# CCNV Space-Times as Potential Supergravity Solutions

Authors: A. Coley; D. McNutt; N. Pelavas
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

It is of interest to study supergravity solutions preserving a nonminimal fraction of supersymmetries. A necessary condition for supersymmetry to be preserved is that the space-time admits a Killing spinor and hence a null or time-like Killing vector field. Any space-time admitting a covariantly constant null vector (CCNV) field belongs to the Kundt class of metrics and more importantly admits a null Killing vector field. We investigate the existence of additional non-space-like isometries in the class of higher-dimensional CCNV Kundt metrics in order to produce potential solutions that preserve some supersymmetries.


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## Body

## 1. Introduction

Supersymmetric supergravity solutions have been studied in the context of the AdS/CFT conjecture, the microscopic properties of black hole entropy, and the interconnection of string theory dualities. For example, in five dimensions, solutions preserving various fractions of supersymmetry ofN=2 gauged supergravity have been studied. The Killing spinor equations imply that supersymmetric solutions preserve 2, 4, 6, or 8 of the supersymmetries. The AdS5 solution with vanishing gauge field strengths and constant scalars preserves all of the supersymmetries. Half supersymmetric solutions in gauged five-dimensional supergravity with vector multiplets possess two Dirac Killing spinors and hence two time-like or null Killing vector fields. These solutions have been fully classified, using the spinorial geometry method, in [1–3]. Indeed, in a number of supergravity theories [4, 5], in order to preserve some supersymmetry, it is necessary that the space-time admits a Killing spinor which then yields a null or time-like Killing vector field (isometry) from its Dirac current. Therefore, a necessary (but not sufficient) condition for supersymmetry to be preserved is that the space-time admits a null or time-like Killing vector field.In this short communication we study supergravity solutions preserving a nonminimal fraction of supersymmetries by determining the existence of additional non-space-like isometries in the class of higher-dimensional Kundt space-time admitting a covariantly constant null vector field (CCNV) [6, 7]. CCNV space-time belongs to the Kundt class because it contains a null Killing vector field which is geodesic, nonexpanding, shear-free, and nontwisting. The existence of an additional isometry puts constraints on the metric functions and the vector field components. Killing vector fields that are null or time-like locally or globally (for all values of the coordinate v) are of particular importance. As an illustration we present two explicit examples in this paper.A constant scalar invariant (CSI) space-time is a space-time such that all of the polynomial scalar invariants constructed from the Riemann tensor and its covariant derivatives are constant. In three and four dimensions, it has been proven that all CSI space-times are either locally homogeneous or belongs to the degenerate Kundt class [8–10]; it is conjectured that this is true in higher dimensions as well. The VSI space-times are CSI space-times for which all of these polynomial scalar invariants vanish. The subset of CCNV space-times which are also VSI are of interest. Indeed, it has been shown previously that the higher-dimensional VSI space-time with fluxes and dilaton is solutions of type IIB supergravity [11–13]. The higher-dimensional Weyl typeN pp-wave space-times (Weyl and Ricci type are defined in terms of the alignment classification [14]) are known to be solutions in type IIB supergravity for an appropriate choice of an R-R five-form or an NS-NS form field [15–19].In fact, all Ricci typeN VSI space-times are solutions to supergravity and, moreover, there are VSI space-time solutions of type IIB supergravity which are of Ricci typeIII, including the string gyratons, assuming that appropriate source fields are provided [11–13]. It has been argued that the VSI supergravity space-time is exact string solutions to all orders in the string tension. Those VSI space-time in which supersymmetry is preserved admits a CCNV. Higher-dimensional VSI space-time solutions to type IIB supergravity preserving some supersymmetry are of Ricci typeN, Weyl typeIII(a), orN [11–13].It is also known thatAdSd×SN-d space-time is supersymmetric CSI solutions of IIB supergravity. There is an amount of another CSI space-time known to be solution of supergravity admitting supersymmetries [8–10], including generalizations of AdS×S [20], the chiral null models [15], and the string gyratons [21]. Some explicit examples of CSICCNV Ricci typeN supergravity space-time have been constructed [11–13]. The CSI space-times also contain the universal space-times [22]; a space-time isuniversal if every symmetric conserved rank 2 tensor which can be constructed from the metric, the Riemann tensor, and its covariant derivatives is proportional to the metric. This ensures that any quantum correction to such a classical solution is proportional to the metric and hence is a solution to all quantum gravity theories defined in terms of a gravitational Lagrangian. Therefore, a subset of the CSI space-time will be solutions to all quantum gravity theories defined in this manner. This suggests that the CSICCNV space-times presented in this paper will contain supergravity solutions.


