Circle Fibration Research Articles

Static M-horizons

We determine the geometry of all static black hole horizons of M-theory preserving at least one supersymmetry. We demonstrate that all such horizons are either warped products R^{1,1} *_w S or AdS_2 *_w S, where S admits an appropriate Spin(7) or SU(4) structure respectively; and we derive the conditions imposed by supersymmetry on these structures. We show that for electric static horizons with Spin(7) structure, the near horizon geometry is a product R^{1,1} * S, where S is a compact Spin(7) holonomy manifold. For electric static solutions with SU(4) structure, we show that the horizon section S is a circle fibration over an 8-dimensional Kahler manifold which satisfies an additional condition involving the Ricci scalar and the length of the Ricci tensor. Solutions include AdS_2 * S^3 * CY_6 as well as many others constructed from taking the 8-dimensional Kahler manifold to be a product of Kahler-Einstein and Calabi-Yau spaces.

Five-dimensional SYM from undeformed ABJM

We expand undeformed ABJM theory around the vacuum solution that was found in arxiv:0909.3101. This solution can be interpreted as a circle-bundle over a two-dimensional plane with a singularity at the origin. By imposing periodic boundary conditions locally far away from the singularity, we obtain a local fuzzy two-torus over which we have a circle fibration. By performing fluctuation analysis we obtain five-dimensional SYM with the precise value on the coupling constant that we would obtain by compactifying multiple M5 branes on the vacuum three-manifold. In the resulting SYM theory we also find a coupling to a background two-form.

States of a chiral 2D CFT

We study the dual description of the self-dual orbifold, a locally AdS3 spacetime which is a circle fibration over AdS2 and arises as the near-horizon limit of the extreme BTZ black hole. The geometry has two boundaries; we argue that this should correspond to a saddle-point for two copies of a chiral CFT living on these two boundaries in an entangled state. This picture arises naturally in the near-horizon limit, but there is a potential inconsistency with the bulk physics because of causal connections between the boundaries. We discuss a possible resolution of this puzzle. We also construct geometries which asymptotically approach the self-dual orbifold on a single boundary. These geometries (which contain mild singularities) enable us to explore other states of the dual chiral CFT. One of the geometries corresponds to the ground state of this CFT and can be obtained as a particular near-horizon limit of the BTZ M = 0 black hole. The self-dual orbifold is a finite temperature version of this geometry.

Rigidity for Multi-Taub-NUT metrics

This paper provides a classification result for gravitational instantons with cubic volume growth and cyclic fundamental group at infinity. It proves that a complete hyperkahler manifold asymptotic to a circle fibration over the Euclidean three-space is either the standard R 3 ◊ S 1 or a Multi-Taub-NUT manifold. In par- ticular, the underlying complex manifold is either C ◊ C/Z or a minimal resolution of a cyclic Kleinian singularity.

On the asymptotic geometry of gravitational instantons

We investigate the geometry at infinity of the so-called instan- tons, i.e. asymptotically flat hyperkahler four-manifolds, in relation with their volume growth. In particular, we prove that gravitational instantons with cubic volume growth are ALF, namely asymptotic to a circle fibration over a Euclidean three-space, with fibers of asymptotically constant length.

Topology change from (heterotic) Narain T-duality

We consider Narain T-duality on a nontrivially fibered n-torus bundle in the presence of a topologically nontrivial NS H flux. The action of the duality group on the topology and H flux of the corresponding type II and heterotic string backgrounds is determined. The topology change is specialized to the case of supersymmetric T 2 -fibered torsional string backgrounds with nontrivial H flux. We prove that it preserves the global tadpole condition in the total space as well as on the base of the torus fibration. We find that some of these T-dualities exchange half of the field strength of an unbroken U ( 1 ) gauge symmetry with the anti-selfdual part of the curvature of a physical circle fibration. We verify that such T-dualities indeed exchange the supersymmetry condition for the circle bundle with that of the gauge bundle.

