Geometry of all supersymmetric four-dimensional N=1 supergravity backgrounds
aa r X i v : . [ h e p - t h ] J un Geometry of all supersymmetric four-dimensional N = 1 supergravity backgrounds U. Gran , J. Gutowski and G. Papadopoulos Fundamental PhysicsChalmers University of TechnologySE-412 96 G¨oteborg, Sweden DAMTP, Centre for Mathematical SciencesUniversity of CambridgeWilberforce Road, Cambridge, CB3 0WA, UK Department of MathematicsKing’s College LondonStrandLondon WC2R 2LS, UK
Abstract
We solve the Killing spinor equations of N = 1 supergravity, with four super-charges, coupled to any number of vector and scalar multiplets in all cases. We findthat backgrounds with N = 1 supersymmetry admit a null, integrable, Killing vec-tor field. There are two classes of N = 2 backgrounds. The spacetime in the firstclass admits a parallel null vector field and so it is a pp-wave. The spacetime of theother class admits three Killing vector fields, and a vector field that commutes withthe three Killing directions. These backgrounds are of cohomogeneity one with ho-mogenous sections either R , or AdS and have an interpretation as domain walls.The N = 3 backgrounds are locally maximally supersymmetric. There are N = 3backgrounds which arise as discrete identifications of maximally supersymmetricones. The maximally supersymmetric backgrounds are locally isometric to either R , or AdS . Introduction
Four-dimensional supergravity coupled to vector and scalar multiplets with N = 1 su-persymmetry, four supercharges, is a minimal supersymmetric extension of the standardmodel. Because of this, it has widespread applications in particle physics phenomenol-ogy. The theory has been developed in stages beginning from the construction of puresupergravity [1, 2]. The couplings to the vector and scalar multiplets were added later ,see e.g. [3] and references within.In recent years and following the work of Paul Tod [4], there has been much interestin the systematic understanding of supersymmetric configurations of supergravity the-ories. In lower-dimensional supergravities, the focus has been on the classification ofsupersymmetric solutions of four- and five-dimensional theories with more than 8 super-charges, see e.g. [5, 6, 7]. Special supersymmetric solutions of N = 1 four-dimensionaltheories are also known. These include the stringy cosmic strings [8, 9, 10], domain walls[11, 12, 13] and pp-waves.In this paper, we solve the Killing spinor equations of four-dimensional N = 1 super-gravity coupled to any number of vector and scalar multiplets in all cases . For this weuse the spinorial geometry approach of [14]. We find that there are backgrounds with N = 1, N = 2, N = 3 and N = 4 supersymmetry. The spacetime metric of backgroundswith N = 1 supersymmetry admits an integrable, null, Killing vector field. Adaptingappropriate coordinates, the metric is given in (3.11) and (3.13). There are two kinds of N = 2 backgrounds. One admits a parallel null, Killing vector field and the metric is thatof a pp-wave. The other admits three Killing vector fields and an additional vector fieldthat commutes with the three Killing ones. The metric is given in special coordinates(4.18). These backgrounds are of cohomogeneity one with homogeneous sections either R , or AdS . The N = 3 backgrounds are locally maximally supersymmetric. How-ever, we have shown by adapting the results of [16] that there are N = 3 backgroundswhich arise as discrete quotients of maximally supersymmetric ones. The maximallysupersymmetric backgrounds are locally isometric to either R , or AdS .This paper has been organized as follows. In section two, we state the Killing spinorequations which arise from the supersymmetry variation of the fermions of the supergrav-ity theory. In section three, we solve the Killing spinor equations of N = 1 backgroundsand describe the geometry of spacetime. In section four, we investigate the solution ofthe Killing spinor equations for N = 2 backgrounds. In section five, we show that the N = 3 backgrounds are locally maximally supersymmetric and that the N = 4 back-grounds are locally isometric to either R , or AdS . In section six, we give an exampleof an N = 3 background which can be constructed as discrete identification of AdS andin section seven we give our conclusions. In appendix A, we present the integrabilityconditions of the Killing spinor equations. The theory has appeared in the literature in many different conventions. We shall mostly followthose of [3], page 212. Killing spinor equations
