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.


Introduction
in gravitational physics. In particular, they are used in the integrability of the geodesic systems, Hamilton-Jacobi equation, Klein-Gordon equation and Dirac equation on black hole and other spacetimes, for some selected works see [5]- [12] and for a concise review see [13] and references within. The KY and CKY equations are also used to find the conserved charges of supersymmetric relativistic and non-relativistic particles [14,15,16] systems.
Generalizations of the KY and CKY have also been considered. In the context of supersymmetric relativistic and non-relativistic particles, a generalization of the KY equations has been introduced in [22,20] that includes skew-symmetric torsion. A similar generalization of CKY equation has been considered in the context of gravitational physics in [17,18]. The relation between KY and G-structures has been explored in [19,20,21]. Further generalizations of KY and CKY equations have been investigated in [16].

Twisted covariant form hierarchies 2.1 CKY equations and covariantly constant spinors
Let M be a n-dimensional manifold with (r, s) signature equipped with a metric g. The CKY condition on a k-form 2 ω on M is where ∇ is the Levi-Civita connection of g, α X (Y ) = g(X, Y ) and δ is the adjoint operation of the exterior derivative d. If δω = 0, the remaining condition is the KY equation. The condition (1) on 1-forms implies that the associated vector field generates a conformal motion on M. A generalization of (1) is to replace the Levi-Civita connection ∇ with a connection ∇ H = ∇ + 1 2 H, where H is a skew-symmetric torsion, and appropriately replace d and δ with d H and δ H , respectively.
It has been known for sometime that the Killing spinor bi-linears constructed from the solutions ǫ of the KSE 3 satisfy the CKY equation, where λ is a complex constant. Such KSEs arise on spheres S n , de-Sitter space dS n and Anti-de-Sitter AdS n spaces. To see this define the spinor bi-linears where ·, · D is the Spin-invariant Dirac inner product. Although ·, · D has been used here for definiteness, any other Spin-invariant bi-linear can be used, see e.g. appendix B of [1] for a discussion. Next suppose that ǫ is a Killing spinor, i.e. D M ǫ = 0, then one can show that for Lorentzian signature manifolds while for Euclidean signature manifolds A consequence of the two equations above is that the right-hand-side can be rewritten in terms of the left-hand-side yielding (1). As a result the form τ k satisfies the CKY equation.
Although CKY equations are suitable to describe the geometry of some manifolds that admit Killing spinors, as established above, the conditions imposed by generic KSEs on the Killing spinor bi-linears are far more involved. Therefore a suitable generalization of the CKY condition is required to proceed further.

Twisted form hierarchies and CKY equations
To give the definition of a TCFH, let M be a manifold with a metric g and signature (r, s), Λ * c (M) be the complexified bundle of all forms on M and F be a multi-form, i.e. be a section of Λ * , with possibly different degrees p, which satisfy where P, Q : Γ(Λ * c (M)) → Γ(Λ * c (M)) are constructed from exterior derivative d, the wedge product with F and inner derivation with respect to F viewed as a multi-vector valued multi-form and their adjoints with respect to the standard inner product in Γ(Λ * c (M)). Furthermore ∇ F , the covariant hierarchy connection, is a connection in Λ * c (M) constructed from the Levi-Civita connection and F . ∇ F is not necessarily degree preserving.
The condition (6) is analogous to the conditions (4) and (5) given in the previous section. Additional examples will be presented below. The TCFHs have been used in [4] to explore the geometry of manifolds that arise in certain eigenvalue estimates for a class of modified Dirac operators.
As we have seen (4) and (5) are associated with a CKY equation. Indeed after comparing the left-hand and right-hand sides of (6), one finds that where (. . . )| p denotes a restriction of the expression to p-forms, d F is the exterior derivative constructed using ∇ F X and similarly δ F is an adjoint constructed using again ∇ F X . This follows immediately from (6) by shew-symmetrizing all indices first to derive the d F term and then contracting with the metric to derive the δ F term. Incidentally this gives the definitions of d F and δ F operations.
The equation (7) can be seen as a generalization of the CKY equations. Though there are several differences between (7) and previous generalizations of CKY equation. The new connection used in the relation ∇ F X may not degree preserving. Moreover (7) is a relation between a collection of forms with possibly different degrees while typically CKY equation and other generalization are conditions on a form with a definite degree.
Note that although (7) is implied by (6), the converse it not necessarily the case unless one imposes in addition that Thus (7) is a coarser relation than that of (6). Therefore, it is expected that there are solution of (7) which are not solutions of (6). Although of course all solutions of (6) are also solutions of (7).

