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We study the general requirement for supersymmetric AdS $$_6$$ solutions in type IIB supergravity. We employ the Killing spinor technique and study the differential and algebraic relations among various Killing spinor bilinears to find the canonical form of the solutions. Our result agrees precisely with the work of Apruzzi et al. (JHEP 1411:099, 2014), which used the pure spinor technique. Hoping to identify the geometry of the problem, we also computed four-dimensional theory through the dimensional reduction of type IIB supergravity on AdS $$_6$$ . This effective action is essentially a non-linear sigma model with five scalar fields parametrizing $$\text {SL}(3,\mathbb {R})/\text {SO}(2,1)$$ , modified by a scalar potential and coupled to Einstein gravity in Euclidean signature. We argue that the scalar potential can be explained by a subgroup CSO(1,1,1) $$\subset \text {SL}(3,\mathbb {R})$$ in a way analogous to gauged supergravity.

Using an off-shell Killing spinor analysis we perform a systematic investigation of the supersymmetric background and black hole solutions of the $$ \mathcal{N}=\left(1,\;1\right) $$ Cosmological New Massive Gravity model. The solutions with a null Killing vector are the same pp-wave solutions that one finds in the $$ \mathcal{N}=1 $$ model but we find new solutions with a time-like Killing vector that are absent in the $$ \mathcal{N}=1 $$ case. An example of such a solution is a Lifshitz spacetime. We also consider the supersymmetry properties of the so-called rotating hairy BTZ black holes and logarithmic black holes in an AdS 3 background. Furthermore, we show that under certain assumptions there is no supersymmetric Lifshitz black hole solution.

We investigate the second Dirac eigenvalue on Riemannian manifolds admitting a Killing spinor. In small dimensions the whole Dirac spectrum depends on special eigenvalues on functions and 1-forms. We compute and discuss the formulas in dimension $n = 7$., Comment: 10 pages

We initiate a systematic study of the solutions of three-dimensional matter-coupled half-maximal ( N = 8 ) supergravities which admit a Killing spinor. To this end we analyze in detail the invariant tensors built from spinor bilinears, a technique originally developed and applied in higher dimensions. This reveals an intriguing interplay with the scalar target space geometry SO ( 8 , n ) / ( SO ( 8 ) × SO ( n ) ) . Another interesting feature of the three-dimensional case is the implementation of the duality between vector and scalar fields in this framework. For the ungauged theory with timelike Killing vector, we explicitly determine the scalar current and show that its integrability relation reduces to a covariant holomorphicity equation, for which we present a number of explicit solutions. For the case of a null Killing vector, we give the most general solution which is of pp-wave type.

We give a spinorial characterization of isometrically immersed hypersurfaces into 4-dimensional space forms and product spaces $\M^3(\kappa)\times\R$, in terms of the existence of particular spinor fields, called generalized Killing spinors or equivalently solutions of a Dirac equation. This generalizes to higher dimensions several recent results for surfaces by T. Friedrich, B. Morel and the two authors. The main argument is the interpretation of the energy-momentum tensor of a generalized Killing spinor as the second fundamental form, possibly up to a tensor depending on the ambient space. As an application, we deduce a non-existence result for hypersurfaces in the 4-dimensional Euclidean space., Comment: 21 pages, application added

We show that the heterotic supersymmetry (Killing spinor equations) and the anomaly cancellation imply the heterotic equations of motion in dimensions five, six, seven, eight if and only if the connection on the tangent bundle is an instanton. For heterotic compactifications in dimension six this reduces the choice of that connection to the unique SU(3) instanton on a manifold with stable tangent bundle of degree zero., LaTeX 2e, 11 pages, exposition clarified, reference added, final version to appear in Phys. Lett. B

Killing tensors have been of interest historically primarily for generating first integrals for the geodesic equation and for their use in finding separable coordinate systems. A related notion, that of Killing spinors, has recently been shown to be important in the study of generalized symmetries of Maxwell's equations. In a given spacetime, the generalized symmetries depend on the existence of Killing spinors of the spacetime of certain valences. The existence of Killing spinors for the curved metric of Gödel's Universe is investigated. There are five (1,1) Killing spinors, 14 (2,2) and five (1,5) Killing spinors of the spacetime, in addition to the unique (0,2) and (0,4) Killing spinors which are exhibited here as well.