The Killing spinor equations can be read off from the supersymmetry transformationsof N = 1 supergravity. There are three Killing spinor equations associated with thesupersymmetry transformations of the fermions of the gravitational, gauge and scalarmultiplets, respectively. After some apparent changes in notation from that of [3], theKilling spinor equations of N = 1 supergravity can be written as follows:The gravitino Killing spinor equation is2[ ∇ µ ǫ L + 14 ( ∂ i K D µ φ i − ∂ ¯ i K D µ φ ¯ i ) ǫ L ] + ie K W γ µ ǫ R = 0 , (2.1)the gaugino Killing spinor equation is F aµν γ µν ǫ L − iµ a ǫ L = 0 , (2.2)and the Killing spinor equation associated with the scalar multiplet is iγ µ ǫ R D µ φ i − e K G i ¯ j D ¯ j ¯ W ǫ L = 0 , (2.3)where ∇ is the spin connection, φ i is a complex scalar field, K = K ( φ i , φ ¯ j ) is the K¨ahlerpotential of the (K¨ahler) scalar or sigma model manifold S , G i ¯ j = ∂ i ∂ ¯ j K , W = W ( φ i )is a (local) holomorphic function on S , D i W = ∂ i W + ∂ i KW , D µ φ i = ∂ µ φ i − A aµ ξ ia , (2.4) ξ a are holomorphic Killing vector fields on S , A a is the gauge connection with fieldstrength F a and µ a is the moment map, i.e. G i ¯ j ξ ¯ ja = i∂ i µ a . (2.5)We mostly follow the metric and spinor conventions of [3]. In particular, the spacetimemetric has signature mostly plus, ǫ is a Majorana spinor and ǫ L,R = (1 ± γ ) ǫ , where γ = 1. We have set the gauge coupling to 1.The gravitino Killing spinor equation is a parallel transport equation for a connectionwhich, apart from the Levi-Civita part, contains additional terms that depend on thematter couplings. The gauge group of the Killing spinor equations is Spin c (3 ,
1) =
Spin (3 , × Z U (1). The Spin (3 ,
1) acts on ǫ with the Majorana representation while U (1) acts on the chiral component ǫ L with the standard 1-dimensional representationand on the anti-chiral ǫ R with its conjugate. The additional U (1) gauge transformation isdue to the coupling of the spinor ǫ to the U (1) connection constructed from the K¨ahlerpotential K associated with the matter couplings. In what follows, we use only the Spin (3 ,
1) component of the gauge group to choose the representatives of the Killingspinors. Incidentally, the holonomy of the supercovariant connection is contained in
P in c (3 , U (1) component inthe holonomy group is again due the the K¨ahler potential coupling mentioned above.2 N=1 backgrounds
The starting point in the spinorial geometry approach [14] to solving Killing spinorequations is the choice of a normal form for the Killing spinors up to gauge transforma-tions. We have already mentioned that the gauge group is
Spin c (3 , Spin (3 ,
1) = SL (2 , C ) and the chiral (Weyl) representation is identified with the stan-dard representation of SL (2 , C ) on C . The Majorana representation which is relevanthere is simply ⊕ ¯ with ¯ the complex conjugate of . Using the explicit realization ofspinors in terms of forms, the chiral representation is identified with even forms, Λ ev ( C ),and the anti-chiral with odd ones, Λ odd ( C ). Introducing a Hermitian basis ( e , e ) in C with respect to a Hermitian inner product < · , · > , a basis in Λ ev ( C ) is (1 , e ), e = e ∧ e , and a basis in Λ odd ( C ) is ( e , e ). In particular, the gamma matrices acton Λ ev ( C ) and Λ odd ( C ) asΓ = − e ∧ + e y , Γ = e ∧ + e y , Γ = e ∧ + e y , Γ = i ( e ∧ − e y ) , (3.1)where y is the adjoint operation of the form skew-product. For later use, we also adopta light-cone Hermitian basis in the space of spinors as γ + = √ e y , γ − = √ e ∧ ,γ = √ e ∧ , γ ¯1 = √ e y . (3.2)There is one orbit of SL (2 , C ) on Λ ev ( C ), and so the chiral component of ǫ can be chosenas 1. In this basis, the Majorana inner product is given by B ( η , η ) = < Γ η ∗ , η > , (3.3)where < · , · > is the Hermitian inner product on C extended on Λ ⋆ ( C ), and η , η ∈ Λ ⋆ ( C ). Observe that B is a bi-linear. The spacetime forms constructed as spinor bi-linears are defined as τ µ ...