### 1.1. Kundt Metrics and CCNV Space-Times

AnyN-dimensional space-time possessing a CCNV, l, must necessarily belong to the class of Kundt metrics. Local coordinates (u,v,xe) can be chosen, where l=∂v, so that the metric can be written [6, 7](1)ds2=2dudv+Hu,xedu+W^eu,xfdxe+gefu,xgdxedxf,where the metric functions are independent of the light-cone coordinate v.A Kundt metric admitting a CCNV is CSI if and only if the transverse metricgef is locally homogeneous [8–10]. Due to the local homogeneity of gef a coordinate transformation can be performed so that mie in (2) are independent of u; this implies that the Riemann tensor is of typeII or less [14]. If a CSI-CCNV metric satisfies RabRab=0, then the metric is VSI, and the Riemann tensor will be of typesIII,N, orO and the transverse metric is flat (i.e., gef=δef). The constraints on a CSICCNV space-time to admit an additional Killing vector field are obtained as subcases of the cases analyzed below where the transverse metric is a locally homogeneous Riemannian manifold.


## 1.1. Kundt Metrics and CCNV Space-Times

AnyN-dimensional space-time possessing a CCNV, l, must necessarily belong to the class of Kundt metrics. Local coordinates (u,v,xe) can be chosen, where l=∂v, so that the metric can be written [6, 7](1)ds2=2dudv+Hu,xedu+W^eu,xfdxe+gefu,xgdxedxf,where the metric functions are independent of the light-cone coordinate v.A Kundt metric admitting a CCNV is CSI if and only if the transverse metricgef is locally homogeneous [8–10]. Due to the local homogeneity of gef a coordinate transformation can be performed so that mie in (2) are independent of u; this implies that the Riemann tensor is of typeII or less [14]. If a CSI-CCNV metric satisfies RabRab=0, then the metric is VSI, and the Riemann tensor will be of typesIII,N, orO and the transverse metric is flat (i.e., gef=δef). The constraints on a CSICCNV space-time to admit an additional Killing vector field are obtained as subcases of the cases analyzed below where the transverse metric is a locally homogeneous Riemannian manifold.