We analyze the AdS_3 x M_7 type supersymmetric solutions, including non-trivial fluxes, of the Killing spinor equations in the heterotic supergravity. We classify these solutions by their G-structures and intrinsic torsions, for the cases that the number of seven-dimensional Killing spinors N are equal to 1,2,3 and 4. We find that the solutions cannot have non-trivial warp factor and the seven dimensional manifold M_7 is charactrized by G_2(SU(3))-structures for N=1 (2) case and SU(2)-structure for N=3 and 4 cases. They are further classified by their non-trivial intrinsic torsions. It is shown, including the leading order \alpha'-corrections, that if we impose the Bianchi identities, the integrability conditions of the Killing spinor equations imply all the field equations., Comment: 32 pages

In this paper we prove that a (2, n − 2)-manifold admitting an imaginary Killing spinor with nontrivial Dirac current, is at least locally a codimension one warped product with a special warping function. Hence, a (2, 2)-manifold admitting such an imaginary Killing spinor is Einstein iff it is conformally flat.

Let (M^n,g) be a Riemannian spin manifold. The basic equations in supergravity models of type IIa string theory with 4-form flux involve a 3-form T, a 4-form F, a spinorial covariant derivative \nabla depending on \nabla^g, T, F, and a \nabla-parallel spinor field \Psi. We classify and construct many explicit families of solutions to this system of spinorial field equations by means of non-integrable special geometries. The latter include \alpha-Sasakian structures in dimensions 5 and 7, almost Hermitian structures in dimension 6 and cocalibrated G_2-structures in dimension 7. We show that there are several examples also satisfying an additional constraint for the energy-momentum tensor., Comment: 28 pages; revised, more concise

We investigate all N=2 supersymmetric IIB supergravity backgrounds with non-vanishing five-form flux. The Killing spinors have stability subgroups $Spin(7)\ltimes\bR^8$, $SU(4)\ltimes\bR^8$ and $G_2$. In the $SU(4)\ltimes\bR^8$ case, two different types of geometry arise depending on whether the Killing spinors are generic or pure. In both cases, the backgrounds admit a null Killing vector field which leaves invariant the $SU(4)\ltimes \bR^8$ structure, and an almost complex structure in the directions transverse to the lightcone. In the generic case, the twist of the vector field is trivial but the almost complex structure is non-integrable, while in the pure case the twist is non-trivial but the almost complex structure is integrable and associated with a relatively balanced Hermitian structure. The $G_2$ backgrounds admit a time-like Killing vector field and two spacelike closed one-forms, and the seven directions transverse to these admit a co-symplectic $G_2$ structure. The $Spin(7)\ltimes\bR^8$ backgrounds are pp-waves propagating in an eight-dimensional manifold with holonomy $Spin(7)$. In addition we show that all the supersymmetric solutions of simple five-dimensional supergravity with a time-like Killing vector field, which include the $AdS_5$ black holes, lift to $SU(4)\ltimes\bR^8$ pure Killing spinor IIB backgrounds. We also show that the LLM solution is associated with a co-symplectic co-homogeneity one $G_2$ manifold which has principal orbit $S^3\times S^3$., 39 pages, typos corrected and references amended

We prove the existence of 3/4-BPS preons in four- and five-dimensional gauged supergravities by explicitly constructing them as smooth quotients of the AdS_4 and AdS_5 maximally supersymmetric backgrounds, respectively. This result illustrates how the spacetime topology resurrects a fraction of supersymmetry previously ruled out by the local analysis of the Killing spinor equations., Comment: 10 pages (a minor imprecision has been corrected)

We classify the simply-connected Spin q manifolds admitting either a non-zero parallel or a non-zero real Killing spinor.

The K = 0 Friedmann–Robertson–Walker metric, which satisfies the Killing spinor equation and approximates a solution to the field equations of the quadratic gravity theory derived from the heterotic string effective action, generates a discontinuity in the derivative of the scale factor. The resulting change in the Hamiltonian is conjectured to be the origin of energy in the very early Universe.

