The supersymmetric solutions and extensions of ungauged matter-coupled N=1,d=4 supergravity
aa r X i v : . [ h e p - t h ] M a r IFT-UAM/CSIC-08-08 arXiv:0802.1799
February 13 th The supersymmetric solutions andextensions of ungauged matter-coupled N = 1 , d = 4 supergravity Tom´as Ort´ın
Instituto de F´ısica Te´orica UAM/CSIC,Facultad de Ciencias C-XVI, C.U. Cantoblanco, E-28049 Madrid, Spain
Abstract
We find the most general supersymmetric solutions of ungauged N = 1 , d =4 supergravity coupled to an arbitrary number of vector and chiral supermulti-plets, which turn out to be essentially pp -waves and strings. We also introducemagnetic 1-forms and their supersymmetry transformations and 2-forms asso-ciated to the isometries of the scalar manifold and their supersymmetry trans-formations. Only the latter can couple to BPS objects (strings), in agreementwith our results. Introduction
Supersymmetric classical solutions of supergravity theories (low-energy superstring the-ories) are a key tool in the current research on many topics ranging from
AdS/CF T correspondence to stringy black-hole physics. Not all locally supersymmetric solutions arenecessarily interesting or useful in the end, but, clearly, it is an important goal to find themall for every possible supergravity theory.This goal has been pursued and reached in several lower-dimensional theories and fam-ilies of theories. The pioneering work [1] was done in 1983 by Tod in pure, ungauged, N = 2 , d = 4 supergravity. It has been subsequently extended to the gauged case inRef. [2], to include the coupling to general (ungauged) vector multiplets and hypermulti-plets in Refs. [3] and [4], respectively and some partial results on the theory with gaugedvector multiplets have been recently obtained [5]. Research on pure N = 4 , d = 4 super-gravity was started in Ref. [6] and completed in Ref. [7].In d = 5, the minimal N = 1 (sometimes referred as N = 2) theory was worked outin Ref. [8] and the results were extended to the gauged case in Ref. [9]. The coupling toan arbitrary number of vector multiplets and their Abelian gaugings was considered inRefs. [10, 11] . The inclusion of (ungauged) hypermultiplets was considered in [14] andthe extension to the most general gaugings with vector multiplets and hypermultiplets wasworked out in [18].The minimal d = 6 SUGRA was dealt with in Refs. [19, 20], some gaugings wereconsidered in Ref. [21] and the coupling to hypermultiplets has been fully solved in Ref. [22].All these works are essentially based on the method pioneered by Tod and generalizedby Gauntlett et al. in Ref. [8], which we will use here. An alternative method is that ofspinorial geometry, developed in Ref. [23]. Further works on this subject in 4 or higherdimensions are Refs. [24].It is somewhat surprising that the simpler N = 1 , d = 4 theories have not yet beenstudied. The purpose of this paper is to start filling this gap. We will find all the super-symmetric configurations and solutions of ungauged N = 1 , d = 4 supergravity and we willrelate them to supersymmetric solutions of N = 2 , d = 4 supergravity theories that we cantruncate to N = 1 , d = 4 theories following [25, 26]. As we are going to see, there are notimelike supersymmetric solutions such as charged, extreme, black holes in these theoriesand in the null class we find essentially pp -waves, cosmic strings and combinations of both.This is, precisely, the kind of supersymmetric solutions of N = 2 , d = 4 supergravity thatwould survive the truncation to N = 1.We are also going to study the extension of the set of standard bosonic fields of N =1 , d = 4 supergravity along the lines of Ref. [27]. We are going to show that we can addconsistently (we can define supersymmetry transformations for them such that the localsupersymmetry algebra closes) the magnetic vectors and also 2-forms which are associatedto the isometries of the scalar manifold. The electric and magnetic vectors