Euclidean Taub-NUT Space Research Articles

Atiyah–Hitchin in five-dimensional Einstein–Gauss–Bonnet gravity

We construct a new class of stationary exact solutions to five-dimensional Einstein–Gauss–Bonnet gravity. The solutions are based on four-dimensional self-dual Atiyah–Hitchin geometry. We find analytical solutions to the five-dimensional metric function that are regular everywhere. We find some constraints on the possible physical solutions by investigating the solutions numerically. We also study the behavior of the solutions in the extremal limits of the Atiyah–Hitchin geometry. In the extremal limits, the Atiyah–Hitchin metric reduces to a bolt structure and Euclidean Taub–NUT space, respectively. In these limits, the five-dimensional metric function approaches to a constant value and infinity, respectively. We find that the asymptotic metrics are regular everywhere.

Spectral properties of Schwarzschild instantons

We study spectral properties of the Dirac and scalar Laplace operator on the Euclidean Schwarzschild space, both twisted by a family of abelian connections with anti-self-dual curvature. We show that the zero-modes of the gauged Dirac operator, first studied by Pope, take a particularly simple form in terms of the radius of the Euclidean time orbits, and interpret them in the context of geometric models of matter. For the gauged Laplace operator, we study the spectrum of bound states numerically and observe that it can be approximated with remarkable accuracy by that of the exactly solvable gauged Laplace operator on the Euclidean Taub-NUT space.

Hidden symmetries and killing tensors on curved spaces

Higher-order symmetries corresponding to Killing tensors are investigated. The intimate relation between Killing-Yano tensors and nonstandard supersymmetries is pointed out. In the Dirac theory on curved spaces, Killing-Yano tensors generate Dirac-type operators involved in interesting algebraic structures as dynamical algebras or even infinite dimensional algebras or superalgebras. The general results are applied to space-times which appear in modern studies. One presents the infinite dimensional superalgebra of Dirac type operators on the 4-dimensional Euclidean Taub-NUT space that can be seen as a twisted loop algebra. The existence of the conformal Killing-Yano tensors is investigated for some spaces with mixed 3-Sasakian structures.

Non-standard symmetries and Killing tensors

Higher order symmetries corresponding to Killing tensors are investigated. The intimate relation between Killing-Yano tensors and non-standard supersymmetries is pointed out. The gravitational anomalies are absent if the hidden symmetry is associated with a Killing-Yano tensor. In the Dirac theory on curved spaces, Killing-Yano tensors generate Dirac type operators involved in interesting algebraic structures as dynamical algebras or even infinite dimensional algebras or superalgebras. The general results are applied to space-times which appear in modern studies. The 4-dimensional Euclidean Taub-NUT space and its generalizations introduced by Iwai and Katayama are analyzed from the point of view of hidden symmetries. One presents the infinite dimensional superalgebra of Dirac type operators on Taub-NUT space that can be seen as a twisted loop algebra. The axial anomaly, interpreted as the index of the Dirac operator, is computed for the generalized Taub-NUT metrics. The existence of the conformal Killing-Yano tensors is investigated for some spaces with mixed Sasakian structures.

Geodesic Motions in Euclidean Taub-NUT Spinning Spaces

We investigate the geodesic motions of pseudo-classical spin- \(\frac{1}{2}\) point particles in the Euclidean Taub-NUT space. We derive the constants of motion from the solutions of the generalized Killing equations for spinning spaces and exploiting those the motion of pseudo-classical Dirac fermions are analyzed on a cone and plane.

