Killing Spinors Research Articles

Non-Riemannian isometries from double field theory

We explore the notion of isometries in non-Riemannian geometries. Such geometries include and generalise the backgrounds of non-relativistic string theory, and they can be naturally described using the formalism of double field theory. Adopting this approach, we first solve the corresponding Killing equations for constant flat non-Riemannian backgrounds and show that they admit an infinite-dimensional algebra of isometries which includes a particular type of supertranslations. These symmetries correspond to known worldsheet Noether symmetries of the Gomis-Ooguri non-relativistic string, which we now interpret as isometries of its non-Riemannian doubled background. We further consider the extension to supersymmetric double field theory and show that the corresponding Killing spinors can depend arbitrarily on the non-Riemannian directions, leading to “supersupersymmetries” that square to supertranslations.

Spinors of real type as polyforms and the generalized Killing equation

We develop a new framework for the study of generalized Killing spinors, where every generalized Killing spinor equation, possibly with constraints, can be formulated equivalently as a system 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 Sigma of real type as a real algebraic variety in the Kähler–Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of text {AdS}_4 space-time.

Skew Killing spinors in four dimensions

This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor psi is a spinor that satisfies the equation nabla _Xpsi =AXcdot psi 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 {mathbb {R}}times N with N having a skew Killing spinor and we explain under which conditions on the spinor the special case of a local isometry to {mathbb {S}}^2times {mathbb {R}}^2 occurs. In the non-degenerate case, the existence of skew Killing spinors is related to doubly warped products whose defining data we will describe.

Symmetries of mathcal{N} = (1, 0) supergravity backgrounds in six dimensions

General mathcal{N} = (1, 0) supergravity-matter systems in six dimensions may be described using one of the two fully fledged superspace formulations for conformal supergravity: (i) SU(2) superspace; and (ii) conformal superspace. With motivation to develop rigid supersymmetric field theories in curved space, this paper is devoted to the study of the geometric symmetries of supergravity backgrounds. In particular, we introduce the notion of a conformal Killing spinor superfield ϵα, which proves to generate extended superconformal transformations. Among its cousins are the conformal Killing vector ξa and tensor ζa(n) superfields. The former parametrise conformal isometries of supergravity backgrounds, which in turn yield symmetries of every superconformal field theory. Meanwhile, the conformal Killing tensors of a given background are associated with higher symmetries of the hypermultiplet. By studying the higher symmetries of a non-conformal vector multiplet we introduce the concept of a Killing tensor superfield. We also analyse the problem of computing higher symmetries for the conformal d’Alembertian in curved space and demonstrate that, beyond the first-order case, these operators are defined only on a limited class of backgrounds, including all conformally flat ones.

Ground state Dirac bubbles and Killing spinors

We prove a classification result for ground state solutions of the critical Dirac equation on \(\mathbb {R}^n\), \(n\geqslant 2\). By exploiting its conformal covariance, the equation can be posed on the round sphere \(\mathbb {S}^n\) and the non-zero solutions at the ground level are given by Killing spinors, up to conformal diffeomorphisms. Moreover, such ground state solutions of the critical Dirac equation are also related to the Yamabe equation for the sphere, for which we crucially exploit some known classification results.

Neutral signature gauged supergravity solutions

We classify all supersymmetric solutions of minimal D = 4 gauged supergravity with (2) signature and a positive cosmological constant which admit exactly one Killing spinor. This classification produces a geometric structure which is more general than that found for previous classifications of N = 2 supersymmetric solutions of this theory. We illustrate how the N = 2 solutions which consist of a fibration over a 3-dimensional Lorentzian Gauduchon-Tod base space can be written in terms of this more generic geometric structure.

Super Yang-Mills theories with inhomogeneous mass deformations

We construct the 4-dimensional mathcal{N}=frac{1}{2} and mathcal{N} = 1 inhomogeneously mass-deformed super Yang-Mills theories from the mathcal{N} = 1* and mathcal{N} = 2* theories, respectively, and analyse their supersymmetric vacua. The inhomogeneity is attributed to the dependence of background fluxes in the type IIB supergravity on a single spatial coordinate. This gives rise to inhomogeneous mass functions in the mathcal{N} = 4 super Yang-Mills theory which describes the dynamics of D3-branes. The Killing spinor equations for those inhomogeneous theories lead to the supersymmetric vacuum equation and a boundary condition. We investigate two types of solutions in the mathcal{N}=frac{1}{2} theory, corresponding to the cases of asymptotically constant mass functions and periodic mass functions. For the former case, the boundary condition gives a relation between the parameters of two possibly distinct vacua at the asymptotic boundaries. Brane interpretations for corresponding vacuum solutions in type IIB supergravity are also discussed. For the latter case, we obtain explicit forms of the periodic vacuum solutions.