We study the intrinsic geometry of hypersurfaces in Calabi-Yau manifolds of real dimension 6 and, more generally, SU(2)-structures on 5-manifolds defined by a generalized Killing spinor. We prove that in the real analytic case, such a 5-manifold can be isometrically embedded as a hypersurface in a Calabi-Yau manifold in a natural way. We classify nilmanifolds carrying invariant structures of this type, and present examples of the associated metrics with holonomy SU(3)., 30 pages. v2: corrected the statement and proof of Theorem 14; added a comment on the embedding property in the non-real-analytic case

We classify the supersymmetric vacua of N=4, d=5 supergravity in terms of G-structures. We identify three classes of solutions: with R^3, SU(2) and generic SO(4) structure. Using the Killing spinor equations, we fully characterize the first two classes and partially solve the latter. With the N=4 graviton multiplet decomposed in terms of N=2 multiplets: the graviton, vector and gravitino multiplets, we obtain new supersymmetric solutions corresponding to turning on fields in the gravitino multiplet. These vacua are described in terms of an SO(5) vector sigma-model coupled with gravity, in three or four dimensions. A new feature of these N=4 vacua, which is not seen from an N=2 point of view, is the possibility for preserving more exotic fractions of supersymmetry. We give a few concrete examples of these new supersymmetric (albeit singular) solutions. Additionally, we show how by truncating the N=4, d=5 set of fields to minimal supergravity coupled with one vector multiplet we recover the known two-charge solutions.

We review some aspects of the spinorial geometry approach to the classification of supersymmetric solutions of supergravity theories. In particular, we explain how spinorial geometry can be used to express the Killing spinor equations in terms of a linear system for the fluxes and the geometry of spacetime. The solutions of this linear system express some of the fluxes in terms of the spacetime geometry and determine the conditions on the spacetime geometry imposed by supersymmetry. We also present some of the recent applications like the classification of maximally supersymmetric G-backgrounds in IIB, this includes the most general pp-wave solution preserving 1/2 supersymmetry, and the classification of N=31 backgrounds in ten and eleven dimensions., 17 pages. Invited review for Modern Physics Letters A

We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact K\"ahler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of K\"ahlerian Killing ${\rm spin}^c$ spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a K\"ahlerian Killing ${\rm spin}^c$ spinor field vanishes. This extends to the ${\rm spin}^c$ case the result of A. Moroianu stating that, on a compact K\"ahler-Einstein manifold of complex dimension $4\ell+3$ carrying a complex contact structure, the Clifford multiplication between an effective harmonic form and a K\"ahlerian Killing spinor is zero.

We consider $$ \mathcal{N}=2 $$ supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a U(1) isometry. This is used to explicitly compute the supersymmetric path integral on S2 × S2 via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity.

We first classify all supersymmetric solutions of the 3-dimensional half-maximal ungauged supergravity that possess a timelike Killing vector coming from the Killing spinor bilinear by considering their identification under the complexification of the local symmetry of the theory. It is found that only solutions that preserve $16/2^n, 1 \leq n \leq 3$ real supersymmetries are allowed. We then classify supersymmetric solutions under the real local symmetry of the theory and we are able to solve the equations of motion for all of them. It is shown that all such solutions can be expressed as a direct sum of solutions of the integrable Liouville and SU(3) Toda systems. This completes the construction of all supersymmetric solutions of the model since the null case has already been solved., Comment: 38 pages + appendices, v2:published version, some clarifications

We classify the supersymmetric solutions of minimal $N=2$ gauged supergravity in four dimensions with neutral signature. They are distinguished according to the sign of the cosmological constant and whether the vector field constructed as a bilinear of the Killing spinor is null or non-null. In neutral signature the bilinear vector field can be spacelike, which is a new feature not arising in Lorentzian signature. In the $\Lambda0$ non-null case, the manifold is a fibration over a Lorentzian Gauduchon-Tod base space. Finally, in the $\Lambda>0$ null class, the metric is contained in the Kundt family, and it turns out that the holonomy is reduced to ${\rm Sim}(1)\times{\rm Sim}(1)$. There appear no self-dual solutions in the null class for either sign of the cosmological constant., Comment: 40 pages, uses JHEP3.cls. v2: Appendix and ref. added. v3: Published version

