Yang-Mills instantons on ALE gravitational instantons were constructed by Kronheimer and Nakajima in terms of matrices satisfying algebraic equations. These were conveniently organized into a quiver. We construct generic Yang-Mills instantons on ALF gravitational instantons. Our data are formulated in terms of matrix-valued functions of a single variable, that are in turn organized into a bow. We introduce the general notion of a bow, its representation, its associated data and moduli space of solutions. For a judiciously chosen bow the Nahm transform maps any bow solution to an instanton on an ALF space. We demonstrate that this map respects all complex structures on the moduli spaces, so it is likely to be an isometry, and use this fact to study the asymptotics of the moduli spaces of instantons on ALF spaces.
We study the space of square-integrable harmonic forms over ALF gravitational instantons of type $A _{ K -1 } $ and of type $D _K $. We first calculate its dimension making use of a result by Hausel, Hunsicker and Mazzeo which relates the Hodge cohomology of a gravitational instanton $M$ to the singular cohomology of a particular compactification $X _M $ of $M$. We then exhibit an explicit basis, exact for $A _{ K -1 } $ and approximate for $D _K $, and interpret geometrically the relations between $M$, $X _M $ and their cohomologies.
We investigate the thermodynamical properties of solutions to the 5D Einstein and Einstein-Maxwell-dilaton equations describing static black holes on ALF gravitational instantons. Some general expressions regarding the nut charge and potential are derived, as well as Smarr-like relations. The thermodynamics of two particular solutions representing a static black hole on Taub-Nut and Taub-Bolt instantons are examined in detail. Two dierent approaches for computation of the mass and the tension - the Komar and counterterm method, are applied and compared. A new charged black hole on Taub-Bolt in…
Unitary anti-self-dual connections on Asymptotically Locally Flat (ALF) hyperkähler spaces are constructed in terms of data organized in a bow. Bows generalize quivers, and the relevant bow gives rise to the underlying ALF space as the moduli space of its particular representation -- the small representation. Any other representation of that bow gives rise to anti-self-dual connections on that ALF space. We prove that each resulting connection has finite action, i.e. it is an instanton. Moreover, we derive the asymptotic form of such a connection and compute its topological class.
In this paper we prove that over an asymptotically locally flat (ALF) Riemannian four-manifold the energy of an "admissible" SU(2) Yang--Mills is always integer. This result sharpens the previously known energy identity for such Yang--Mills instantons over ALF geometries. Furthermore we demonstrate that this statement continues to hold for the larger gauge group U(2). Finally we make the observation that there might be a natural relationship between 4 dimensional Yang--Mills theory over an ALF space and 2 dimensional conformal field theory. This would provide a further support for the existenc…
We study splitting and joining interactions of giant gravitons with angular momenta $N^{1/2}\ll J\ll N$ in the type IIB string theory on $AdS_5 \times S^5$ by describing them as instantons in the tiny graviton matrix model introduced by Sheikh-Jabbari. At large $J$ the instanton equation can be mapped to the four-dimensional Laplace equation and the Coulomb potential for $m$ point charges in an $n$-sheeted Riemann space corresponds to the $m$-to-$n$ interaction process of giant gravitons. These instantons provide the holographic dual of correlators of all semi-heavy operators and the instanton…
We show that cosmological acceleration, Dark Energy (DE) effect is a consequence of the zero rest mass, conformal non-invariance of gravitons, and 1-loop finiteness of quantum gravity (QG). The effect is due to graviton-ghost condensates arising from the interference of quantum coherent states. The theory is constructed as follows: Faddeev-Popov-De Witt gauged path integral -> factorization of classical and quantum variables -> transition to the 1-loop approximation -> choice of ghost sector, satisfying 1-loop finiteness of the theory off the mass shell. The Bogolyubov-Born-Green-Kirckwood-Yvo…