Proof of main result
To prove that the KSEs of a supergravity theory are associated with a TCFH, let us first recall that the supercovariant connection of a supergravity theory has the structure where c denotes the Clifford algebra element associated with the multi-forms i X H = where H p 's and G p 's are the p-form field strengths of a supergravity theory. ∇ X is typically the Levi-Civita connection but it can also be twisted with a gauge connection.
Notice that we have not imposed any restrictions on the signature of the spacetime. The argument below applies to the KSEs of all standard supergravities defined on Lorentzian signature manifolds as well as to those of non-standard supergravities defined on (r,s)-signature manifolds, e.g Euclidean signature manifolds.
As the argument that follows is linear in the field strengths H p and G p , it is sufficient to take H and G to be single forms H and G of degree ℓ, H ℓ = H and G ℓ = G. Next consider the form bi-linears {χ p } constructed from a Killing spinors ǫ, D X ǫ = 0, with respect to some Spin-invariant inner product say ·, · s , i.e.
where {e A } is a (pseudo)-orthonormal frame adapted to the spacetime metric. See e.g. appendix B in [1] for a discussion on Spin-invariant bi-linears. Then one has that After using the Hermiticity properties of the inner product and the Clifford algebra relation as well as the definition of the form bi-linears, one finds that whereH is the complex conjugate of H, c 1 q andc 1 q are combinatorial coefficients which depend on p, ℓ and the inner product ·, · s whose values are not essential for the proof that follows. Although they are computed explicitly in the examples presented below. Moreover i X H · χ q denotes multi-index contractions between the i X H and χ q forms and similarly for i XH ·χ q , where the indices are raised with respect to the metric. For example ψ k · ω m denotes any of the contractions of the forms ψ k and ω m and p = k + m − 2s. Furthermore, one can show after using the Clifford algebra relations and the properties of ·, · s that where again the c's are combinatorial coefficients that their value is not essential for the proof. Note that in the above expression the terms i X G·χ q and i XḠ ·χ q do not contribute. This is one of the key observations required for the proof of the statement and it is a consequence of the Clifford algebra relation. The last term in the above expression can be rewritten as for some combinatorial coefficients c 4 q , c 5 q andc 4 q ,c 5 q . The last term in the above expression has the same structure as that in (12). Using this, (11) can be written after some rearrangement as Clearly this equation defines a TCFH (6) with and F = {H, G} ind are the linearly independent form field strengths. Note that ∇ F X as defined above is a connection in the space of forms-it satisfies all the axioms of a connection-but it is not necessarily degree preserving unless all combinatorial coefficients c 0 q andc 0 q vanish apart from those of q = p. This completes the proof of the main result. The proof above utilizes the form bi-linears (10) of a single Killing spinor ǫ. However, it can be easily generalize to include the bi-linears of any number of Killing spinors with respect to any Spin-invariant inner product. The proof is essentially the same. The only difference is that instead of considering (10), one should replace them with where both η and ǫ are Killing spinors. The theorem can also be generalized to effective theories that include supergravity with higher order corrections, like for example those that emerge as low energy effective theories of superstrings and M-theory. This is because the general structure of the supercovariant connection after including higher order corrections is expected to have the general form of (9). In fact for the theorem to apply, it is not necessary to assume that H is a form. Instead H can be a section of Λ 1 c (M) × Λ * c (M) provided that that i X H which appears in the expression for the supercovariant derivative (9) is a multi-form.
It should also be noted that there is an ambiguity in the definition of ∇ F . To see this observe that there may be terms F · χ p which have the property that e.g. terms for which all indices of χ p are contracted to indices of F . Such terms can either contribute to ∇ F or to P. If all such terms are included in ∇ F , then such a covariant hierarchy connection will be called maximal and denoted with ∇ F . On the other hand if all such terms are included in P, then ∇ F will be called minimal and denoted with D F . It is clear that there may be many intermediate cases.
The TCFHs that arise in supergravity theory can always be chosen such that the Hodge duality operation on {χ p } is an automorphism of the hierarchy. This can always be achieved by choosing in the set {χ p } all bi-linears and their Hodge duals. In such a case the hierarchy will be twisted by F as originally has been indicated. However in many of the examples below the set of bi-linears {χ p } is chosen up to a Hodge duality operation. This simplifies the selection and so the final result. In such a case, the Hodge duality operation may not be an automorphism of the TCFH. In addition such a TCFH will be twisted with respect to both F and its dual * F .

