Eguchi-Hanson Metrics Research Articles

Geodesics on metrics of Eguchi–Hanson type

Geodesic equations are solved when at least two of theta , phi and psi are constant, or r is constant, on scalar flat metrics of Eguchi–Hanson type. They can also be solved also on Eguchi–Hanson metrics which are Ricci flat if only phi is constant. However, the explicit solution of the geodesic equations is not available yet if only psi is constant.

The Dirac equation on metrics of Eguchi–Hanson type

We investigate parallel spinors on the Eguchi–Hanson metrics and find the space of complex parallel spinors is complex two-dimensional. For the metrics of Eguchi–Hanson type with the zero scalar curvature, we separate variables for the harmonic spinors and obtain the solutions explicitly.

Infinitely many new families of complete cohomogeneity one G$_2$-manifolds: G$_2$ analogues of the Taub–NUT and Eguchi–Hanson spaces

We construct infinitely many new 1-parameter families of simply connected complete non-compact G _2 -manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter values: at one special value we obtain a unique member of each family with asymptotically conical (AC) geometry; on approach to the other special parameter value the family of metrics collapses to an AC Calabi–Yau 3-fold. Our infinitely many new diffeomorphism types of AC G _2 -manifolds are particularly noteworthy: previously the three examples constructed by Bryant and Salamon in 1989 furnished the only known simply connected AC G _2 -manifolds. We also construct a closely related conically singular G _2 -holonomy space: away from a single isolated conical singularity, where the geometry becomes asymptotic to the G _2 -cone over the standard nearly Kähler structure on the product of a pair of 3-spheres, the metric is smooth and it has ALC geometry at infinity. We argue that this conically singular ALC G2-space is the natural G _2 analogue of the Taub–NUT metric in 4-dimensional hyperKähler geometry and that our new AC G _2 -metrics are all analogues of the Eguchi–Hanson metric, the simplest ALE hyperKähler manifold. Like the Taub–NUT and Eguchi–Hanson metrics, all our examples are cohomogeneity one, i.e. they admit an isometric Lie group action whose generic orbit has codimension one.

Higher order obstructions to the desingularization of Einstein metrics

We find new obstructions to the desingularization of compact Einstein orbifolds by smooth Einstein metrics. These new obstructions, specific to the compact situation, raise the question of whether a compact Einstein $4$-orbifold which is limit of Einstein metrics bubbling out Eguchi-Hanson metrics has to be Kahler. We then test these obstructions to discuss if it is possible to produce a Ricci-flat but not Kahler metric by the most promising desingularization configuration proposed by Page in 1981. We identify $84$ obstructions which, once compared to the $57$ degrees of freedom, indicate that almost all flat orbifold metrics on $\mathbb{T}^4/\mathbb{Z}_2$ should not be limit of Ricci-flat metrics with generic holonomy while bubbling out Eguchi-Hanson metrics. Perhaps surprisingly, in the most symmetric situation, we also identify a $14$-dimensional family of desingularizations satisfying all of our $84$ obstructions.

Metrics of Eguchi–Hanson types with the negative constant scalar curvature

We construct two types of Eguchi–Hanson metrics with the negative constant scalar curvature. The type I metrics are Kähler. The type II metrics are ALH whose total energy can be negative.

ALE spaces from noncommutative [formula omitted] instantons via exact Seiberg–Witten map

The exact Seiberg–Witten (SW) map of a noncommutative (NC) gauge theory gives the commutative equivalent as an ordinary gauge theory coupled to a field dependent effective metric. We study instanton solutions of this commutative equivalent whose self-duality equation turns out to be the exact SW map of NC instantons. We derive general differential equations governing U(1) instantons and we explicitly get an exact solution corresponding to the single NC instanton. Remarkably the effective metric induced by the single U(1) instanton is related to the Eguchi–Hanson metric—the simplest gravitational instanton. Surprisingly the instanton number is not quantized but depends on an integration constant. Our result confirms the expected non-perturbative breakdown of the SW map. However, the breakdown of the map arises in a consistent way: the instanton number plays the role of a parameter giving rise to a one-parameter family of Eguchi–Hanson metrics.

Axial Anomaly for Eguchi–Hanson Metrics with Nonzero Total Mass

We compute the Dirac indexes for the two spinstructures κ0 and κ1 for Eguchi–Hanson metricswith nonzero total mass. It shows that the Dirac indexes do notvanish in general, and axial anomaly exists. When the metric haszero total mass, the Dirac index vanishes for the spin structureκ0, and no axial anomaly exists inthis case.