## 2. CCNV Space-Time with Additional Isometries

Let us choose the coframema(2)m1=n=dv+Hdu+W^edxe,m2=l,mi=miedxe,where miemif=gef and miemje=δij. The frame derivatives are given by (3)l=D1=∂v,n=D2=∂u-H∂v,mi=Di=mie∂e-W^e∂v.The Killing vector field can be written as X=X1n+X2l+Ximi. A coordinate transformation is made to eliminate W^3 in (1) and we rotate the frame in order to set X3≠0 and Xm=0 [6]. X is now given by (4)X=X1n+X2l+χm3.Without loss of generality, we assume that the matrix mie is upper-triangular. As a note, the indices e,f,g,…, range from 3 to N while the indices m,n,r,p,… range from 4 to N.A subset of the Killing equation can then be written as follows:(5)X1,v=0,X1,u+X2,v=0,m3eX1,e+X3,v=0,mneX1,e=0,which imply (6)X1=F1u,xe,X2=-D2X1v+F2u,xe,X3=-D3X1v+F3u,xe,and the remaining Killing equations are (7)D2X2+∑iJiXi=0,(8)DiX2+D2Xi-JiX1-∑jAji+BijXj=0,(9)DjXi+DiXj+2BijX1-2∑kΓkijXk=0,where (10)Bij=mie,umje,Wi=mieW^e,Dijk≡2mie,fmjemkf,Ji≡Γ2i2=DiH-D2Wi-BjiWj,Aij≡DjWi+DkijWk,Γikj=-12Dijk+Djki-Dkij=ΓikjS.Further information can be found by taking the Killing equations and applying the commutation relations. This produces two cases: (1)  D3X1=0 or (2)  Γ3n2=Γ3n3=Γ3nm=0.Case 1 (D3X1=0).
Using (7) and the definition of F2 from (6), we have the fact that X1=c1u+c2. If c1≠0 we may always choose coordinates to set X1=u; while if c1=0 we may choose c2=1.Subcase 1 (F3=0).
(i)c1≠0, X1=u; F2 must be of the form (11)F2=f2xeu+g2uu.H and Wm are given in terms of these two functions (where g′≡dg/du) (12)H=f2xeu2-g2′uu+g2uu2,Wm=Bmxeu.
(ii)c1=0, X1=1; F2,u=0, and H and Wn are (13)H=F2xe+A0u,xr,Wn=∫DnA0du+Cnxe.In either case, the only requirement on the transverse metric is that it has to be independent of u. The arbitrary functions in this case are F2 and the functions arising from integration.Subcase 2 (F3≠0).
The transverse metric is now determined by(14)m33=-∫1X1F3,3du+A1x3,xr,(15)mnr,u=-mnr,3F3m33X1,m3r,u=-F3,rX1-m3r,3m33F3X1.
(i)c1≠0, X1=u; Fiu,xei=1,2 are arbitrary functions, H is given by (16)H=-D2F2-D2F322u-F3D3F2u-F3D3F322u2,and Wn is determined by(17)D2uWn+F3D3Wn+DnF2-uH=0.
(ii)c1=0, c2≠0, and X1=1; F2 and F3 satisfy (18)D2F2+F3D3F2+12D2F32+12F3D3F32=0.H may be written as (19)H=∫m33D2F3dx3+F2+12F32+A2u,xr. The only equation for Wn is (20)F3D3Wn+D2Wn=DnH.
(iii)X1=0.
We have the following constraints on the functionsmie:(21)F3,3=0,mnr,3=0,D2log⁡m33=-D3F2F3-D2log⁡F3,while the metric functions must be of the form (22)Wn=-∫m33DnF2F3dx3+Enu,xr,H-∫m33D2F2F3dx3+A3u,xr.There are two further subcases depending upon whether m33,r=0 or not; whence we may further integrate to determine the transverse metric.Case 2 (Γ3ia=0).
This implies that the upper-triangular matrixmie takes the form (23)m33=M,3u,x3,m3r=0,mnr=mnru,xr,while Wn must satisfy D3(Wn)=0. The remaining Killing equations simplify; in particular, BmnX1=0, which leads to two subcases: (1)  X1=0 or (2)  Bmn=0.Subcase 1 (X1=0, Bmn≠0).
F 2 , r = 0, F3,e=0, with mie, H, and Wn given by (21) and (22).Subcase 2 (B(mn)=0, X1≠0).
This case is similar to the subcases dealt with in Subcase 1 (see (11)–(14), (20)–(22)). For n<p, the vanishing of Bnp implies that mnr,u=0, the special form of mie implies that mr3=0, and the only nonzero component of the tensor B is B33.
If we assume thatF1,3≠0 and F1 is independent of xr(24)m33,3m33=F1,33F1,3,m33,um33=F1,3uF1,3,thus m33(u,x3) is entirely defined by F1. We may now solve for H and Wn(25)H=D3D2F1D3F12F3-D22F1D3F12F1-2D2F3D3F1,Wn=-DnF3D3F1.F3 is of the form (26)F3=∫m33F1D3D2F1D3F1dx3+A6u,xr.There are differential equations for F2 in terms of the arbitrary functions F1u,x3 and A6(u,xr). These solutions are summarized in Table 5.2 in [7].


