Real supersymmetric solutions of (3,2) signature five-dimensional supergravity

We classify supersymmetric solutions of D = 5 (3,2) signature supergravity with either vanishing or imaginary gauge coupling constant preserving the minimal N = 2 supersymmetry. We prove that the geometry of such solutions is characterized by a nilpotent integrable endomorphism, and obtain the necessary and sufficient conditions on the fluxes imposed by supersymmetry. We also construct examples of supersymmetric domain wall solutions for which the nilpotent integrable endomorphism is associated with a Killing–Yano two-form, as well as a new descendant ‘preon’ solution which preserves N = 6 supersymmetry. This is notable as such N = 6 descendant solutions do not exist in the standard signature D = 5 supergravity.


Introduction
The classification and analysis of supersymmetric solutions of five-dimensional N = 2 supergravity theories coupled to vector multiplets [1,2] have attracted considerable attention in recent years.A large class of these theories can be obtained via the dimensional reduction of eleven-dimensional supergravity of [3] on a Calabi-Yau threefold (CY3) [4].The first systematic classification of supersymmetric solutions of five-dimensional ungauged minimal supergravity in Lorentzian signature (and without vector multiplets) was given in [5].The generalizations of the results of [5] to solutions of supergravity theories with vector multiplets was later performed in [6].Spinorial geometry techniques were then developed in the context of the classification of supersymmetric supergravity solutions in ten and eleven dimensions [7].Such methods have been applied in the classification of many different types of solutions in numerous theories (see [8] and references therein).
In recent years, there has been renewed interest in the construction and study of supergravity theories in various space-time signatures.Theories with exotic spacetime signatures arise naturally in the study of the web of duality symmetries in string and M-theory.Such exotic theories are obtained through a time-like T-duality of ordinary string theories and have unconventional effective field theory description which involves RR fields with wrong sign of kinetic energy terms.Therefore the existence of such theories as a component of the full M-theory is linked to allowing Tduality transformations along a time-like circle [9,10,11].Non-Lorentzian signatures are also relevant to the study of non-perturbative effects as well as the study of quantum gravity.
Euclidean versions of the ungauged supergravity theories of [1] were considered in [12].It was concluded that the kinetic terms of the gauge fields in the Euclidean theory have the non-conventional sign in the Lagrangian of the theory.Five-dimensional theories with various space-time signature can be obtained through the reduction of Hull's exotic eleven-dimensional supergravity on CY3.We shall denote the space-time signature by (s, t) where s refers to spatial dimensions and t refers to time dimensions.The reduction of supergravity theories with signatures (10, 1), (6,5) and (2,9) produces five-dimensional theories with signatures (4, 1), (0, 5) and (2, 3) with standard sign of gauge kinetic terms.The reduction of theories with signatures (1, 10), (5,6) and (9,2) produces five-dimensional theories with signatures (1,4), (5,0) and (3,2) with the non-conventional sign of the gauge kinetic terms [13].For a detailed analysis of off-shell supersymmetry transformations and associated supersymmetric Lagrangians for five-dimensional vector multiplets in arbitrary space-time dimensions, we refer the reader to [14].Exotic gauged supergravity theories can be obtained via non-linear Kaluza Klein reduction of exotic type IIB and M theory [15].
Solutions for a particular choice of a Dirac Killing spinor were considered for all signatures in ungauged five-dimensional supergravity theories in [16].It was found that the four-dimensional base space of the Euclidean theories are given, as in the Lorentzian cases, in terms of hyper-Kähler manifolds.In terms of supersymmetry, the ungauged (3, 2) signature theory admits Majorana Killing spinors, though the spinor ǫ considered in [16] was not Majorana.Consequently, such solutions actually preserved an extended N = 4 supersymmetry, and could be described in terms of a fibration over a 4-dimensional base space which is equipped with a hypersymplectic structure [17].Supersymmetric solutions with Majorana Killing spinors in four-dimensional minimal ungauged (2, 2) signature supergravity were classified in [18].
The goal of our present work is to classify minimally supersymmetric (N = 2) solutions of D = 5, (3, 2) supergravity with either vanishing or imaginary gauge coupling constant, for which the Killing spinor is Majorana.In such a case, we shall determine the geometric structures common to all such solutions, and which also underpin the hypersymplectic structures found for the case of enhanced N = 4 supersymmetry considered in [16].We shall show that the geometric structure associated with minimal N = 2 supersymmetry in these theories is characterized by a nilpotent integrable endomorphism, and we fully determine the necessary and sufficient conditions on the fluxes for supersymmetry.We remark that this situation, with respect to the amount of supersymmetry preserved, is novel to theories with such mixed signature, and does not hold for the ungauged (4, 1) signature theory considered in [5], where the solutions preserve either N = 0, N = 4, or N = 8 supersymmetry, and there are no N = 2 supersymmetric solutions.
We also construct a number of examples of supersymmetric solutions, including domain wall solutions, and a novel class of "descendant" supersymmetric solutions in the non-minimal ungauged (3, 2) theory for which the gravitino Killing spinor equation preserves N = 8 maximal supersymmetry, but the N = 8 supersymmetry is broken down to N = 6 by the gaugino Killing spinor equation.Descendant solutions were first classified in heterotic supergravity [19,20].In five dimensions, the existence of such solutions preserving N = 6 supersymmetry is again particular to the (3, 2) signature; it is absent in the case of (4, 1) signature.
The plan of this paper is as follows.In Section 2 we summarize some key properties of the (3, 2) signature theories whose supersymmetric solutions we classify.We also use spinorial geometry techniques to determine a simple canonical form for a single Majorana Killing spinor, the corresponding gauge-invariant spinor bilinears, and the associated stabilizer subgroup.In Section 3 we obtain the necessary and sufficient conditions for N = 2 supersymmetry obtained from the gravitino and gaugino Killing spinor equations, and prove that the geometric structure underpinning all such solutions is characterized by a nilpotent integrable endomorphism.In Section 4, we construct some examples, considering solutions for which the spinor bilinear is parallel, and also domain wall solutions for which all the gauge field terms vanish.We also construct the novel N = 6 descendant solution mentioned above, which is also a special case of a solution with parallel spinor bilinear.In Section 5 we present our conclusions, as well as a brief discussion on integrability conditions of the Killing spinor equations, and enhanced supersymmetry.There are four Appendices.In Appendix A we present our Clifford algebra conventions.In Appendix B we list the linear system associated with the gravitino Killing spinor equation, which is used to establish that the necessary conditions which are obtained from the covariant derivative of the spinor bilinear are in fact also sufficient for N = 2 supersymmetry in the gravitino equation.In Appendix C we present some further steps in the analysis of the necessary and sufficient conditions for N = 2 supersymmetry in the gaugino equation, and which are also useful in considering the integrability conditions of the Killing spinor equations.In Appendix D we present the analysis which is used to construct simple canonical forms for solutions with enhanced N = 4 supersymmetry, which relates to the classification constructed for a class of N = 4 supersymemtric solutions in [16].