Examples
To illustrate the proof given above, we shall present some examples and explore their properties. It is not the purpose here to give a complete description of the TCFHs of all supergravity theories-this will be presented elsewhere. To begin the supercovariant connection of the heterotic and ungauged N = (1, 0) d = 6 supergravity theories is induced from the connection on the tangent bundle of the spacetime which has skew-symmetric torsion H. Denoting this connection with ∇ H = ∇ + 1 2 H, it is straightforward to observe that the twisted covariant hierarchy connection ∇ F coincides with ∇ H . Furthermore the Killing spinor bi-linears are covariantly constant.
The same conclusion holds for the heterotic strings for up and including 2-loop corrections in the sigma model perturbation theory. In the gauged N = (1, 0) d = 6 supergravity some of the form Killing spinor bi-linears are twisted with respect to the gauge connection. Again ∇ F is given by ∇ F = ∇ H + A r D r , where A r is the gauge connection and D r is the representation of the Lie algebra of the gauge group acting on some of the form-bi-linears. The holonomy of these connections has been explored in to classify all the supersymmetric solutions of heterotic [23,24] and a N = (1, 0) d = 6 [25,26] supergravities coupled to matter multiplets.

Minimal N = 2 supergravity in four dimensions
As another example consider the N = 2 d = 4 minimal supergravity. To begin consider the bi-linears constructed from the Dirac inner product where γ 5 = iΓ 0123 . These are the algebraically independent bi-linears up to a Hodge duality operation. The supercovariant derivative of the theory is Assuming the ǫ is a Killing spinor, D M ǫ = 0, one can easily compute the covariant derivative of the bi-linear and re-arrange the terms as a TCFH. In particular, the TCFH with respect to the minimal connection reads where * F M N = 1 2 ǫ M N P Q F P Q with ǫ 0123 = −1. Observe that the above equations can be arranged to be real with an appropriate redefinition of the form bi-linears as in the basis chosen some of them are imaginary.
Similarly the TCFH with respect to maximal connection is It is clear that in both cases the covariant form hierarchy is twisted by the multi-form F = {F, * F }. Note that if the basis in the space of bi-linears included the Hodge dual forms, then the hierarchy would have been twisted just with F = {F }.
The associated generalized CKY equations with respect to the minimal connection are The associated generalized CKY equations with respect to the maximal connection are Observe that if for some background * F M [N Y R] = 0, then ω satisfies the KY equation. Of course the vector field associated to K is Killing as expected.

