The double copy is a map from non-abelian gauge theories to gravity, that has been demonstrated both for scattering amplitudes and exact classical solutions. In this study, we reconsider the double copy for exact solutions that are self-dual in either the gauge or gravity theory. In this case, one may formulate a general double copy in terms of a certain differential operator, which generates the gauge and gravity solutions from a harmonic function residing in a biadjoint scalar theory. As an illustration, we examine the single copy of the well-known Eguchi-Hanson instanton in gravity. The gauge field thus obtained represents an abelian-like object whose field is dipole-like at large distances, and which has no magnetic or electric charge.
Eguchi-Hanson solitons are odd-dimensional generalizations of the four-dimensional Eguchi-Hanson metric and are asymptotic to AdS$_5$\mathbb{Z}_p$ when the cosmological constant is either positive or negative. We find soliton solutions to Lovelock gravity in 5 dimensions that are generalizations of these objects.
Using the numerical method, we study dynamics of coalescing black holes on the Eguchi-Hanson base space. Effects of a difference in spacetime topology on the black hole dynamics is discussed. We analyze appearance and disappearance process of marginal surfaces. In our calculation, the area of a coverall black hole horizon at the creation time in the coalescing black holes solutions on Eguchi-Hanson space is larger than that in the five-dimensional Kastor-Traschen solutions. This fact suggests that the black hole production on the Eguchi-Hanson space is easier than that on the flat space.
We consider a model of $F(R)$ gravity in which exponential and power corrections to Einstein-$Λ$ gravity are included. We show that this model has 4-dimensional Eguchi-Hanson type instanton solutions in Euclidean space. We then seek solutions to the five dimensional equations for which space-time contains a hypersurface corresponding to the Eguchi-Hanson space. We obtain analytic solutions of the $F(R)$ gravitational field equations, and by assuming certain relationships between the model parameters and integration constants, find several classes of exact solutions. Finally, we investigate the…
We construct higher-dimensional generalizations of the Eguchi-Hanson gravitational instanton in the presence of higher-curvature deformations of general relativity. These spaces are solutions to Einstein gravity supplemented with the dimensional extension of the quadratic Chern-Gauss-Bonnet invariant in arbitrary even dimension $D=2m\geq 4$, and they are constructed out of non-trivial fibrations over $(2m-2)$-dimensional Kähler-Einstein manifolds. Different aspects of these solutions are analyzed; among them, the regularization of the on-shell Euclidean action by means of the addition of topol…
We use the approach used by Eguchi-Hanson in constructing four-dimensional instanton metrics and construct a class of regular six-dimensional instantons which are nothing but $S^2\times S^2$ resolved conifolds. We then also obtain D3-brane solutions on these EH-resolved conifolds.
In this note, we review various seemingly different ways of obtaining Eguchi-Hanson metric with or without a cosmological constant term. Interestingly, the conformal class of metric corresponding to hyperbolic $n$-monopole solution obtained from the generalized Gibbons-Hawking ansatz, reduces to the Eguchi-Hanson metric in a particular limit. These results, though known from an algebraic geometry point of view, are useful while dealing with rotational killing symmetry of self-dual metrics in general theory of relativity as well as in the context of duality symmetry in string theory.
We show that it is possible to construct a supersymmetric mechanics with four supercharges possessing not conformally flat target space. A general idea of constructing such models is presented. A particular case with Eguchi--Hanson target space is investigated in details: we present the standard and quotient approaches to get the Eguchi--Hanson model, demonstrate their equivalence, give a full set of nonlinear constraints, study their properties and give an explicit expression for the target space metric.
We obtain new charged rotating multi-black hole solutions on the Eguchi-Hanson space in the five-dimensional Einstein-Maxwell system with a Chern-Simons term and a positive cosmological constant. In the two-black holes case, these solutions describe the coalescence of two rotating black holes with the spatial topologies of S^3 into a single rotating black hole with the spatial topology of the lens space S^3/Z_2. We discuss the differences in the horizon areas between our solutions and the two-centered Klemm-Sabra solutions which describe the coalescence of two rotating black holes with the spa…
We construct new supersymmetric black ring solutions on the Eguchi-Hanson base space as solutions of five-dimensional minimal supergravity. The solutions have the same two angular momentum components and the asymptotic structure on timeslices is asymptotically locally Euclidean. The S^1-direction of the black ring is along the equator on a S^2-bolt on the Eguchi-Hanson space. We also investigate the limit to a black hole, which describes the BMPV black hole with the topology of the lens space L(2;1)=S^3/Z_2.