µ k = B ( η , γ µ ...µ k η ) , k = 0 , . . . , . (3.4)The Dirac inner product in the (3.1) basis is D ( η , η ) = < Γ η , η > . Equating theDirac and Majorana conjugates, one finds that the complex conjugation operation isimposed by the anti-linear map, C = − Γ ∗ , C = 1. Applying this to the spinor 1, onefinds that a Majorana representative for the orbit is ǫ = 1 + e , ǫ L = 1 , ǫ R = e . (3.5)This can be chosen as the first Killing spinor of the theory. The isotropy group of thespinor 1 in SL (2 , C ) is C . This will be used later to choose the second Killing spinor.3 .2 Solution to the Killing spinor equations Evaluating the gravitino equation on the Killing spinor ǫ = 1 + e , we find that − Ω + , + − + Ω + , + 12 ( ∂ i K D + φ i − ∂ ¯ i K D + φ ¯ i ) = 0 , Ω + , +1 = 0 , − Ω − , + − + Ω − , + 12 ( ∂ i K D − φ i − ∂ ¯ i K D − φ ¯ i ) = 0 , − , +¯1 + √ ie K W = 0 − Ω , + − + Ω , + 12 ( ∂ i K D φ i − ∂ ¯ i K D φ ¯ i ) = 0 , Ω , +¯1 = Ω ¯1 , +¯1 = 0 , − Ω ¯1 , + − + Ω ¯1 , + 12 ( ∂ i K D ¯1 φ i − ∂ ¯ i K D ¯1 φ ¯ i ) + √ ie K W = 0 , (3.6)where Ω is the spin connection of the four-dimensional spacetime metric.The gaugino equation (2.2) acting on 1 + e gives F a +1 = F a + − = 0 , F a − iµ ( a ) = 0 , (3.7)and similarly the Killing spinor equation of the scalar multiplet (2.3) gives D + φ i = 0 , √ i D φ i = e K G i ¯ j D ¯ j ¯ W . (3.8)The equations (3.6)-(3.8) is the linear system associated with the N = 1 supersymmetricbackgrounds.To solve the linear system, substitute D + φ i = 0 into (3.6) to find that the gravitinoequations can be rewritten asΩ + , + − = Ω + , = Ω + , +1 = Ω − , − + = Ω , +¯1 = Ω , +1 = Ω − , +1 + Ω , + − = 0 , (3.9)and Ω − , + 12 ( ∂ i K D − φ i − ∂ ¯ i K D − φ ¯ i ) = 0 ,i √ e K W + 2Ω − , +¯1 = 0 , Ω − , +1 + Ω , + 12 ( ∂ i K D φ i − ∂ ¯ i K D φ ¯ i ) = 0 . (3.10)In what follows, we explore the consequences of the above conditions on the geometryof spacetime. To proceed, write the metric in a light-cone Hermitian frame as ds = 2 e − e + + 2 e e ¯1 . (3.11)4he spacetime form bilinears associated with the Killing spinor, see (3.4), are propor-tional to e − and e − ∧ ( e + e ¯1 ), and their spacetime duals. Setting e − = X µ dy µ , it is easyto see that (3.9) implies that ∇ ( µ X ν ) = 0 , e − ∧ d e − = 0 , e − ∧ e ¯1 ∧ d e = 0 . (3.12)Observe also that e − ∧ e ∧ d e = 0.The first condition in (3.12) implies that the metric admits a null Killing vector field.While the second implies that the distribution defined by X is integrable. As a resultthe metric can be written as in (3.11) with e − = Hdu , e + = dv + V du + w i dx i , e = β dx + β dx , (3.13)where u, v, x i , i = 1 ,
2, are real coordinates,
H, V, w i are real spacetime functions inde-pendent of v and β , β are complex spacetime functions. Substituting these into thelast condition in (3.12), we find that the frame e and so its complex conjugate e ¯1 canbe chosen independent of v .In fact, the basis given in (3.13) can be simplified further; one can work in a gaugefor which w = w = 0 in e + . To see how such a gauge may be chosen, consider the Spin (3 ,