of the theory Previous work on these theories can be found in Refs. [12, 13]. Previous partial results on that problem were presented in Refs. [15, 16, 17]. κ -symmetric action for 0-branes, in agreement with the absence of supersymmetric black-hole solutions in the theory. The 2-forms do transform into the gravitino and one can, inprinciple, construct κ -symmetric actions for 1-branes, which agrees with the existence ofsupersymmetric string solutions.This paper is organized as follows: in Section 2 we introduce ungauged N = 1 , d =4 supergravity coupled to vector and chiral supermultiplets. We obtain this theory bytruncation of ungauged N = 2 , d = 4 supergravity coupled to vector supermultiplets andhypermultiplets in Appendix A. This helps us to fix the conventions and to relate thesolutions to N = 2 , d = 4 solutions. In Section 3 we set up the problem we aim to solve.In Section 4.1 we find all the bosonic field configurations that admit Killing spinors (as wecheck in Section 4.2) and in Section 4.3 we identify amongst them those that satisfy theclassical equations of motion, which solves our problem. In Section 5 we find the bosonicfield extensions of the theory. Finally, in Section 6 we discuss our results and give ourconclusions.After completion of this work we became aware that a similar results have been obtainedby U. Gran, J. Gutowski and G. Papadopoulos and are about to be published [28]. N = 1 , d = 4 supergrav-ity In this section we describe briefly the theory [30], which is obtained by truncation of N = 2 , d = 4 theories in the appendix. Our conventions are derived from those we use inthe study of N = 2 , d = 4 theories [3, 4, 5]. It contains a supergravity multiplet with onegraviton e aµ and one chiral gravitino ψ • µ , n C chiral multiplets with as many chiral dilatini χ • i and complex scalars Z i , i = 1 , · · · n C that parametrize a K¨ahler-Hodge manifold withmetric G ij ∗ , and n V vector multiplets with as many vector fields A Λ and chiral gaugini λ • Λ Λ = 1 , · · · , n V .The action for the bosonic fields is S = Z d x p | g | (cid:2) R + 2 G ij ∗ ∂ µ Z i ∂ µ Z ∗ j ∗ − ℑ m f ΛΣ F Λ µν F Σ µν − ℜ e f ΛΣ F Λ µν ⋆ F Σ µν (cid:3) , (2.1)where f ΛΣ ( Z ) is a n V × n V matrix with entries which are holomorphic functions of thecomplex scalars and with definite positive imaginary part.The supersymmetry transformation rules for the bosonic fields are3 ǫ e aµ = − i ¯ ψ • µ γ a ǫ • + c . c . , (2.2) δ ǫ A Λ µ = i ¯ λ • Λ γ µ ǫ • + c . c . , (2.3) δ ǫ Z i = ¯ χ • i ǫ • , (2.4)and those of the fermions, for vanishing fermions, are δ ǫ ψ • µ = D µ ǫ • = (cid:0) ∇ µ + i Q µ (cid:1) ǫ • , (2.5) δ ǫ λ • Λ = F Λ + ǫ • , (2.6) δ ǫ χ • i = i ∂Z i ǫ • , (2.7)where Q µ is the pullback of the K¨ahler 1-form connection Q ≡ i ( dz i ∂ i K − dz ∗ i ∗ ∂ i ∗ K ) , (2.8)where K is the K¨ahler potential from which the K¨ahler metric can be derived in thestandard fashion, namely G ij ∗ = ∂ i ∂ j ∗ K . (2.9)For convenience, we denote the bosonic equations of motion by E aµ ≡ − p | g | δSδe aµ , E i ≡ − G ij ∗ p | g | δSδZ ∗ j ∗ , E Λ µ ≡ p | g | δSδA Λ µ . (2.10)and the Bianchi identities for the vector field strengths by B Λ µ ≡ ∇ ν ⋆ F Λ νµ , ⋆ B Λ ≡ − dF Λ . (2.11)Then, using the action Eq. (2.1), we find4 µν = G µν + 2 G ij ∗ [ ∂ µ Z i ∂ ν Z ∗ j ∗ − g µν ∂ ρ Z i ∂ ρ Z ∗ j ∗ ] − ℑ m f ΛΣ F Λ + µρ F Σ − νρ , (2.12) E i = G ij ∗ D µ ∂ µ Z ∗ i ∗ + ∂ i [ F Λ µν ⋆ F Λ µν ] (2.13)= G ij ∗ D µ ∂ µ Z ∗ i ∗ − i ∂ i f ΛΣ F Λ + µν F Σ + µν , (2.14) E Λ µ = ∇ ν ⋆ F Λ νµ , (2.15)where we have defined the dual vector field strength F Λ by F Λ µν ≡ − p | g | δSδ ⋆ F Λ µν = ℜ e f ΛΣ F Σ µν − ℑ m f ΛΣ ∗ F Σ µν = 2 ℜ e ( f ΛΣ F Σ + ) . (2.16)The Maxwell equations can be read as Bianchi identities for these dual field strengthsensuring the local existence of n V dual vector potentials A Λ such that F Λ = dA Λ . (2.17)It is convenient to combine the standard, electric, field strengths and potentials andtheir