Minimal N = 1 supergravity in five dimensions
Next let us turn to N = 1 supergravity in five dimensions. Consider the algebraically independent Killing spinor bi-linears up to a Hodge duality operation where now ǫ is a Spin(4, 1) spinor and Γ 4 = Γ 4 = iΓ 0123 . The supercovariant connection of the theory is Assuming the ǫ is a Killing spinor, D M ǫ = 0, the conditions on the bi-linears imposed by the gravitino KSE have been found in [2]. These conditions have been given in a non-TCFH expression. Putting them into the TCFH form with respect to the minimal connection, one finds that Similarly, the TCFH with respect to the maximal connection is Note that there are additional form bi-linears that can be added to (26) as the theory always preserves even number of supersymmetries. However the choice made above suffices to demonstrate the general theorem. The associated generalizations of the CKY equations are and respectively. Observe that both the minimal and maximal TCFH connections on ω are connections with skew-symmetric torsion H = 2 √ 3 * F . In turn the associated generalization of the CKY equation is that for which the Levi-Civita connection ∇ is replaced with ∇ H . As a result all supersymmetric solutions of N = 1 d = 5 supergravity with ω = 0 admit a CKY 2-form associated with a connection with skew-symmetric torsion. This includes electrically and magnetically charged black holes as well pp-wave backgrounds. Furthermore again K is associated with a Killing vector field.

11-dimensional supergravity
As a final example, let us consider the N = 1 supergravity in eleven dimensions and backgrounds that admit one Killing spinor. Such backgrounds admit a 1-form, K, 2form, ω, and 5-form, τ , Killing spinor bi-linears up to a Hodge duality operation. The supercovariant connection of 11-dimensional supergravity is Using D M ǫ = 0, the covariant derivative of these bi-linears has been computed in [27]. Again the equations are not in TCFH form. Rewriting the expressions as a TCFH with respect to the minimal connection these read where ǫ 01...9(10) = −1. While the same expression with respect to the maximal connection is Clearly the TCFHs are twisted with respect to F = {F, * F }. The associated twisted CKY equations are and respectively. K is associated with a Killing vector field. It is expected that in many special backgrounds the bi-linears will satisfy the CKY equations as some components of the fluxes will vanish and so the TCFH connections will be simplified.

Conclusions
We have demonstrated that the gravitino KSE of all supergravity theories, irrespective of spacetime signature and including higher order corrections, are associated with a family of TCFHs. Each TCFH in the family is characterized with a connection ∇ F X on the space of forms which may not be degree preserving. In turn each TCFH in the family gives rise to a twisted CKY structure on all supersymmetric solutions of a supergravity theory. This result establishes the close relation between KSEs and a suitable generalizations of the CKY equation. It is clear that the TCFH connections ∇ F X characterize the underlying geometric structure of a supersymmetric background of a supergravity theory. It is likely that their holonomy for a generic background is of general linear type in analogy with the holonomy of the supercovariant connections [28,29,30]. Nevertheless there is a ∇ F X connection for each member in the family of TCFHs associated with a supergravity theory. Because of this, one expects that the holonomy of the connections ∇ F X represents a refinement of that of the supercovariant connection. Some insight in this direction may be obtained by computing the holonomy of the ∇ F X for some well-known supersymmetric backgrounds. In turn this may give some insight into the symmetries of M-theory. It may turn out that supersymmetric backgrounds can be characterized with the holonomy of TCFH connections.
The proof presented here that all supersymmetric solutions admit a suitable generalization of the CKY equation opens another avenue towards understanding the geometric structure of supersymmetric backgrounds. It is likely that for many special supersymmetric backgrounds the twisted CKY equations simplify to the more standard CKY equations possibly twisted with a connection which is form degree preserving. Furthermore the geometry of supersymmetric backgrounds can be understood in parallel with that of some non-supersymmetric ones that have been known for sometime that they admit CKY forms.
In a similar theme, as the twisted CKY condition is more coarse than that of an associated TCFH, it is likely that there are solutions of the twisted CKY equation that are not solutions of that of TCFH. Such non-supersymmetric solutions will be very closely related to supersymmetric ones. It would be of interest to construct examples of such backgrounds.
It is well known the the CKY equation is associated with the conserved charges of supersymmetric relativistic and non-relativist particle actions. Therefore the question arises whether this is the case for the twisted CKY equations associated with supersymmetric backgrounds demonstrated in this work. The construction of such particle or possibly string actions that exhibit such symmetries may give some insights into these theories.