Majorana Killing Spinors
In this section we examine the Killing Spinor Equations (KSE) of five-dimensional gauged supergravity in signature (+, +, +, −, −) coupled to arbitrary many vector multiplets.The gravitino equation is where k, g are constants and ∇ denotes the Levi-Civita connection.The gaugino KSE are 2) The 2-form H is related to the 2-form field strengths F I by H = X I F I , with F I = dA I , V J are constants, with A = V I A I and X = V I X I .The scalar fields X I are real, and satisfy the condition where X I = 1 6 C IJK X J X K and the (real) intersection numbers C IJK are symmetric in I, J, K. Further details of this formalism can be found in [21].
We shall proceed to consider the constants k and g appearing in these KSE.We first remark that the constants appearing in (2.1) have already been fixed in such a way as to ensure that the integrability conditions obtained by computing γ µ [D µ , D ν ]ǫ can be expanded out solely in terms of field equations and Bianchi identities.In particular, modifying the coefficient of the A µ term in (2.1) would produce from such a calculation a term linear in the gauge field strength which would not correspond to such a field equation or Bianchi identity.Furthermore, it is also straightforward to show that one must also take k ∈ R. To see this, it is necessary to consider the integrability conditions in more detail.These have already been computed for gauged supergravity coupled to vector multiplets in [21] for the case of the theory with signature (+, −, −, −, −); moreover the gamma matrices in that theory were also taken to satisfy the same conditions given in (A.8).It follows that the integrability conditions for the theory considered in this paper can be directly read off from those given in [21], on making the replacements F I → kF I , A I → kA I , A → kA, and χ → −g.In particular, the integrability conditions, assuming the Bianchi identities dF I = 0, imply that and where E µν and S I correspond to Einstein and scalar field equation terms respectively, and As we require that G Iµ = 0 should correspond to a gauge field equation obtained from a real Einstein-Maxwell-Chern-Simons action, we therefore impose that k ∈ R, and without loss of generality we therefore set k = 1.It then remains to consider g.If g ∈ R with g = 0, then requiring that (2.2) admits a Majorana Killing spinor ǫ implies that (2.7) In turn, this implies that ∂ µ XX I + X∂ µ X I = 0, and on contracting with X I this implies that X is constant, so in this case dH = 0. Furthermore, the gravitino equation also factorizes and implies that and hence it follows that the geometric conditions, and the conditions on H, obtained from the gravitino equation correspond to a special case of those obtained by considering those obtained from the analysis of the minimal ungauged supergravity (with additional conditions on the flux).
Conversely, if ig ∈ R (including the case g = 0), then [C * , D µ ] = 0 and [C * , A I ] = 0.In particular, for such choices of g, if ǫ is a Killing spinor then so is C * ǫ.Hence it follows that for solutions preserving the minimal N = 2 supersymmetry, the Killing spinors must be of the form {ǫ, iǫ} where ǫ is a Majorana spinor satisfying C * ǫ = ǫ.We shall in this work concentrate on the case for which ig ∈ R (including the case g = 0), and for which solutions with minimal supersymmetry are described by a single Majorana Killing spinor.

