We compute the scalar curvature of 7-dimensional ${G}_2$-manifolds admitting a connection with totally skew-symmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3-form field. In dimension n=7 the dilation function involved in the second fermionic string equation has an interpretation as a conformal change of the underlying integrable ${G}_2$-structure into a cocalibrated one of pure type $W_3$.
We outline the solution of the Killing spinor equations of the heterotic supergravity. In addition, we describe the classification of all half supersymmetric solutions.
This paper is a survey on special geometric structures that admit conformal Killing spinors based on lectures, given at the ``Workshop on Special Geometric Structures in String Theory'', Bonn, September 2001 and at ESI, Wien, November 2001. We discuss the case of Lorentzian signature and explain which geometries occur up to dimension 6.
We study the spinorial Killing equation of supergravity involving a torsion 3-form $\T$ as well as a flux 4-form $\F$. In dimension seven, we construct explicit families of compact solutions out of 3-Sasakian geometries, nearly parallel $\G_2$-geometries and on the homogeneous Aloff-Wallach space. The constraint $\F \cdot Ψ= 0$ defines a non empty subfamily of solutions. We investigate the constraint $\T \cdot Ψ= 0$, too, and show that it singles out a very special choice of numerical parameters in the Killing equation, which can also be justified geometrically.
We propose a way to classify all supersymmetric configurations of D=11 supergravity using the G-structures defined by the Killing spinors. We show that the most general bosonic geometries admitting a Killing spinor have at least a local SU(5) or an (Spin(7)\ltimes R^8)x R structure, depending on whether the Killing vector constructed from the Killing spinor is timelike or null, respectively. In the former case we determine what kind of local SU(5) structure is present and show that almost all of the form of the geometry is determined by the structure. We also deduce what further conditions mus…
In this paper, we extend the study of generalized Killing spinors on Riemannian Spin$^c$ manifolds started by Moroianu and Herzlich to complex Killing functions. We prove that such spinor fields are always real Spin$^c$ Killing spinors or imaginary generalized Spin$^c$ Killing spinors, providing that the dimension of the manifold is greater or equal to 4. Moreover, we classify Riemannian Spin$^c$ manifolds carrying imaginary and imaginary generalized Killing spinors.
We study generalized Killing spinors on round spheres $\mathbb{S}^n$. We show that on the standard sphere $\mathbb{S}^8$ any generalized Killing spinor has to be an ordinary Killing spinor. Moreover we classify generalized Killing spinors on $\mathbb{S}^n$ whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola--Friedrich's canonical spinor on 3-Sasakian manifolds of dimension 7. Finally we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors.
We introduce a notion of Ricci flow in generalized geometry, extending a previous definition by Gualtieri on exact Courant algebroids. Special stationary points of the flow are given by solutions to first-order differential equations, the Killing spinor equations, which encompass special holonomy metrics with solutions of the Hull-Strominger system. Our main result investigates a method to produce new solutions of the Ricci flow and the Killing spinor equations. For this, we consider T-duality between possibly topologically distinct torus bundles endowed with Courant structures, and demonstrat…
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 $D= \nabla + Ω$ in a purely algebraic and algorithmic way, where $Ω: TM \rightarrow Λ^*(TM)$ 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 s…
The basic first-order differential operators of spin geometry that are Dirac operator and twistor operator are considered. Special types of spinors defined from these operators such as twistor spinors and Killing spinors are discussed. Symmetry operators of massless and massive Dirac equations are introduced and relevant symmetry operators of twistor spinors and Killing spinors are constructed from Killing-Yano (KY) and conformal Killing-Yano (CKY) forms in constant curvature and Einstein manifolds. The squaring map of spinors gives KY and CKY forms for Killing and twistor spinors respectively…
We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations preserving 2, 1, or none of the 3 independent Killing spinors. Toric 3-Sasakian manifolds provide non-trivial examples with integrable deformation preserving precisely 2 Killing spinors. Thus the dimension of Killing spinors is not preserved under Einstein deformations but is only …
Manifolds admitting Killing spinors are Einstein manifolds. Thus, a deformation of a Killing spinor entails a deformation of Einstein metrics. In this paper, we study infinitesimal deformations of Killing spinors on nearly parallel $\mathrm{G}_2$-manifolds. Since there is a one-to-one correspondence between nearly parallel $\mathrm{G}_2$-structures and Killing spinors on 7-dimensional spin manifolds, our results imply that infinitesimal deformations of nearly parallel $\mathrm{G}_2$-structures are examined in terms of Killing spinors. Applying the same technique, we identify that the space of …
This text is dedicated to the real Killing equation on 3-dimensional Weyl manifolds. Any manifold admitting a real Killing spinor of weight 0 satisfies the conditions of a Gauduchon-Tod geometry. Conversely, any simply connected Gauduchon-Tod geometry has a 2-dimensional space of solutions of the real Killing equation on the spinor bundle of weight 0.
