Vector bundles over three-dimensional spherical space forms

We consider the problem of enumerating the G-bundles over low-dimensional manifolds (dimension ≤3) and in particular vector bundles over the three-dimensional spherical space forms. We give a complete answer to these questions and we give tables for the possible vector bundles over the 3-dimensional spherical space forms

Generalized Dirac monopoles in non-Abelian Kaluza-Klein theories

A method is proposed for generalizing the Euclidean Taub-NUT space, regarded as the appropriate background of the Dirac magnetic monopole, to non-Abelian Kaluza-Klein theories involving potentials of generalized monopoles. Usual geometrical methods combined with a recent theory of the induced representations governing the Taub-NUT isometries lead to a general conjecture where the potentials of the generalized monopoles of any dimensions can be defined in the base manifolds of suitable principal fiber bundles. Moreover, in this way one finds that apart from the monopole models which are of a space-like type, there exists a new type of time-like models that can not be interpreted as monopoles. The space-like models are studied pointing out that the monopole fields strength are particular solutions the Yang-Mills equations with central symmetry producing the standard flux of $4\pi$ through the two-dimensional spheres surrounding the monopole. Examples are given of manifolds with Einstein metrics carrying SU(2) monopoles.Comment: 32 page

Spherical T-duality II: An infinity of spherical T-duals for non-principal SU(2)-bundles

Recently we initiated the study of spherical T-duality for spacetimes that are principal SU(2)-bundles. In this paper, we extend spherical T-duality to spacetimes that are oriented non-principal SU(2)-bundles. There are several interesting new examples in this case and a new phenomenon appearing in the non-principal case is the existence of infinitely many spherical T-duals.Comment: 14 pages, spherical T-duality of Aloff-Wallach spaces with 7-flux included in Section

Cone spherical metrics and stable vector bundles

Cone spherical metrics are conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces. A cone spherical metric is called irreducible if each developing map of the metric does not have monodromy lying in ${\rm U(1)}$. We establish on compact Riemann surfaces of positive genera a correspondence between irreducible cone spherical metrics with cone angles being integral multiples of $2\pi$ and line subbundles of rank two stable vector bundles. Then we are motivated by it to prove a theorem of Lange-type that there always exists a stable extension of $L^*$ by $L$, for $L$ being a line bundle of negative degree on each compact Riemann surface of genus greater than one. At last, as an application of these two results, we obtain a new class of irreducible spherical metrics with cone angles being integral multiples of $2\pi$ on each compact Riemann surface of genus greater than oneComment: 22 pages, Submitte

Examples of noncommutative manifolds: complex tori and spherical manifolds

We survey some aspects of the theory of noncommutative manifolds focusing on the noncommutative analogs of two-dimensional tori and low-dimensional spheres. We are particularly interested in those aspects of the theory that link the differential geometry and the algebraic geometry of these spaces.Comment: Survey article. Final version. To appear in the proceedings volume of the "International Workshop on Noncommutative Geometry", IPM, Tehran 200

Flux compactification on smooth, compact three-dimensional toric varieties

Three-dimensional smooth, compact toric varieties (SCTV), when viewed as real six-dimensional manifolds, can admit G-structures rendering them suitable for internal manifolds in supersymmetric flux compactifications. We develop techniques which allow us to systematically construct G-structures on SCTV and read off their torsion classes. We illustrate our methods with explicit examples, one of which consists of an infinite class of toric CP^1 bundles. We give a self-contained review of the relevant concepts from toric geometry, in particular the subject of the classification of SCTV in dimensions less or equal to 3. Our results open up the possibility for a systematic construction and study of supersymmetric flux vacua based on SCTV.Comment: 27 pages, 10 figures; v2: references, minor typos & improvement