Nonlinear symmetries on spaces admitting Killing tensors

Nonlinear symmetries corresponding to Killing tensors are investigated. The intimate relation between Killing–Yano tensors and non-standard supersymmetries is pointed out. The gravitational anomalies are absent if the hidden symmetry is associated with a Killing–Yano tensor. In the case of the nonlinear symmetries the dynamical algebras of the Dirac-type operators is more involved and could be organized as infinite dimensional algebras or superalgebras. The general results are applied to some concrete spaces involved in theories of modern physics. As a first example it is considered the 4-dimensional Euclidean Taub-NUT space and its generalizations introduced by Iwai and Katayama. One presents the infinite dimensional superalgebra of Dirac type operators on Taub-NUT space that could be seen as a graded loop superalgebra of the Kac-Moody type. The axial anomaly, interpreted as the index of the Dirac operator, is computed for the generalized Taub-NUT metrics. Finally the existence of the conformal Killing–Yano tensors is investigated for some spaces with mixed Sasakian structures.

Atiyah-Hitchin space in five-dimensional Einstein-Maxwell theory

We construct exact solutions to five-dimensional Einstein-Maxwell theory based on Atiyah-Hitchin space. The solutions cannot be written explicitly in a closed form, so their properties are investigated numerically. The five-dimensional metric is regular everywhere except on the location of original bolt in four-dimensional Atiyah-Hitchin base space. On each time-fixed slices, the metric, asymptotically approaches an Euclidean Taub-NUT space.

Non-standard supersymmetries on spaces admitting Killing–Yano tensors

We attempt to point out the intimate relation between Killing–Yano tensors and non-standard supersymmetries, along the way reviewing some recent results. The Euclidean Taub–NUT space is taken to illustrate some of the aspects of the general discussion about the connection between spacetime higher order geometrical symmetries and supersymmetries. The dynamical symmetry of this space leads in a natural way to an infinite dimensional twisted loop superalgebra of the conserved operators.

Dirac-type operators on curved spaces admitting Killing-Yano tensors

We attempt to point out the relation between Killing-Yano tensors and non-standard supersymmetries. The Killing-Yano tensors play an important role in the Dirac theory on curved spaces where they produce Dirac-type operators. The Euclidean Taub-NUT space is taken to illustrate some of the aspects of the general discussion about the connection between spacetime higher order geometrical symmetries and supersymmetries. The dynamical symmetry of this space leads in a natural way to an infinite dimensional twisted loop superalgebra of the conserved operators.

Infinite loop superalgebras of the Dirac theory on the Euclidean Taub–NUT space

The Dirac theory in the Euclidean Taub–NUT space gives rise to a large collection of conserved operators associated with genuine or hidden symmetries. They are involved in interesting algebraic structures as dynamical algebras or even infinite-dimensional algebras or superalgebras. One presents here the infinite-dimensional superalgebra specific to the Dirac theory in manifolds carrying the Gross–Perry–Sorkin monopole. It is shown that there exists an infinite-dimensional superalgebra that can be seen as a twisted loop superalgebra.

Remarks on a five-dimensional Kaluza-Klein theory of the massive Dirac monopole

The Gross-Perry-Sorkin spacetime, formed by the Euclidean Taub-NUT space with the time trivially added, is the appropriate background of the Dirac magnetic monopole without an explicit mass term. One remarks that there exists a very simple five-dimensional metric of spacetimes carrying massive magnetic monopoles that is an exact solution of the vacuum Einstein equations. Moreover, the same isometry properties as the original Euclidean Taub-NUT space are preserved. This leads to an Abelian Kaluza-Klein theory whose metric appears as a combinations between the Gross-Perry-Sorkin and Schwarzschild ones. The asymptotic motion of the scalar charged test particles is discussed, now by accounting for the mixing between the gravitational and magnetic effects.

Dirac-Type Operators on Curved Spaces and the Role of Killing-Yano Tensors

We consider the Dirac equation in curved backgrounds and investigate the role of Killing-Yano tensors in constructing Dirac-type operators. The general results are applied to the case of the four-dimensional Euclidean Taub-NUT space. We investigate the gravitational anomalies for generalized Euclidean Taub-NUT metrics that admit hidden symmetries analogous to the Runge-Lenz vector of the Kepler-type problem.

