Conformal Symmetries for Extremal Black Holes with General Asymptotic Scalars in STU Supergravity
aa r X i v : . [ h e p - t h ] F e b UPR-1308-T MI-TH-212
Conformal Symmetries for Extremal Black Holes with GeneralAsymptotic Scalars in STU Supergravity
M. Cvetiˇc , , C.N. Pope , and A. Saha Department of Physics and Astronomy, and Department of Mathematics,University of Pennsylvania, Philadelphia, PA 19104, USA Center for Applied Mathematics and Theoretical Physics,University of Maribor, SI2000 Maribor, Slovenia George P. & Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,Texas A&M University, College Station, TX 77843, USA DAMTP, Centre for Mathematical Sciences, Cambridge University,Wilberforce Road, Cambridge CB3 OWA, UK
Abstract
We present a construction of the most general BPS black holes of STU supergravity( N = 2 supersymmetric D = 4 supergravity coupled to three vector super-multiplets)with arbitrary asymptotic values of the scalar fields. These solutions are obtained byacting with a subset of of the global symmetry generators on STU BPS black holes withzero values of the asymptotic scalars, both in the U-duality and the heterotic frame.The solutions are parameterized by fourteen parameters: four electric and four magneticcharges, and the asymptotic values of the six scalar fields. We also present BPS blackhole solutions of a consistently truncated STU supergravity, which are parameterizedby two electric and two magnetic charges and two scalar fields. These latter solutionsare significantly simplified, and are very suitable for further explicit studies. We alsoexplore a conformal inversion symmetry of the Couch-Torrence type, which maps anymember of the fourteen-parameter family of BPS black holes to another member ofthe family. Furthermore, these solutions are expected to be valuable in the studies ofvarious swampland conjectures in the moduli space of string compactifications. ontents T Reduction From Six Dimensions 48
Many intriguing non-perturbative aspects of string theory and M-theory have been broughtto light by studying the black hole and higher p -brane solutions (see, for example, [1,2]). Especially important in this context are the supersymmetric BPS solutions, which areexpected to be protected in the face of stringy corrections to the leading-order effectiveaction. Thus by studying the BPS solutions in the low-energy supergravity limit one canexpect to gain insights that could remain relevant in the full theory.2he full four-dimensional supergravity theories resulting from the dimensional reductionof the ten-dimensional heterotic or type II superstring are quite complicated, with manyfield strengths and scalar fields in their bosonic sectors. For many purposes, however, itsuffices to focus on the black hole solutions residing within a truncation of the theoriesto the so-called STU supergravity, which comprises N = 2 supergravity coupled to threevector supermultiplets. Solutions within the STU theory can be rotated using the globalsymmetries of the full heterotic or type II theories to “fill out” solution sets of the largertheories.There are different ways to present the STU supergravity, which are related by dualisa-tions of one or more of the four gauge field strengths. These different presentations of thetheory arise naturally in different contexts. For example, one of the duality complexions,which we refer to as the heterotic formulation, is the one that arises naturally when oneperforms a toroidal reduction from the ten-dimensional heterotic theory and then makes aconsistent truncation to the STU supergravity. A different duality complexion, which werefer to as the U-duality formulation, arises if one makes a consistent reduction of eleven-dimensional supergravity on the 7-sphere, truncates further to the subsector of fields thatare invariant under the U (1) maximal torus of the SO (8) isometry group of the 7-sphere,and then turns off the gauge coupling by sending the radius of the sphere to infinity. Thisformulation of STU supergravity is related to the heterotic formulation by a dualisation oftwo out of the four field strengths. There is also an intermediate duality complexion whichcorresponds to dualising any one of the four fields in the U-duality formulation. We referto this as the 3+1 formulation of the theory.As a preliminary to subsequent calculations, one of the purposes of this paper is toobtain fully explicit relations between the fields in the three above-mentioned formulationsof the STU supergravity theory. These will then be utilised later in the paper, when weconstruct explicit expressions for the most general static BPS black hole solutions in the STUsupergravity theory. The most general such solutions are characterised by a total of eightcharges, corresponding to four electric and four magnetic charges carried by the four gaugefield strengths. One can use the compact U (1) subgroup of the SL (2 , R ) global symmetryof the STU theory to rotate these solutions to ones involving only 5 independent charges,but for our purposes it is more useful to present the solutions in the more symmetrical8-charge characterisation. In fact, we obtain this symmetrical form by using the U (1) symmetry in the opposite direction, to go from a 5-charge to an 8-charge expression. Thisprocedure was originally partially implemented in [3]; for later purposes we also wish to3eneralise the solutions by allowing for arbitrary asymptotic values of the six scalar fieldsof the STU supergravity. This can be done by acting with the remaining six generators ofthe coset SL (2 , R ) /U (1) within the global symmetry group.One of our reasons for constructing the most general static BPS black hole solutionsexplicitly is to study in detail a phenomenon that was first noticed by Couch and Torrence in1984 in the case of the extremal Reissner-Nordstr¨om black hole [4]. This solution, which isa special case [5] of the static BPS black holes of STU supergravity in which all four electriccharges are set equal and the magnetic charges are all set to zero, exhibits a conformalinversion symmetry as follows. Writing the extreme Reissner-Nordstr¨om metric as ds = − (cid:16) Qr (cid:17) − dt + (cid:16) Qr (cid:17) ( dr + r d Ω ) , (1.1)where d Ω is the metric on the unit 2-sphere, the horizon is located at r = 0. Defined aninverted coordinate ˆ r and a conformally-related metric d ˆ s byˆ r = Q r , d ˆ s = Q r ds , (1.2)one finds that metric d ˆ s takes exactly the same form as the original metric ds , writtennow in terms of the inverted coordinate: d ˆ s = − (cid:16) Q ˆ r (cid:17) − dt + (cid:16) Q ˆ r (cid:17) ( d ˆ r + ˆ r d Ω ) . (1.3)The Couch-Torrence symmetry therefore maps the near-horizon region to the asymptoticregion near infinity, and vice versa. This symmetry has been employed in more recent timesto related conserved quantities on the horizon to conserved quantities at infinity. (See, forexample, [6, 7, 8].)Recently, generalisations of the Couch-Torrence symmetry have been studied for thestatic black hole solutions of STU supergravity. In [8] it was shown that the set of solutionscarrying four electric charges and no magnetic charges maps into itself under a similarconformal inversion. The mapping is no longer a symmetry, in that a black hole withcharges ( Q , Q , Q , Q ) maps into a different member of the 4-charge family, with differentcharges ( ˆ Q , ˆ Q , ˆ Q , ˆ Q ), related by ˆ Q i = Q − i ( Q Q Q Q ) / [8]. More recently, westudied the problem for the general 8-charge static BPS black holes [9]. Our focus wason the metrics where the asymptotic values of the scalar fields were taken to vanish. Weestablished that any such 8-charge black hole will certainly be related to other membersof this family under a conformal inversion, although it does not seem to be possible anylonger to present an explicit mapping of the charges in the way that could be done for the4-charge solutions. 4 slightly different approach was developed in [10], which could be applied not only tothe static BPS black holes of STU supergravity, but more generally in Einstein-Maxwell-scalar theories of type E [10]. An important aspect of this approach is that it is appliedin the context of the general class of static BPS black holes including the parameters as-sociated with the asymptotic values of the scalar fields. We exhibit this description of theconformal inversion mapping in detail for the STU supergravity case, where it allows anexplicit mapping of the general 14-parameter family of solutions (comprising 8 charges plus6 asymptotic scalar values). At the price of the conformal inversion always entailing amapping from one set of asymptotic scalar values to a different set of values, the approachallows one to give fully explicit transformations under the inversion.The organisation of this paper is as follows. In section 2, we discuss how the four-dimensional STU supergravity can be obtained by Kaluza-Klein reduction from higher di-mensions, where the internal space is a torus. In particular, it suffices for our purposes tostart from the pure N = 2 non-chiral supergravity in six dimensions (which itself can beobtained from a toroidal reduction of supergravity in ten dimensions), and then perform areduction on a 2-torus to four dimensions. In addition to discussing the resulting formu-lation of STU supergravity, known sometimes as the heterotic formulation, we also carryout explicitly the dualisation of two of the four gauge fields. This results in the formulationknown as the U-duality basis. This is the form in which the theory would arise if one beganwith the gauged N = 8 supergravity coming from the reduction of eleven-dimensional su-pergravity on S , truncated this to N = 2 supergravity plus three vector supermultiplets,and then took the ungauged limit. We also discuss the STU supergravity in a third dualitycomplex, which we refer to as the 3+1 formulation, where just one of the field strengths