We numerically study the event horizons of two kinds of five-dimensional coalescing black hole solutions with different asymptotic structures: the five-dimensional Kastor-Traschen solution (5DKT) and the coalescing black hole solution on Eguchi-Hanson space (CBEH). Topologies of the spatial infinity are ${\rm S}^3$ and $L(2;1)={\rm S}^3/{\mathbb Z}_2$, respectively. We show that the crease sets of event horizons are topologically ${\rm R}^1$ in 5DKT and ${\rm R}^1\times {\rm S}^1$ in CBEH, respectively. If we choose the time slices which respect space-time symmetry, the first contact points of…
We construct multi-black hole solutions in the five-dimensional Einstein-Maxwell theory with a positive cosmological constant on the Eguchi-Hanson space, which is an asymptotically locally Euclidean space. The solutions describe the physical process such that two black holes with the topology of S^3 coalesce into a single black hole with the topology of the lens space L(2;1)=S^3/Z_2. We discuss how the area of the single black hole after the coalescence depends on the topology of the horizon.
We construct new supersymmetric multi-black ring solutions on the Eguchi-Hanson base space as solutions of the five-dimensional minimal supergravity. The space-time has an asymptotically locally Euclidean time slice, i.e., it has the spatial infinity with the topology of the lens space $L(2;1)=S^3/Z_2$. The configurations of black rings are restricted by the requirement of the absence of a Dirac-Misner string everywhere outside horizons. Especially, in the case of two black rings, the solutions have the limit to a pair of rotating black holes with the horizon topology of S^3.
BPS wall solutions are obtained for N=2 supersymmetric nonlinear sigma model with Eguchi-Hanson target manifold in a manifestly supersymmetric manner. The model is constructed by a massive hyper-Kahler quotient method both in the N=1 superfield and in the N=2 superfield (harmonic superspace). We describe the model in simple terms and give relations between various parameterizations which are useful to describe the model and the solution. Some more details can be found in our previous paper [hep-th/0211103]. This article is dedicated to Professor Hiroshi Ezawa on the occasion of his seventieth …
We consider the eigenvalue equation for the Laplace-Beltrami operator acting on scalar functions on the non-compact Eguchi-Hanson space. The corresponding differential equation is reducible to a confluent Heun equation with Ince symbol [0,2,1_2]. We construct approximations for the eigenfunctions and their asymptotic scattering phases with the help of the Liouville-Green approximation (WKB). Furthermore, for specific discrete eigenvalues obtained by a continued T-fraction we construct the solution by the Frobenius method and determine its scattering phase by a monodromy computation.
We present a new class of solutions in odd dimensions to Einstein's equations containing either a positive or negative cosmological constant. These solutions resemble the even-dimensional Eguchi-Hanson-(A)dS metrics, with the added feature of having Lorentzian signatures. They are asymptotic to (A)dS$_{d+1}/Z_p$. In the AdS case their energy is negative relative to that of pure AdS. We present perturbative evidence in 5 dimensions that such metrics are the states of lowest energy in their asymptotic class, and present a conjecture that this is generally true for all such metrics. In the dS cas…
We investigate the behaviour of a particle moving on the quotient manifold $M=C^2/Z_$ which is derived from the EH metric as the two centers approach each other. In the classical region of the configuration space we specify the physically acceptable solutions and observe a tendency of the radial wave function to concentrate around the conical singularity. Fot the quantum case, using Schrödinger's equation, we determine the energy spectra and the radial eigenfunctions for a class of potentials.