We study ten-dimensional supersymmetric vacua with NSNS non-geometric fluxes, in the framework of $\beta$-supergravity. We first provide expressions for the fermionic supersymmetry variations. Specifying a compactification ansatz to four dimensions, we deduce internal Killing spinor equations. These supersymmetry conditions are then reformulated in terms of pure spinors, similarly to standard supergravity vacua admitting an SU(3)xSU(3) structure in Generalized Complex Geometry. The standard d-H acting on the pure spinors is traded for a generalized Dirac operator D, depending here on the non-geometric fluxes. Rewriting it with an exponential of the bivector $\beta$ leads us to discuss the geometrical characterisation of the vacua in terms of a $\beta$-twist, in analogy to the standard twist by the b-field. Thanks to D, we also propose a general expression for the superpotential to be obtained from standard supergravities or $\beta$-supergravity, and verify its agreement with formulas of the literature. We finally comment on the Ramond-Ramond sector, and discuss a possible relation to intermediate or dynamical SU(2) structure solutions., Comment: 25 pages + appendices and references; v2: minor changes, published version

We show how the theory of invariant principal bundle connections for reductive homogeneous spaces can be applied to determine the holonomy of generalised Killing spinor covariant derivatives of the form $D= \nabla + \Omega$ in a purely algebraic and algorithmic way, where $\Omega : TM \rightarrow \Lambda^*(TM)$ is a left-invariant homomorphism. Specialising this to the case of symmetric M-theory backgrounds (i.e. $(M,g,F)$ with $(M,g)$ a symmetric space and $F$ an invariant closed 4-form), we derive several criteria for such a background to preserve some supersymmetry and consequently find all supersymmetric symmetric M-theory backgrounds., Comment: Updated abstract for clarity. Added missing geometries to section 6. Main result stands

We show that higher degree Dirac currents of twistor and Killing spinors correspond to the hidden symmetries of the background spacetime which are generalizations of conformal Killing and Killing vector fields respectively. They are the generalizations of 1-form Dirac currents to higher degrees which are used in constructing the bosonic supercharges in supergravity theories. In the case of Killing spinors, we find that the equations satisfied by the higher degree Dirac currents are related to Maxwell-like and Duffin-Kemmer-Petiau equations. Correspondence between the Dirac currents and harmonic forms for parallel and pure spinor cases is determined. We also analyze the supergravity twistor and Killing spinor cases in 10 and 11-dimensional supergravity theories and find that although different inner product classes induce different involutions on spinors, the higher degree Dirac currents still correspond to the hidden symmetries of the spacetime., Comment: 18 pages, published version

Suppose that $\Sigma=\partial\Omega$ is the $n$-dimensional boundary, with positive (inward) mean curvature $H$, of a connected compact $(n+1)$-dimensional Riemannian spin manifold $(\Omega^{n+1},g)$ whose scalar curvature $R\ge -n(n+1)k^2$, for some $k\textgreater{}0$. If $\Sigma$ admits an isometric and isospin immersion $F$ into the hyperbolic space ${\mathbb{H}^{n+1}\_{-k^2}}$, we define a quasi-local mass and prove its positivity as well as the associated rigidity statement. The proof is based on a holographic principle for the existence of an imaginary Killing spinor. For $n=2$, we also show that its limit, for coordinate spheres in an Asymptotically Hyperbolic (AH) manifold, is the mass of the (AH) manifold., Comment: in Journal of Geometry and Physics, Elsevier, 2015

We classify all the supersymmetric configurations of ungauged N = 2, d = 4 supergravity coupled to n vector multiplets and determine under which conditions they are also classical solutions of the equations of motion. The supersymmetric configurations fall into two classes, depending on the timelike or null nature of the Killing vector constructed from Killing spinor bilinears. The timelike class configurations are ∗ ]

We summarize our recent work on supergravity backgrounds dual to part of the Coulomb branch of N=1 theories constructed as marginal deformations of N=4 Yang-Mills. In particular, we present a summary of the behaviour of the heavy quark-antiquark potential which shows confining behaviour in the IR as well as of the spectrum of the wave equation. The reduced supersymmetry is due to the implementation of T-duality in the construction of the deformed supergravity solutions. As a new result we analyze and explicitly solve the Killing spinor equations of the N=1 background in the superconformal limit., 12 pages, 2 figures, Proceedings of the RTN Corfu workshop, September 2005