duals Eq. (2.16) into a single 2 n V -dimensional symplectic vector F ≡ (cid:18) F Λ F Λ (cid:19) = d A ≡ d (cid:18) A Λ A Λ (cid:19) . (2.18)The global symmetries of these theories will be the isometries of the scalar manifoldthat can be embedded in Sp (2 n V , R ) [31]. Our first goal is to find all the bosonic field configurations { g µν , F Λ µν , Z i } for which theKilling spinor equations (KSEs): δ ǫ ψ • µ = D µ ǫ • = 0 , (3.1) δ ǫ λ • Λ = F Λ + ǫ • = 0 , (3.2) δ ǫ χ • i = i ∂Z i ǫ • = 0 , (3.3)5dmit at least one solution. It must be stressed that the configurations considered neednot be classical solutions of the equations of motion. Furthermore, we will not assume thatthe Bianchi identities are satisfied by the field strengths of a configuration.Our second goal will be to identify among all the supersymmetric field configurationsthose that satisfy all the equations of motion (including the Bianchi identities).Let us initiate the analysis of the KSEs by studying their integrability conditions. Using the supersymmetry transformation rules of the bosonic fields Eqs. (2.2–2.4) and usingthe results of Refs. [32, 33] we can derive following relations (
Killing spinor identities , KSIs)between the (off-shell) equations of motion of the bosonic fields Eqs. (2.12–2.15) that aresatisfied by any field configuration { e aµ , A Λ µ , Z i } admitting Killing spinors: E µa γ a ǫ • = 0 , (3.4) E Λ µ γ µ ǫ • = 0 , (3.5) E i ǫ • = 0 . (3.6)In this way of finding the KSIs the Bianchi identities are assumed to be satisfied. It isconvenient to have KSIs in which they appear explicitly. These can be found through the in-tegrability conditions of the KSEs. The only KSI in which we expect the Bianchi identitiesto appear is the second one above, which involves the Maxwell equations. The Bianchi iden-tities should combine with the Maxwell equations in a electric-magnetic duality-invariantway. Then, the second KSI above should be replaced by( E Λ µ − f ΛΣ B Σ µ ) γ µ ǫ • = 0 . (3.7)This can be explicitly checked via the following integrability condition of the gaugini: D δ ǫ λ • Λ = ( ℑ m f ) − | ΛΣ ( Σ − f ∗ ΣΩ Ω ) ǫ • + i ( ℑ m f ) − | ΛΣ ∂f ΣΩ δ ǫ λ • Ω − F Λ − δ ǫ χ • i ∗ + γ µ F Λ + δ ǫ ψ • µ . (3.8)From these identities one can derive identities that involve tensors constructed as bi-linears of the Killing spinors. In N = 1 supergravity there is only one chiral spinor ǫ • .With it, we can only construct a real null vector l µ = i √ ǫ • γ µ ǫ • , one self-dual 2-formΦ µν = ¯ ǫ • γ µν ǫ • and no scalars. In the N > N = 1 , d = 4 there is no timelike case. It is convenientto introduce an auxiliary chiral spinor η • with normalization6 ǫ • η • = , (3.9)and with the same chirality but opposite K¨ahler weight as ǫ • . With both spinors weconstruct the null tetrad l µ = i √ ǫ • γ µ ǫ • , n µ = i √ η • γ µ η • ,m µ = i √ ǫ • γ µ η • , m ∗ µ = i √ ǫ • γ µ η • . (3.10) l and n have 0 U (1) charges but m has − ǫ and m ∗ has +2 times thecharges of ǫ . E µν l ν = E µν m ν = 0 , (3.11)( E Λ µ − f ΛΣ B Σ µ ) l µ = ( E Λ µ − f ΛΣ B Σ µ ) m µ = 0 , (3.12) E i = 0 . (3.13)This means that the only independent equations of motion that we have to impose onsupersymmetric configurations are E µν n µ n ν = 0 , (3.14)( E Λ µ − f ΛΣ B Σ µ ) n µ = 0 , (3.15)( E Λ µ − f ΛΣ B Σ µ ) m ∗ µ = 0 . (3.16) Our first goal is to derive from the KSEs consistency conditions expressed in terms of thenull tetrad vectors.Acting on the KSE Eq. (3.2) with ¯ ǫ • γ µ and ¯ η • γ µ we get, respectively F Λ + µν l ν = 0 , (4.1) F Λ + µν m ∗ ν = 0 , (4.2)7hich imply that F Λ + = φ Λ ˆ l ∧ ˆ m ∗ , (4.3)for some functions φ Λ to be determined. This form of F Λ + solves the KSE Eq. (3.2) byvirtue of the Fierz identities l µ γ µν ǫ • = l ν ǫ • , m ∗ µ γ µν ǫ • = m ∗ ν ǫ • . (4.4)Acting now on the KSE Eq. (3.3) with ¯ ǫ • and ¯ η • we get, respectively l µ ∂ µ Z i = 0 , (4.5) m µ ∂ µ Z i = 0 , (4.6)which imply dZ i = A i ˆ l + B i ˆ m , (4.7)for some functions