The Einstein vacuum equations in five dimensions with negative cosmological constant are studied in biaxial Bianchi IX symmetry. We show that if initial data of Eguchi-Hanson type, modelled after the four-dimensional Riemannian Eguchi-Hanson space, have negative mass, the future maximal development does not contain horizons, i.e. the complement of the causal past of null infinity is empty. In particular, perturbations of Eguchi-Hanson-AdS spacetimes within the biaxial Bianchi IX symmetry class cannot form horizons, suggesting that such spacetimes are potential candidates for a naked singularit…
We consider heterotic strings on a warped Eguchi-Hanson space with five-brane and line bundle gauge fluxes. The heterotic string admits an exact CFT description in terms of an asymmetrically gauged SU(2)xSL(2,R) WZW model, in a specific double scaling limit in which the blow-up radius and the string scale are sent to zero simultaneously. This allows us to compute the perturbative 6D spectra for these models in two independent fashions: i) Within the supergravity approximation we employ a representation dependent index; ii) In the double scaling limit we determine all marginal vertex operators …
We construct infinitely many new 1-parameter families of simply connected complete noncompact 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-ma…
We present new M2 and M5 brane solutions in M-theory based on transverse self-dual Bianchi type IX space. All the other recently M2 and M5 branes constructed on transverse self-dual Taub-NUT, Egughi-Hanson and Atiyah-Hitchin spaces are special cases of this solution. The solution provides a smooth transition from Eguchi-Hanson type I based M branes to corresponding branes based on Eguchi-Hanson type II space. All the solutions can be reduced down to ten dimensional fully localized intersecting brane configurations.
We present novel regular Euclidean solutions to General Relativity in presence of Maxwell and conformally coupled scalar fields. In particular, we consider metrics of the Eguchi-Hanson and Taub-NUT families to solve the field equations analytically. The solutions have nontrivial topology labeled by the Hirzebruch signature and Euler characteristic that we compute explicitly. We find that, although the solutions are locally inequivalent with the original (anti-)self-dual Eguchi-Hanson metric, their asymptotically locally Euclidean limit leads to the same global properties. We revisit the Taub-N…
We construct modular invariant partition functions for type II strings on a conifold and a singular Eguchi-Hanson instanton by means of the SL(2,R)/U(1) version of Gepner models. In the conifold case, we find an extra massless hypermultiplet in the IIB spectrum and argue that it may be identified as a soliton. In the Eguchi-Hanson case, our formula is new and different from the earlier result, in particular does not contain graviton. The lightest IIB fields are combined into a six-dimensional (2,0) tensor multiplet with a negative mass square. We give an interpretation to it as a doubleton-lik…
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 gravitati…
We study non-Einstein Bach-flat gravitational instanton solutions that can be regarded as the generalization of the Taub-NUT/Bolt and Eguchi-Hanson solutions of Einstein gravity to conformal gravity. These solutions include non-Einstein spaces which are either asymptotically locally flat spacetimes (ALF) or asymptotically locally Anti-de Sitter (AlAdS). Nevertheless, solutions with different asymptotic conditions exist: we find geometries that present a weakened AlAdS asymptotia, exhibiting the typical low decaying mode of conformal gravity. This permits to identify the simple Neumann boundary…
In this paper we examine a small but detailed test of the emergent gravity picture with explicit solutions in gravity and gauge theory. We first derive symplectic U(1) gauge fields starting from the Eguchi-Hanson metric in four-dimensional Euclidean gravity. The result precisely reproduces the U(1) gauge fields of the Nekrasov-Schwarz instanton previously derived from the top-down approach. In order to clarify the role of noncommutative spacetime, we take the Braden-Nekrasov U(1) instanton defined in ordinary commutative spacetime and derive a corresponding gravitational metric. We show that t…
We consider heterotic string theory on Eguchi-Hanson space, as a local model of a resolved A_1 singularity in a six-dimensional flux compactification, with an Abelian gauge bundle turned on and non-zero torsion. We show that in a suitable double scaling limit, that isolates the physics near the non-vanishing two-cycle, a worldsheet conformal field theory description can be found. It contains a heterotic coset whose target space is conformal to Eguchi-Hanson. Starting from the blow-down limit of the singularity, it can be viewed as a dynamical deformation of the near-horizon fivebrane backgroun…
We obtain singularity resolutions for various overlapping brane configurations, including those of two heterotic 5-branes, type II 5-branes or D4-branes. In these solutions, the ``harmonic'' function H for each brane component depends only on the associated four-dimensional relative transverse space. The resolution is achieved by replacing these transverse spaces with Eguchi-Hanson or Taub-NUT spaces, both of which admit a normalisable self-dual (or anti-self-dual) harmonic 2-form. Due to the manner in which the interaction terms for the form fields modify their Bianchi identities or equations…
We study instanton effects along the Coulomb branch of an N=2 supersymmetric Yang-Mills theory with gauge group SU(2) on Asymptotically Locally Euclidean (ALE) spaces. We focus our attention on an Eguchi-Hanson gravitational background and on gauge field configurations of lowest Chern class.