ofthe heterotic formulation is dualised. We shall make use of all three of these formulations ofSTU supergravity, and the explicit relations between them, in the remainder of the paper.In section 3, we consider the general extremal BPS static black hole solutions of STU su-pergravity in the U-duality formulation. These solutions carry 8 charges in total, comprisingelectric and magnetic charges for each of the four gauge fields. These are really equivalent tosolutions with just 5 independent charges, since the U (1) subgroup of the global SL (2 , R ) symmetry group of the STU theory allows 3 of the 8 charges to be transformed away (whilemaintaining the vanishing of the asymptotic values of the scalar fields at infinity). In factthe 8-charge solutions were constructed in [3], in the heterotic formulation, by starting froma 5-charge “seed solution” and then acting with the U (1) transformations. The analogousprocess was employed recently in [9] to construct the 8-charge solution in the U-duality5ormulation of STU supergravity. The new feature that we implement now in section 3 is toallow also for arbitrary asymptotic values of the six scalar fields in the 8-charge black holesolutions. This can be achieved by employing the action of the remaining six symmetriesin the coset SL (2 , R ) /U (1) .In section 4, we carry out the same steps of generating the general static BPS black holesolutions in the heterotic formulation. Again, we first “fill out” the 5-charge seed solutionto the full set of 8 charges by acting with the U (1) subgroup of the global symmetrygroup. We highlight similarities and also differences with the description in the U-dualityformulation.In section 5 we describe the consistent truncation of the STU supergravity to a su-persymmetric theory whose bosonic sector comprises the metric, two gauge fields and twoscalar fields (a dilaton and an axion). The theory is considerably simpler to work with thanthe full STU supergravity, and as we show, the general static BPS black hole solutions,characterised by four charges and asymptotic values for the two scalar fields, are very muchsimpler.In section 6, we consider the static BPS black holes in the 3+1 formulation, using thedescription of the theory in terms of the K¨ahler geometry of the scalar manifold. We thenuse this to investigate the behaviour of the black hole metrics under an inversion of the radialcoordinate, which, together with a conformal rescaling, maps the horizon to infinity andvice versa. In this description of the conformal inversion, which was discussed previously in[10], one can show how any given member of the general 14-parameter family of static BPSblack holes (characterised by the 4 electric charges, 4 magnetic charges and 6 asymptoticscalar values) is mapped by the conformal inversion to another member of the 14-parameterfamily. In this description, unlike one considered previously in [8, 9], the asymptotic valuesof the scalar fields are always different in the original and the conformally-inverted metrics.Section 7 contains a summary of our conclusions, and also further discussion. Somedetails of the Kaluza-Klein reduction from six to four dimensions are relegated to appendixA. In this section, we shall provide explicit expressions for the bosonic sector of four-dimensionalungauged STU supergravity in the three different formulations we shall be using in this pa- Partial results including the scalars were obtained in [3]. T , together with appropriate trunca-tions. The truncated theory has N = 2 supersymmetry, and comprises N = 2 supergravitycoupled to three vector multiplets. The theory was presented in this form in [11]. Weshall also consider the formulation of STU supergravity that one would obtain by reducingeleven-dimensional supergravity on S , performing an appropriate truncation from N = 8to N = 2 supersymmetry, and also turning off the gauge coupling constant in the four-dimensional theory. This formulation of STU supergravity is sometimes referred to as the U -duality invariant formulation. The two formulations are related in four dimensions byperforming an appropriate dualisation of two of the four gauge fields. We shall also considerSTU supergravity in an intermediate formulation which we refer to as the 3+1 formulation.In this case, instead of dualising two field strengths in the heterotic formulation only one isdualised. In the heterotic formulation one can in fact conveniently describe the STU theory by firstreducing from ten dimensions to six dimensions on T and truncating to pure N = 2non-chiral supergravity. The STU theory is then obtained by reducing this on T with nofurther truncation. (Except, of course, the usual Kaluza-Klein truncation in which onlythe singlets under the U (1) isometry of the T are retained.) The bosonic sector of thenon-chiral supergravity in six-dimensional supergravity is described by the Lagrangian L = ˆ R ˆ ∗ − ˆ ∗ d ˆ φ ∧ d ˆ φ − e −√ φ ˆ ∗ ˆ H (3) ∧ ˆ H (3) , (2.1)where ˆ H (3) = d ˆ B . The reduction down to four dimensions is described in detail in appendixA. After dualising the 2-form potential for the field H (3) to an axion χ , the resultingLagrangian for the bosonic sector of STU supergravity is given by (A.14): L = R ∗ − X i =1 , ( ∗ dϕ i ∧ dϕ i + e ϕ i ∗ dχ i ∧ dχ i ) − ∗ d ˜ ϕ ∧ d ˜ ϕ − e ϕ ∗ d e χ ∧ d e χ − e − ϕ h e ˜ ϕ − ϕ ∗ e F ∧ e F + e − ˜ ϕ + ϕ ∗ e F ∧ e F (2.2)+ e − ˜ ϕ − ϕ ∗ F ∧ F + e ˜ ϕ + ϕ ∗ F ∧ F i + χ ( e F ∧ e F + F ∧ F ) , e F = d e A , e F = d e A , F = dA and F = dA are the “raw” field strengths, and e F = e F − e χ F , e F = e F − χ F ,F = F , F = F + χ e F + e χ e F − χ χ F , (2.3)with overbars, denote the “dressed” field strengths that appear in the kinetic terms in (2.2).The gauge fields numbered 1 and 2 are denoted with tildes here; later on, we shall dualisethese fields in order to obtain the STU supergravity theory in the formalism in which the SL (2 , R ) symmetry is manifest. In this formulation the raw field strengths ( e F , e F , F , F ) are organised into the columnvector F ≡ F F F F = F e F F e F , (2.4)and the ( ϕ , χ ) dilaton/axion pair are redefined asΦ = ϕ , Ψ = χ . (2.5)The Lagrangian (2.2) can then be written as L = R ∗
1l + Tr( ∗ dM ∧ LdM L ) − ∗ d Φ ∧ d Φ − e ∗ d Ψ ∧ d Ψ − e − ( LM L ) ij ∗F i ∧ F j + Ψ L ij F i ∧ F j . (2.6)where L = σ ⊗ I = I I . (2.7)The scalar matrix M can be read off by comparing the kinetic terms for the gauge fields F i in (2.6) with the kinetic terms in (2.2). It is straightforward to see that M can then bewritten as M = G − − G − BB G − G − B G − B , (2.8)where G = e − ϕ e − ˜ ϕ + e χ e ˜ ϕ − e χ e ˜ ϕ − e χ e ˜ ϕ e ˜ ϕ , B = − χ χ . (2.9)8ne can recognise G as being associated with the internal metric on the 2-torus in theKaluza-Klein reduction (A.1), ( i.e. ds = G ij dz i dz j is the metric enclosed in squarebrackets in (A.1), in the 2-torus directions), and B as being associated with the internalcomponent A (0) of the 2-form potential in (A.2).The four-dimensional Lagrangian (2.6) is in the form that was obtained in [3]. It isinvariant under the O (2 ,
2) transformations (the T -duality of the 2-torus compactification): M −→ Ω M Ω T , A i −→ Ω ij A j , (2.10)where g µν and S are inert, and A i are the potentials for the field strengths F i ; that is, F i = d A i . The transformation matrix Ω ∈ O (2 ,
2) preserves the O (2 , L : Ω T L Ω =
L , L = I I . (2.11) L is in fact the metric tensor in the (2 , O (2 ,
2) trans-formations Ω act. L has components L ij with L = L = L = L = 0 and theremainder being zero. The inverse metric L − has components L ij , and for these too L = L = L = L = 0 with the remainder being zero. Note that one also has L M L = M − . (2.12)Note that M has components M ij and its inverse M − has components ( M − ) ij = L ik L jℓ M kℓ .The equations of motion and Bianchi identities are in addition invariant under the SL (2 , R ) transformations (electromagnetic S -duality): S −→ aS + bcS + d , F i → ( c Ψ + d ) F i + c e − ( M L ) ij ∗F j , (2.13)where g µν and the scalar matrix M are inert.The equations of motion for the electromagnetic fields that follow from (2.6) are d G i = 0 , where G i ≡ − e − ( LM L ) ij ∗F j + Ψ L ij F j , (2.14)which implies that the canonical electric charges ~α will be given by α i = 14 π Z G i . (2.15) We have changed the overall sign of the definition of the dual field strength G i in (2.14), relative to theone in [3]. This is for consistency with our conventions for the Hodge dualisation of differential forms (whichis made explicit in eqn (3.65), and our conventions in the rest of the paper. ~p are given by p i = 14 π Z F i . (2.16)It is convenient for later purposes to define also magnetic charges ~β with a lowered O (2 , β i = L ij p j . (2.17)These will be referred to later as the “canonical” magnetic charges. If the asymptoticvalues of the scalars are taken to be zero, one has S ∞ = i and M ∞ = I . We arrive at this formulation by dualising the gauge fields e A and e A appearing in theLagrangian (2.2). To do this, we employ the standard procedure of introducing dual poten-tials A and A as Lagrange multipliers, and adding the terms A ∧ d e F + A ∧ d e F to theLagrangian (A.14). Up to total derivatives, this is equivalent to adding L LM = F ∧ e F + F ∧ e F , (2.18)where F = dA and F = dA are the raw dualised field strengths. Varying the totalLagrangian with respect to e F and e F , now treated as independent fields, shows that e F and e F satisfy F + χ e F = e − ϕ + ˜ ϕ ( e − ϕ ∗ e F + χ e ϕ ∗ F ) ,F + χ e F = e − ϕ + ϕ ( e − ˜ ϕ ∗ e F + e χ e ˜ ϕ ∗ F ) , (2.19)and hence, using (2.3), F + χ e F + χ χ F = e − ϕ + ˜ ϕ ( e − ϕ ∗ e F + χ e ϕ ∗ F ) ,F + χ e F + χ e χ F = e − ϕ + ϕ ( e − ˜ ϕ ∗ e F + e χ e ˜ ϕ ∗ F ) , (2.20)These two equations, together with their Hodge duals, can be solved algebraically for e F and e F , with the results expressed, using (2.3) again, in terms of ( F , F , F , F ) and In toroidally compactified heterotic string theory the canonical electric charges ~α are quantised and spanan even self-dual lattice, subject to the constraint: ~α T L~α = − , , , · · · . For BPS-saturated configurationsone further requires ~α T L~α >
0. By S-duality the same conditions are satisfied for quantised magnetic charges ~β , along with the the condition that when ~α ∝ ~β with magnetic and electric charge vector components beingco-prime integers [12]. ∗ F , ∗ F , ∗ F , ∗ F ). Since the expressions are a little unwieldy, we shall not present themhere.Having solved for e F and e F these may be substituted back into the Lagrangian (2.2)plus (2.18), to give the theory in the dualised form, with the gauge fields ( A , A , A , A ).After one final step, in which the ˜ ϕ and e χ scalars are subjected to the discrete involutive SL (2 , R ) transformation˜ τ = − τ , where ˜ τ = e χ + i e − ˜ ϕ , τ = χ + i e − ϕ , (2.21)one finds that the Lagrangian is precisely equal to the one described in appendix B of[14] and appendix A of [9]. (The dualisation to go from the heterotic to the U-dualityformulation has also been discussed in [13].) The kinetic terms involving the field strengthsgiven, as in eqn (A.2) of [9], by L F = − f RAB ∗ F A ∧ F B − f IAB F A ∧ F B , (2.22)where f RAB and f IAB are the real and imaginary parts of the scalar matrix given in eqn(A.9) of [9]. This is written in the formalism and notation described in Freedman and VanProeyen [15], in which, defining G A = − f RAB ∗ F B − f IAB F B , (2.23)the field equations dG A = 0 and Bianchi identities dF A = 0 are invariant under the trans-formations FG −→ S FG , S = A BC D , (2.24)where the constant 4 × A , B , C and D obey A T C − C T A = 0 , B T D − D T B = 0 , A T D − C T B = I , (2.25)and we have defined F = ( F , F , F , F ) T and G = ( G , G , G , G ) T . The equations(2.25) are precisely the conditions for the matrix S to be an element of Sp (8 , R ), obeying[15] S T Ω S = Ω , where Ω = I − I . (2.26)The scalar field Lagrangian L scal = −
12 3 X i =1 ( ∗ dϕ i ∧ dϕ i + e ϕ i ∗ dχ i ∧ dχ i ) (2.27)11s invariant under SL (2 , R ) , which is a subgroup of Sp (8 , R ), and so in this formulation theSTU supergravity theory has a manifest SL (2 , R ) symmetry, at the level of the equationsof motion. The explicit forms of the A , B , C and D matrices corresponding to the SL (2 , R ) subgroup of Sp (8 , R ) are given in [9].The conserved electric and magnetic charges in the U-duality formalism are given by Q A = 14 π Z G A , P A = 14 π Z F A . (2.28)If we define the charge vectors P = ( P , P , P , P ) T and Q = ( Q , Q , Q , Q ) then theaction of SL (2 , R ) on the charges takes the same form as for the fields and their duals in(2.24), namely PQ −→ S PQ , (2.29)where again the 4 × A , B , C and D that form S are restricted to the SL (2 , R ) subgroup of Sp (8 , R ), as given in [9].There is in fact a simpler way to present the action of the SL (2 , R ) global symmetrieson the charges, by introducing the charge tensor γ aa ′ a ′′ , where a , a ′ and a ′′ are doubletindices of the three SL (2 , R ) factors in the symmetry group (see, for example, [11]). If wemake the assignments γ = − P , γ = P , γ = P , γ = P ,γ = − Q , γ = Q , γ = Q , γ = Q , (2.30)then the SL (2 , R ) transformation of the charges is given by γ aa ′ a ′′ −→ ( S ) ab ( S ) a ′ b ′ ( S ) a ′′ b ′′ γ bb ′ b ′′ , (2.31)where S i = a i b i c i d i , a i d i − b i c i = 1 for each i . (2.32)This gives exactly the same SL (2 , R ) transformation as in appendix A of [9]. It furthermoredemonstrates that γ aa ′ a ′′ is covariant with respect to SL (2 , R ) transformations.One can, of course, conveniently write the SL (2 , R ) transformation (2.24) in the anal-ogous way, by defining the field strength tensorΦ = − F , Φ = F , Φ = F , Φ = F , Φ = − G , Φ = G , Φ = G , Φ = G , (2.33)12hich transforms according toΦ aa ′ a ′′ −→ ( S ) ab ( S ) a ′ b ′ ( S ) a ′′ b ′′ Φ bb ′ b ′′ , (2.34)Furthermore, if the dilaton/axion pairs of scalar fields ( ϕ , χ ), ( ϕ , χ ) and ( ϕ , χ ) areassembled into the matrices M i = e ϕ i − χ i e ϕ i − χ i e ϕ i e − ϕ i + χ i e ϕ i , for i = 1 , , , (2.35)whose components are ( M ) ab , ( M ) a ′ b ′ and ( M ) a ′′ b ′′ respectively, then the scalars trans-form under the SL (2 , R ) factors according to M i −→ M ′ i = ( S Ti ) − M i S − i , (2.36)(with the i ’th scalars transforming only under the i ’th SL (2 , R ) group). As we saw in section 2.2, the scalar fields in the heterotic formulation comprise the dila-ton/axion pairs (Φ , Ψ), ( ϕ , χ ) and ( ϕ , χ ). These come from the Kaluza-Klein reductionfrom six dimensions, as described in appendix A, with Φ = ϕ and Ψ = χ . The pairs( ϕ , χ ) and ( ϕ , χ ) are packaged into the O (2 , /U (1) scalar coset matrix M given by(2.8) and (2.9), while (Φ , Ψ) parameterise the SL (2 , R ) /U (1) scalar coset.In the U-duality formulation the scalars comprise the same set ( ϕ , χ ), ( ϕ , χ ) and( ϕ , χ ), except that now the ( ϕ , χ ) pair are subjected to the involution τ → − /τ ,where τ = χ + ie − ϕ , as given in (2.21). Thus, in total, the mapping of fields from thoseof the heterotic formulation and to those of the U-duality formulation is as follows:Φ −→ ϕ , Ψ −→ χ , ϕ −→ ϕ , χ −→ χ ,e ˜ ϕ −→ (1 + χ e ϕ ) e − ϕ , e χ −→ − χ e ϕ (1 + χ e ϕ ) − . (2.37)The relation between the electric and magnetic charges in the two formulations followsfrom the relations between the field strengths, which we discussed earlier. The dual fieldstrengths G i in the heterotic formulation, given by (2.14), can be seen, after making use ofthe equations (2.20) to express the fields e F and e F in (2.4) in terms of the dual fields F and F , to be related to the fields F A and dual fields G A of the U-duality formulation by G = G (cid:12)(cid:12)(cid:12) τ →− /τ , G = − F , G = G (cid:12)(cid:12)(cid:12) τ →− /τ , G = − F , (2.38)13here in addition to the involution of ( ϕ , χ ) indicated here, we also make the replacementsΦ → ϕ and Ψ → χ , as given in (2.37). Similarly, one finds that the fields F i of theheterotic formulation are related to F A and G A by F = F , F = G (cid:12)(cid:12)(cid:12) τ →− /τ , F = F , F = G (cid:12)(cid:12)(cid:12) τ →− /τ . (2.39)From the definitions (2.15) and (2.16) for the electric and magnetic charges α i and p i in theheterotic formulation, and the definitions (2.28) for the electric and magnetic charges Q A and P A in the U-duality formulation we therefore have ~α = ( α , α , α , α ) = ( Q , − P , Q , − P ) (2.40)and ~p = ( p , p , p , p ) = ( P , Q , P , Q ) . (2.41)It is also useful to record the mapping from the magnetic charges with lowered index in theheterotic formulation, β i = L ij p j , for which we therefore have ~β = ( β , β , β , β ) = ( P , Q , P , Q ) . (2.42) formulation There is a third choice of duality frame for STU supergravity that is useful for some purposes.In this frame, which we refer to as the “3 + 1 formalism,” the starting point is again theSTU supergravity Lagrangian given in eqns (2.2) and (2.3). We then dualise just the e F field. in this formalism, therefore, the simple 4-charge BPS solution would carry 3 electriccharges and 1 magnetic (or vice versa).To perform the dualisation of e F , we add a Lagrange multiplier term L LM = F ∧ e F ,where F = dA is the dualised field, and then solve the algebraic equation of motionobtained by varying F . This allows us to solve for the dressed field e F = e F − χ F ,finding e F = − e χ e ϕ ) h e ϕ + ˜ ϕ + ϕ ( ∗ F + χ ∗ e F ) + e χ e ϕ ( F + χ e F ) i . (2.43)Substituting back into the original Lagrangian plus the Lagrange multiplier term, we obtainthe bosonic STU supergravity Lagrangian written in the 3 + 1 formulation. It is convenientto define the new ( ϕ , χ ) scalars related to ( ˜ ϕ , e χ ) by the involution ˜ τ = − /τ , exactlyas in (2.37), and the three “dressed” field strengths:ˆ F = F + χ e F , ˆ F = F + χ e F , ˆ F = F + χ e F . (2.44)14n terms of these, the 3 + 1 STU Lagrangian takes the form L = R ∗ −
12 3 X i =1 ( ∗ dϕ i ∧ dϕ i + e ϕ i ∗ dχ i ∧ dχ i ) − e − ϕ − ϕ − ϕ ∗ e F ∧ e F − e ϕ − ϕ − ϕ ∗ ˆ F ∧ ˆ F − e − ϕ + ϕ − ϕ ∗ ˆ F ∧ ˆ F − e − ϕ − ϕ + ϕ ∗ ˆ F ∧ ˆ F + χ F ∧ F + χ F ∧ F + χ F ∧ F +( χ χ F + χ χ F + χ χ F ) ∧ e F + χ χ χ e F ∧ e F . (2.45)Note that e F , F , F and F are the bare field strengths: e F = d e A , F = dA , F = dA , F = dA . (2.46)Note also that the scalar fields here are exactly the same as the ones in the U-dualityformulation. In this 3 + 1 formulation there is a permutation symmetry among the sets offields ( ϕ , χ , F ) , ( ϕ , χ , F ) , ( ϕ , χ , F ) . (2.47)We shall return to a discussion of the 3+1 formulation of STU supergravity in section6. The most general extremal BPS static black hole solution in STU supergravity will carry 8independent charges, since each of the four field strengths can carry an electric charge anda magnetic charge. Since the theory has an SL (2 , R ) global symmetry, this can be used inorder to map one solution into another solution that is equivalent under the group action.The 3-parameter compact subgroup U (1) leaves the asymptotic values of the six scalarfields unchanged, and this means that it suffices to consider an 8 − U (1) . If ones starts with a 5-charge seed solution in which the dilatonicand axionic scalars vanish asymptotically at infinity, then the 8-charge solutions obtained inthis way will all have vanishing asymptotic scalars. One can also then choose to fill out thesolution set further by then acting with the remaining 6-parameter coset SL (2 , R ) /U (1) transformations, thereby giving arbitrary asymptotic values to the six scalar fields. A dualisation to obtain the STU theory in the 3+1 formulation can also be found in [16].
15n the subsections below, we shall present the general 8-charge static BPS black holesolutions in the U-duality formulation, both for vanishing values of the asymptotic scalarfields and for arbitrary values of the asymptotic scalar fields.