We review how the classification of all supersymmetric backgrounds of IIB supergravity can be reduced to the evaluation of the Killing spinor equations and their integrability conditions, which contain the field equations, on five types of spinors. This is an extension of the work [hep-th/0503046] to IIB supergravity. By using the explicit expressions for the Killing spinor equations evaluated on the five types of spinors the Killing spinor equations become a linear system in terms of the fluxes, the geometry and the spacetime derivatives of the functions that determine the Killing spinors. This system can be solved to express the fluxes in terms of the geometry and to determine the conditions on the geometry of any supersymmetric background. Similarly, the integrability conditions of the Killing spinor equations are turned into a linear system. This can be used to determine the field equations that are implied by the Killing spinor equations for any supersymmetric background. These linear systems simplify for generic backgrounds with maximal and half-maximal number of H-invariant Killing spinors, H ⊂ Spin(9, 1). In the maximal case, the Killing spinor equations factorise, whereas in the half-maximal case they do not.

From the existence of parallel spinor fields on Calabi-Yau, hyper-Kahler or complex flat manifolds, we deduce the existence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the submanifolds are compact, we obtain sharp estimates on their Betti numbers which generalize those obtained by Smoczyk in [49]. When the ambient manifold is Kahler-Einstein with positive scalar curvature, and especially if it is a complex contact manifold or the complex projective space, we prove the existence of Kahlerian Killing spinor fields for some particular spin c structures. Using these fields, we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangian submanifolds and give some estimates for their spectra. These results also generalize some theorems by Smoczyk in [50]. Finally, applications on the Morse index of minimal Lagrangian submanifolds are obtained.

The quartic-curvature corrections derived from string theory have a specific impact on the geometry of target-space manifolds of special holonomy. In the cases of Calabi-Yau manifolds, D = 7 manifolds of G2 holonomy and D = 8 manifolds of Spin7 holonomy, string theory α' corrections conspire to preserve the unbroken supersymmetry of these backgrounds despite the fact that the α' corrections cause the Riemannian holonomy to lose its special character. We show how this supersymmetry preservation is expressed in the language of generalized holonomy for the Killing spinor operator.

This is a review article of eleven dimensional supergravity in which we present all necessary calculations, namely the Noether procedure, the equations of motion (without neglecting the fermions), the Killing spinor equation, as well as some simple and less simple supersymmetric solutions to this theory. All calculations are printed in much detail and with explicit comments as to how they were done. Also contained is a simple approach to Clifford algebras to prepare the grounds for the harder calculations in spin space and Fierz identities., 96 pages, corrected and extended version

We study the properties of the transversal Killing and twistor spinors for a Riemannian foliation with a transverse spin structure. And we investigate the relations between them. As an application, we give a new lower bound for the eigenvalues of the basic Dirac operator by using the transversal twistor operator.

We give a lower bound for the eigenvalues of the Dirac operator on a compact domain of a Riemannian spin manifold under the $\MIT$ bag boundary condition. The limiting case is characterized by the existence of an imaginary Killing spinor., 10 pages

We investigate the Killing spinor equations of IIB supergravity for one Killing spinor. We show that there are three types of orbits of Spin(9,1) in the space of Weyl spinors which give rise to Killing spinors with stability subgroups $Spin(7)\ltimes \bR^8$, $SU(4)\ltimes \bR^8$ and $G_2$. We solve the Killing spinor equations for the $Spin(7)\ltimes \bR^8$ and $SU(4)\ltimes \bR^8$ invariant spinors, give the fluxes in terms of the geometry and determine the conditions on the spacetime geometry imposed by supersymmetry. In both cases, the spacetime admits a null, self-parallel, Killing vector field. We also apply our formalism to examine a class of $SU(4)\ltimes \bR^8$ backgrounds which admit one and two pure spinors as Killing spinors and investigate the geometry of the spacetimes., 48 pages. v2: formulae simplified and reference added

We show that on every Spin(7)-manifoldthere always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor andthe Spin(7) structure. We express its torsion andthe Riemannian scalar cur- vature in terms of the fundamental 4-form. We present an explicit formula for the Riemannian covariant derivative of the fundamental 4-form in terms of its exterior differential. We show the vanishing of the ˆ A-genus andobtain a linear relation between Betti numbers of a compact Spin(7) manifoldwhich is locally but not globally conformally equivalent to a space with closedfund amental 4-form. A general solution to the Killing spinor equations is presented.