In this paper, we study the existence of a skew Killing spinor (see the definition below) on 2 and 3-dimensional Riemannian spin manifolds. We establish the integrability conditions and prove that these spinor fields correspond to twistor spinors in the two dimensional case while, up to a conformal change of the metric, they correspond to parallel spinors in the three dimensional case.
Exact solutions to an Einstein-Maxwell(E-D-M) system with an electric current are simple models of global cosmic magnetic phenomena in the universe. In this paper, we consider an E-D-M system with two chiral spinors $ ψ^\pm$ coupled with a gauge field $ A$ and an electromagnetic field $F=dA$. We construct a family of exact solutions to the E-D-M system on four-dimensional static Sasakian spacetimes. The electromagnetic field in the solutions is a contact magnetic field, and the chiral spinors are induced from Sasakian quasi-Killing spinors. We find closed universe models whose weak energy cond…
We study a Killing spinor type equation on spin Riemannian flows. We prove integrability conditions and partially classify those Riemannian flows $M$ carrying non-trivial solutions to that equation in case $M$ is a local Riemannian product, a Sasakian manifold or 3-dimensional.
This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor $ψ$ is a spinor that satisfies the equation $\nabla$X$ψ$ = AX $\times$ $ψ$ with a skew-symmetric endomorphism A. We consider the degenerate case, where the rank of A is at most two everywhere and the non-degenerate case, where the rank of A is four everywhere. We prove that in the degenerate case the manifold is locally isometric to the Riemannian product R x N with N having a skew Killing spinor and we explain under which conditions on the spinor the spec…
Semi-Riemannian $\text{spin}^c$ manifolds carrying generalized Killing spinors and the classification of Riemannian $\text{spin}^c$ manifolds admitting a type I imaginary generalized Killing spinor
We develop a new framework for the study of generalized Killing spinors, where generalized Killing spinor equations, possibly with constraints, can be formulated equivalently as systems of partial differential equations for a polyform satisfying algebraic relations in the Kähler-Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module $Σ$ of real type as a real algebraic variety in the Kähler-Atiyah…
The complex projective space $\mathbb C P^2$ of complex dimension $2$ has a Spin$^c$ structure carrying Kählerian Killing spinors. The restriction of one of these Kählerian Killing spinors to a surface $M^2$ characterizes the isometric immersion of $M^2$ into $\mathbb C P^2$ if the immersion is either Lagrangian or complex.
A supermanifold M is canonically associated to any pseudo Riemannian spin manifold (M_0,g_0). Extending the metric g_0 to a field g of bilinear forms g(p) on T_p M, p\in M_0, the pseudo Riemannian supergeometry of (M,g) is formulated as G-structure on M, where G is a supergroup with even part G_0\cong Spin(k,l); (k,l) the signature of (M_0,g_0). Killing vector fields on (M,g) are, by definition, infinitesimal automorphisms of this G-structure. For every spinor field s there exists a corresponding odd vector field X_s on M. Our main result is that X_s is a Killing vector field on (M,g) if and o…
We study generalized Killing spinors on compact Einstein manifolds with positive scalar curvature. This problem is related to the existence compact Einstein hypersurfaces in manifolds with parallel spinors, or equivalently, in Riemannian products of flat spaces, Calabi-Yau, hyperkaehler, G_2 and Spin(7) manifolds.
Let $M$ be a pseudo-Riemannian spin manifold of dimension $n$ and signature $s$ and denote by $N$ the rank of the real spinor bundle. We prove that $M$ is locally homogeneous if it admits more than ${3/4}N$ independent Killing spinors with the same Killing number, unless $n\equiv 1 \pmod 4$ and $s\equiv 3 \pmod 4$. We also prove that $M$ is locally homogeneous if it admits $k_+$ independent Killing spinors with Killing number $λ$ and $k_-$ independent Killing spinors with Killing number $-λ$ such that $k_++k_->{3/2}N$, unless $n\equiv s\equiv 3\pmod 4$. Similarly, a pseudo-Riemannian manifold …
Riemannian manifolds with non-zero Killing spinors are Einstein manifolds. Klaus Kröncke 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 Kähler-Einstein manifold. We prove that if the ba…
We study the geometric structure of Lorentzian spin manifolds, which admit imaginary Killing spinors. The discussion is based on the cone construction and a normal form classification of skew-adjoint operators in signature $(2,n-2)$. Derived geometries include Brinkmann spaces, Lorentzian Einstein-Sasaki spaces and certain warped product structures. Exceptional cases with decomposable holonomy of the cone are possible.