Generalized Dirac monopoles in non-Abelian Kaluza–Klein theories

A method is proposed for constructing the geometries of the non-Abelian Kaluza–Klein theories of generalized monopoles in arbitrary dimensions. These represent a natural generalization of the Euclidean Taub-NUT space, regarded as the appropriate background of the Dirac magnetic monopole. A recent theory of induced representations governing the isometries of the Euclidean Taub-NUT space is combined with usual geometrical methods obtaining a conjecture in which the potentials of the generalized monopoles can be written down without to solve explicitly the Yang–Mills equations. 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 cannot be interpreted as monopoles. The space-like models are studied pointing out that their monopole fields strength are particular solutions of the Yang–Mills equations with central symmetry, producing the standard flux of 4 π through the two-dimensional spheres surrounding the monopole. Examples are given of manifolds with Einstein metrics carrying SU ( 2 ) monopoles.

THE INDUCED REPRESENTATION OF THE ISOMETRY GROUP OF THE EUCLIDEAN TAUB–NUT SPACE AND NEW SPHERICAL HARMONICS

It is shown that the SO(3) isometries of the Euclidean Taub–NUT space combine a linear three-dimensional representation with one induced by an SO(2) subgroup, giving the transformation law of the fourth coordinate under rotations. This explains the special form of the angular momentum operator on this manifold which leads to a new type of spherical harmonics and spinors.

Symmetries of the Dirac operators associated with covariantly constant Killing–Yano tensors

The continuous and discrete symmetries of the Dirac-type operators produced by particular Killing–Yano tensors are studied in manifolds of arbitrary dimensions. The Killing–Yano tensors considered are covariantly constant and realize certain square roots of the metric tensor. Such a Killing–Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. The Dirac operators are related among themselves through continuous or discrete transformations. It is shown that the groups of continuous symmetry can be only U(1) and SU(2), specific to (hyper-)Kähler spaces, but arising even in cases when the requirements for these special geometries are not fulfilled. The discrete symmetries are also studied obtaining the discrete groups and . The briefly presented examples are the Euclidean Taub–NUT space and the Minkowski spacetime.

FERMIONS IN TAUB-NUT BACKGROUND

We discuss the relativistic spin-[Formula: see text] particles in pseudo-classical models involving anticommuting Grassmann variables for the spin degrees of freedom. The constants of motion are expressed in terms of Killing vectors and Killing-Yano tensors. Passing from the spinning spaces to the Dirac equation in curved backgrounds we point out the role of the Killing-Yano tensors in the construction of the Dirac-type operators. The general results are applied to the case of the four-dimensional Euclidean Taub-NUT space.

Runge–Lenz operator for Dirac field in Taub-NUT background

Fermions in D=4 self-dual Euclidean Taub-NUT space are investigated. Dirac-type operators involving Killing–Yano tensors of the Taub-NUT geometry are explicitly given showing that they anticommute with the standard Dirac operator and commute with the Hamiltonian as it is expected. They are connected with the hidden symmetries of the space allowing the construction of a conserved vector operator analogous to the Runge–Lenz vector of the Kepler problem. This operator is written down pointing out its algebraic properties.

On the Taub-NUT Spinning Space

The geodesic motion of pseudo-classical spinning particles in Euclidean Taub-NUT space is analyzed. Investigating the generalized Killing equations for spinning spaces, we express the constants of motion in terms of Killing-Yano tensors. Some previous results from the literature are corrected.

Spinning particles in Taub-NUT space

The geodesic motion of pseudo-classical spinning particles in Euclidean Taub-NUT space is analysed. The constants of motion are expressed in terms of Killing-Yano tensors. Some previous results from the literature are corrected.

Kaluza-Klein Dyons in String Theory

S-duality of hetertotic / type II string theory compactified on a six dimensional torus requires the existence of Kaluza-Klein dyons, carrying winding charge. We identify the zero modes of the Kaluza-Klein monopole solution which are responsible for these dyonic excitations, and show that we get the correct degeneracy of dyons as predicted by S-duality. The self-dual harmonic two form on the Euclidean Taub-NUT space plays a crucial role in this construction.