The recently found non-BPS multi-wall configurations in the N=1 supergravity in four dimensions is shown to have no tachyonic scalar fluctuations without additional stabilization mechanisms. Mass of radion (lightest massive fluctuation) is found to be proportional to $Lambda {\rm e}^{-\pi\Lambda R/2}$, where $\Lambda $ is the inverse width of the wall and $ R$ is the radius of compactified dimension. We obtain localized massless graviton and gravitino forming a supermultiplet with respect to the Killing spinor. The relation between the bulk energy density and the boundary energy density (cosmological constants) is an automatic consequence of the field equation and Einstein equation. In the limit of vanishing gravitational coupling, the Nambu-Goldstone modes are reproduced., Comment: 30 pages, 7 figures

We present classical solutions of Dp-branes $(p \ge 5)$ in plane wave spacetime with nonconstant R-R 3-form flux. We also show the existence of a system of D3-branes in this background. We further analyze the supersymmetric properties of these branes by solving type II Killing spinor equations explicitly., Comment: 13 pages, references added, to appear in Phys. Letts. B

We briefly review the hierarchy for the hyper-K\"ahler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-K\"ahler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyper-K\"ahler metric admits a hidden symmetry if it admits a certain Killing spinor. We show that if the hidden symmetry is tri-holomorphic, then this is equivalent to requiring symmetry along a higher time and the hidden symmetry determines a `twistor group' action as introduced by Bielawski \cite{B00}. This leads to a construction for the solution to the hierarchy in terms of linear equations and variants of the generalised Legendre transform for the hyper-K\"ahler metric itself given by Ivanov & Rocek \cite{IR96}. We show that the ALE spaces are examples of hyper-K\"ahler metrics admitting three tri-holomorphic Killing spinors. These metrics are in this sense analogous to the 'finite gap' solutions in soliton theory. Finally we extend the concept of a hierarchy from that of \cite{DM00} for the four-dimensional hyper-K\"ahler equations to a generalisation of the conformal anti-self-duality equations and briefly discuss hidden symmetries for these equations., Comment: Final version. To appear in the August 2003 special issue of JMP on `Integrability, Topological Solitons, and Beyond'

A complete and explicit classification of all locally constructed conserved currents and underlying conserved tensors is obtained for massless linear symmetric spinor fields of any spin s 1/2 in fo...

In heterotic string theory compactified to four dimensions with N=2 supersymmetry, string-loop corrections to the universal sector of the low-energy effective action are studied. Within the framework of N=2 supersymmetric formulation of the theory, in the first order in string coupling constant, we solve the system of the loop-corrected Maxwell and Killing spinor equations. Taking as the in-put the tree-level dyonic black hole solution, we calculate string-loop corrections to the string tree-level metric and moduli of dyonic black hole., Comment: Improved version. Discussion of the ambiguity of the prepotential added. Mixing of the dilaton with other moduli properly taken into account. Explanatory notes added

The aim of this paper is to describe some results concerning the geometry of Lorentzian manifolds admitting Killing spinors. We prove that there are imaginary Killing spinors on simply connected Lorentzian Einstein–Sasaki manifolds. In the Riemannian case, an odd-dimensional complete simply connected manifold (of dimension n ≠7) is Einstein–Sasaki if and only if it admits a non-trivial Killing spinor to λ=± 1 2 . The analogous result does not hold in the Lorentzian case. We give an example of a non-Einstein Lorentzian manifold admitting an imaginary Killing spinor. A Lorentzian manifold admitting a real Killing spinor is at least locally a codimension one warped product with a special warping function. The fiber of the warped product is either a Riemannian manifold with a real or imaginary Killing spinor or with a parallel spinor, or it again is a Lorentzian manifold with a real Killing spinor. Conversely, all warped products of that form admit real Killing spinors.