Majorana Spinor Orbits, Bilinears, and Stabilizer
To proceed, we next consider how to apply Spin(3, 2) gauge transformations in order to simplify a (single) Majorana spinor.A general spinor ǫ satisfying ǫ = C * ǫ is given by However as e π 2 γ 51 (e 1 + e 2 ) = 1 + e 12 , it follows that e 1 + e 2 is in the same orbit as 1 + e 12 .Furthermore, we also have and so it follows that 1 + e 12 ± (e 1 + e 2 ) is also in the same orbit as 1 + e 12 .So, there is a single Majorana spinor orbit.Hence, without loss of generality, by using appropriately chosen Spin(3, 2) gauge transformations, we can take a single Majorana spinor to be written as (2.11) We remark that there are only two Spin(3, 2) gauge-invariant spinor bilinears associated with this spinor.There is a 2-form ω with components where the B is defined by (A.17).One finds that The other spinor bilinear is the 3-form corresponding to the Hodge dual of ω, ⋆ω = −e 5 ∧ ω . (
It is also useful to consider the stability subgroup of Spin(3, 2) which leaves invariant this spinor.We shall solve the condition (2.15) The following conditions on λ µν are obtained: λ 51 + λ 54 = 0, λ 52 + λ 53 = 0, λ 12 − λ 34 + λ 13 − λ 24 = 0, λ 14 + λ 23 = 0 . (2.16) Hence, the stabilizer subgroup is 6-dimensional, and it will be convenient to use the following basis (2.17) and It is straightforward to show that the S i satisfy the Heisenberg algebra, n 3 , with the only non-zero [S, S] commutator being given by and the K j satisfy the sl(2, R) algebra, with the non-zero [K, K] commutators given by (2.20) The remaining non-zero [S, K] commutators are given by Hence, the stabilizer subgroup is the semidirect sum sl(2, R) ⋉ n 3 .

Analysis of the KSE
In this section we analyse the necessary and sufficient conditions for supersymmetry obtained from the KSE (2.1) and (2.2), for the case k = 1, ig ∈ R, taking the spinor ǫ to be Majorana, with ǫ = 1 + e 12 .As noted in the previous section, there is a single nontrivial spinor bilinear ω (2.12) (as well as the Hodge dual of ω).
To proceed, consider (3.19).On contracting with ξp , one obtains an expression for U p , and also on contracting with ξ p one obtains W p .On imposing that U p is imaginary, one finds the following condition on substituting this expression back into (3.19), the remaining content of this condition is ξ p τ r ξ n τ q ∇ q τn + ξn τ q ∇ q τ n − 1 2 ξ p τr ξn τ q ∇ q τ n + ξ n τ q ∇ q τn .
(3.28) Furthermore, the condition (3.28) is equivalent to 6igτ ℓ A ℓ = τ p ξq ∇ q τ p − ξ p τ q ∇ q τp − ξp τ q ∇ q τ p (3.29) and τ p τ q ∇ p τq = 0 . (3.30) Next, consider the condition (3.21), which is equivalent to and Further conditions are obtained by substituting the expressions for β 1 , β 2 , β 3 into (3.20):We remark that (3.35) implies that β 3 as given in (3.32) is imaginary, as required.A further geometric condition is obtained from the condition ∇ 5 τ 5 = τ ℓ H ℓ5 , together with (3.27) and also (3.29):To proceed further, it is useful to utilize a specific gauge choice associated with (3.26).In particular, one may without loss of generality choose a gauge with respect to which Furthermore, the condition (3.37) is equivalent to and the condition (3.30) is equivalent to requiring that the Lee form Θ of ω vanish, Further covariantization of the geometric conditions may be obtained by defining 1 If V is a 1-form, then we denote by V ♯ the vector field dual to V with respect to the metric.
It is straightforward to show that all of the geometric conditions are equivalent to N = 0, together with (3.39).However, we also note that (3.39) implies that and conversely, if (3.43) holds, there exists a gauge in which (3.39) holds.Hence, the geometric conditions are also equivalent to N = 0 together with (3.43).
It is also useful to consider the endomorphism J : T M → T M generated by ω given by for X, Y ∈ T M, which is nilpotent, J 2 = 0 (though ImJ = KerJ).It is straightforward to show that the geometric conditions imply that J is integrable, in the sense that if X, Y ∈ KerJ then [X, Y ] ∈ KerJ.In comparison, for integrability of an almost complex structure I, one requires that if X, Y in Ker(1+iI) then [X, Y ] ∈ Ker(1+iI).
Here, the integrability of J differs, presumably due to the fact that J 2 = 0 in contrast to I 2 = −1.Furthermore, the conditions N = 0 are equivalent to for all vector fields X and Y .The Nijenhuis tensor of J [22] is defined by on using J 2 = 0. Hence the conditions N = 0 are equivalent to As has been mentioned, this condition, together with (3.39) encodes all of the geometric conditions necessary and sufficient for the supersymmetry.Notably, the associated Haantjes tensor [23] vanishes identically, on using J 2 = 0, together with (3.47).This implies that certain linear combinations of eigenvectors of J generate integrable distributions [22,23]; it would be interesting to explore this further.
Next we consider the components of H. On using (3.27),H r5 can be simplified to where we have made use of the identity In addition, the conditions (3.22) and (3.32) imply that where F is a real function which is not determined by the KSE.