## 3. Killing Lie Algebras

There are three particular forms for the Killing vector fields in CCNV space-times admitting an additional isometry:(A)
XA=cn+F2(u,xe)l+F3(u,xe)m3(B)
XB=un+F2u,xe-vl+F3u,xem3(C)
XC=F1u,x3n+F2u,xe-D2F1vl+F3u,xe-D3F1vm3.To determine if this space-time admits even more isometries we examine the commutator ofX with l in each case. In case (A) XA,l=0 and in case (B) XB,l=-l, implying that there are no additional Killing vector fields.In the most general caseYC≡XC,l can yield a new Killing vector field; YC=D2F1l+D3F1m3. However, this will always be space-like since D3F12>0. Note that YC,l=0, while, in general, YC,XC≠0.


### 3.1. Globally Non-Space-Like Killing Vector Fields

Let us consider the set of CCNV space-times admitting an additional non-space-like isometry. Equation (6) implies that the norm of this vector field must satisfy (27)D3X12v2+2D2X1X1-D3X1F3v+F32-2X1F2≤0.If the Killing vector field is non-space-like for all values of v, then D3X1 must vanish and X1 is constant. Therefore, those subcases with X1 nonconstant are excluded.We need only consider the Killing vector field of the formXA. In the time-like case, the subcases with X1=0 are no longer valid as this would imply that F32<0.In the case thatXA is null, if c=0, F3 must vanish and F2 must be constant, implying that X is a scalar multiple of l. If c≠0 we can rescale n so that 2F2=F32; we can then integrate out the various cases:(i)
IfF3=0, F2 must vanish as well and the Killing vector field, X, is proportional to n. The remaining metric functions are now H=A0u,xr and Wn=∫‍DnA0du+Cnxe. The transverse metric is unaffected.(ii)
IfF3≠0, the metric functions must satisfy (28)H=A2u,xr,D2Wn+D3WnF3=DnA2,log⁡m33,u=D2log⁡F3.In this section we have shown that if we require that the Killing vector field is non-space-like for all values ofv, this puts strong conditions on the permitted form of the Killing vector field. If we only require that the Killing vector field is non-space-like in a subset of the space-time there is greater diversity in the choice of Killing vector fields permitted. We present two explicit examples in the following subsections, one which is non-space-like in a region of the space-time and the other which is globally non-space-like. The CSICCNV space-times, admitting globally non-space-like Killing vector fields, are the above cases where the transverse space is locally homogeneous.


### 3.2. Example 1

We first present an explicit example for the case whereX1=u and F3≠0. Assuming that F3u,xi=ϵum33 and ϵ is a nonzero constant, we obtain(29)mis,u+ϵmis,3=0and the transverse metric is thus given by(30)mis=misx3-ϵu,xn.We have the algebraic solution(31)W^3=-1ϵH+F2,u-F2,3-ϵm332,where F2u,xi is an arbitrary function and H is given by(32)Hu,xi=1u-∫Sz,x3-ϵu+ϵz,xndz+Ax3-ϵu,xn,where A is an arbitrary function and S is given by(33)Su,x3,xn=uF2,uu+ϵuF2,3u+ϵ2um332u.Furthermore, the solution for W^n, n=4,…,N, is(34)W^nu,xi=1u-∫Tnz,x3-ϵu+ϵz,xmdz+Bnx3-ϵu,xm,where Bn are arbitrary functions and Tn is given by(35)Tnu,x3,xm=uF2u+ϵuF2,3+ϵ2um332,n+ϵm3nm33.In this example, the Killing vector field and its magnitude are given by(36)X=un+-v+F2l+ϵum33m3,XaXa=-2uv+2uF2+ϵum332.Clearly, the causal character of X will depend on the choice of F2u,xi and for any fixed u,xiX is time-like or null for appropriately chosen values of v. Moreover, (36) is an example of case (B); therefore the commutator of X and l gives rise to a constant rescaling of l and, in general, there are no more Killing vector fields.The additional isometry is only time-like or null locally (for a restricted range of coordinate values). However, the solutions can be extended smoothly so that the vector field is time-like or null on a physically interesting part of space-time. For example, a solution valid onu>0, v>0 (with F2<0), can be smoothly matched across u=v=0 to a solution valid on u<0, v<0 (with F2>0), so that the Killing vector field is time-like on the resulting coordinate patch.As an illustration, suppose thatm3s are separable as follows:(37)m3s=x3-ϵupshsxnand F2 has the form(38)F2=-ϵ2p3+1x3-ϵu2p3+1h32+gu,xn,where ps are constants and hs are g arbitrary functions. Thus, from (32)(39)H=-ϵ2x3-ϵu2p3-1x3-ϵp3+1uh32-g,u+u-1Ax3-ϵu,xn,and hence from (31)(40)W^3=-ϵ2p3ux3-ϵu2p3-1h32-ϵu-1Ax3-ϵu,xn.Lastly, (34) gives(41)W^n=ϵx3-ϵup3h32x3-ϵup32p3+1x3-ϵp3+32uh3,n-x3-ϵupnhn-g,n+u-1Bnx3-ϵu,xm.