A Mass for ALF Manifolds

We prove positive mass theorems on ALF manifolds, i.e. complete noncompact manifolds that are asymptotic to a circle fibration over a Euclidean base, with fibers of asymptotically constant length.

Taub–NUT black holes in third order Lovelock gravity

We consider the existence of Taub–NUT solutions in third order Lovelock gravity with cosmological constant, and obtain the general form of these solutions in eight dimensions. We find that, as in the case of Gauss–Bonnet gravity and in contrast with the Taub–NUT solutions of Einstein gravity, the metric function depends on the specific form of the base factors on which one constructs the circle fibration. Thus, one may say that the independence of the NUT solutions on the geometry of the base space is not a robust feature of all generally covariant theories of gravity and is peculiar to Einstein gravity. We find that when Einstein gravity admits non-extremal NUT solutions with no curvature singularity at r=N, then there exists a non-extremal NUT solution in third order Lovelock gravity. In 8-dimensional spacetime, this happens when the metric of the base space is chosen to be CP3. Indeed, third order Lovelock gravity does not admit non-extreme NUT solutions with any other base space. This is another property which is peculiar to Einstein gravity. We also find that the third order Lovelock gravity admits extremal NUT solution when the base space is T2×T2×T2 or S2×T2×T2. We have extended these observations to two conjectures about the existence of NUT solutions in Lovelock gravity in any even-dimensional spacetime.

Branch structure of 𝐽–holomorphic curves near periodic orbits of a contact manifold

Let M M be a three–dimensional contact manifold, and ψ ~ : D ∖ { 0 } → M × R \tilde {\psi }:D\setminus \{0\}\to M\times {\mathbb R} a finite–energy pseudoholomorphic map from the punctured disc in C {\mathbb C} that is asymptotic to a periodic orbit of the contact form. This article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such that ψ ~ \tilde {\psi } resembles a holomorphic curve, invoking comparison with the theory of topological linking of plane complex algebroid curves near a singular point. Examples of this behaviour, which are studied in some detail, include pseudoholomorphic maps into E p , q × R {\mathbb E}_{p,q}\times {\mathbb R} , where E p , q {\mathbb E}_{p,q} denotes a rational ellipsoid (contact structure induced by the standard complex structure on C 2 {\mathbb C}^{2} ), as well as contact structures arising from non-standard circle–fibrations of the three–sphere.

Higher-dimensional Kaluza–Klein monopoles

It is well known that the Kaluza–Klein monopole of Sorkin, Gross and Perry can be obtained from the Euclidean Taub–NUT solution with an extra compact fifth spatial dimension via Kaluza–Klein reduction. In this paper we consider Taub–NUT-like solutions of the vacuum Einstein field equations, with or without cosmological constant, in five dimensions and higher, and similarly perform Kaluza–Klein reductions to obtain new magnetic KK brane solutions in higher dimensions, as well as further sphere reductions to magnetic monopoles in four dimensions. In six dimensions we also employ spatial and timelike Hopf dualities to untwist the circle fibration characteristic to these spaces and obtain charged strings. A variation of these methods in ten dimensions leads to a non-uniform electric string in five dimensions.

Some Geometric Characterizations of the Hopf Fibrations of the Three-Sphere

The Frechet manifold of all embeddings (up to orientation preserving reparametrizations) of the circle in S 3 has a canonical weak Riemannian metric. We use the characterization obtained by H. Gluck and F. Warner of the oriented great circle fibrations of S 3 to prove that among all such fibrations p:S 3?B, the manifold B consisting of the oriented fibers is totally geodesic in , or has minimum volume or diameter with the induced metric, exactly when p is a Hopf fibration

Geometric transitions, flops and non-Kähler manifolds: I

We construct a duality cycle which provides a complete supergravity description of geometric transitions in type II theories via a flop in M-theory. This cycle connects the different supergravity descriptions before and after the geometric transitions. Our construction reproduces many of the known phenomena studied earlier in the literature and allows us to describe some new and interesting aspects in a simple and elegant fashion. A precise supergravity description of new torsional manifolds that appear on the type IIA side with branes and fluxes and the corresponding geometric transition are obtained. A local description of new G 2 manifolds that are circle fibrations over non-Kähler manifolds is presented.