Gaugino KSE
we consider the gaugino KSE (2.2).These imply the following conditions: The condition (3.54) is equivalent to The case for which λ = 5 holds automatically as a consequence of (3.56).The case for which λ = p implies that (ω 1 ) p q (F I − X I H) 5q + ∂ q X I = 0 (3.58) and hence On setting σ = 5, ρ = p one obtains (3.58).Finally, we set σ = q, ρ = p in (3.60) we obtain On setting the condition (3.61) is equivalent to Furthermore τ q Φ I q = τ q Φ I q = 0 as a consequence of (3.56).It follows that where Υ I − ῩI = 0. Hence (3.61) is equivalent to The gaugino conditions are therefore equivalent to where α I are 1-forms satisfying and Λ I are 2-forms satisfying

Examples
In this section, we briefly consider some special classes of supersymmetric solutions.

Solutions with ∇ω = 0
To begin, consider the special case when ∇ω = 0.For such a case, the geometric condition N = 0 holds automatically.Furthermore, the geometric condition (3.43) implies, on using dω = 0, that gX = 0. Thus, generically, solutions with parallel ω only exist for the ungauged theory, g = 0.There is also considerable simplification to the H-flux conditions (3.50) and (3.52) which are equivalent to where F is a real function.Furthermore the gaugino conditions (3.66) also simplify to give where α I and Λ I are 1-and 2-forms satisfying (3.67) and (3.68) respectively.

Domain Wall Solutions
We next consider consider domain wall solutions for which all the gauge fields and potentials are set to zero.The geometric conditions are equivalent to which is equivalent to requiring that ω be a Killing-Yano 2-form also satisfying The conditions on the scalars are equivalent to which can be rewritten as In the ungauged theory, ω must be parallel, and the scalars satisfy ω ∧ dX I = 0.

A N = 6 Supersymmetric Descendant Preon Solution
Next, we shall explicitly construct a solution preserving exactly N = 6 supersymmetry.For the case of N = 6 supersymmetry, we take three linearly independent (over R) Majorana spinors {ǫ 1 , ǫ 2 , ǫ 3 }.These three spinors must be orthogonal to a normal Majorana spinor ψ with respect to B. By using Spin(3, 2) gauge transformations, one may choose the normal spinor to be ψ = 1 + e 12 , and consequently The solution we shall construct is an example of a descendant solution [19,20].In particular, for our solution, the gravitino equation preserves the maximal N = 8 supersymmetry, however the gaugino equation breaks the supersymmetry down from N = 8 to N = 6.For simplicity, we will work in the ungauged theory, setting g = 0, and we also set the scalars X I to be constant.However, we do not work in the minimal theory; in particular we take constants Y I such that X I Y I = 0, and set F I = Y I χ, for a real 2-form χ, so that H = 0.With this choice the gravitino equation simplifies to ∇ǫ = 0, and we choose a trivial geometry consistent with this, which is R 3,2 , with Imposing (4.9) for ǫ = ǫ i , i = 1, 2, 3, we find that there exists a non-zero solution for χ corresponding to taking χ = iqω for q ∈ R constant, q = 0, and where ω is given by (2.13).So, it is clear that the gaugino equation admits a nontrivial flux F I = 0 consistent with N = 6 supersymmetry.Furthermore, the amount of (gaugino) supersymmetry preserved by such a solution is exactly N = 6, because imposing (4.9) for ǫ = ǫ i , i = 1, 2, 3, 4 would force F I = 0. Furthermore, it is also straightforward to verify that the all components of the Einstein and gauge field equations hold, because in this case ω is parallel, and furthermore ω ∧ ω = 0, ω µλ ω ν λ = 0 (4.10) and the scalar field equations follow as a consequence of the integrability condition (2.4).Hence this solution, with F I = iqY I ω, preserves N = 8 supersymmetry from the perspective of the gravitino equation, but it is broken to exactly N = 6 supersymmetry by the gaugino equation.