### 3.3. Example 2

A second example belonging to Case1, that is, with X1=1, gives the same solutions to (30) for the transverse metric by assuming that F3(u,xi)=ϵm33 (although, in this case, the additional isometry is globally time-like or null). In addition, we have(42)W^3=∫H,3du+ϵ-1F2+f,where (u,xi), F2x3-ϵu,xn, and fxi are arbitrary functions. Last, the metric functions W^n are(43)W^nu,xi=∫uLnz,x3-ϵu+ϵz,xmdz+Enx3-ϵu,xm,with En arbitrary and Ln given by(44)Lnu,x3,xm=H,n+ϵ∫H,3ndu+f,n.The Killing vector field and its magnitude are(45)X=n+F2l+ϵm33m3,XaXa=2F2+ϵm332.Since F2 and m33 have the same functional dependence there always exists an F2 such that X is everywhere time-like or null. The Killing vector field (45) is an example of case (A) and thus X and l commute and hence no additional isometries arise. For instance, suppose that H=Hx3-ϵu,xn and f is analytic at x3=0 then (42) and (43) simplify to give (46)W^3=-ϵ-1H-F2-f,W^n=ϵ-1∑p=0∞∂n∂3pf0,xmx3p+1p+1!+Enx3-ϵu,xm.This explicit solution is an example of a space-time admitting 2 global null or time-like Killing vector fields, and thus it preserves a nonminimal fraction of supersymmetries.


## 3.1. Globally Non-Space-Like Killing Vector Fields

Let us consider the set of CCNV space-times admitting an additional non-space-like isometry. Equation (6) implies that the norm of this vector field must satisfy (27)D3X12v2+2D2X1X1-D3X1F3v+F32-2X1F2≤0.If the Killing vector field is non-space-like for all values of v, then D3X1 must vanish and X1 is constant. Therefore, those subcases with X1 nonconstant are excluded.We need only consider the Killing vector field of the formXA. In the time-like case, the subcases with X1=0 are no longer valid as this would imply that F32<0.In the case thatXA is null, if c=0, F3 must vanish and F2 must be constant, implying that X is a scalar multiple of l. If c≠0 we can rescale n so that 2F2=F32; we can then integrate out the various cases:(i)
IfF3=0, F2 must vanish as well and the Killing vector field, X, is proportional to n. The remaining metric functions are now H=A0u,xr and Wn=∫‍DnA0du+Cnxe. The transverse metric is unaffected.(ii)
IfF3≠0, the metric functions must satisfy (28)H=A2u,xr,D2Wn+D3WnF3=DnA2,log⁡m33,u=D2log⁡F3.In this section we have shown that if we require that the Killing vector field is non-space-like for all values ofv, this puts strong conditions on the permitted form of the Killing vector field. If we only require that the Killing vector field is non-space-like in a subset of the space-time there is greater diversity in the choice of Killing vector fields permitted. We present two explicit examples in the following subsections, one which is non-space-like in a region of the space-time and the other which is globally non-space-like. The CSICCNV space-times, admitting globally non-space-like Killing vector fields, are the above cases where the transverse space is locally homogeneous.