1) gauge transformation generated by Rγ +1 + ¯ Rγ +¯1 for R ∈ C , which leavesinvariant 1 + e . It is straightforward to see that this gauge transformation correspondsto the following change of basis e − → e − e + → e + − | R | e − − R e − R e ¯1 e → e + 2 R e − e ¯1 → e ¯1 + 2 ¯ R e − . (3.14)By making such a gauge transformation, one can set w = w = 0 in e + . Finally, aco-ordinate transformation in x , x can be used to eliminate the du term from e . Thebasis is then given by (3.13), with w = w = 0.The last two conditions in (3.10) can be rewritten as √ e K W e − − ⋆ ( e ∧ d e − ) = 0 ,⋆d ( e − ∧ e ¯1 ) − √ e K ¯ W e − − i ∂ i K D φ i − ∂ ¯ i K D φ ¯ i ) e − = 0 , (3.15)where the orientation of the spacetime is chosen as ǫ − +1¯1 = − i . The first condition in(3.10) cannot be written in a more covariant form. However, if one takes the fields to beindependent of u , then the connection part vanishes.To solve (3.7), one can locally always choose the gauge A a + = 0. The first twoconditions in (3.7) will then imply that the remaining components of A are independentof v . There is no general procedure to give an explicit solution for the last condition(3.7).Next turn into the conditions (3.8) that arise from the Killing spinor equations of thematter multiplet. In the gauge A a + = 0, the first condition in (3.8) implies that the scalarfields can be taken to be independent of v , ∂ v φ = 0. The last condition in (3.8) can beinterpreted as a holomorphic flow equation. The construction of explicit solutions willdepend on the form of the K¨ahler potential and W , and so of the details of the model.5 N=2 backgrounds
The first Killing spinor is the same as that of the N = 1 case investigated above. So weset ǫ = ǫ , where ǫ is given in (3.5). To choose the second Killing spinor, consider themost general Majorana spinor ǫ = a be + C ( a be ) , a, b ∈ C . (4.1)The isotropy group of ǫ in Spin (3 ,
1) is C . This can be used to simplify the expressionfor ǫ . There are two cases to consider. If b = 0, the C isotropy transformation leaves ǫ invariant and ǫ = a ae . (4.2)Linear independence of ǫ and ǫ requires that Im a = 0.Next suppose that b = 0. After a C transformation with parameter λ , one has ǫ ′ = ( a + λb )1 + be + C [( a + λb )1 + be ] . (4.3)Setting λ = − ab , one can choose the normal form of ǫ as ǫ = be − ¯ be . (4.4)So the second Killing spinor can be chosen either as in (4.2) or as in (4.4) with a, b promoted to complex spacetime functions. ǫ = a ae Consider first the case for which ǫ = a ae . The linear system is easy to read offfrom that of the N = 1 case. In particular, the supercovariant connection along the − light-cone direction gives 2 a Ω − , +¯1 + i √ ae K W = 0 . (4.5)Comparing this condition with those of the N = 1 case, one concludes that either W = 0on the field configurations φ of the solution or a = ¯ a . If the latter is the case, then itturns out that a is also constant and so ǫ is not linearly independent from ǫ . It remainsto choose W = 0. In such a case, one finds that the parameter a is constant, i.e. a ∈ C ,and the additional conditions to those of N = 1 areΩ − , +1 = 0 , D φ i = 0 , W = 0 . (4.6) This does not imply that W vanishes. It means that W vanishes on the solution for φ . N = 1 backgrounds, we find that the gravitino and matterKilling spinor equations giveΩ + , + − = Ω + , = Ω + , +1 = Ω − , − + = Ω , +¯1 = Ω , +1 = Ω − , +1 = Ω , + − = 0 , (4.7)and Ω − , + 12 ( ∂ i K D − φ i − ∂ ¯ i K D − φ ¯ i ) = 0 , Ω , − ∂ ¯ i K D φ ¯ i = 0 ,W = ∂ j W = 0 , D φ i = D + φ i = 0 . (4.8)There are no additional conditions that arise from the gaugino Killing spinor equationapart from those that we have found in the N = 1 case (3.7). ǫ = be − ¯ be Next consider the case where ǫ = be − ¯ be . The gravitino Killing spinor equation gives ∂ + b = 0 , b Ω + , − + ¯ b Ω − , +¯1 = 0 ,∂ − b − Ω − , b = 0 , Ω − , − = 0 ,∂ b − b (Ω , − + + Ω + , − + Ω , ) = 0 , Ω , − = 0 ,∂ ¯1 b − b Ω ¯1 , = 0 , Ω ¯1 , − = 0 , (4.9)where we have used the N = 1 relations to simplify the expressions. Moreover thegaugino Killing spinor equation gives F a − = 0 , F a + iµ a = 0 . (4.10)In addition, the Killing spinor equation associated with the matter multiplet gives D − φ i = 0 , i √ b D ¯1 φ i + be K G i ¯ j D ¯ j ¯ W = 0 . (4.11) ǫ = a ae The geometric constraints (4.7) imply that X = e − is covariantly constant with respectto the Levi-Civita connection. So the spacetime admits a parallel null Killing vector field.Such a spacetime has an interpretation as a pp-wave. Note, however, that the cosmicstring solutions [8] and their generalizations [9, 10] also admit a null parallel vector fieldand so belong to this class of solutions. In particular, one can choose co-ordinates v, u such that X = ∂∂v is a Killing vector, and e − = du , i.e. the frame can be chosen as in(3.13) with H = 1. We have used the same symbol X to denote the one-form and thedual vector field.The investigation of remaining conditions is similar to that of the N = 1 case. Inparticular the first condition in (4.8) does not have a straightforward interpretationunless one takes the fields to be independent of u . In such a case the connection termvanishes. The second condition in (4.8) can be written as ⋆ d ( e − ∧ e ) − i ∂ i K D φ i − ∂ ¯ i K D φ ¯ i ) e − = 0 . (4.12)7he conditions on W in (4.8) imply that the solution for the scalars should be chosensuch that the superpotential W and its first derivative vanish.The restrictions on φ in (4.8) can be interpreted as light-cone pseudo-holomorphicityconditions. However notice that the light-cone almost-hermitian distribution ( e + , e ) isnot integrable in general. However if one takes the fields to be independent of u , ( e + , e )is integrable and D + φ i = D φ i = 0 are light-cone holomorphicity conditions. Moreoverin such a case, one can always choose a gauge locally such that A a + = A a = 0, since F +1 = 0, and so write ∂ + φ i = ∂ φ i = 0. ǫ = be − ¯ be To analyze the conditions (4.9) which arise from the Killing spinor equations in this case,it is convenient to define the 1-forms X = e − , Y = | b | e + , Z = ¯ b e + b e ¯1 , W = i ¯ b e − ib e ¯1 . (4.13)Observe that Z is orthogonal to X, Y, W , and W is orthogonal to X, Y, Z . Then it isstraightforward to show that the Killing spinor equations imply that X , Y and Z are allKilling vectors. Furthermore, W is closed, dW = 0. In addition, one finds the followingconstraints on the vector field commutators:[ W, X ] = [
W, Y ] = [
W, Z ] = 0 (4.14)and [
X, Y ] = cZ, [ X, Z ] = − cX, [ Y, Z ] = 2 cY , (4.15)where c = b (Ω − , +1 − Ω + , − ) and we use the same symbols to denote the vector fieldstheir dual one-forms.Consider the commutator [ X, Y ] = cZ . Since W commutes with the other threevector field, the Jacobi identity implies that W c = 0. Similarly, the Jacobi identity for
Z, X and Y together with the linear independence of these three vector field imply that Xc = Y c = Zc = 0. So c can be taken to be a constant.Next, since Z and W commute one can choose coordinates x, y such that Z = ∂ x and W = ∂ y . Moreover, the rest of the commutators imply that there are additionalcoordinates u , v such that X = e cx ∂ v , Y = e − cx (cid:18) ( c v + 2 cλ ( u ) v + ν ( u )) ∂ v + ( cv + λ ( u )) ∂ x + ρ ( u ) ∂ y + ∂ u (cid:19) , (4.16)where λ, ν and ρ are arbitrary functions of u . The functions λ and ρ can be eliminatedusing a u -depedent shift transformation in v and y . The resulting expression for Y is asin (4.16) with l = ρ = 0. The rest of the vector fields remain unchanged. Using (4.13),one can compute the frame in terms of the coordinates x, y, v, u to find e − = e cx | b | du , e + = e − cx ( dv − ( c v + ν ) du ) , e = b [( dx − idy ) − cvdu ] , e ¯1 = ¯ b [( dx + idy ) − cvdu ] . (4.17)8ence the spacetime metric can be written as ds = 2 | b | [ ds ( M ) + dy ] , (4.18)where ds ( M ) = du ( dv − ( c v + ν ) du ) + ( dx − cvdu ) , (4.19)and ν is a function of u , ν = ν ( u ). However, by direct examination of the Riemanncurvature tensor, we find that the 3-manifold with metric ds ( M ) is either R , if c = 0,or AdS if c = 0.The function b depends only on y , satisfying dbdy = √ | b | e K W + 1 √ e K b (cid:0) b∂ i KG i ¯ j D ¯ j ¯ W − ¯ b∂ ¯ i KG ¯ ij D j W (cid:1) . (4.20)If