Conclusion
We have obtained the necessary and sufficient conditions for solutions of (3, 2) signature supergravity, with ig ∈ R, coupled to arbitrary many vector multiplets, to preserve the minimal N = 2 supersymmetry: (i) The geometric conditions are N = 0 where N is defined by (3.42), as well as the condition dω = 3ig(Xe 5 − A) ∧ ω . (5.1) We have shown that the condition N = 0 is equivalent to where J is the nilpotent endomorphism J : T M → T M, satisfying J 2 = 0, given by g(JX, Y ) = −iω(X, Y ) .
(ii) The components of the H-flux are given by (3.50) and (3.52).In particular, we note that not the entirety of H is fixed by the gravitino KSE as there is a term proportional to ω which is projected out by the KSE.
In terms of the integrability conditions of the KSE, it is clear from the integrability condition (2.4), that if the gauge field equations hold, then supersymmetry implies that the scalar field equations also hold.Furthermore, the integrability condition (2.5) implies that if the gauge field equations hold, then we have where E = 0 is equivalent to the Einstein equations.As noted in Appendix C, the condition (2.5) is equivalent to which is not sufficient to impose vanishing of all components of the Einstein equation.Specifically, all components of the Einstein equation are forced to vanish, with the exception of the 3 (real) components associated with E τ τ and E τ τ .We note that this is in contrast to the cases found for Lorentzian signature supergravity [5], where for the null spinor orbit only one real component of the Einstein equations was unfixed by KSE integrability conditions.Due to the preponderance of additional null directions in the (3, 2) signature theory, it is perhaps unsurprising that more components of the Einstein equation are unfixed when compared to the Lorentzian theory.
Having classified the N = 2 solutions, it remains to examine the cases of enhanced supersymmetry for N = 4, N = 6 and N = 8.In the case of maximal supersymmetry N = 8, the gaugino equation (2.2) implies that and consequently such solutions reduce to solutions of the minimal supergravity.It remains to then consider the integrability conditions of (2.1) in the minimal theory, which can be read off from the integrability calculation in [24].If g = 0 then maximal supersymmetry requires that the coefficient multiplying the identity matrix in this condition vanish, which sets all of the fluxes to zero.Such a solution is therefore maximally symmetric.In the case g = 0 the analysis of the gravitino integrability conditions is more complicated, and one might expect that there be (3, 2) signature analogues of solutions such as AdS 2 × S 3 , as well as more esoteric solutions such as the Gödel solution found in [5].
For the case of N = 6 supersymmetry, we have already constructed a simple descendant solution which preserves exactly N = 6 supersymmetry.This was possible because the conditions imposed on the fluxes by the gaugino equation in (3, 2) signature turn out to be weaker than the gaugino conditions found in the Lorentzian theory.Again, as mentioned previously in the context of integrability of the gravitino equation, the fact that such an algebraic condition imposes weaker conditions for N = 6 supersymmetry when compared to the case of standard signature supergravities is unsurprising, due to the additional null directions present in the (3, 2) signature.In the case of standard Lorentzian supergravity, it has been shown that all solutions preserving N = 6 supersymmetry are locally isometric to N = 8 maximally supersymmetric solutions [25], following [26].However, it is possible to break this (global) N = 8 supersymmetry by taking appropriate quotients [27].It would be interesting to fully classify the N = 6 solutions in (3, 2) signature.We remark in particular that the N = 6 supersymmetric solution we constructed only exists in the non-minimal theory (e.g for the STU model).It remains to be determined if there exist N = 6 solutions in the minimal theory.