Choices of spinor inner products on M-theory backgrounds

M-theory backgrounds in the form of unwarped compactifications with or without fluxes are considered. We construct the bilinear forms of supergravity Killing spinors for different choices of spinor inner products on these backgrounds. The equations satisfied by the bilinear forms and their decompositions into product manifolds are obtained for different inner product choices in the special case for which the spinors factorize. It is found that the AdS solutions can only appear for some special choices of spinor inner products on product manifolds. The reduction of bilinears of supergravity Killing spinors into the hidden symmetries of product manifolds which are Killing-Yano and closed conformal Killing-Yano forms for AdS solutions is shown. These hidden symmetries are lifted to eleven-dimensional backgrounds to find the hidden symmetries on them. The relation between the choices of spinor inner products, AdS solutions and hidden symmetries on M-theory backgrounds are investigated.

Totally umbilical hypersurfaces of Spinc manifolds carrying special spinor fields

Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spin[Formula: see text] manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spin[Formula: see text] case the result of Kowalski stating that, every totally umbilical hypersurface of an Einstein manifold of dimension greater or equal to [Formula: see text] is of constant mean curvature. As an application, we prove that there are no extrinsic hypersheres in complete Riemannian [Formula: see text] manifolds of non-constant sectional curvature carrying a parallel, Killing or imaginary Killing spinor.

Supersymmetric traversable wormholes

We study traversable wormhole solutions in pure gauged N = 2 supergravity with and without electromagnetic fields, which are locally isometric under SO(2, 1) × SO(1, 1). The model allows for 1/2-BPS wormhole solutions whose corresponding globally defined Killing spinors are presented. A non-contractible cycle can be obtained by compactifying one of the coordinates which leaves the residual supersymmetry unaffected, although not all the isometries will remain realized globally. The wormholes connect two asymptotic, locally AdS4 regions and depend on certain electric and magnetic charge parameters and, implicitly, on the range of the compact coordinate around the throat. We provide an analysis of the boundary of the spacetime and show that it can be either disconnected or not, depending on the values of the parameters in the metric. Finally, we show how a class of these space-times avoid a topological censorship theorem.

Some Properties of Dirac–Einstein Bubbles

We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac-Einstein equations on $\mathbb{R}^3$, which appear in the bubbling analysis of conformal Dirac-Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin-Talenti functions, while the spinorial part is the conformal image of $-\frac{1}{2}$-Killing spinors on the round sphere $\mathbb{S}^3$.

Killing spinor-valued forms and their integrability conditions

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.

Nonexistence of locally but not globally supersymmetric orbifolds

Motivated by the smallness of the cosmological constant we investigate whether it is possible to have vanishing one-loop heterotic string partition functions for six-dimensional non-supersymmetric toroidal orbifolds. A straightforward way to realize this presents itself, when each orbifold sector separately preserves some Killing spinors, but none of them survives in all sectors combined. By applying some representation theory to the abstract finite point groups underlying toroidal orbifolds it turns out, that this is never possible. This leads to a nonexistence proof of locally but not globally supersymmetric orbifolds.

Twisted form hierarchies, Killing-Yano equations and supersymmetric backgrounds

We show that the Killing spinor equations of all supergravity theories which may include higher order corrections on a (r,s)-signature spacetime are associated with twisted covariant form hierarchies. These hierarchies are characterized by a connection on the space of forms which may not be degree preserving. As a consequence we demonstrate that the form Killing spinor bi-linears of all supersymmetric backgrounds satisfy a suitable generalization of conformal Killing-Yano equation with respect to this connection. To illustrate the general proof the twisted covariant form hierarchies of some supergravity theories in 4, 5, 6, 10 and 11 dimensions are also presented.

Non-relativistic supersymmetry on curved three-manifolds

We construct explicit examples of non-relativistic supersymmetric field theories on curved Newton-Cartan three-manifolds. These results are obtained by performing a null reduction of four-dimensional supersymmetric field theories on Lorentzian manifolds and the Killing spinor equations that their supersymmetry parameters obey. This gives rise to a set of algebraic and differential Killing spinor equations that are obeyed by the supersymmetry parameters of the resulting three-dimensional non-relativistic field theories. We derive necessary and sufficient conditions that determine whether a Newton-Cartan background admits non-trivial solutions of these Killing spinor equations. Two classes of examples of Newton-Cartan backgrounds that obey these conditions are discussed. The first class is characterised by an integrable foliation, corresponding to so-called twistless torsional geometries, and includes manifolds whose spatial slices are isomorphic to the Poincaŕe disc. The second class of examples has a non-integrable foliation structure and corresponds to contact manifolds.