A i and B i to be determined. This form of dZ i solves the KSE Eq. (3.3)by virtue of the Fierz identities lǫ ∗ = mǫ ∗ = 0 . (4.8)Now, , from the normalization condition of the auxiliary spinor η • we find the condition D µ η • + a µ ǫ • = 0 , (4.9)for some a µ with U (1) charges − ǫ , i.e. D µ a ν = ( ∇ µ − i Q µ ) a ν , (4.10)to be determined by the requirement that the integrability conditions of this differentialequation have to be compatible with those of the differential equation for ǫ .Taking the covariant derivative of the null tetrad vectors and using the KSE Eq. (3.1),we find D µ l ν = ∇ µ l ν = 0 , (4.11) D µ n ν = ∇ µ n ν = − a ∗ µ m ν − a µ m ∗ ν , (4.12) D µ m ν = ( ∇ µ − i Q µ ) m ν = − a µ l ν . (4.13)8he first of these equations is solved by identifying the most general metric compatiblewith it: a Brinkmann pp -wave metric [34, 35]. One introduces the coordinates u and v such that ˆ l = l µ dx µ ≡ du , (4.14) l µ ∂ µ ≡ ∂∂v , (4.15)and defines a complex coordinate z by ˆ m = e U dz , (4.16)where U may depend on z, z ∗ and u . The most general form that ˆ n can take in this case isˆ n = dv + Hdu + ˆ ω , ˆ ω = ω z dz + ω z ∗ dz ∗ , (4.17)where all the functions in the metric are independent of v and where either H or the 1-formˆ ω could, in principle, be removed by a coordinate transformation but we have to check thatthe tetrad integrability equations (4.11)-(4.13) are satisfied by our choices of e U , H and ˆ ω The above choice of coordinates leads to the metric ds = 2 du ( dv + Hdu + ˆ ω ) − e U dzdz ∗ . (4.18)It also implies that the complex scalars Z i are functions of z and u but not of z ∗ and v .The same is true for A i and B i .Let us consider the tetrad integrability equations (4.11)-(4.13): the first equation issolved because the metric does not depend on v . The third equation, with the choice ofcoordinate z , Eq. (4.16), implies ˆ a = n µ ( ∂ µ U − i Q µ ) ˆ m + D ˆ l , (4.19) m µ ∂ µ ( U − i Q µ ) = 0 , (4.20)where D is a function to be determined.The second equation can be written using the definition of the K¨ahler connection andthe dependence Z i ( z, u ) in the form ∂ z ∗ ( U + K /
2) = 0 ⇒ U = −K / h ( u ) , (4.21)where h ( u ) can be eliminated by a coordinate redefinition that does not change the generalform of the Brinkmann metric.The second tetrad integrability equation (4.12) implies9 = e − U ( ∂ z ∗ H − ˙ ω z ∗ ) , (4.22)( dω ) zz ∗ = 2 ie U n µ Q µ , (4.23)whence ˆ a is given by ˆ a = [ ˙ U − e − U ( dω ) zz ∗ ] ˆ m + e − U ( ∂ z ∗ H − ˙ ω z ∗ )ˆ l . (4.24) We are now going to see that field configurations given by a metric of the form (Eqs. (4.18)where ˆ ω satisfies (Eq. (4.23)) and U satisfies Eq. (4.21), field strengths given by Eqs. (4.3)and scalars of the form (4.7) are always supersymmetric, even though we derived theseequations as necessary conditions for supersymmetry.With the above form of the scalars and vector field strengths the KSE δ ǫ χ • i = 0 takesthe form i [ A i l + B i m ] ǫ • = 0 . (4.25)This equation is solved by imposing two conditions on the spinors: lǫ • = 0 , mǫ • = 0 . (4.26)As shown in Ref. ([3]) these two constraints are not just compatible but equivalent andonly half of the supersymmetries are broken by them.Let us now consider the KSE δ ǫ ψ • a = 0. It takes the form { ∂ a − ω abc γ bc + i Q a } ǫ • = 0 . (4.27)The v component is automatically satisfied for v -independent Killing spinors. The z and z ∗ components take, after use of the constraints Eq. (4.26) and their consequence γ zz ∗ ǫ • = ǫ • the form { ∂ z + ∂ z ( U + K / } ǫ • = 0 , (4.28) { ∂ z ∗ + ∂ z ∗ ( U + K / } ǫ • = 0 . (4.29)They are solved for z - and z ∗ -independent spinors once Eq. (4.21) is taken into account.The u component simply implies that the Killing spinors are also u -independent.Thus, all the configurations identified are supersymmetric with Killing spinors whichare constant spinors satisfying Eqs. (4.26). Thus, they generically preserve 1 / .3 Solutions The Bianchi identities take, in differential-form language, the formˆ B Σ = − dF Λ = d ( φ Σ ˆ m + c . c) ∧ ˆ l , (4.30)and are solved by A Λ = ϕ Λ ( z, u ) du + c . c . , e K / ∂ z ϕ Λ ( z, u ) = φ ∗ Λ . (4.31)The Maxwell equations take the formˆ E Λ = d ( f ΛΣ F Λ + + c . c . ) = − d ( f ΛΣ φ Σ ˆ m ∗ + c . c) ∧ ˆ l , (4.32)which is solved by holomorphic functions ϕ Λ ( z, u ) such that ∂ z ϕ Λ ( z, u ) = f ∗ ΛΣ φ ∗ Σ e −K / . (4.33)Using the solution of the Bianchi identities, we get ∂ z ϕ Λ ( z, u ) = f ∗ ΛΣ ∂ z ϕ Σ ( z, u ) . (4.34)Taking now into account that f ΛΣ is a holomorphic function of the Z i s which are, them-selves, holomorphic functions of z (and standard functions of u ), we arrive to the conclusionthat the above equation can be solved in two ways: either the Z i s are z -independent or ∂ z ϕ Λ ( z, u ) = f ∗ ΛΣ ∂ z ϕ Σ ( z, u ) = 0 . (4.35)In general f ΛΣ will not have null eigenvectors and, therefore, the only generic solutions are z -independent ϕ Σ and, therefore, trivial vector fields.Taking into account Eq. (4.35), the only non-automatically satisfied component of theEinstein equations is ∂ z ∂ z ∗ H − e −K / ∂ u e −K / − e −K G ij ∗ ∂ u Z i ∂ u Z ∗ j ∗ − ℑ m f ΛΣ ∂ z ϕ Λ ∂ z ∗ ϕ ∗ Σ = 0 . (4.36)There are two cases to be considered: • When the Z i s are z -independent. Then H = ℜ e f ( z ) + [ e −K / ∂ u e −K / + e −K G ij ∗ ∂ u Z i ∂ u Z ∗ j ∗ ] | z | + ℑ m f ΛΣ ϕ Λ ϕ ∗ Σ . (4.37)These solutions describe gravitational, electromagnetic and scalar pp waves. For simplicity we choose the gauge ω = 0. When the Z i s are not z -independent. The vector fields are trivial, but the above equa-tion is not easy to integrate. In the special case in which the Z i s are u -independentholomorphic functions of z H = ℜ e f ( z ) . (4.38)These solutions describe a superposition of a pp -wave and cosmic strings such as thosestudied in Refs. [36, 37, 38, 27] and found in N = 4 , d = 4 [6, 33] and N = 2 , d = 4 [3, 4]theories. In this section we are going to explore the possible extensions of the standard formulationof N = 1 , d = 4 supergravity, using our previous results on the supersymmetric solutionsof the theory. These suggest the possible addition of 2-forms associated to the isometriesof the K¨ahler scalar manifold. These should couple to the cosmic string solutions exactlyin the form discussed in Ref. [27] for N = 2 , d = 4 supergravity. Since one can definemagnetic potentials from the Maxwell equations, it should also be possible to add dual,magnetic, 1-forms. These, however, may not couple to any standard 0-brane since all1-forms transform into gaugini (and not the gravitino) under supersymmetry. Given the supersymmetry transformation rule of the standard (electric) potentials Eq. (2.3)and the definition of the dual field strengths Eq. (2.16), the simplest Ansatz for the trans-formation of the dual (magnetic) potentials A Λ would be δ ǫ A Λ µ = i f ∗ ΛΣ ¯ ǫ • γ µ λ • Σ + c . c . . (5.1)[ δ η , δ ǫ ] A Λ µ = − ℜ e[ af ∗ ΛΣ F Σ − µν ] ξ ν , (5.2)where ξ ν ≡ i ¯ ǫ • γ ν η • + c . c . . (5.3)In absence of the functions f ΛΣ , we have[ δ η , δ ǫ ] A Λ µ = − ℜ e[ F Λ − µν ] ξ ν = − F Λ µν ξ ν = [ δ g . c . t . ( ξ ) + δ gauge (Λ Λ )] A Λ µ , (5.4)where δ g . c . t . ( ξ ) A Λ µ = ξ ν ∂ ν A Λ µ + ∂ µ ξ ν A Λ ν , (5.5)and 12 gauge (Λ) A Λ µ = ∂ µ Λ Λ , Λ Λ ≡ − ξ ν A Λ ν . (5.6)In presence of the functions f ΛΣ , we have[ δ η , δ ǫ ] A Λ µ = − ℜ e[ f ∗ ΛΣ F Σ − µν ] ξ ν = − F Λ µν ξ ν = [ δ g . c . t . ( ξ ) + δ gauge (Λ Λ )] A Λ µ , (5.7)where the g.c.t.s and gauge transformations have the same form and the parameter of thegauge transformations is now Λ Λ ≡ ξ ν A Λ ν . (5.8) Sp (2 n V , R ) [31]. The action of these symmetrieson the fields is δZ i = α A k Ai ( Z ) , (5.9) δ F = α A T A F , (5.10)(5.11)where F is defined in Eq. (2.16) and where the T A are matrices of sp (2 n V ) that generatethe Lie algebra of the symmetry group:[ k A , k B ] = − f AB C k C , [ T A , T B ] = + f ABC T C . (5.12)The computation of the Noether current proceeds as in Ref. [27] and the result isidentical, up to the difference between the period matrix and f ΛΣ : J N µ = α A J N A µ , J