We discuss a special class of quantum gravity phenomena that occur on the scale of the Universe as a whole at any stage of its evolution. These phenomena are a direct consequence of the zero rest mass of gravitons, conformal non-invariance of the graviton field, and one-loop finiteness of quantum gravity. The effects are due to graviton-ghost condensates arising from the interference of quantum coherent states. Each of coherent states is a state of gravitons and ghosts of a wavelength of the order of the horizon scale and of different occupation numbers. The state vector of the Universe is a c…
We investigate Yang--Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these spaces introduced by Hausel--Hunsicker--Mazzeo. First referring to the codimension 2 singularity removal theorem of Sibner--Sibner and Rade we prove that given a smooth, finite energy, self-dual SU(2) connection over a complete ALF space, its energy is congruent to a Chern--Simons invariant of the boundary three-manifold if the connection satisfies a certain holo…
The equivalence of NS5-branes and ALF spaces under T-duality is well known. However, a naive application of T-duality transforms the ALF space into a smeared NS5-brane, de-localized on the dual, transverse, circle. In this paper we re-examine this duality, starting from a two-dimensional N=(4,4) gauged linear sigma model describing Taub-NUT space. After dualizing the circle fiber, we find that the smeared NS5-brane target space metric receives corrections from multi-worldsheet instantons. These instantons are identified as Nielsen-Olesen vortices. We show that their effect is to break the isom…
In this paper we consider features of graviton scattering in Matrix theory compactified on a 2-torus. The features which interest us can only be determined by nonperturbative effects in the corresponding 2+1 dimensional super Yang Mills theory. We show that the superconformal symmetry of strongly coupled Super Yang Mills Theory in 2+1 dimensions almost determines low energy, large impact parameter ten dimensional graviton scattering at zero longitudinal momentum in the Matrix model of IIB string theory. We then show that amplitudes involving arbitrary transverse momentum transfer are governed …
In this paper we explicitly calculate the analogue of the 't Hooft SU(2) Yang--Mills instantons on Gibbons--Hawking multi-centered gravitational instantons which come in two parallel families: the multi-Eguchi--Hanson, or A_k ALE gravitational instantons and the multi-Taub--NUT, or A_k ALF gravitational instantons. We calculate their action and find the reducible ones. Following Kronheimer we also exploit the U(1) invariance of our solutions and study the corresponding explicit singular SU(2) magnetic monopole solutions on Euclidean three-space.
The construction of Atiyah, Drinfeld, Hitchin, and Manin [ADHM78] provided complete description of all instantons on Euclidean four-space. It was extended by Kronheimer and Nakajima to instantons on ALE spaces, resolutions of orbifolds $\mathbb{R}^4/Γ$ by a finite subgroup $Γ\subset SU(2).$ We consider a similar classification, in the holomorphic context, of instantons on some of the next spaces in the hierarchy, the ALF multi-Taub-NUT manifolds, showing how they tie in to the bow solutions to Nahm's equations [Che09] via the Nahm correspondence. Recently in [Nak18a] and [NT17], based on [Nak0…
This talk is based on the recent work in collaboration with M. Azreg-A\"ınou and G. Clément devoted to extremal instantons in the one-vector truncation of the Euclidean $\mathcal{N}=4,\, D=4$ theory. Extremal solutions satisfying the no-force condition can be associated with null geodesic curves in the homogeneous target space of the three-dimensional sigma model arising in toroidal reduction of the four-dimensional theory. Here we (preliminarily) discuss the case of two vector fields sufficient to find all relevant metrics in the full $\mathcal{N}=4,\, D=4$ theory. Classification of instanton…
ALE and Taub-NUT (or ALF) hyper-Kahler four-manifolds can be naturally constructed as hyper-Kahler quotients. In the ALE case, this construction has long been understood in terms of D-branes; here we give a D-brane derivation in the Taub-NUT case. Likewise, instantons on ALE spaces and on Taub-NUT spaces have ADHM-like constructions related to hyper-Kahler quotients. Here we refine the analysis in the Taub-NUT case by making use of a D-brane probe, and give an application to M-theory.