## 3.2. Example 1

We first present an explicit example for the case whereX1=u and F3≠0. Assuming that F3u,xi=ϵum33 and ϵ is a nonzero constant, we obtain(29)mis,u+ϵmis,3=0and the transverse metric is thus given by(30)mis=misx3-ϵu,xn.We have the algebraic solution(31)W^3=-1ϵH+F2,u-F2,3-ϵm332,where F2u,xi is an arbitrary function and H is given by(32)Hu,xi=1u-∫Sz,x3-ϵu+ϵz,xndz+Ax3-ϵu,xn,where A is an arbitrary function and S is given by(33)Su,x3,xn=uF2,uu+ϵuF2,3u+ϵ2um332u.Furthermore, the solution for W^n, n=4,…,N, is(34)W^nu,xi=1u-∫Tnz,x3-ϵu+ϵz,xmdz+Bnx3-ϵu,xm,where Bn are arbitrary functions and Tn is given by(35)Tnu,x3,xm=uF2u+ϵuF2,3+ϵ2um332,n+ϵm3nm33.In this example, the Killing vector field and its magnitude are given by(36)X=un+-v+F2l+ϵum33m3,XaXa=-2uv+2uF2+ϵum332.Clearly, the causal character of X will depend on the choice of F2u,xi and for any fixed u,xiX is time-like or null for appropriately chosen values of v. Moreover, (36) is an example of case (B); therefore the commutator of X and l gives rise to a constant rescaling of l and, in general, there are no more Killing vector fields.The additional isometry is only time-like or null locally (for a restricted range of coordinate values). However, the solutions can be extended smoothly so that the vector field is time-like or null on a physically interesting part of space-time. For example, a solution valid onu>0, v>0 (with F2<0), can be smoothly matched across u=v=0 to a solution valid on u<0, v<0 (with F2>0), so that the Killing vector field is time-like on the resulting coordinate patch.As an illustration, suppose thatm3s are separable as follows:(37)m3s=x3-ϵupshsxnand F2 has the form(38)F2=-ϵ2p3+1x3-ϵu2p3+1h32+gu,xn,where ps are constants and hs are g arbitrary functions. Thus, from (32)(39)H=-ϵ2x3-ϵu2p3-1x3-ϵp3+1uh32-g,u+u-1Ax3-ϵu,xn,and hence from (31)(40)W^3=-ϵ2p3ux3-ϵu2p3-1h32-ϵu-1Ax3-ϵu,xn.Lastly, (34) gives(41)W^n=ϵx3-ϵup3h32x3-ϵup32p3+1x3-ϵp3+32uh3,n-x3-ϵupnhn-g,n+u-1Bnx3-ϵu,xm.


## 3.3. Example 2

A second example belonging to Case1, that is, with X1=1, gives the same solutions to (30) for the transverse metric by assuming that F3(u,xi)=ϵm33 (although, in this case, the additional isometry is globally time-like or null). In addition, we have(42)W^3=∫H,3du+ϵ-1F2+f,where (u,xi), F2x3-ϵu,xn, and fxi are arbitrary functions. Last, the metric functions W^n are(43)W^nu,xi=∫uLnz,x3-ϵu+ϵz,xmdz+Enx3-ϵu,xm,with En arbitrary and Ln given by(44)Lnu,x3,xm=H,n+ϵ∫H,3ndu+f,n.The Killing vector field and its magnitude are(45)X=n+F2l+ϵm33m3,XaXa=2F2+ϵm332.Since F2 and m33 have the same functional dependence there always exists an F2 such that X is everywhere time-like or null. The Killing vector field (45) is an example of case (A) and thus X and l commute and hence no additional isometries arise. For instance, suppose that H=Hx3-ϵu,xn and f is analytic at x3=0 then (42) and (43) simplify to give (46)W^3=-ϵ-1H-F2-f,W^n=ϵ-1∑p=0∞∂n∂3pf0,xmx3p+1p+1!+Enx3-ϵu,xm.This explicit solution is an example of a space-time admitting 2 global null or time-like Killing vector fields, and thus it preserves a nonminimal fraction of supersymmetries.