N A µ = 2 k ∗ A i ∂ µ Z i + c . c . − h ⋆ F µν | T A A ν i . (5.13)These Noether currents are covariantly conserved, i.e. d ⋆ J N A = 0 , (5.14)which implies the local existence of 2-forma B A such that dB A ≡ ⋆J N A = 2 k ∗ A i ⋆ dZ i + c . c . − h F | T A A i . (5.15)The second term in the r.h.s. is not invariant under the gauge transformations of the vectorpotentials, and the same is therefore true for the 2-forms B A , which transform as13 gauge A = d Λ , (5.16) δ gauge (Λ , Λ A ) B A = d Λ A − h F | T A Λ i . (5.17)One, then, defines the gauge-invariant 3-form field strengths H A ≡ dB A + 2 h F | T A A i = 2 k ∗ A i ⋆ dZ i + c . c . . (5.18)Inspired by the results of Ref. [27] it is not difficult to guess the form of the supersym-metry transformation rules of these 2-forms: δ ǫ B A µν = − i k ∗ A i ¯ ǫ • γ µν χ • i + c . c . + i P A ¯ ǫ • γ [ µ | ψ •| ν ] + c . c . − h A [ µ | | T A δ ǫ A | ν ] i , (5.19)where P A is the momentum map associated to the Killing vector k A .We find [ δ η , δ ǫ ] B A µν = [ δ g . c . t . ( ξ ) + δ gauge (Λ , Λ A )] B A µν , (5.20)where ξ is defined in Eq. (5.3), Λ in Eqs. (5.6) and (5.8) and Λ A is given byΛ A µ ≡ − P A ξ µ . (5.21)A shown in Ref. [27] in N = 2 , d = 4 supergravity theories, these 2-forms can be coupledto strings of different species labeled by A whose tensions would be proportional to P A . We have found all the supersymmetric configurations and solutions of ungauged N =1 , d = 4 with arbitrary couplings to vector and chiral supermultiplets. It is clear that,qualitatively, these are those of ungauged N = 2 , d = 4 supergravity whose fields andKilling spinors survive the N = 2 → N = 1 truncation explained in the appendix, althoughthe scalar manifolds of the N = 1 theory are more general. In particular, all the N = 2supersymmetric configurations in the timelike class (typically black holes) do not surviveto this truncation since their supersymmetry projectors ǫ I + iǫ IJ γ ǫ J = 0 , (6.1)involve necessarily the two supersymmetry parameters and one of them is eliminated inthe truncation. The fields of extreme, supersymmetric N = 2 , d = 4 black holes may stillsurvive the truncation to N = 1, but they will not be BPS in this theory.14he Killing spinors supersymmetric configurations of the null class obey projections ofthe form γ u ǫ I = 0 , I = 1 , , (6.2)and, thus, they always survive the truncation.It is likely that the situation in the most general (gauged) N = 1 , d = 4 theory is thesame, and, again qualitatively, the supersymmetric solutions can be obtained by truncationfrom the N = 2 , d = 4 theory on which some partial results are already available [5]. Ofcourse, a direct calculation is necessary and, anyway, the most general supersymmetricsolutions of gauged N = 2 , d = 4 supergravity are not known, although progress in thisdirection is being made [39]. Work in this direction is already in progress [40].Further extensions (3- and 4-forms) are clearly possible and a more general study ofthe possibilities in more general (gauged) N = 1 , d = 4 supergravities has to be performed[41] to compare the results with those of the Kac-Moody approach. Acknowledgments
This work has been supported in part by the Spanish Ministry of Science and Educationgrants FPA2006-00783 and PR2007-0073, the Comunidad de Madrid grant HEPHACOSP-ESP-00346, the Spanish Consolider-Ingenio 2010 Program CPAN CSD2007-00042, andby the EU Research Training Network
Constituents, Fundamental Forces and Symmetriesof the Universe
MRTN-CT-2004-005104. The author would like to thank the the StanfordInstitute for Theoretical Physics for its hospitality, E. Bergshoeff, J. Hartong, M. H¨ubscherand P. Meessen for useful discussions and and M.M. Fern´andez for her continuous support.