b is taken to be real, the above equation can be further simplified to write d log bdy = √ e K Re W , i Im W + 12 (cid:0) ∂ i KG i ¯ j D ¯ j ¯ W − ∂ ¯ i KG ¯ ij D j W (cid:1) = 0 . (4.21)Clearly, the spacetime is of cohomogeneity one with homogenous section either AdS or R , . So this class of N = 2 solutions can be thought of as domain wall spacetimes.The gaugino Killing spinor equation implies that F a = 0 , µ a = 0 . (4.22)So the gauge connection is flat and can locally be set to zero. The vanishing of themoment map restricts the scalars to lie on a K¨ahler quotient of S .The scalars φ i are independent of v . Since we have set A = 0 locally, the additionalconstraints on D φ i imply that ∂ x φ i = ∂ u φ i = 0. Moreover, the remaining Killing spinorequations of the scalar multiplet (4.11) gives dφ i dy = −√ be K G i ¯ j D ¯ j ¯ W . (4.23)Observe that this expression depends on b . This is again a flow equation driven bythe holomorphic potential W . One can change parameterisation to simplify the flowequations (4.20) and (4.23). The construction of explicit solutions depends on the detailsof the models. To find the Killing spinors of N = 3 backgrounds, we use the gauge group to bringthe normal to the Killing spinors to a canonical form as in [15]. Since there is a singleorbit of Spin (3 ,
1) on the space of Majorana spinors, we can always choose the normaldirection to the three Killing spinors to be i ( e + e ) with respect to the Majorana inner9roduct, A ( ζ , η ) = < Γ ζ ∗ , η > , where <, > is the standard Hermitian inner product.The orthogonal directions to i ( e + e ) are { η r } = { e , e − e , i ( e + e ) } . So thethree Killing spinors can be chosen as ǫ r = X s f rs η s , r, s = 1 , , , (5.1)where ( f rs ) is a real 3 × ǫ = f η .In the N = 4 backgrounds, the Killing spinors can again be written as a linearcombination of the basis { e , i (1 − e ) , e − e , i ( e + e ) } of Majorana spinors withreal spacetime functions as coefficients. Next we shall solve the Killing spinor equationsfor both cases. Let us begin with the N = 3 case. We shall first solve the Killing spinor equationslocally. To proceed observe that (5.1) implies that schematically ǫ L = f η L and ǫ R = f η R .Substituting this into the gaugino (2.2) and chiral (2.3) Killing spinor equations, onefinds that the dependence on ( f ) can be eliminated, because f is invertible. Moreoverthe conditions that one obtains are those of (3.7), (3.8), and (4.10) and (4.11) for b = 1and b = i . These imply that F aµν = D µ φ i = D i W = µ a = 0 . (5.2)Since the gauge connection is flat, we can locally set the gauge potential to vanish, A aµ = 0. As a result the second equation implies that φ are constant. Substituting thesedata into the gravitino Killing spinor equation, and taking its integrability condition, wefind that R µν,ρσ γ ρσ η L + 2 e K W ¯ W γ µν η L = 0 . (5.3)Clearly the integrability condition takes values in spin (3 , Spin (3 ,
1) is the identity, (5.3) implies that R µν,ρσ = − e K W ¯ W ( g µρ g νσ − g µσ g νρ ) . (5.4)It is easy to see that (5.2) and (5.4) are precisely the conditions that one gets for back-grounds that admit N = 4 supersymmetries. So one concludes that N = 3 backgroundsadmit locally an additional supersymmetry and so are locally maximally supersymmet-ric. Furthermore (5.4) implies that the spacetime is either R , or AdS . In the formercase, e K | W | = 0 and in the latter e K | W | = 0 when evaluated at the constant maps φ ,respectively.The moment map condition in (5.2), µ a = 0, together with the remaining constantgauge transformations imply that the constant maps φ take values in a K¨ahler quotientof the sigma model target space S . It remains to investigate D i W = 0. Suppose thatwe have chosen some constant maps φ = φ . If W ( φ ) = 0, then D i W = 0 implies that ∂ i W ( φ ) = 0. So W and its first derivative vanish at φ = φ . On the other hand if W ( φ ) = 0, D i W = 0 relates the value of the first derivative of W to that of the K¨ahlerpotential at φ = φ . 10 Supersymmetric Quotients
