the real differential geometry of Yarg‐Milis field equations are indicated based on the fact that a principal SU(2)‐bundle over a compactified realized space‐time can be given the structure of an almost contact metric 3‐submersion.

Matter couplings in N = 2 supergravity

We determine the general coupling of an arbitrary number of massless pin (0, 1 2 ) multiplets to N = 2 supergravity. We find that the matter coupling has a natural description in the language of the supersymmetric non-linear sigma model, where the sigma model is based on a quaternionic manifold. This implies that the matter self-couplings that are allowed in global N = 2 supersymmetry are forbidden in supergravity, and vice versa.

Signature of quaternionic Kaehler manifolds

On montre que la signature d'une variete de Kahler quaternionique compacte M est egale au nombre de Betti b 2n (M), dim(M)=4n et on etablit des proprietes de sa cohomologie

Quaternionic Kaehler Manifolds

The topological classification of $4$- and $8$- (real) dimensional compact quaternionic Kaehler manifolds is given. There is only the torus in dimension 4. In dimension 8, there are 12 homeomorphism classes; representatives are given explicitly.

Quaternionic Kaehler manifolds

The topological classification of 4 4 - and 8 8 - (real) dimensional compact quaternionic Kaehler manifolds is given. There is only the torus in dimension 4. In dimension 8, there are 12 homeomorphism classes; representatives are given explicitly.