We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing-Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing-Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of conformal Killing vectors and twistor spinors and play important roles in supersymmetric field theories in conformal backgrounds are extended to more general superalgebras by using the graded Lie algebra structure of conformal Killing-Yano forms and the symmetry operators of twis…
In this review, basic definitions of spin geometry are given and some of its applications to supersymmetry, supergravity and condensed matter physics are summarized. Clifford algebras and spinors are defined and the first-order differential operators on spinors which lead to the definitions of twistor and Killing spinors are discussed. Holonomy classification for manifolds admitting parallel and Killing spinors are given. Killing-Yano and conformal Killing-Yano forms resulting from the spinor bilinears of Killing and twistor spinors are introduced and the symmetry operators of special spinor e…
We investigate instantons on manifolds with Killing spinors and their cones. Examples of manifolds with Killing spinors include nearly Kaehler 6-manifolds, nearly parallel G_2-manifolds in dimension 7, Sasaki-Einstein manifolds, and 3-Sasakian manifolds. We construct a connection on the tangent bundle over these manifolds which solves the instanton equation, and also show that the instanton equation implies the Yang-Mills equation, despite the presence of torsion. We then construct instantons on the cones over these manifolds, and lift them to solutions of heterotic supergravity. Amongst our s…
We generalize the symmetry superalgebras of isometries and geometric Killing spinors on a manifold to include all the hidden symmetries of the manifold generated by Killing spinors in all dimensions. We show that bilinears of geometric Killing spinors produce special Killing-Yano and special conformal Killing-Yano forms. After defining the Lie algebra structure of hidden symmetries generated by Killing spinors, we construct the symmetry operators as the generalizations of the Lie derivative on spinor fields. All these constructions together constitute the structure of generalized symmetry supe…
We generalize the well-known lower estimates for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold proved by Th. Friedrich (1980) and O. Hijazi (1986, 1992). The special solutions of the Einstein-Dirac equation constructed recently by Friedrich/Kim are examples for the limiting case of these inequalities. The discussion of the limiting case of these estimates yields two new field equations generalizing the Killing equation as well as the weak Killing equation for spinor fields. Finally, we discuss the 2- and 3-dimensional case in more detail.
We study invariant systems of PDEs defining Killing vector-valued forms, and then we specialize to Killing spinor-valued forms. We give a detailed treatment of their prolongation and integrability conditions by relating the point-wise values of solutions to the curvature of the underlying manifold. As an example, we completely solve the equations on model spaces of constant curvature producing brand new solutions which do not come from the tensor product of Killing spinors and Killing-Yano forms.
We study generalized Killing spinors on the standard sphere $\mathbb{S}^3$, which turn out to be related to Lagrangian embeddings in the nearly Kähler manifold $S^3\times S^3$ and to great circle flows on $\mathbb{S}^3$. Using our methods we generalize a well known result of Gluck and Gu concerning divergence-free geodesic vector fields on the sphere and we show that the space of Lagrangian submanifolds of $S^3\times S^3$ has at least three connected components.
On a pseudo-Riemannian manifold $\mathcal{M}$ we introduce a system of partial differential Killing type equations for spinor-valued differential forms, and study their basic properties. We discuss the relationship between solutions of Killing equations on $\mathcal{M}$ and parallel fields on the metric cone over $\mathcal{M}$ for spinor-valued forms.
Let $(M,ω)$ be a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure) and a torsion-free symplectic connection $\nabla.$ Symplectic Killing spinor fields for this structure are sections of the symplectic spinor bundle satisfying a certain first order partial differential equation and they are the main topic of this paper. We derive a necessary condition satisfied by a symplectic Killing spinor field. The advantage of this condition consists in the fact that it is expressed by a zeroth order operator. This condition helps us substantiona…
We determine the geometry of supersymmetric heterotic string backgrounds for which all parallel spinors with respect to the connection $\hat\nabla$ with torsion $H$, the NS$\otimes$NS three-form field strength, are Killing. We find that there are two classes of such backgrounds, the null and the timelike. The Killing spinors of the null backgrounds have stability subgroups $K\ltimes\bR^8$ in $Spin(9,1)$, for $K=Spin(7)$, SU(4), $Sp(2)$, $SU(2)\times SU(2)$ and $\{1\}$, and the Killing spinors of the timelike backgrounds have stability subgroups $G_2$, SU(3), SU(2) and $\{1\}$. The former admit…
We prove the instability of some families of Riemannian manifolds with non-trivial real Killing spinors. These include the invariant Einstein metrics on the Aloff-Wallach spaces $N_{k, l}={\rm SU}(3)/i_{k, l}(S^{1})$ (which are all nearly ${\rm G}_2$ except $N_{1,0}$), and Sasaki Einstein circle bundles over certain irreducible Hermitian symmetric spaces. We also prove the instability of most of the simply connected non-symmetric compact homogeneous Einstein spaces of dimensions $5, 6, $ and $7$, including the strict nearly Kähler ones (except ${\rm G}_2/{\rm SU}(3)$).