The self-dual solution to lattice Euclidean gravity is constructed. In contrast to the well known Eguchi-Hanson solution to continuous Euclidean Gravity, the lattice solution is asymptotically {\it{globally}} Euclidean, i.e., the boundary of the space as $r\longrightarrow\infty$ is $S^3=SU(2)$.
$α'$ corrections to self-dual gravitational instantons
Instituto de Física Teórica UAM/CSIC（UAM/CSIC理论物理研究所）；Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez（阿道夫·伊巴涅斯大学自由艺术学院科学系）
The double copy is by now a firmly-established correspondence between amplitudes and classical solutions in biadjoint scalar, gauge and gravity theories. To date, no strongly coupled examples of the double copy in four dimensions have been found, and previous attempts based on exact non-linear solutions of biadjoint theory in Lorentzian signature have failed. In this paper, we instead look for biadjoint solutions in Euclidean signature, which may be relatable to Yang-Mills or gravitational instantons. We show that spherically symmetric power-like Euclidean solutions do not exist in precisely f…
We construct eight-dimensional gravitational instantons by solving appropriate self-duality equations for the spin-connection. The particular gravitational instanton we present has $Spin(7)$ holonomy and, in a sense, it is the eight-dimensional analog of the Eguchi-Hanson 4D space. It has a removable bolt singularity which is topologically S^4 and its boundary at infinity is the squashed S^7. We also lift our solutions to ten and eleven dimensions and construct fundamental string and membrane configurations that preserve 1/16 of the original supersymmetries.
Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California at Los Angeles（加州大学洛杉矶分校物理与天文学系马尼·L·布豪米克理论物理研究所）；Centre for Research in String Theory, School of Physics and Astronomy, Queen Mary University of London（伦敦玛丽女王大学物理与天文学院弦理论研究中心）；Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh（爱丁堡大学物理与天文学院希格斯理论物理中心）
In type IIB string theory, we consider fractional D3-branes in the orbifold background dual to four-dimensional N=2 supersymmetric Yang-Mills theory. We find the gravitational dual description of the generation of a non-trivial field theory potential on the Coulomb branch. When the orbifold singularity is softened to the smooth Eguchi-Hanson space, the resulting potential induces spontaneous partial supersymmetry breaking. We study the N=1 theory arising in those vacua and we see how the resolved conifold geometry emerges in this way. It will be natural to identify the size of the blown-up two…
We describe the explicit construction of Yang-Mills instantons on ALE spaces, following the work of Kronheimer and Nakajima. For multicenter ALE metrics, we determine the abelian instanton connections which are needed for the construction in the non-abelian case. We compute the partition function of Maxwell theories on ALE manifolds and comment on the issue of electromagnetic duality. We discuss the topological characterization of the instanton bundles as well as the identification of their moduli spaces. We generalize the 't Hooft ansatz to SU(2) instantons on ALE spaces and on other hyper-Ka…
We employ the ADHM method to derive the moduli space of two instantons in U(1) gauge theory on a noncommutative space. We show by an explicit hyperKähler quotient construction that the relative metric of the moduli space of two instantons on $R^4$ is the Eguchi-Hanson metric and find a unique threshold bound state. For two instantons on $R^3\times S^1$, otherwise known as calorons, we give the asymptotic metric and conjecture a completion. We further discuss the relationship of caloron moduli spaces of A, D and E groups to the Coulomb branches of three dimensional gauge theory. In particular, …
In this note, we discuss the consistency conditions which a discrete $Z_N$ symmetry should satisfy in order that it is not violated by gauge and gravitational instantons. As examples, we enlist all the $Z_N$ ${\cal R}$-symmetries as well as non-${\cal R}$ $Z_N$ symmetries (N=2,3,4) in the minimally supersymmetric standard model (MSSM) that are free from gauge and gravitational anomalies. We show that there exists non-anomalous discrete symmetries that forbid Baryon number violation up to dimension 6 level (in superspace). We also observe that there exists no non-anomalous $Z_3$ ${\cal R}$-symm…
We derive a superpotential for null electromagnetic fields in which the field line structure is in the form of an arbitrary torus knot. These fields are shown to correspond to single copies of a class of anti-self-dual Kerr-Schild spacetimes containing the Sparling-Tod metric. This metric is the pure Weyl double copy of the electromagnetic Hopfion, and we show that the Eguchi-Hanson metric is a mixed Weyl double copy of this Hopfion and its conformally inverted state. We formulate two conditions for electromagnetic fields, generalizing torus knotted fields and linked optical vortices, that, vi…