A Truncating N = 2 to N = 1 supergravity in d = 4 The purpose of this appendix is to show, following Refs. [25, 26], how ungauged N = 2 , d =4 supergravity coupled to vector multiplets can be truncated to ungauged N = 1 , d = 4supergravity by decoupling the N = 1 supermultiplet that contains the second gravitino ψ µ . We will only deal with the leading terms in fermions. In doing so, we will obtain N = 1 , d = 4 supergravity in suitable conventions and the relations between the fields ofboth theories. A.1 Ungauged matter-coupled N = 2 , d = 4 supergravity We start by a very brief description of ungauged N = 2 , d = 4 supergravity coupledto vector multiplets referring the reader to Refs. [3, 7] for detailed description of theconventions and further references to the literature.The gravity multiplet of the N = 2 , d = 4 theory consists of the graviton e aµ , a pairof gravitinos ψ I µ , ( I = 1 ,
2) which we describe as Weyl spinors, and a vector field A µ .15ach of the n vector supermultiplets of N = 2 , d = 4 supergravity that we are going tocouple to the pure supergravity theory contains complex scalar Z i , ( i = 1 , · · · , n V ), apair of gauginos λ I i , which we also describe as Weyl spinors and a vector field A iµ . Inthe coupled theory, the n V + 1 vectors can be treated on the same footing and they aredescribed collectively by an array A Λ µ (Λ = 1 , · · · , n V + 1). The coupling between thecomplex scalars is described by a non-linear σ -model with K¨ahler metric G ij ∗ ( Z, Z ∗ ) , andthe coupling to the vector fields by a complex scalar-field-valued matrix N ΛΣ ( Z, Z ∗ ). Thesetwo couplings are related by a structure called special K¨ahler geometry, described in thereferences.Each hypermultiplet consists of 4 real scalars q ( hyperscalars ) and 2 Weyl spinors ζ called hyperinos . The 4 m hyperscalars are collectively denoted by q u , u = 1 , · · · , m andthe 2 n H hyperinos are collectively denoted by ζ α , α = 1 , · · · , n H . The 4 n H hyperscalarsparametrize a quaternionic K¨ahler manifold with metric H uv ( q ).The action for the bosonic fields of the theory is S = Z d x p | g | (cid:2) R + 2 G ij ∗ ∂ µ Z i ∂ µ Z ∗ j ∗ + 2 H uv ∂ µ q u ∂ µ q v +2 ℑ m N ΛΣ F Λ µν F Σ µν − ℜ e N ΛΣ F Λ µν ⋆ F Σ µν (cid:3) , (A.1)In these conventions ℑ m N ΛΣ is negative definite.For vanishing fermions, the supersymmetry transformation rules of the fermions are δ ǫ ψ I µ = D µ ǫ I + ǫ IJ T + µν γ ν ǫ J , (A.2) δ ǫ λ Ii = i ∂Z i ǫ I + ǫ IJ G i + ǫ J , (A.3) δ ǫ ζ α = − i C αβ U βI u ε IJ ∂q u ǫ J , (A.4)where D µ , the Lorentz- and K¨ahler- and SU (2)-covariant derivative acts on the spinors ǫ I as D µ ǫ I = ( ∇ µ + i Q µ ) ǫ I + A µ I J ǫ J . (A.5)and Q µ is the pullback of the K¨ahler 1-form defined in Eq. (2.8)and A µ I J is the pullback of the SU (2) connection A I J .The 2-forms T and G i are the combinations T µν ≡ T Λ F Λ µν , (A.6) G iµν ≡ T i Λ F Λ µν , (A.7)where, in turn, T Λ and T i Λ are, respectively, the graviphoton and the matter vector fieldsprojectors, defined by 16 Λ ≡ i L Λ = 2 i L Σ ℑ m N ΣΛ , (A.8) T i Λ ≡ − f ∗ Λ i = −G ij ∗ f ∗ Σ j ∗ ℑ m N ΣΛ . (A.9)The supersymmetry transformations of the bosons are δ ǫ e aµ = − i ( ¯ ψ I µ γ a ǫ I + ¯ ψ I µ γ a ǫ I ) , (A.10) δ ǫ A Λ µ = ( L Λ ∗ ǫ IJ ¯ ψ I µ ǫ J + L Λ ǫ IJ ¯ ψ I µ ǫ J )+ i ( f Λ i ǫ IJ ¯ λ Ii γ µ ǫ J + f Λ ∗ i ∗ ǫ IJ ¯ λ I i ∗ γ µ ǫ J ) , (A.11) δ ǫ Z i = ¯ λ Ii ǫ I , (A.12) δ ǫ q u = U αI u ( ¯ ζ α ǫ I + C αβ ǫ IJ ¯ ζ β ǫ J ) . (A.13) A.2 Truncation to N = 1 , d = 4 supergravity The truncation to N = 1 , d = 4 supergravity consists in setting to zero the supermultipletthat contains the second gravitino ψ µ and the graviphoton. The remaining fields in thesupergravity multiplet { e aµ , ψ µ } will become those of the N = 1 , d = 4 supergravitymultiplet and the n V N = 2 , d = 4 vector multiplets will be split into n V chiral multiplets,each of them containing one complex scalar and the first component of one N = 2 gaugino λ i and n V vector multiplets, each of them containing one vector and the second componentof one N = 2 gaugino λ i . However, not all of them can simultaneously. Finally, only halfof the hyperscalars, parametrizing a K¨ahler manifold will survive the truncation.We relabel the N = 2 indices Λ → Λ and i, i ∗ → i , i ∗ to label the N = 1 vectormultiplets with Λ and the chiral multiplets with i . We set ψ µ = δ ǫ ψ µ = ǫ = 0 , (A.14)and define ψ • µ ≡ ψ µ , ǫ • ≡ ǫ . (A.15)The supersymmetry transformations of the two gravitini