Supersymmetric solutions of N = 1 four-dimensional supergravity theories can be con-structed by taking quotients of maximally supersymmetric solutions with respect to adiscrete subgroup of the isometry group. Here we shall not investigate all possible cases,instead we shall present an explicit construction of an N = 3 background from a discretequotient of AdS . A similar question has been raised in [16] in the context of N = 2supergravity theory. To proceed, consider the gravitino Killing spinor equation equationfor an N = 3 solution which is locally isometric to AdS . We take the gauge connectionto be trivial and so the scalars to be constant. As W and K are constant, it is convenientto set W = − iRe iθ (6.1)for real R , θ , with R >
0. Furthermore, define ℓ by ℓ = e − K R (6.2)and set ˆ ǫ = e − iθ ǫ L + e iθ ǫ R . (6.3)Observe that ˆ ǫ is Majorana. Then the Killing spinor equation implies that ∇ µ ˆ ǫ + 12 ℓ γ µ ˆ ǫ = 0 . (6.4)The general solution to this equation has been constructed in [16] using the same nota-tion. In particular, one defines the following real basis for AdS : e = ℓ cosh ρ ( dt + 12 r dx ) , e = ℓ r cosh ρdx , e = ℓdρ , e = ℓr cosh ρdr , (6.5)for x, ρ ∈ R , t ∈ [0 , π ), r >
0. The smooth quotient is obtained by making theidentification x ∼ x + 2 k . In order to demonstrate how taking this quotient breaks thesupersymmetry from N = 4 to N = 3, it suffices to exhibit four Majorana spinors whichare globally well-defined on AdS , such that three of these spinors remain globally well-defined in the quotient geometry, whereas the fourth fails to be globally well-defined.These Majorana spinors can be read off directly from equation (24) of [16]:ˆ ǫ = e iπ (cid:18) r (cosh ρ − i sinh ρ e ) + 2 r (sinh ρ − i cosh ρ e − e ) (cid:19) , ˆ ǫ = 2 e it (cosh ρ i sinh ρ − e it (sinh ρ i cosh ρ e + 2 e − it (cosh ρ − i sinh ρ e + 2 e − it (sinh ρ − i cosh ρ e , ǫ = 2 ie it (cosh ρ i sinh ρ − ie it (sinh ρ i cosh ρ e − ie − it (cosh ρ − i sinh ρ e − ie − it (sinh ρ − i cosh ρ e , ˆ ǫ = ie iπ (cid:18) r (1 − ir x )(cosh ρ − i sinh ρ − r (1 + ir x )(sinh ρ − i cosh ρ e − r (1 − ir x )(sinh ρ − i cosh ρ e − r (1 + ir x )(cosh ρ − i sinh ρ e (cid:19) . (6.6)Clearly, ˆ ǫ , ˆ ǫ and ˆ ǫ remain well-defined on making the identification x ∼ x + 2 k .However, as ˆ ǫ contains terms linear in x , ˆ ǫ fails to be globally well-defined in thisquotient of AdS , and hence this solution is an N = 3 solution. It may worth re-investigating the number of supersymmetries preserved by this solutions after introducingappropriate flat but no trivial gauge and scalar fluxes. We have solved the Killing spinor equations of N = 1 supergravity coupled to anynumber of vector and scalar multiplets. In particular, we have determined the geometryof spacetime in all cases. We have shown that there are backgrounds with any numberof supersymmetries ranging from N = 1 to N = 4. N = 1 backgrounds admit asingle null, integrable, Killing vector. N = 2 backgrounds admit either a single parallel,null, vector field or three Killing vector fields. In the former case, the spacetime hasan interpretation as a pp-wave. In the latter, the metric can be written in specialcoordinates, and the spacetime is of co-homogeneity one with homogenous section either R , or AdS . Such backgrounds can be thought of as domain walls. N = 3 backgroundsare locally maximally supersymmetric. In addition there are backgrounds which admit N = 3 supersymmetry which can be constructed as discrete identifications of maximallysupersymmetric ones. The maximally supersymmetric backgrounds are locally isometricto either R , or AdS .We have not been able to solve explicitly all the equations. Supersymmetry imposesstrong restrictions in all backgrounds which admit more than one supersymmetry, N > N = 2 domain wall back-grounds are associated with standard flow equations. Explicit solutions can be obtainedfor special models. Although, we have given an example of an N = 3 background whichcan be constructed as discrete identification of a maximally supersymmetric one basedon [16], we have not investigated all N = 3 backgrounds that can be obtained in thisway. It may be possible to construct all such backgrounds utilizing the results of [17]. Acknowledgements