Tension Between a Vanishing Cosmological Constant and Non‐Supersymmetric Heterotic Orbifolds

AbstractWe investigate under which conditions the cosmological constant vanishes perturbatively at the one‐loop level for heterotic strings on non‐supersymmetric toroidal orbifolds. To obtain model‐independent results, which do not rely on the gauge embedding details, we require that the right‐moving fermionic partition function vanishes identically in every orbifold sector. This means that each sector preserves at least one, but not always the same Killing spinor. The existence of such Killing spinors is related to the representation theory of finite groups, i.e. of the point group that underlies the orbifold. However, by going through all inequivalent (Abelian and non‐Abelian) point groups of six‐dimensional toroidal orbifolds we show that this is never possible: For any non‐supersymmetric orbifold there is always (at least) one sector, that does not admit any Killing spinor. The underlying mathematical reason for this no‐go result is formulated in a conjecture, which we have tested by going through an even larger number of finite groups. This conjecture could be applied to situations beyond symmetric toroidal orbifolds, like asymmetric orbifolds.

On the supersymmetric solutions of the Heterotic Superstring effective action

We consider the effective action of the Heterotic Superstring to first order in α′ and derive the necessary and sufficient conditions that a field configuration has to satisfy in order to admit at least one Killing spinor using the spinor bilinear method in an arbitrary spinorial basis and corresponding arbitrary gamma matrices. As a previous step in this derivation, we compute the complete spinor bilinear algebra using the Fierz identities, obtaining as a by-product the algebra satisfied by the Spin(7) structure contained in the bilinears in an arbitrary basis. We find the off-shell relations existing between the bosonic equations of motion evaluated on supersymmetric field configurations using the Killing Spinor Identities instead of the (far more complicated) integrability conditions of the Killing Spinor Equations as it is common in the literature. We show how to include the Kalb-Ramond’s Bianchi identity in the Killing Spinor Identities.

Symmetries of supergravity backgrounds and supersymmetric field theory

In four spacetime dimensions, all mathcal{N} = 1 supergravity-matter systems can be formulated in the so-called U(1) superspace proposed by Howe in 1981. This paper is devoted to the study of those geometric structures which characterise a background U(1) superspace and are important in the context of supersymmetric field theory in curved space. We introduce (conformal) Killing tensor superfields {mathrm{ell}}_{left({alpha}_1dots {alpha}_mright)left({overset{cdot }{alpha}}_1dots {overset{cdot }{alpha}}_nright)} , with m and n non-negative integers, m + n > 0, and elaborate on their significance in the following cases: (i) m = n = 1; (ii) m − 1 = n = 0; and (iii) m = n > 1. The (conformal) Killing vector superfields {mathrm{ell}}_{alpha overset{cdot }{alpha }} generate the (conformal) isometries of curved superspace, which are symmetries of every (conformal) supersymmetric field theory. The (conformal) Killing spinor superfields ℓα generate extended (conformal) supersymmetry transformations. The (conformal) Killing tensor superfields with m = n > 1 prove to generate all higher symmetries of the (massless) massive Wess-Zumino operator.

Non-compact manifolds with Killing spinors

We present here a quick result for non-compact manifolds with invertible Dirac operator, where we link the presence of a massless Killing spinor, with a harmonic, closed conformal Killing–Yano tensor, if one exists for the specific manifold. A couple of examples are introduced.

Spacetimes with all Penrose limits diagonalisable**I would like to dedicate this paper to the memory of Stephen Siklos, who first derived some of the metrics discussed here. He was a droll and cultured colleague, with the true pedagogic and academic values.

We consider the problem of finding all space-time metrics for which all plane-wave Penrose limits are diagonalisable plane waves. This requirement leads to a conformally invariant differential condition on the Weyl spinor which we analyse for different algebraic types in the Petrov–Pirani–Penrose classification. The only vacuum examples, apart from actual plane waves which are their own Penrose limit, are some of the nonrotating type D metrics, but some nonvacuum solutions are also identified. The condition requires the Weyl spinor, whenever it is nonzero, to be proportional to a valence-4 Killing spinor with a real function of proportionality.