become δ ǫ ψ • µ = (cid:0) ∇ µ + i Q µ + A µ (cid:1) ǫ • , (A.16) δ ǫ ψ µ = A µ ǫ • − T + µν γ ν ǫ • = 0 . (A.17)17his means that the component A µ of the SU (2) connection has to be integrated intothe K¨ahler connection and the component A µ and the graviphoton field strength has tobe set to zero A µ = 0 , (A.18) T + µν = 0 . (A.19)The supersymmetry transformation rule of the graviton becomes, simply δ ǫ e aµ = − i ¯ ψ • µ γ a ǫ • + c . c . (A.20)Let us now consider the N = 2 vector multiplets.The most general solution to the constraint Eq. (A.19) is to see it as an orthogonalitycondition between the graviphoton projector and the vector fields [25, 26]. The N = 2vector index is split Λ = (Λ , X ), where Λ = 1 , · · · , n and X = 0 , , · · · , n V − n V ≡ n C and T Λ = 2 i L Σ ℑ m N Σ Λ = 0 , F X + µν = 0 . (A.21)The N = 2 vector multiplets in the range Λ give only N = 1 vector multiplets (thechiral multiplets have to be truncated) and those in the range X give only chiral N = 1multiplets (the N = 1 vector multiplets must be truncated). Since the dual vector fieldstrengths F X + µν = N X Λ F Λ + µν = N X Λ F Λ+ µν + N XY F Y + µν , (A.22)must also vanish for consistency, the off-diagonal blocks of the period matrix must alsovanish N X Λ = 0 , (A.23)and, therefore T Λ = 2 i L Σ ℑ m N ΣΛ = 0 ⇒ L Λ = 0 . (A.24)Only the components L X survive, and, together with the period matrix N XY , define aspecial K¨ahler manifold of dimension n V − n V = n C and with K¨ahler metric G ij ∗ = − ℑ m N XY f X i f Y j ∗ , i, j = 1 , · · · , n C . (A.25)The diagonal block N ΛΣ ≡ f ∗ ΛΣ , (A.26)determines the couplings of the scalars of the chiral multiplets to the vectors. It can beshown that f ΛΣ is a holomorphic function of the Z i s.18he consistency of these conditions leads to several conditions that the special K¨ahlermanifold has to satisfy on order to be reducible to N = 1 and can be found in [25, 26].It is convenient to study the supersymmetry transformations of the two gaugini in theform f Λi δ ǫ λ I i = if Λi ∂Z i ǫ I + F Λ + ǫ IJ ǫ J , (A.27)where we have used the constraint Eq. (A.19). Then, splitting the index i = ( α, i ) with α = 1 , · · · , n and i = 1 , · · · , n C , the above equation splits as follows f Λ i δ ǫ λ i = 0 , (A.28) f X i δ ǫ λ i = if X i ∂Z i ǫ • , (A.29) f Λ α δ ǫ λ α = F Λ + ǫ • , (A.30) f X i δ ǫ λ i = 0 . (A.31)Then, we define the N = 1 gaugini and dilatini λ • Λ ≡ − f Λ α λ α , (A.32) χ • i ≡ λ i , (A.33)and set to zero all the other components. Their resulting supersymmetry transformationrules are δ ǫ λ • Λ = F Λ + ǫ • , (A.34) δ ǫ χ • i = i ∂Z i ǫ • . (A.35)The supersymmetry transformation rules of the vector fields are split in δ ǫ A Λ µ = i ¯ λ • Λ γ µ ǫ • + c . c . , (A.36) δ ǫ A X µ = 0 . (A.37)Finally, the supersymmetry transformation rules of the scalars split into19 ǫ Z i = ¯ χ • i ǫ • , (A.38) δ ǫ Z α = 0 . (A.39)Let us now consider the truncation in the hypermultiplet sector. The 4 n H real di-mensional quaternionic-K¨ahler manifold has to be truncated to a n H complex dimensionalK¨ahler manifold [25, 26]. The truncation can only be done in some quaternionic-K¨ahlermanifold: if we split the Sp (2 n H ) index α into A, ˙ A = 1 , · · · , n H and the undotted indicescorrespond to the sector which will survive, the components Ω ˙ A ˙ B ˙ C ˙ D must vanish identically.If this condition is satisfied, then one can set U A = A = A = ∆ A ˙ B = ζ ˙ A = 0 , (A.40)consistently. The surviving components of the Quadbein are U A and its complex conjugate U A which can be expressed in terms of just n H holomorphic coordinates w s .The independent non-vanishing supersymmetry transformation rules of the hyperscalarsand the hyperinos are ( U Au ) ∗ δ ǫ q u = ¯ ζ • A ǫ • , (A.41) U Au δ ǫ ζ • A = i ∂q u ǫ • . (A.42)Using the holomorphic coordinates w s we now define the n H N = 1 dilatini ζ s ζ • s ≡ U As ζ • A , (A.43)and the above supersymmetry transformation rules take the standard form δ ǫ w s = ¯ ζ • s ǫ • , (A.44) δ ǫ ζ • s = i ∂w s ǫ • . (A.45)The quaternionic K¨ahler manifolds that can be truncated to N = 1 chiral multipletsare precisely those in which one can construct cosmic string solutions ( hyperstrings ): inRef. [4] the supersymmetry equations were solved by choosing a metric of the form ds = dt − ( dx ) − e Φ( z,z ∗ ) dzdz ∗ , (A.46)hyperscalars which are real functions of the complex coordinate z and its complex conjugate q u ( z, z ∗ ). 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