The Blaschke conjecture and great circle fibrations of spheres

We construct an explicit diffeomorphism taking any fibration of a sphere by great circles into the Hopf fibration. We use elementary differential geometry, and no surgery or K -theory, to carry out the construction—indeed the diffeomorphism is a local (differential) invariant, algebraic in derivatives. This result is new only for 5 dimensional spheres, but our new method of proof is elementary.

AFFINE MAXIMAL TORUS FIBRATIONS OF A COMPACT LIE GROUP

By a generalization of the method developed by Gluck and Warner to characterize the oriented great circle fibrations of the three-sphere, we give, for any compact connected semisimple Lie group G, a general procedure to obtain the continuous fibrations of G by Weyl-oriented affine maximal tori, find conditions for smoothness and provide infinite dimensional spaces of concrete examples.

Smooth Great Circle Fibrations and an Application to the Topological Blaschke Conjecture

Smooth Great Circle Fibrations and an Application to the Topological Blaschke Conjecture

Smooth great circle fibrations and an application to the topological Blaschke conjecture

We study great smooth circle fibrations of round spheres and Blaschke manifolds of the homotopy type of complex projective spaces.

Great sphere fibrations of manifolds

Abstract : There are three questions that guide our study: (1) given two fibrations are they topologically equivalent, (2) is it possible to deform one fibration to another through a one parameter family of fibrations, and (3) what is the homotopy type of the space of all fibrations? We address these questions for great circle fibrations of odd dimensional round spheres and arbitrary great k-sphere fibrations.

A note on skew-Hopf fibrations

Each great circle fibration of the unit 3 3 -sphere in 4 4 -space can be identified with a subset of the Grassmann manifold of oriented 2 2 -planes in 4 4 -space by associating each great circle fiber with the 2 2 -plane it lies in. This Grassmann manifold can be identified with the space S 2 × S 2 {S^2} \times {S^2} . H. Gluck and F. Warner, in Great circle fibrations of the three sphere, Duke Math. J. 50 (1983), 107-132, have shown that the subsets of this Grassmann manifold which correspond to great circle fibrations can be interpreted as the graphs of distance decreasing maps from S 2 {S^2} and S 2 {S^2} and that Hopf fibrations correspond to constant maps. This note characterizes explicitly the maps which correspond to "skew-Hopf" fibrations: those fibrations of the 3 3 -sphere obtained from Hopf fibrations by applying a linear transformation of 4 4 -space followed by projection of the fibers back to the unit 3 3 -sphere.

A Note on Skew-Hopf Fibrations

Each great circle fibration of the unit $3$-sphere in $4$-space can be identified with a subset of the Grassmann manifold of oriented $2$-planes in $4$-space by associating each great circle fiber with the $2$-plane it lies in. This Grassmann manifold can be identified with the space ${S^2} \times {S^2}$. H. Gluck and F. Warner, in Great circle fibrations of the three sphere, Duke Math. J. 50 (1983), 107-132, have shown that the subsets of this Grassmann manifold which correspond to great circle fibrations can be interpreted as the graphs of distance decreasing maps from ${S^2}$ and ${S^2}$ and that Hopf fibrations correspond to constant maps. This note characterizes explicitly the maps which correspond to "skew-Hopf" fibrations: those fibrations of the $3$-sphere obtained from Hopf fibrations by applying a linear transformation of $4$-space followed by projection of the fibers back to the unit $3$-sphere.

Great circle fibrations of the three-sphere

Great circle fibrations of the three-sphere