The work of UG is funded by the Swedish Research Council.12 ppendix A Integrability conditions
There are three integrability conditions that can be derived from the Killing spinorequations in section 2. The first is obtained by commuting two gravitino variations, h R µν,ρσ γ ρσ + 2( ∂ i K D [ µ D ν ] φ i − ∂ ¯ i K D [ µ D ν ] φ ¯ i ) + 2 e K W ¯ W γ µν i ǫ L +4 ie K/ D i W D [ µ φ i γ ν ] ǫ R = 0 , (A.1)the second by commuting the gravitino and gaugino variations,2 ∇ µ ( F aρσ γ ρσ − iµ a ) ǫ L − ie K/ W ( F aρσ γ ρσ − iµ a ) γ µ ǫ R = 0 , (A.2)and the third by commuting the gravitino and scalar variations,2( D µ D ρ φ i ) γ ρ ǫ R + e K G i ¯ j ( D ¯ j ¯ W ) W γ µ ǫ R + D ρ φ i γ ρ (cid:16) ( ∂ i K D µ φ i − ∂ ¯ i K D µ φ ¯ i ) ǫ R + ie K/ ¯ W γ µ ǫ L (cid:17) +2 ie K/ h ( ∂ l K D µ φ l − ∂ ¯ l K D µ φ ¯ l ) G i ¯ j D ¯ j ¯ W + ∂ l G i ¯ j D µ φ l D ¯ j ¯ W + ∂ ¯ l G i ¯ j D µ φ ¯ l D ¯ j ¯ W + G i ¯ j D µ D ¯ j ¯ W i ǫ L = 0 . (A.3)It is clear from the integrability condition of the gravitino that the holonomy of thesupercovariant connection is included in P in c (3 , References [1] D. Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, “Progress Toward A Theory OfSupergravity,” Phys. Rev. D (1976) 3214.[2] S. Deser and B. Zumino, “Consistent Supergravity,” Phys. Lett. B , 335 (1976).[3] J. Wess and J. Bagger, “Supersymmetry and supergravity,” Princeton, USA: Univ. Pr.(1992) 259 p [4] K. P. Tod, “All Metrics Admitting Supercovariantly Constant Spinors,” Phys. Lett. B (1983) 241.[5] J. P. Gauntlett, J. B. Gutowski, C. M. Hull, S. Pakis and H. S. Reall, “All supersymmetricsolutions of minimal supergravity in five dimensions,” Class. Quant. Grav. (2003) 4587[arXiv:hep-th/0209114].[6] J. Bellorin and T. Ortin, “Characterization of all the supersymmetric solutions of gaugedN=1,d=5 supergravity,” JHEP (2007) 096 [arXiv:0705.2567 [hep-th]].[7] S. L. Cacciatori, M. M. Caldarelli, D. Klemm, D. S. Mansi and D. Roest, “Geometry offour-dimensional Killing spinors,” JHEP (2007) 046 [arXiv:0704.0247 [hep-th]].[8] B. R. Greene, A. D. Shapere, C. Vafa and S. T. Yau, “Stringy Cosmic Strings AndNoncompact Calabi-Yau Manifolds,” Nucl. Phys. B (1990) 1.
9] J. Gutowski and G. Papadopoulos, “Magnetic cosmic strings of N = 1, D = 4 supergravitywith cosmological constant,” Phys. Lett. B (2001) 371 [arXiv:hep-th/0102165].[10] G. Dvali, R. Kallosh and A. Van Proeyen, “D-term strings,” JHEP (2004) 035[arXiv:hep-th/0312005].[11] M. Cvetic, S. Griffies and S. J. Rey, “Static domain walls in N=1 supergravity,” Nucl.Phys. B , 301 (1992) [arXiv:hep-th/9201007].M. Cvetic and H. H. Soleng, “Supergravity domain walls,” Phys. Rept. , 159 (1997)[arXiv:hep-th/9604090].[12] H. Lu, C. N. Pope and P. K. Townsend, “Domain walls from anti-de Sitter spacetime,”Phys. Lett. B (1997) 39 [arXiv:hep-th/9607164].[13] J. Gutowski, “Stringy domain walls of N = 1, D = 4 supergravity,” Nucl. Phys. B (2002) 381 [arXiv:hep-th/0109126].[14] J. Gillard, U. Gran and G. Papadopoulos, “The spinorial geometry of supersymmetricbackgrounds,” Class. Quant. Grav. (2005) 1033 [arXiv:hep-th/0410155].[15] U. Gran, J. Gutowski, G. Papadopoulos and D. Roest, “N = 31 is not IIB,” JHEP (2007) 044 [arXiv:hep-th/0606049].“N = 31, D = 11,” JHEP (2007) 043 [arXiv:hep-th/0610331].[16] J. Figueroa-O’Farrill, J. Gutowski and W. Sabra, “The return of the four- and five-dimensional preons,” Class. Quant. Grav. (2007) 4429 [arXiv:0705.2778 [hep-th]].[17] J. Figueroa-O’Farrill and J. Simon, “Supersymmetric Kaluza-Klein reductions of AdSbackgrounds,” Adv. Theor. Math. Phys. (2004) 217 [arXiv:hep-th/0401206].(2004) 217 [arXiv:hep-th/0401206].