Massive nonlinear sigma models and BPS domain walls in harmonic superspace

Four-dimensional massive N=2 nonlinear sigma models and BPS wall solutions are studied in the off-shell harmonic superspace approach in which N=2 supersymmetry is manifest. The general nonlinear sigma model can be described by an analytic harmonic potential which is the hyper-Kähler analog of the Kähler potential in N=1 theory. We examine the massive nonlinear sigma model with multi-center four-dimensional target hyper-Kähler metrics and derive the corresponding BPS equation. We study in some detail two particular cases with the Taub-NUT and double Taub-NUT metrics. The latter embodies, as its two separate limits, both Taub-NUT and Eguchi–Hanson metrics. We find that domain wall solutions exist only in the double Taub-NUT case including its Eguchi–Hanson limit.

U(1)× U(1) quaternionic metrics from harmonic superspace

We construct, using harmonic superspace and the quaternionic quotient approach, a quaternionic-Kähler extension of the most general two centres hyper-Kähler metric. It possesses U(1)× U(1) isometry, contains as special cases the quaternionic-Kähler extensions of the Taub-NUT and Eguchi–Hanson metrics and exhibits an extra one-parameter freedom which disappears in the hyper-Kähler limit. Some emphasis is put on the relation between this class of quaternionic-Kähler metrics and self-dual Weyl solutions of the coupled Einstein–Maxwell equations. The relation between our explicit results and the recent general ansatz of Calderbank and Pedersen for quaternionic-Kähler metrics with U(1)× U(1) isometries is traced in detail.

Yang–Mills instantons in the gravitational instanton backgrounds

The simplest and the most straightforward new algorithm for generating solutions to (anti) self-dual Yang-Mills (YM) equation in the typical gravitational instanton backgrounds is proposed. When applied to the Taub-NUT and the Eguchi-Hanson metrics, the two best-known gravitational instantons, the solutions turn out to be the rather exotic type of instanton configurations carrying finite YM action but generally fractional topological charge values.

Quaternionic metrics from harmonic superspace: Lagrangian approach and quotient construction

Starting from the most general harmonic superspace action of self-interacting Q + hypermultiplets in the background of N=2 conformal supergravity, we derive the general action for the bosonic sigma model with a generic 4 n-dimensional quaternionic-Kähler (QK) manifold as the target space. The action is determined by the analytic harmonic QK potential and supplies an efficient systematic procedure of the explicit construction of QK metrics by the given QK potential. We find out this action to have two flat limits. One gives the hyper-Kähler (HK) sigma model with a 4 n-dimensional target manifold, while another yields a conformally-invariant sigma model with 4( n+1)-dimensional HK target. We work out the harmonic superspace version of the QK quotient construction and use it to give a new derivation of QK extensions of four-dimensional Taub–NUT and Eguchi–Hanson metrics. We analyze in detail the geometrical and symmetry structure of the second metric. The QK sigma model approach allows us to reveal the enhancement of its SU(2)⊗ U(1) isometry to SU(3) or SU(1,2) at the special relations between its free parameters: the Sp(1) curvature (“Einstein constant”) and the “mass”.

Gauging N = 2 sigma models in harmonic superspace

We construct the general minimal coupling of N = 2 Yang-Mills fields to N = 2 sigma models in flat and curved harmonic superspace. We show that the complete symmetry group of the matter action can always be gauged in curved space. In flat space, we find that the gauging can be blocked by a local obstruction. We give a number of examples based on homogeneous quaternionic spaces and on quaternionic generalizations of the Taub-NUT and Eguchi-Hanson metrics.

Geometry of the Z-fold

The author computes the normalisation matrix and Yukawa couplings on the Z-manifold in the geometric approximation where the singularities on the Z-orbifold are blown up with Eguchi-Hanson metrics (1980). The orbifold limit is discussed in detail from a geometric and algebraic point of view, and the results are compared with those obtained using algebraic geometry and conformal field theory.

Eguchi-Hanson metrics with cosmological constant

The author constructs a solution to Einstein's equations with a cosmological constant. The metric contains two parameters (a,b). When b=0 the metric is recognized as the Eguchi-Hanson I (II) solution with anti-self-dual Weyl tensor W-. When a=0 it is seen to be the (pseudo) Fubini Study metric with a self-dual Weyl tensor W+. The solution has a Weyl tensor W++W-, is a Kahler metric, and is of Petrov type D. The author shows that in some cases the metric is complete.

A physical picture of the K3 gravitational instanton

The self-dual metric on K3 (the manifold of Kummer's quartic surface) has 58 parameters. A physical interpretation of these is given in terms of an approximate construction of the K3 gravitational instanton from 16 Eguchi-Hanson metrics arranged in a lattice with certain identifications.