We give exact solutions for a recently developed ~$N=1$~ locally supersymmetric self-dual gauge theories in $~(2+2)\-$dimensions. We give the exact solutions for an $~SL(2)$~ self-dual Yang-Mills multiplet and what we call ``self-dual tensor multiplet'' on the gravitational instanton background by Eguchi-Hanson. We use a general method to get an $~SL(2)$~ self-dual Yang-Mills solution from any known self-dual gravity solution. Our result is the first example of exact solutions for the coupled system of these $~N=1$~ locally supersymmetric self-dual multiplets in ~$(2+2)\-$dimensions, which is …
Instantons on Eguchi-Hanson spaces provide explicit examples of stable bundles on non-compact four dimensional C^2/Z_n orbifold resolutions with non-Abelian structure groups. With this at hand, we can consider compactifications of ten dimensional SO(32) supergravity (arising as the low energy limit of the heterotic string) on the resolved spaces in the presence of non-Abelian bundles. We provide explicit examples in the resolved C^2/Z_3 case, and give a complete classification of all possible effective six dimensional models where the instantons are combined with Abelian gauge fluxes in order …
We present the gravity dual to a class of three-dimensional N=2 supersymmetric gauge theories on a biaxially squashed three-sphere, with a non-trivial background gauge field. This is described by a 1/2 BPS Euclidean solution of four-dimensional N=2 gauged supergravity, consisting of a Taub-NUT-AdS metric with a non-trivial instanton for the graviphoton field. The holographic free energy of this solution agrees precisely with the large N limit of the free energy obtained from the localized partition function of a class of Chern-Simons quiver gauge theories. We also discuss a different supersymm…
We show that the classical equations of motion of the low-energy effective field theory describing the massless modes of the heterotic (or type I) string admit two classes of supersymmetric self--dual backgrounds. The first class, which was already considered in the literature, consists of solutions with a (conformally) flat metric coupled to axionic instantons. The second includes Asymptotically Locally Euclidean (ALE) gravitational instantonic backgrounds coupled to gauge instantons through the so--called ``standard embedding''. We show that some elements of these two classes of solutions ar…
We show that $N=8$ {\it self-dual} supergravity theory, which is the consistent background for $N=2$ closed superstring theory in $2+2\-$dimensions, can accommodate the recently discovered two-dimensional dilaton gravity black hole solution, {\it via} appropriate dimensional reductions and truncations. Interestingly, the usual dilaton field in this set of solutions emerges from the scalar field in the ${\bf 70}\-$dimensional representation of an intrinsic global $SO(8)$ group. We also give a set of exact solutions, which can be interpreted as the dilaton field on Eguchi-Hanson gravitational in…
I develop a formalism for solving topological field theories explicitly, in the case when the explicit expression of the instantons is known. I solve topological Yang-Mills theory with the $k=1$ Belavin {\sl et al.} instanton and topological gravity with the Eguchi-Hanson instanton. It turns out that naively empty theories are indeed nontrivial. Many unexpected interesting hidden quantities (punctures, contact terms, nonperturbative anomalies with or without gravity) are revealed. Topological Yang-Mills theory with $G=SU(2)$ is not just Donaldson theory, but contains a certain {\sl link} theor…
Thus far, there seem to be no complete criteria that can settle the issue as to what the correct generalization of the Dirac-Born-Infeld (DBI) action, describing the low-energy dynamics of the D-branes, to the non-abelian case would be. According to recent suggestions, one might pass the issue of worldvolume solitons from abelian to non-abelian setting by considering the stack of multiple, coincident D-branes and use it as a guideline to construct or censor the relevant non-abelian version of the DBI action. In this spirit, here we are interested in the explicit construction of SU(2) Yang-Mill…
We explore various aspects of the self-dual Plebański-Demiański family in the Euclidean Einstein-Maxwell-$Λ$ system. The Killing-Yano tensor which was recently found by Yasui and one of the present authors allows us to prove that the self-dual Plebański-Demiański metric can be brought into the self-dual Carter metric by an orientation-reversing coordinate transformation. We show that the self-dual Plebański-Demiański solution admits two independent Killing spinors in the framework of $N=2$ minimal gauged supergravity, whereas the non-self-dual solution admits only a single Killing spinor. This…