In this paper, the Dirac, twistor and Killing equations on Weyl manifolds with CSpin structures are investigated. A conformal Schr"odinger-Lichnerowicz formula is presented and used to show integrability conditions for these equations. By introducing the Killing equation for spinors of arbitrary weight, the result of Andrei Moroianu in [9] is generalized in the following sense. The only non-closed Weyl manifolds of dimension greater than 3 that admit solutions of the real Killing equation are 4-dimensional and non-compact. Any Weyl manifold of these dimensions admitting a real Killing spinor h…
We review the actions of the supergravity theory in eleven dimensions as well as the type IIA and IIB supergravities in ten dimensions and derive the bosonic equations of motion in a coordinate-free notation. We also consider the existence of supersymmetries and the associated generalized Killing spinor equations. The aim of this note is to serve as a formulary and make the equations of supergravity more easily accessible to mathematicians.
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).
This paper is devoted to the systematic investigation of the cone construction for Riemannian $G$ manifolds M, endowed with an invariant metric connection with skew torsion $\nabla^c$, a `characteristic connection'. We show how to define a $\bar G$ structure on the cone $\bar M=M\x \R^+$ with a cone metric, and we prove that a Killing spinor with torsion on $M$ induces a spinor on $\bar M$ that is parallel w.\,r.\,t. the characteristic connection of the $\bar G$ structure. We establish the explicit correspondence between classes of metric almost contact structures on $M$ and almost hermitian c…
In a previous article we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kaehler manifolds. In the present article we study the limiting case, i. e. manifolds where the lower bound is attained as an eigenvalue. We give an equivalent formulation in terms of a quaternionic Killing equation and show that the only symmetric quaternionic Kaehler manifolds with smallest possible eigenvalue are the quaternionic projective spaces.
We determine the holonomy of generalized Killing spinor covariant derivatives of the form $D= \nabla + Ω$ on pseudo-Riemannian reductive homogeneous spaces in a purely algebraic and algorithmic way, where $Ω: TM \rightarrow Λ^*(TM)$ is a left-invariant homomorphism. This is essentially an application of the theory of invariant principal bundle connections defined over homogeneous principal bundles. Moreover, the algorithm allows a computation of the associated Killing superalgebra in certain cases. The procedure is demonstrated by determining the supersymmetries of certain homogeneous M2 duals…
Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西奥多·斯托伊洛瓦数学研究所）；Scuola Internazionale Superiore di Studi Avanzati（国际高等研究院）；Departamento de Matemáticas, Universidad UNED - Madrid（马德里国立远程教育大学数学系）
We prove that, for M theory or type II, generic Minkowski flux backgrounds preserving $\mathcal{N}$ supersymmetries in dimensions $D\geq4$ correspond precisely to integrable generalised $G_{\mathcal{N}}$ structures, where $G_{\mathcal{N}}$ is the generalised structure group defined by the Killing spinors. In other words, they are the analogues of special holonomy manifolds in $E_{d(d)} \times\mathbb{R}^+$ generalised geometry. In establishing this result, we introduce the Kosmann-Dorfman bracket, a generalisation of Kosmann's Lie derivative of spinors. This allows us to write down the internal…
We construct the space of infinitesimal variations for the Strominger system and an obstruction space to integrability, using elliptic operator theory. We initiate the study of the geometry of the moduli space, describing the infinitesimal structure of a natural foliation on this space. The associated leaves are related to generalized geometry and correspond to moduli spaces of solutions of suitable Killing spinor equations on a Courant algebroid. As an application, we propose a unifying framework for metrics with holonomy $\SU(3)$ and solutions of the Strominger system.
Suppose that $Σ=\partialΩ$ is the $n$-dimensional boundary, with positive (inward) mean curvature $H$, of a connected compact $(n+1)$-dimensional Riemannian spin manifold $(Ω^{n+1},g)$ whose scalar curvature $R\ge -n(n+1)k^2$, for some $k\textgreater{}0$. If $Σ$ 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,…