## 4. Discussion

The CCNV space-times discussed in this paper will preserve a nonminimal fraction of supersymmetries, if they are solutions of some supergravity theory. To show that there are indeed CCNV space-times that are solutions to a supergravity theory, we note that there exist VSI and CSI space-times which are solutions to supergravity theories [11–13]. The CSI and VSI solutions admit a covariantly constant null vector (i.e., they are CCNV-CSI space-times) or are constructed from the warped product of a CCNV-VSI space-time and a locally homogeneous Riemannian manifold.The construction of CSI solutions for supergravity theories was motivated by the observation thatAdSd×SD-d is a supersymmetric exact solution of supergravity (for certain values of (D,d) and for particular ratios of the radii of curvature of the two space forms; in particular, d=5,D=10, and AdS5×S5). The more general D-dimensional product space-time Md×K(D-d) (in brief M×K) can be considered as a Freund–Rubin background. For example, for (D,d)=(11,4),(11,7), and (5,5) it is sufficient that M and K are Einstein. Since M×K is a Freund–Rubin background, if M is any Lorentzian Einstein manifold and K is any Riemannian Einstein manifold (with the same ratio of the radii of curvature as in the AdS×S case), then M×K will be a solution of some supergravity theory without any consideration of preservation of supersymmetry, where the fluxes are given purely in terms of the volume forms of the relevant factor(s). In general, M must have negative scalar curvature and K will have positive scalar curvature in order to satisfy the supergravity equations of motion.There are many examples of CSI space-times in the Freund–RubinM×K supergravity set. K could be a homogeneous space or a space of constant curvature. One must ask whether these CSI solutions preserve any supersymmetry. The condition for preservation of supersymmetry demands that M and K admit Killing spinors which imply the existence of Killing vectors. In this paper we have examined the class of space-time that will potentially admit more than one Killing spinor without requiring that it is solution to some supergravity theory. Noting that a CCNV space-time will be CSI if the transverse metric is locally homogeneous we can relate particular instances of the two examples presented in Section 3 to subcases of known CCNV-CSI supergravity solutions.For example, requiring that the transverse space is flat leads to the condition thatm33=1 and the two CCNV examples in Section 3 will contain the subclass of CCNV-VSIϵ=0 space-time admitting two time-like or null Killing vectors [11]. The Ricci type NCCNV-VSIϵ=0 space-time has been shown to be solution to superstring and heterotic string theory [15]. Similarly, imposing the condition that the transverse metric is locally homogeneous will yield CCNV-CSI space-time admitting two non-space-like Killing vectors which can be related to subcases of known CCNV-CSI solutions of supergravity theories. As a simple example in five dimensions, we may choose the transverse space to be S3 with unit radius; then for an appropriate choice of metric functions, the example in Section 3.3 corresponds to the metric given by (10) and (11) in [12] admitting an additional null or time-like Killing vector field. The metric, together with a constant dilaton and appropriate antisymmetric field, is an exact solution of bosonic string theory.Motivated by the examples of CCNV-CSI space-times given in the literature, it is worthwhile to ask if the CCNV space-time admitting additional Killing vector fields will contain supergravity solutions beyond the CSI or VSI examples. By removing the condition for the transverse space to be locally homogeneous, such a solution would preserve a nonminimal fraction of supersymmetries while reducing the number of space-like symmetries. This will be investigated in future work in the context of five-dimensional bosonic string theory.



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