The general 8-charge solution for the case of vanishing asymptotic scalars was constructedin this formulation in [9]. The starting point was the solution with 5 independent chargesthat was constructed in [3], after translating it into the U-duality formulation. The fiveindependent charges could be taken to be Q = ( Q , Q , Q , Q ) , P = (¯ p, − ¯ p, , . (3.1)(We are placing bars on the charges in the 5-charge seed solution.) For this solution, themetric is given by [3] ds = − r √ V dt + √ Vr ( dr + r d Ω ) , (3.2) V = r + α r + β r + γ r + ∆ , (3.3)where α = X i Q i , β = X i Ψ = − ¯ p ( Q − Q )2( r + Q )( r + Q ) , (3.15) e ˜ ϕ = ( r + Q )( r + Q ) √ V , e χ = ¯ p [ r + ( Q + Q )]( r + Q )( r + Q ) , (3.16) e ϕ = ( r + Q )( r + Q ) √ V , χ = − ¯ p [ r + ( Q + Q )]( r + Q )( r + Q ) . (3.17) The labelling of the torus coordinates is opposite in [3] to the labelling we are using in this paper, withour coordinates ( z , z ) in appendix A being equal to ( y , y ) in [3]. This means that the torus metric and2-form components G , G , G and B in eqns (42) in [3] should be interpreted as our G , G , G and − B respectively (see eqns (2.8) and (2.9)). e ϕ = ( r + Q )( r + Q ) √ V , χ = − ¯ p ( Q − Q )2( r + Q )( r + Q ) , (3.18) e ϕ = ( r + Q )( r + Q ) √ V , χ = − ¯ p [ r + ( Q + Q )]( r + Q )( r + Q ) , (3.19) e ϕ = ( r + Q )( r + Q ) √ V , χ = − ¯ p [ r + ( Q + Q )]( r + Q )( r + Q ) . (3.20)To find the expressions for the scalar fields in the general 8-charge solution, we followthe same strategy that we used previously for the metric. Namely, we take the expressionsabove for the scalar fields in the 5-charge seed solution, and then fill these out to 8-chargesolutions by acting with the U (1) ∈ SL (2 , R ) global symmetry transformations. A newfeature that arises here is that the scalar fields, unlike the metric, themselves transformunder the global symmetries, and so these transformations must be included also in thecalculation. To be precise, the dilaton/axion pair ( ϕ , χ ) transforms under SL (2 , R ) butis inert under SL (2 , R ) and SL (2 , R ) . Analogous statements apply to ( ϕ , χ ) and to( ϕ , χ ).As in our construction of the metric functions for the 8-charge solution, we find itconvenient to first obtain the scalar solutions for the case where the asymptotic values ofthe dilatons and axions are all zero. A very simple replacement at the final stage of thecalculation allows the introduction of arbitrary values for the asymptotic scalars.The process of promoting the 5-charge seed solution to a full 8-charge solution proceedsin the same we that we described in [9]. Because we have made a small adjustment in theconventions in this paper, in order to allow a direct mapping to equivalent the results inthe heterotic formulation, we need first to record the explicit results for the U (1) rotationsthat yield the 8-charge solution. Specifically, it was convenient in this paper to change the5-charge configuration in the U-duality formalism from the one specified in eqn (B.4) of[9] to the one specified in eqn (3.1) of this paper. (That is, P = ( p, − p, , 0) rather than P = (0 , , p, − p ).) This modifies the expressions that were found in [9] when solving for the5 charges and the three U (1) angles θ i in terms of the 8 generic charges Q i and P i . Thuswe solve the 8 equations contained in¯ γ aa ′ a ′′ = ( U ) ab ( U ) a ′ b ′ ( U ) a ′′ b ′′ γ bb ′ b ′′ , (3.21)where ¯ γ aa ′ a ′′ is the charge tensor for the 5-charge seed configuration in (3.1); γ aa ′ a ′′ is thecharge tensor of the generic 8-charge configuration, and U i denote the three U (1) matrices19ith θ , θ and θ as parameters: U i = cos θ i sin θ i − sin θ i cos θ i . (3.22)(See eqn (2.30) for a definition of γ aa ′ a ′′ .) Eqn (3.21) thus yields expressions for the threeangles θ i and the 5 charges of the seed solution, expressed in terms of the 8 arbitrary chargesof the generic 8-charge solution. We findtan θ + = ( P + P ) − ( Q + Q ) tan θ ( Q + Q )+( P + P ) tan θ , tan θ − = ( P − P )+( Q − Q ) tan θ ( Q − Q ) − ( P − P ) tan θ , (3.23)tan 2 θ = 2( P + P )( Q + Q ) − P + P )( Q + Q )( P + P + P + P )( P + P − P − P ) − ( Q + Q + Q + Q )( Q + Q − Q − Q ) , where θ ± = θ ± θ , and¯ p = − ¯ γ , Q = − ¯ γ , Q = ¯ γ , Q = ¯ γ , Q = ¯ γ , (3.24)where we have placed bars on the 5 seed charges Q i and ¯ p .To begin, let us consider the dilaton/axion pair ( ϕ , χ ). The first step is to promotethe right-hand sides of eqns (3.18) to 8-charge expressions, by using eqns (3.24) togetherwith (3.24). We shall write the expressions in (3.18) as e − ϕ = p V ( r ) D ( r ) χ = δD ( r ) , (3.25)where D ( r ) = r + α D r + β D , (3.26) α D = ( Q + Q ) , β D = Q Q , δ = ¯ p ( ¯ Q − ¯ Q ) . (3.27)We first promote these 5-charge expressions to 7-charge expressions, by acting with the U (1) and U (1) rotations, while keeping θ = 0. From (3.24), this latter requirementmeans the 7-charge restriction implies the Q A and P A must obey( P + P )( Q + Q ) − ( P + P )( Q + Q ) = 0 . (3.28)This leads straightforwardly to the 7-charge augmentations α D (7) = ( Q + Q ) + ( P + P ) , β D (7) = Q Q + P P ,δ (7) = [( P − P )( Q − Q ) − ( P − P )( Q − Q )] . (3.29)20he augmentation from 7 to 8 charges, relaxing the constraint (3.28), is achieved byfurther rotating the quantities in (3.29) under the remaining U (1) transformation (nowwith θ and θ set to zero). It is helpful at this point to establish a general result for the U (1) rotation of a general 2-index symmetric tensor W ab ; this obeys the transformation W ab −→ f W ab = ( S ) ac ( S ) bd W cd , (3.30)where S is the U (1) ∈ SL (2 , R ) matrix S = cos θ sin θ − sin θ cos θ . (3.31)If we define the U (1) × U (1) -invariant symmetric tensor Z ab = ( ǫ a ′ b ′ ǫ a ′′ b ′′ − δ a ′ b ′ δ a ′′ b ′′ ) γ aa ′ a ′′ γ bb ′ b ′′ , (3.32)where the charge tensor γ aa ′ a ′′ is defined in (2.30), then the expression for tan 2 θ in eqn(3.24) can be written in the compact formtan 2 θ = 2 Z Z − Z . (3.33)From this we find cos 2 θ = Z − Z Ξ , sin 2 θ = 2 Z Ξ , (3.34)where Ξ is the U (1) -invariant quantity defined byΞ = Z ab Z cd ( δ ac δ bd − ǫ ac ǫ bd ) . (3.35)It then follows straightforwardly from (3.30) that f W = h δ ab W ab + G ( W )Ξ i , f W = h δ ab W ab − G ( W )Ξ i , f W = 1Ξ Z ab W cd δ ac ǫ bd , (3.36)where G ( W ) ≡ W ab Z cd ( δ ac δ bd − ǫ ac ǫ bd ) . (3.37)(Note from (3.35) that G ( Z ) = Ξ .)To proceed with the augmentation of the 7-charge expressions (3.29), we note that if wedefine the two U (1) × U (1) -invariant tensors X ab = ǫ a ′ b ′ ǫ a ′′ b ′′ γ aa ′ a ′′ γ bb ′ b ′′ , Y ab = δ a ′ b ′ δ a ′′ b ′′ γ aa ′ a ′′ γ bb ′ b ′′ , (3.38)21here the charge tensor γ aa ′ a ′′ is defined in eqn (2.30), then the 7-charge expressions in(3.29) can be written as α D (7) = − e Z , β D (7) = − e X , δ (7) = ( e X + e Y ) . (3.39)Using the U (1) transformations (3.36), we therefore obtain the 8-charge expressions α D = − ( δ ab Z ab − Ξ) ,β D = − h δ ab X ab − G ( X )Ξ i ,δ = − 14Ξ ( X ab + Y ab ) Z cd δ ac ǫ bd , (3.40)and G ( X ) is given by substituting W ab = X ab in (3.37). These expressions are all manifestlyinvariant under U (1) . It is useful to note that α D can be written in terms of α (given in(3.7)) and Ξ as α D = α + Ξ2 α . (3.41)It is also worth noting that if we define the U (1) invariants Z = − δ ab ǫ a ′ b ′ ǫ a ′′ b ′′ γ aa ′ a ′′ γ bb ′ b ′′ ,Z = − ǫ ab δ a ′ b ′ ǫ a ′′ b ′′ γ aa ′ a ′′ γ bb ′ b ′′ ,Z = − ǫ ab ǫ a ′ b ′ δ a ′′ b ′′ γ aa ′ a ′′ γ bb ′ b ′′ ,Z = δ ab δ a ′ b ′ δ a ′′ b ′′ γ aa ′ a ′′ γ bb ′ b ′′ , (3.42)then Ξ , defined in (3.35), can be written in the factorised formΞ = ( Z + Z + Z + Z )( Z + Z − Z − Z ) . (3.43)It remains to implement the U (1) transformations that augment the 5 charges to 8charges on the scalar fields. Since we are looking specifically at ϕ and χ , which transformonly under U (1) , this means that we just have to make the replacement χ + i e − ϕ −→ ( χ + i e − ϕ ) cos θ + sin θ cos θ − ( χ + i e − ϕ ) sin θ . (3.44)In other words, the scalars ϕ and χ will be given by e ϕ = D D ( r ) p V ( r ) , χ = CD , (3.45) The transformation (3.36) is viewed here as the inverse , taking us from the general 8-charge (untilded)configuration to the 7-charge (tilded) configuration. C = δD ( r ) cos 2 θ − h V ( r ) + δ D ( r ) − i sin 2 θ , D = 12 h V ( r ) + δ D ( r ) i − δD ( r ) sin 2 θ + 12 h − V ( r ) + δ D ( r ) i cos 2 θ , (3.46)where V ( r ) in (3.2) is calculated using the 8-charge expressions (3.7)–(3.10), the quantities δ , α D and β D appearing in D ( r ) are given by the 8-charge expressions in (3.40), and cos 2 θ and sin 2 θ are given by (3.34).The expressions for ϕ and χ are in fact considerably simpler than is immediatelyevident from (3.45) and (3.46). This is because the quadratic function D ( r ) is actually adivisor of the quartic function V ( r ) + δ that appears in (3.46). In fact we can show that V ( r ) + δ = D ( r ) e D ( r ) , e D ( r ) = r + ˜ α D r + ˜ β D , (3.47)where the coefficients in the quadratic function e D ( r ) are given by˜ α D = − ( δ ab Z ab + Ξ) , ˜ β D = − h δ ab X ab + G ( X )Ξ i . (3.48)(Note that ˜ α D and ˜ β D are closely related to the coefficients α D and β D in the function D ( r ) = r + α D r + β D given in (3.40).) Thus we see that ϕ and χ are given by e ϕ = r + b r + b p V ( r ) , χ = c r + c r + b r + b , (3.49)where b = α D + ˜ α D α D − ˜ α D )( Z − Z )2Ξ ,b = β D + ˜ β D − δ Z Ξ + ( β D − ˜ β D )( Z − Z )2Ξ ,c = ( α D − ˜ α D ) Z Ξ , c = δ ( Z − Z ) + ( β D − ˜ β D ) Z Ξ , (3.50)Using our previous expressions for the various quantities appearing in (3.50), we find thatin terms of the eight charges of the general static BPS black holes, b = 1 α h ( P + P ) X A P A + ( Q + Q ) X A Q A i ,b = P P + Q Q ,c = 1 α h ( P + P )( Q + Q ) − ( P + P )( Q + Q ) i ,c = ( P Q + P Q − P Q − P Q ) , (3.51)where as usual α = p ( P A P A ) + ( P A Q A ) .23inally, it remains to introduce non-vanishing asymptotic values for the scalar fields.As before when we discussed the metric, this is accomplished by making SL (2 , R ) /U (1) coset transformations on the charges, and now also on the scalar fields. Thus we introducean asymptotic coset vielbein V i for each of the SL (2 , R ) i /U (1) i coset factors, with V i = e 12 ¯ ϕ i − ¯ χ i e 12 ¯ ϕ i e − 12 ¯ ϕ i , (3.52)where ( ¯ ϕ i , ¯ χ i ) denotes the asymptotic values of the scalar fields. The corresponding trans-formation of the charge tensor is therefore γ aa ′ a ′′ −→ V b a V b ′ a ′ V b ′′ a ′′ γ bb ′ b ′′ . (3.53)The effect of this is that all U (1) invariants, such as α , β , γ , α D , β D , δ , ˜ α D , ˜ β D will receivemodification, namely that each Kronecker delta in the expressions (3.7)–(3.9), (3.32), (3.42),(3.50), etc., will be replaced by the corresponding asymptotic scalar matrix M i = V Ti V i , (3.54)as in eqn (3.12). Additionally, the components of the tensor Z ab appearing in the expressions(3.50) will be transformed by making the replacement Z ab −→ V c a V d b Z cd . (3.55)As already observed, the quantity ∆ receives no modification because its construction in(3.10) uses only the SL (2 , R )-invariant epsilon tensors, and no Kronecker deltas.It is worth recording that even when the asymptotic scalars are non-zero, the quantityΞ, defined now by (3.35) with all Kronecker deltas replaced by the asymptotic scalar M matrices as in (3.12), is again simply factorisable, asΞ = α ˜ α , (3.56)where α is given in (3.13) and ˜ α is the closely related quantity given by˜ α = (cid:16) M ab M a ′ b ′ M a ′′ b ′′ γ aa ′ a ′′ γ bb ′ b ′′ − M ab ǫ a ′ b ′ ǫ a ′′ b ′′ + ǫ ab M a ′ b ′ ǫ a ′′ b ′′ + ǫ ab ǫ a ′ b ′ M a ′′ b ′′ (cid:17) γ aa ′ a ′′ γ bb ′ b ′′ . (3.57)In other words Ξ , which is a quartic multinomial in the charges with coefficients that aremultinomial in the ¯ χ i and e ¯ ϕ i asymptotic scalar values, factorises as the product of the twoquadratic multinomials α and ˜ α . 24t is also useful to note that the quantities α D and ˜ α D , given by (3.40) and (3.48) afterthe replacements (3.12), are related to α and ˜ α by α D + ˜ α D = α , α D − ˜ α D = ˜ α . (3.58)Thus after introducing the asymptotic scalar values the coefficients b and c in eqns (3.50)are particularly simple, and are given by b = 12 α h α + e Z − e Z i , c = 1 α e Z , (3.59)where e Z ab = V c a V d b Z cd and all Kronecker deltas involved in the construction of α and Z ab are as usual replaced by M matrices according to (3.12).The action of the coset transformations on the scalar fields themselves will be given bysending M i −→ V Ti M i V i , for i = 1 , , , (3.60)where the M i scalar matrices are given in (2.35). These transformations amount to e − ϕ i −→ e − ϕ i − ¯ ϕ i , χ i −→ ¯ χ i + χ i e − ¯ ϕ i . (3.61)The U (1) rotation angle θ will be modified by the coset transformations, as dictated bysubstituting the γ aa ′ a ′′ transformations (3.53) into the expression (3.32) for Z ab , and thensubstituting these transformed Z ab components into (3.33). However, since we have alreadyre-expressed the expressions in (3.45) and (3.46) for ϕ and χ in the simpler forms given by(3.49) and (3.50), we no longer need to implement the coset transformation on θ explicitly.The procedure we have described above provides specifically the expressions for thedilaton/axion pair ( ϕ , χ ) in the general 8-charge static BPS black hole solutions. Onecould repeat the discussion, starting from the same 5-charge seed solution, to augment theexpressions (3.19) and (3.20) for the ( ϕ , χ ) and ( ϕ , χ ) dilaton/axion pairs. In fact asimpler way of arriving at the same result is to exploit the triality symmetry of the U-dualityformulation of STU supergravity. The precise statement of this triality can be seen fromthe definition of the SL (2 , R ) charge tensor γ aa ′ a ′′ in (2.30). The first SL (2 , R ) index, a ,is associated with the ( ϕ , χ ) pair, and the charges ( P , Q ); the a ′ index with ( ϕ , χ )and the charges ( P , Q ), and the a ′′ index with ( ϕ , χ ) and the charges ( P , Q ). Thusfrom the expressions we have obtained for ( ϕ , χ ), we just have to permute the labellingsaccording to this triality correspondence, in order to obtain the expressions for ( ϕ , χ ) and( ϕ , χ ). 25cting with the Z ∈ triality symmetry( ϕ , χ ; P , Q ) ←→ ( ϕ , χ ; P , Q ) (3.62)on the results obtained above for the the ϕ and χ scalars will give the expressions for ϕ and χ . Acting instead with the Z ∈ triality symmetry( ϕ , χ ; P , Q ) ←→ ( ϕ , χ ; P , Q ) (3.63)will give the expressions for ϕ and χ .If we specialise for simplicity to the case with vanishing asymptotic values for the scalarfields, acting with the triality transformations (3.62) or (3.63) on the constants in (3.51)will map (3.49) into expressions for ϕ and χ , or ϕ and χ , respectively.As a check on the calculations in this section, we can take the expressions of the ϕ i and χ i scalars and make the 5-charge specialisation given by eqn (3.1). Doing this, itis straightforward to see that we do indeed recover the expressions given in eqns (3.18),(3.19) and (3.20). Expressed conversely, this shows that if we were instead to act withthe U-duality transformations in order to elevate the 5-charge expressions for ( ϕ , χ ) and( ϕ , χ ) to general 8-charge expressions, we would indeed obtain the results that follow bymaking the triality transformations on the general 8-charge expressions for ( ϕ , χ ) that wehave constructed. Having obtained the expressions for the metric and the scalar fields, the form of the gaugefields in the general 8-charge static BPS black hole solutions follow straightforwardly fromthe gauge fields equations of motion.In the U-duality formulation, we find F A = P A sin θ dθ ∧ dϕ + 1 √ V (cid:0) ( f R ) − (cid:1) AB ( Q B + f IBC P C ) dt ∧ dr , (3.64)which is consistent with the definitions of the electric and magnetic charges in (2.28).Note that our conventions for Hodge dualisation are such that in the metric (3.2) wehave ∗ (sin θ dθ ∧ dϕ ) = 1 √ V dt ∧ dr , ∗ ( dt ∧ dr ) = −√ V sin θ dθ ∧ dϕ . (3.65)26 General 8-Charge Static BPS Black Holes In The HeteroticFormulation The starting point for writing the general 8-charge static BPS black holes with arbitraryasymptotic values for the scalar fields is again the metric (3.2). The charges in the seedsolution with five independent charges and with vanishing asymptotic scalars take the form ~α = ( q , − ¯ q, q , ¯ q ) , ~p = (0 , p , , p ) (4.1)in the heterotic basis [3]. (The superscripts 2 and 4 refer to the indexed labelling of charges p i . See section 2.2 for the notation.) The quantity ¯ q parameterises the introduction of thefifth, independent, charge) The coefficients α , β , γ and ∆ appearing in the metric function V ( r ) are given by [3] α = q + q + p + p ,β = q q − ¯ q + p p + ( q + q )( p + p ) ,γ = ( q q − ¯ q )( p + p ) + p p ( q + q ) , ∆ = ( q q )( p p ) − ¯ q ( p + p ) . (4.2)(Here ¯ q means just the square of ¯ q .)The general solutions with 8 charges and non-vanishing asymptotic scalars can befilled out by acting with the global O (2 , × SL (2 , R ) symmetry of the heterotic formula-tion, where O (2 , 2) is the T-duality symmetry from the 2-torus and SL (2 , R ) is the elec-tric/magnetic S-duality symmetry. We proceed in three stages; first, acting with the U (1) subgroup of O (2 , 2) to augment the 5-charge solution to 7 charges; then acting with the U (1) subgroup of SL (2 , R ) to augment further to 8 charges; and finally, acting with theremaining O (2 , /U (1) and SL (2 , R ) /U (1) cosets in order to introduce the non-vanishingasymptotic scalars.Acting with U (1) ∈ O (2 , U (1) compact subgroup of O (2 , α , β and γ should be invariant under U (1) but not under the remaining O (2 , /U (1) coset action (that is, when the asymptoticscalars vanish), the 2 additional charges can be introduced by writing α , β and γ as U (1) -invariant expressions in ~α and ~p that reduce to (4.2) under the 5-charge specialisation (4.1).27n fact, rather than using the magnetic charges ~p with components p i we shall insteadfind it convenient to lower the index and work with the magnetic charges ~β , which havethe components β i = L ij p j , as defined in (2.17). The available “building blocks” are thematrix L defined in (2.11), which is fully O (2 , 2) invariant, and the 4 × I ,which is invariant only under the U (1) subgroup. We can also construct the more generalexpression for ∆ along similar lines. Since ∆ is in fact invariant under the full O (2 , I .Defining the matrices ν ± = I ± L − , (4.4)one can straightforwardly establish that α , β , γ and ∆ may be written for the 7-chargesolutions in U (1) -invariant terms as follows: α = ~α T ν + ~α + ~β T ν + ~β + 2Σ , (4.5)2 β = ~α T L − ~α + ~β T L − ~β + 2Σ , (4.6)2 αγ = ( ~α T L − ~α + ~β T L − ~β )Σ+( ~α T L − ~α )( ~β T ν + ~β ) + ( ~β T L − ~β )( ~α T ν + ~α ) , (4.7)4∆ = ( ~α T L − ~α )( ~β T L − ~β ) − ( ~α T L − ~β ) , (4.8)where Σ = ( ~α T ν + ~α )( ~β T ν + ~β ) . (4.9)These expressions reduce to those in (4.2) under the 5-charge specialisation (4.1). Note alsothat in the original 5-charge configuration (4.1), the charges satisfy the constraint ~α T ν + ~β = 0 , (4.10)and that this continues to be true after filling out to 7 charges, as we have done by actingwith U (1) . Actually things are a bit more complicated. There are two further matrices that are invariant under the U (1) subgroup, namely K and K given by K = iσ iσ , K = iσ iσ , (4.3)where σ is the usual Pauli matrix. Note that K and K are antisymmetric. It turns out that by using K = K + K , one can write the quantity Σ that appears later in (4.15) as a manifest perfect square.It then gets dressed up with asymptotic scalars once these are turned on. See later in this section for adiscussion of this point. U (1) ∈ SL (2 , R ):The constraint (4.10) is removed once the 8th and final charge is introduced. This isachieved by acting with the U (1) subgroup of the remaining SL (2 , R ) S-duality symmetry.Its action on ~α and ~β is ~α → ~α cos ψ − ~β sin ψ , ~β → ~α sin ψ + ~β cos ψ , (4.11)where tan 2 ψ = − ~α T ν + ~β )( ~α T ν + ~α ) − ( ~β T ν + ~β ) . (4.12)Thus we have cos 2 ψ = 1Ξ ( ~α T ν + ~α − ~β T ν + ~β ) , sin 2 ψ = − ~α T ν + ~β , (4.13)where Ξ = ( ~α T ν + ~α + ~β T ν + ~β ) − , (4.14)Σ = ( ~α T ν + ~α )( ~β T ν + ~β ) − ( ~α T ν + ~β ) . (4.15)We now apply the transformation (4.11), with cos 2 ψ and sin 2 ψ given by (4.13), to theexpressions (4.5)–(4.8) in order to add in the 8th charge. It is useful to note that for anysymmetric 4 × X we shall therefore have ~α T X ~α −→ ( ~α T X ~α + ~β T X ~β ) + G ( X )2Ξ ,~β T X ~β −→ ( ~α T X ~α + ~β T X ~β ) − G ( X )2Ξ , (4.16) ~α T X ~β −→ h ( ~α T X ~β )( ~α T ν + ~α − ~β T ν + ~β ) − ( ~α T ν + ~β )( ~α T X ~α − ~β T X ~β ) i , where G ( X ) = ( ~α T X ~α − ~β T X ~β )( ~α T ν + ~α − ~β T ν + ~β ) + 4( ~α T X ~β )( ~α T ν + ~β ) . (4.17)Note that G ( ν + ) = Ξ .Applying these results in (4.5)–(4.8), we therefore find that the general 8-charge expres-sions of α , β , γ and ∆ are given by α = ~α T ν + ~α + ~β T ν + ~β + 2Σ , (4.18)2 β = ~α T L − ~α + ~β T L − ~β + 2Σ , (4.19)2 αγ = ( ~α T L − ~α + ~β T L − ~β ) Σ − ~α T L − ~β )( ~α T ν + ~β )+( ~α T L − ~α )( ~β T ν + ~β ) + ( ~β T L − ~β )( ~α T ν + ~α ) , (4.20)4∆ = ( ~α T L − ~α )( ~β T L − ~β ) − ( ~α T L − ~β ) . (4.21)29e may now observe that these 8-charge expressions may be written in a more compactnotation by introducing the SL (2 , R )-valued 8-charge vector ~v a , where ~v = ~α , ~v = ~β . (4.22)We also introduce the SL (2 , R )-invariant antisymmetric tensor ε ab , with ε = 1, and theKronecker delta δ ab , which is invariant only under the U (1) subgroup of SL (2 , R ). Usingthese, the quantities α , β , γ and ∆ in (4.18)–(4.21) may be rewritten as α = δ ab ~v a T ν + ~v b + 2Σ , (4.23)2 β = δ ab ~v a T L − ~v b + 2Σ , (4.24)2 αγ = δ ab ( ~v a T L − ~v b ) Σ + ε ac ε bd ( ~v a T L − ~v b )( ~v c T ν + ~v d ) , (4.25)8∆ = ε ac ε bd ( ~v a T L − ~v b )( ~v c T L − ~v d ) , (4.26)where Σ = ε ac ε bd ( ~v a T ν + ~v b )( ~v c T ν + ~v d ) . (4.27)Introducing the asymptotic scalars:This final step is achieved by acting with the cosets O (2 , /U (1) and SL (2 , R ) /U (1),appropriately parameterised in terms of the asymptotic values of the scalar fields. This canbe done by using a vielbein formulation for both the O (2 , /U (1) scalar coset matrix andthe SL (2 , R ) /U (1) scalar coset matrix.For O (2 , /U (1) , the scalar matrix M given in (2.8) can be written as M = V T V , V = E − − E − B E T , (4.28)where E is the zweibein for the internal 2-torus metric G . Note that V T L − V = L − . Interms of indices we have M ij = δ kℓ V ki V ℓj , L ij = L kℓ V ki V ℓj . (4.29)We shall then make the O (2 , /U (1) transformation F → ( V T ) − F on the gauge fields,which implies the transformations ~α −→ V ~α , ~β −→ V ~β , (4.30)30n the charges, where V denotes the asymptotic value of the scalar vielbein V . Since V T L − V = L and V T V = M , this means that the only change in the expressions (4.18)–(4.27) will be that the matrix I in ν + = I + L − will change to M , and so ν + −→ µ + , µ + = M + L − . (4.31)In a similar way, the scalars Φ and Ψ enter the SL (2 , R ) /U (1) scalar coset in the form N = e − Ψ − Ψ Ψ + e − , (4.32)and this can be written in terms of a vielbein U = U as N = U T U , U = e Φ − Ψ e Φ e − Φ . (4.33)The components U ab of this matrix obey the SL (2 , R ) invariance condition U ca U db ε cd = ε ab , (4.34)while, contracted instead with δ cd , we have U ca U db δ cd = N ab , (4.35)where N ab are the components of the SL (2 , R ) /U (1) scalar matrix (4.32). Thus if wetransform the electric and magnetic charges under the SL (2 , R ) /U (1) transformation ~v a −→ U ab ~v b , (4.36)where U ab is the scalar vielbein (4.33) with the scalars set equal to their asymptotic values,then the only change in the expressions (4.23)–(4.26) and (4.27) will be that the Kroneckerdelta δ ab will be replaced by the corresponding SL (2 , R ) /U (1) scalar matrix N ab , where thescalars in N ab defined in (4.32) are set equal to their asymptotic values.Pulling together the threads of the previous discussion, if we act with O (2 , /U (1) and SL (2 , R ) /U (1) coset elements to introduce asymptotic values for all the scalar fields, thefinal expressions for the quantities α , β , γ and ∆ given previously in (4.23)–(4.26) will begiven by α = N ab ~v a T µ + ~v b + Σ , (4.37) β = N ab ~v a T L − ~v b + Σ , (4.38) γ = 12 α h N ab ( ~v a T L − ~v b ) Σ + ε ac ε bd ( ~v a T L − ~v b )( ~v c T µ + ~v d ) i , (4.39)∆ = ε ac ε bd ( ~v a T L − ~v b )( ~v c T L − ~v d ) , (4.40)31here Σ = ε ac ε bd ( ~v a T µ + ~v b )( ~v c T µ + ~v d ) . (4.41)As was first observed in [3], the quartic invariant ∆ does not depend on the values of theasymptotic scalars.Having obtained these general expressions in the heterotic formulation for the coeffi-cients α , β , γ and ∆ for static BPS black holes with eight independent charges and arbi-trary asymptotic values for the six scalar fields, it is instructive to compare them with theanalogous expressions in the U-duality formulation, which we obtained in (3.13) and (3.10).After making use of the mappings (2.42) and (2.40) between the charges ~α and ~β of theheterotic formulation and the charges ( P i , Q i ) of the U-duality formulation, and also themapping (2.37) between the scalar fields in the two formulations, it is straightforward toverify that the two sets of expressions for α , β , γ and ∆ agree.There is, however, one respect in which the two sets of expressions for the constants α , β and γ ostensibly differ in the two formulations. In the U-duality formulation, the SL (2 , R ) -invariant expressions in (3.13) have the feature that α , β and αγ are manifestlypolynomial in the eight charges. By contrast, in the heterotic formulation the corresponding O (2 , × SL (2 , R )-invariant expressions (4.37)–(4.39) for α , β and αγ are not manifestly polynomial in the eight charges, because they are written using Σ, which is defined via theexpression (4.41) for Σ . In fact one finds that Σ defined in (4.41) turns out to be a perfectsquare when one evaluates it, and so Σ is indeed a quadratic polynomial in the charges.However, one cannot write Σ itself as a manifestly O (2 , × SL (2 , R )-invariant quadraticpolynomial in the charges and asymptotic scalars. The explanation for this phenomenonturns out to be related to the observation made earlier, in footnote 8, namely that thereexists a three-dimensional vector space of matrices that are invariant under the U (1) × U (1)subgroup of O (2 , 2) while being non-invariant under the full O (2 , 2) group.As we proceeded through the steps described above, we first promoted the 5-chargeexpressions for α , β , γ and ∆ to 7-charge expressions. At that stage, we could have usedthe U (1) -invariant matrix (4.3) K = K + K = − − − − (4.42)32n order to write Σ in (4.9) directly, unsquared, asΣ = − ~α T K ~β , (4.43)as may readily be verified. This 7-charge expression can then be augmented to an 8-charge expression by making the replacement (4.11), with ψ given by (4.12). Because thematrix K is antisymmetric, this replacement leaves (4.43) unchanged, and so in the 8-chargeexpressions for α , β , γ and ∆ in (4.18)–(4.21), the expression for Σ following from (4.15)can instead be replaced directly by the unsquared expressionΣ = − ~α T K ~β . (4.44)Finally, when the asymptotic scalars are introduced by acting with the O (2 , /U (1) and SL (2 , R ) /U (1) coset matrices (4.28) and (4.33) as we did previously, the expression (4.44)for Σ will become Σ = ǫ ab ~v a T P ~v b , (4.45)where P = V T K V . (4.46)Thus we haveΣ = − K kℓ V ki V ℓj α i β j = − ( V i + V i )( V j + V j )( α i β j − α j β i ) . (4.47) The derivation of the expressions for the scalar fields in the heterotic formulation proceedsin an analogous fashion. Here, we shall just present the results for the (Φ , Ψ) dilaton/axionpair, associated with the S-duality in the heterotic formalism. They are given in the 5-chargespecial case (4.1) by e − = V ( r ) D ( r ) , Ψ = δD ( r ) , (4.48)where D ( r ) = r + α D r + β D , (4.49)with α D = p + p , β D = p p , δ = q ( p − p ) . (4.50)As in the corresponding derivation in the U-duality formulation, we shall assume the asymp-totic values of the scalar fields are all zero until the final stage in the calculation.33he augmentation to a 7-charge solution is accomplished by acting with the O (2) × O (2)subgroup of the O (2 , 2) T-duality, and this results in expressions for α D , β D and δ that are O (2) × O (2) invariant: α D (7) = ( p + p ) + ( p + p ) = ~β T ν + ~β ,β D (7) = p p + p p = ~β T L − ~β ,δ (7) = − [( p − p )( q − q ) + ( p − p )( q − q )] = ~α T L − ~β . (4.51)These quantities are further augmented to 8-charge expressions by acting with the remain-ing U (1) ∈ SL (2 , R ) symmetry, with angle ψ given by (4.12). Thus the right-hand-mostexpressions in (4.51) are replaced according to the rules (4.16), leading to the 8-chargeexpressions α D = 12 (cid:16) ~α T ν + ~α + ~β T ν + ~β − Ξ (cid:17) ,β D = 14 (cid:16) ~α T L − ~α + ~β T L − ~β − σ Ξ (cid:17) , (4.52) δ = 12Ξ h ( ~α T ν + ~β )( ~α T L − ~α − ~β T L − ~β ) − ( ~α T L − ~β )( ~α T ν + ~α − ~β T ν + ~β ) i , where Ξ is given by (4.14) and (4.15), and σ = G ( L − ) = ( ~α T L − ~α − ~β T L − ~β )( ~α T ν + ~α − ~β T ν + ~β )+4( ~α T L − ~β )( ~α T ν + ~β ) . (4.53)These expressions can be written more compactly in terms of the 8-component charge vector ~v a defined in (4.22), giving α D = 12 (cid:16) δ ab ~v a T ν + ~v b − Ξ (cid:17) ,β D = 14 (cid:16) δ ab ~v a T L − ~v b − σ Ξ (cid:17) ,δ = − δ ac ε bd ( ~v a T L − ~v b )( ~v c T ν + ~v d ) , (4.54)with Ξ and σ given byΞ = ( δ ab ~v a T ν + ~v b ) − , = ( δ ab δ cd − ε ac ε bd ) ( ~v a T ν + ~v b )( ~v c T ν + ~v d ) ,σ = ( δ ab δ cd − ε ac ε bd )( ~v a T ν + ~v b )( ~v c T L − ~v d ) . (4.55)Thus α D , β D and δ are all written for the general 8-charge configuration, in forms that aremanifestly invariant under the U (1) subgroup of the full O (2 , × SL (2 , R ) duality group.As in the analogous earlier discussion in the U-duality formulation, here the dila-ton/axion pair (Φ , Ψ) transforms under the U (1) subgroup of the SL (2 , R ) S-duality group,34nd so to obtain the expressions for Φ and Ψ after the augmentation from the 5-charge seedsolution to the general 8-charge solution, we should transform these too, according to (2.13)with a = d = cos ψ and b = − c = sin ψ , where the U (1) angle is given by (4.12). Thus,analogously to (3.45) the dilaton and axion will then be given by e − = p V ( r ) D D ( r ) , Ψ = CD , (4.56)where C = δD ( r ) cos 2 ψ − h V ( r ) + δ D ( r ) − i sin 2 ψ , D = 12 h V ( r ) + δ D ( r ) i − δD ( r ) sin 2 ψ + 12 h − V ( r ) + δ D ( r ) i cos 2 ψ . (4.57)The function V ( r ) is given by the expression in (3.2) with the coefficients α , β , γ and ∆given by (4.23)–(4.26); the function D ( r ) is given by (3.26) with the coefficients given by(4.54); the coefficient δ is given also in (4.54); and the expressions for cos 2 ψ and sin 2 ψ aregiven, as in (4.13), bycos 2 ψ = 1Ξ ( δ a δ b − δ a δ b ) ~v a T ν + ~v b , sin 2 ψ = 2Ξ δ a δ b ~v a T ν + ~v b . (4.58)(Of course these last expressions should not be invariant under the U (1) subgroup of the SL (2 , R ) S-duality, since ψ is the U (1) angle of the final rotation that introduced the 8thcharge.)Finally, the process of turning on non-vanishing asymptotic values for the scalar fieldscan be accomplished by means of transformations under the [ O (2 , × SL (2 , R )] /U (1) coset,as in section 4.1. This means that in all the expressions above one makes the replacements ν + −→ µ + , δ ab −→ N ab , (4.59)where µ + is defined in eqn (4.31) and N ab is obtained by setting Φ and Ψ to their asymptoticvalues Φ and Ψ in (4.32). Note that δ a and δ a in (4.58) will be replaced by N a and N a also. Finally, the dilaton/axion pair (Φ , Ψ) will transform under the SL (2 , R ) /U (1) cosetalso, undergoing the replacements e − −→ e − − , Ψ −→ Ψ + Ψ e − . (4.60) Having obtained the expressions for the metric and the scalar fields, the form of the gaugefields in the general 8-charge static BPS black hole solutions follow straightforwardly fromthe gauge fields equations of motion. 35n the heterotic formulation we find F i = p i sin θ dθ ∧ dϕ + e √ V M ij ( α j − Ψ β j ) dt ∧ dr . (4.61)As may be verified, these expressions are consistent with the definitions of the electric andmagnetic charges in eqns (2.15) and (2.16). The STU supergravity theory admits a consistent truncation in which the four gauge fieldsare set equal in pairs, and at the same time two of the dilaton/axion pairs are set to zero.Because of the triality symmetry in the U-duality formulation, the three possible ways ofequating pairs of gauge fields are equivalent. We shall choose the pairing A = A , A = A (5.1)in the theory described in section 2.3. It is straightforward to see that this truncation isconsistent provided that at the same time one sets ϕ = ϕ = 0 , χ = χ = 0 . (5.2)The entire bosonic Lagrangian then reduces to L U = R ∗ − ∗ dϕ ∧ dϕ − e ϕ ∗ dχ ∧ dχ (5.3) − e ϕ χ e ϕ h e − ϕ ∗ F ∧ F + χ F ∧ F i − e − ϕ ∗ F ∧ F + χ F ∧ F . Note that this Lagrangian is invariant under the discrete symmetry A −→ A , A −→ A , τ −→ − τ , (5.4)where as usual τ = χ + i e − ϕ , that is, e − ϕ ←→ e ϕ (1 + χ e ϕ ) , χ ←→ − χ e ϕ (1 + χ e ϕ ) . (5.5)In the heterotic formulation, as can be seen from (2.38) and (2.39), the correspondingtruncation amounts to setting A = A , A = A , (5.6)together, again, with ϕ = ϕ = 0 , χ = χ = 0 . (5.7)36he bosonic Lagrangian in the heterotic formulation, described in section 2.2, thereforereduces to L H = R ∗ − ∗ d Φ ∧ d Φ − e ∗ d Ψ ∧ d Ψ − e − ( ∗F ∧ F + ∗F ∧ F ) + Ψ ( F ∧ F + F ∧ F ) . (5.8)The general static BPS black hole solutions in the truncated theory are obtained bysetting the charges associated with the corresponding pairs of equated fields to be equalalso. Thus in the U-duality formulation we set P = P , Q = Q , P = P , Q = Q . (5.9)In the heterotic formulation, this corresponds to setting p = p , q = q , p = p , q = q . (5.10)Not only is the STU theory greatly simplified in the pairwise-equal truncation of the fields,as we saw above, but also the form of the black hole solutions becomes considerably simpler.In particular, the quartic metric function V ( r ) (3.2) now becomes a perfect square, Themetric function V in eqn (3.2) also becomes a perfect square: V ( r ) = (cid:16) r + α r + ∆ / (cid:17) , (5.11)with α and ∆ now given by α = 2 p ( P + P ) + ( Q + Q ) , ∆ = ( P P + Q Q ) . (5.12)(For simplicity, we first consider the case where the asymptotic values of the scalar fieldsare taken to be zero here.)The scalar fields ( ϕ , χ ) themselves can be read off from eqns (3.49) and (3.50), togetherwith the triality-related expressions for ( ϕ , χ ) and ( ϕ , χ ) as detailed in (3.62) and (3.63),after making the pairwise-equal specialisation (5.9). These latter expressions reproduce thevanishing of the ( ϕ , χ ) and ( ϕ , χ ) as in (5.7), and the former give (3.49) e ϕ = r + b r + b r + d r + d , χ = c r + c r + b r + b ,b = 4 α h P ( P + P ) + Q ( Q + Q ) i , b = ( P ) + ( Q ) ,c = 4 α ( P Q − P Q ) , c = P Q − P Q ,d = α , d = P P + Q Q . (5.13)37t is worth noting that the asymmetry of the expressions for ϕ and χ with respect toexchanging ( P , Q ) and ( P , Q ), because of the asymmetry of b and b given in (5.13),is in fact precisely consistent with the exchange symmetry of the pairwise-equal truncationof the STU theory. This symmetry is given in (5.4). One may verify that the scalar fields ϕ and χ given in (5.13) have the property that e ϕ (1 + χ e ϕ ) = r + d r + d r + e r + e , χ e ϕ (1 + χ e ϕ ) = − c r + c r + e r + e , (5.14)where e = 4 α h P ( P + P ) + Q ( Q + Q ) i , e = ( P ) + ( Q ) , (5.15)and so indeed the pairwise-equal solution is compatible with the exchange symmetry givenby (5.5).If the asymptotic values of the scalar fields are taken to be non-vanishing, ϕ → ¯ ϕ and χ → ¯ χ , it is straightforward to check using the results in section 3.2 that the constants b , b , c and c in (5.13) are replaced by b = 4 α n e − ¯ ϕ [( P ) + Q ] + P P + Q Q + ¯ χ ( P Q − P Q )+ ¯ χ e ¯ ϕ h P Q − P Q + ¯ χ [( P ) + ( P ) + Q + Q ] io ,b = e − ¯ ϕ [( P ) + Q ] + ¯ χ ( P Q − P Q )+ ¯ χ e ¯ ϕ h P Q − P Q + ¯ χ [( P ) + ( P ) + Q + Q ] i ,c = 4 α h P Q − P Q + ¯ χ e ¯ ϕ [( P ) + Q ] i ,c = P Q − P Q + ¯ χ e ¯ ϕ [( P ) + Q ] . (5.16)Note that b = α ( b + P P + Q Q ), and c = α c . The quantity α is now given by α = 4 e ¯ ϕ h ( P + ¯ χ Q ) + ( Q − ¯ χ P ) i + 8( P P + Q Q )+4 e − ¯ ϕ [( P ) + Q ] . (5.17)Finally, the scalar fields ϕ and χ themselves should be transformed, as in eqn (3.61). In this section, we shall address a conformal transformation for BPS black holes, studied in[10], in the context of STU theory. In order to study the conformal transformation, we needto go outside the approach of the rest of this paper, and adopt the pre-potential formalism38tudied in [17, 18]. In [18], the authors discuss various stationary solutions of D = 4 , N = 2supergravity, among which the one with pre-potential F ( X I ) = − X X X X (6.1)is of interest to us. The resulting theory, as we shall see later, corresponds to STU super-gravity in the 3+1 formulation that we discussed in section 2.5.Let us consider a K¨ahler manifold with K¨ahler potential given by K = − log[ i ( ¯ X I W I − X I ¯ W I )] (6.2)where X I and W I are related to the “holomorphic section” of the underlying K¨ahlermanifold via the usual