In this letter we demonstrate that the intersection form of the Hausel--Hunsicker--Mazzeo compactification of a four dimensional ALF gravitational instanton is definite and diagonalizable over the integers if one of the Kahler forms of the hyper-Kahler gravitational instanton metric is exact. This leads to the topological classification of these spaces. The proof exploits the relationship between L^2 cohomology and U(1) anti-instantons over gravitational instantons recognized by Hitchin. We then interprete these as reducible points in a singular SU(2) anti-instanton moduli space over the compa…
In this paper we propose ALF gravitational instantons of types A_k and D_k as models for charged particle systems. We calculate the charges of the two families. These are -(k +1) for A_k, which is proposed as a model for k+1 electrons, and 2-k for D_k, which is proposed as a model for either a particle of charge +2 and k electrons or a proton and k-1 electrons. Making use of preferred topological and metrical structures of the manifolds, namely metrically preferred representatives of middle dimension homology classes, we construct two different energy functionals which reproduce the Coulomb in…
We obtain $D_k$ ALF gravitational instantons by a gluing construction which captures, in a precise and explicit fashion, their interpretation as non-linear superpositions of the moduli space of centred $SU(2)$ monopoles, equipped with the Atiyah-Hitchin metric, and $k$ copies of the Taub-NUT manifold. The construction proceeds from a finite set of points in euclidean space, reflection symmetric about the origin, and depends on an adiabatic parameter which is incorporated into the geometry as a fifth dimension. Using a formulation in terms of hyperKähler triples on manifolds with boundaries, we…
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…
Motivated by potential applications to holography on space-times of positive curvature, and by the successful twistor description of scattering amplitudes, we propose a new dual matrix formulation of N = 4 gauge theory on S(4). The matrix model is defined by taking the low energy limit of a holomorphic Chern-Simons theory on CP(3|4), in the presence of a large instanton flux. The theory comes with a choice of S(4) radius L and a parameter N controlling the overall size of the matrices. The flat space variant of the 4D effective theory arises by taking the large N scaling limit of the matrix mo…
We show that five-dimensional anti de-Sitter space remains a solution to low-energy type IIB supergravity when the leading higher-derivative corrections to the classical supergravity (which are non-perturbative in the string coupling) are included. Furthermore, at this order in the low energy expansion of the IIB theory the graviton two-point and three-point functions in $AdS_5 \times S^5$ are shown not to be renormalized and a precise expression is obtained for the four-graviton and related S-matrix elements. By invoking Maldacena's conjectured connection between IIB superstring theory and su…
We propose a model of cosmological evolution of the early and late Universe which is consistent with observational data and naturally explains the origin of inflation and dark energy. We show that the de Sitter accelerated expansion of the FLRW space with no matter fields (hereinafter, empty space) is its natural state, and the model does not require either a scalar field or cosmological constant or any other hypotheses. This is due to the fact that the de Sitter state is an exact solution of the rigorous mathematically consistent equations of one-loop quantum gravity for the empty FLRW space …
We compute the spectrum of primordial gravitational wave perturbations in open de Sitter spacetime. The background spacetime is taken to be the continuation of an O(5) symmetric instanton saddle point of the Euclidean no boundary path integral. The two-point tensor fluctuations are computed directly from the Euclidean path integral. The Euclidean correlator is then analytically continued into the Lorentzian region where it describes the quantum mechanical vacuum fluctuations of the graviton field. Unlike the results of earlier work, the correlator is shown to be unique and well behaved in the …
According to the proposal of Hanany and Witten, Coulomb branches of N=4 SU(n) gauge theories in three dimensions are isometric to moduli spaces of BPS monopoles. We generalize this proposal to gauge theories with matter, which allows us to describe the metrics on their spaces of vacua by means of the hyperkähler quotient construction. To check the identification of moduli spaces a comparison is made with field theory predictions. For SU(2) theory with k fundamental hypermultiplets the Coulomb branch is expected to be the D_k ALF gravitational instanton, so our results lead to a construction of…
We consider the finite part of the leading local interactions in the low energy expansion of the four graviton amplitude from the ladder skeleton diagrams in maximal supergravity on T^2, at three and four loops. At three loops, we express the D^8 R^4 and D^{10} R^4 amplitudes as integrals over the moduli space of an underlying auxiliary geometry. These amplitudes are evaluated exactly for special values of the the moduli of the auxiliary geometry, where the integrand simplifies. We also perform a similar analysis for the D^8 R^4 amplitude at four loops that arise from the ladder skeleton diagr…
We study the four-point function of the lowest-lying half-BPS operators in the ${\cal N} =4$ $SU(N)$ super-Yang-Mills theory and its relation to the flat-space four-graviton amplitude in type IIB superstring theory. We work in a large-$N$ expansion in which the complexified Yang-Mills coupling $τ$ is fixed. In this expansion, non-perturbative instanton contributions are present, and the $SL(2, \mathbb{Z})$ duality invariance of correlation functions is manifest. Our results are based on a detailed analysis of the sphere partition function of the mass-deformed SYM theory, which was previously c…