We give a sharp extrinsic lower bound for the first eigenvaluesof the intrinsic Dirac operator of certain hypersurfaces boundinga compact domain in a spin manifold of negative scalar curvature.Limiting-cases are characterized by the existence, on the domain,of imaginary Killing spinors. Some geometrical applications, as anAlexandrov type theorem, are given.

We discuss the Penrose limit of the Chamseddine-Volkov BPS selfgravitating monopole in four dimensional N=4 supergravity theory with non-abelian gauge multiplets. We analyze the properties of the resulting supersymmetric pp-wave solutions when various Penrose limits are considered. Apart from the usual rescaling of coordinates and fields we find that a rescaling of the gauge coupling constant to zero is required, rendering the theory abelian. We also study the Killing spinor equations showing an enhancement of the supersymmetries preserved by the solutions and discuss the embedding of the pp-wave solution in $d=10$ dimensions., 14 pages, no figures. Minor changes, to appear in Phys. Lett. B

On a compact Riemannian spin manifold with mean-convex boundary, we analyse the ellipticity and the symmetry of four boundary conditions for the fundamental Dirac operator including the (global) APS condition and a Riemannian version of the (local) MIT bag condition. We show that Friedrich's inequality for the eigenvalues of the Dirac operator on closed spin manifolds holds for the corresponding four eigenvalue boundary problems. More precisely, we prove that, for both the APS and the MIT conditions, the equality cannot be achieved, and for the other two conditions, the equality characterizes respectively half-spheres and domains bounded by minimal hypersurfaces in manifolds carrying non-trivial real Killing spinors.

We study string-loop corrections to magnetic black hole. Four-dimensional theory is obtained by compactification of the heterotic string theory on the manifold $K3\times T^2$ or on a suitable orbifold yielding N=1 supersymmetry in 6D. The resulting 4D theory has N=2 local supersymmetry. Prepotential of this theory receives only one-string-loop correction. The tree-level gauge couplings are proportional to the inverse effective string coupling and decrease at small distances from the center of magnetic black hole, so that loop corrections to the gauge couplings are important in this region. We solve the system of spinor Killing equations (conditions for the supersymmetry variations of the fermions to vanish) and Maxwell equations. At the string-tree level, we reproduce the magnetic black hole solution which can be also obtained by solving the system of the Einstein-Maxwell equations and the equations of motion for the moduli. String-loop corrections to the tree-level solution are calculated in the first order in string coupling. The resulting corrections to the metric and dilaton are large at small distances from the center of the black hole. Possible smearing of the singularity at the origin by quantum corrections is discussed., Reference added; minor modifications

In this paper we define the analogue of Calabi–Yau geometry for generic D=4, N=2 flux backgrounds in type II supergravity and M-theory. We show that solutions of the Killing spinor equations are in one-to-one correspondence with integrable, globally defined structures in E7(7)×R+ generalised geometry. Such “exceptional Calabi–Yau” geometries are determined by two generalised objects that parametrise hyper- and vector-multiplet degrees of freedom and generalise conventional complex, symplectic and hyper-Kahler geometries. The integrability conditions for both hyper- and vector-multiplet structures are given by the vanishing of moment maps for the “generalised diffeomorphism group” of diffeomorphisms combined with gauge transformations. We give a number of explicit examples and discuss the structure of the moduli spaces of solutions. We then extend our construction to D=5 and D=6 flux backgrounds preserving eight supercharges, where similar structures appear, and finally discuss the analogous structures in O(d,d)×R+ generalised geometry.

Riemannian manifolds with non-zero Killing spinors are Einstein manifolds. Klaus Kr\"{o}ncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in \cite{Kro15}. In this paper, we obtain a new proof for this stability result by using a Bochner type formula in \cite{DWW05} and \cite{Wan91}. Moreover, existence of real Killing spinors is closely related to the Sasaki-Einstein structure. A regular Sasaki-Einstein manifold is essentially the total space of a certain principal $S^{1}$-bundle over a K\"{a}hler-Einstein manifold. We prove that if the base space is a product of two K\"{a}hler-Einstein manifolds then the regular Sasaki-Einstein manifold is unstable. This provides us many new examples of unstable manifolds with real Killing spinors., Comment: Final version, introduction revised, references added