definition X I W I = e − K L I M I (6.3)where L I M I constitute the “holomorphic section”. The four quantities, W I , are evaluatedfrom the pre-potential mentioned in (6.1) using the definition: W I = ∂F∂X I . Physical scalars z A , A = 1 , , 3, which parameterise the K¨ahler manifold, are given in terms of X I functionsas follows z A = X A X , A = 1 , , . (6.4)It can be shown that the metric function of the 8-charge static BPS black hole shouldbe related to the K¨ahler potential (since the metric has to be duality invariant, it should berelated to a similar duality invariant quantity in special geometry), and X I , W I and theircomplex conjugates should be given by harmonic functions. These are made to ensure thatthe bosonic solutions are supersymmetric (i.e. solutions which render the supersymmetryvariations δψ µ and δλ A equal to zero). We refer to [18] for detailed results. e − U = e − K = i ( ¯ X I W I − X I ¯ W I ) , i ( X I − ¯ X I ) = e H I , where e H I = ˜ h I + p I r , i ( W I − ¯ W I ) = H I , where H I = h I + q I r . (6.5) W I is denoted as F I in most of the literature dealing with supergravity and special geometry, howeverwe choose not to use the notation F I to avoid confusion with field strength. h I and h I are integration constants, while q I and p I are the electric and magneticcharges.Using (6.4) and (6.5), the metric and scalar fields are expressed in terms of theharmonic functions below, z A = (cid:16) e H A H A − e H I H I (cid:17) − i e − U (cid:16) d ABC e H B e H C + 2 e H H A (cid:17) , A = 1 , , ,e − U = 1 r V ( r )= − ( e H I H I ) + ( d ABC e H B e H C d ADE H D H E )+ 4 e H H H H − H e H e H e H . (6.6) d ABC is the “symmetrised ǫ tensor,” d ABC = | ǫ ABC | , so d = 1 = d = (any permutation of 1,2,3 indices) . (6.7)Towards the end of this section we shall relate the charges, scalar fields and the metric men-tioned in the above equation to those in the U-duality formulation, via the 3+1 formulationdiscussed in section 2.5.Not all the eight constants h I and ˜ h I are independent; they are constrained by twoconditions. Since the metric is asymptotically flat, we have the constraint1 = − (˜ h I h I ) + ( d ABC ˜ h B ˜ h C d ADE h D h E )+ 4 ˜ h h h h − h ˜ h ˜ h ˜ h (6.8)The other constraint is [17] ˜ h I q I − h I p I = 0 , (6.9)which arises from a more fundamental requirement of special geometry: h V, D A V i = 0 , where V = L I M I , D A ≡ ∂ A + ( ∂ A K ) , (6.10)where ∂ A = ∂/∂z A . The angular bracket denotes the inner product with respect to thesymplectic metric. We again refer to [18, 17] for details. Thus only six out of the eightintegration constants ˜ h I and h I are independent.We now have all the necessary ingredients for discussing the conformal inversion of thestatic 8-charge BPS metric. Since the metric is in general is parameterised by eight chargesand the asymptotic values of the six scalar fields, the conformal inversion transforms thecharges and the asymptotic values of the scalars into a new set of charges and asymptotic40calar values in the conformally rescaled metric (see, for example, section 4 in [9]). Althoughit is relatively easy to find the relations between constants α , β , γ and ∆ appearing inthe metric function V ( r ) in section 3.1 and those in the conformally rescaled metric, itis quite difficult to find the relation between original and the transformed charges andscalars. It is nonetheless possible to circumvent this difficulty by viewing the metric asbeing parameterised by q I , p I , h I , and ˜ h I . It should again be emphasised that out of thesesixteen parameters, only fourteen (i.e. eight charges and six out of the eight h I and ˜ h I constants) are independent, thus giving us the same number of parameters as we have whenthe metric is written in terms of the eight charges and the asymptotic values of the sixscalars. Following [10], if we implement the transformation h I ˜ h I → ˆ h I ˆ˜ h I = ∆ − q I p I , q I p I → ˆ q I ˆ p I = ∆ h I ˜ h I , (6.11)together with r → ˆ r = √ ∆ /r , it is easy to check that we obtain the correct conformallyrescaled metric, namely V ( r ) = ∆ˆ r ˆ V (ˆ r ) , ds = √ ∆ˆ r ˆ ds ˆ ds = − ˆ r q ˆ V (ˆ r ) dt + q ˆ V (ˆ r )ˆ r ( d ˆ r + ˆ r d Ω ) , (6.12)where ˆ V (ˆ r ) is obtained from the second equation in (6.6) by replacing e H I and H I by ˆ e H I and ˆ H I , with ˆ e H I = ˆ˜ h I + ˆ p I / ˆ r , etc. It is now instructive to see what happens to the asymptotic values of the scalar fieldsunder these transformations. Under the transformations in (6.11), ˆ e H I ˆ H I = r ∆ − e H I H I , e − U = r ∆ − e − U . (6.13)According to (6.6), this implies for scalars thatˆ z A (cid:18) ˆ r, ˆ q I , ˆ p I , ˆ˜ h I , ˆ h I (cid:19) = z A (cid:16) r, q I , p I , ˜ h I , h I (cid:17) . (6.14)Thus the functional form of the scalar fields remain unchanged under the conformal trans-formation. This implies thatlim r →∞ z A = (2˜ h A h A − ˜ h I h I ) − i d ABC ˜ h B ˜ h C + 2 ˜ h h A = lim ˆ r → ˆ z A = 2 ˆ p A ˆ q A − ˆ p I ˆ q I − i q ˆ V (ˆ p I , ˆ q I ) d ABC ˆ p B ˆ p C + 2 ˆ p ˆ q A , (6.15) In [10] and in [9] the quantities after the conformal inversion were denoted using tildes. Here, we areinstead using hats to denote the conformally-inverted quantities, reserving tildes for the functions e H I =˜ h I + p I /r in the notation of [18]. V (ˆ p I , ˆ q I ) is the metric function at the horizon after inversion, which can be obtainedby the replacements e H I → ˆ p I and H I → ˆ q I in the lower equation in (6.6). In other words,the above equation implies that the asymptotic values of the scalar fields in the originalmetric map to the values of the fields at the horizon of the transformed metric.It is important to distinguish this property of the scalar fields undergoing conformalinversion from that considered in [9]. In [9], the asymptotic values of the scalar fieldswere taken to be be zero in the static black hole solution of STU supergravity, and itwas mandated that after the conformal inversion the metric should again be an 8-chargesolution with vanishing asymptotic values for the scalar fields. By contrast, in the presentdiscussion, if we start with zero asymptotic values for the scalar fields, we will end up withthe scalar fields becoming zero at the horizon of the inverted metric , but at spatial infinitythe scalar fields will be non-zero . Therefore, the type of transformation considered in (6.11)clearly does not satisfy the requirements of the conformal inversion discussed in [9], whichinvolved mapping any member of the restricted 8-parameter class of charged black holeswith vanishing asymptotic scalars to another member of this 8-parameter class. Instead itprovides an alternative way of implementing the inversion, describing a mapping from anymember of the 14-parameter general class of black hole solutions to another member of theclass.Although we have in principle demonstrated the effect that the conformal inversionhas on the metric, the charges, the constants ˜ h I , h I , and the scalars, the discussion mightappear abstract to some degree since no real connection has been made between a particulartheory and the particular pre-potential considered in (6.1). To establish that connection,we need to look at two equations which arise as a consequence of holomorphicity of thesection ( L I , M I ) of the underlying K¨ahler manifold, namely, W I = N IJ X J , D A W I = N IJ D A X J , (6.16)where N IJ is the complex matrix describing the coupling of the scalars to the field strengths(see below). Here D A = ∂ A + ( ∂ A K ) is the appropriate covariant derivative with the K¨ahlerconnection. Note that the change of the coefficient of the ( ∂ A K ) term from in eqn (6.10)to 1 here is because the different weights of the differentiands with respect to K , as can beseen in eqn (6.3).Since we know the relation between X A and the scalars z A via (6.4), it is possible tosolve the (4 + 12) = 16 conditions coming from (6.16) to determine N IJ in terms of z A . Thegauge field part of the Lagrangian in the 3+1 formulation considered in section 2.5 has this42tructure, and if we choose z A = − χ A − ie − ϕ A (6.17)then it is straightforward to see that the Lagrangian in (2.45) becomes L = R + L ( ϕ, χ ) − 12 Im( N IJ ) ∗ F I ∧ F J + 12 Re( N IJ ) F I ∧ F J , (6.18)where scalar kinetic terms are given by L ( ϕ, χ ) = − g A ¯ B ∗ dz A ∧ d ¯ z ¯ B = − X i =1 ∗ dϕ i ∧ dϕ i + e ϕ i ∗ dχ i ∧ dχ i ! , (6.19)with g A ¯ B = ∂ A ∂ ¯ B K .Finally, the metric considered in (6.6) as a general function of e H I and H I can be shownto be equal to the static 8-charge metric in the U-duality formulation upon relating thecharges in this section to the charges of U-duality frame via the mapping p A = P A √ , q A = Q A √ , A = 1 , , .p = Q √ , q = − P √ . (6.20)Note that it would also be natural, when mapping from the (3+1) formulation of STUsupergravity used in [18] to the U-duality formulation, to relabel the constants ˜ h and h in the same way as p and q are relabelled in (6.20):˜ h = h , h = − ˜ h . (6.21)Thus the functions e H I and H I with I = 0 , , , I = 1 , , , 4, with e H I = ˜ h I + P I √ r , H I = h I + Q I √ r , I = 1 , , , . (6.22)Although we have given a way to write the metric in terms of the constants ˜ h I , h I andthe charges, and shown how these should transform under conformal inversion, it should beemphasised that the quantities ˜ h I and h I are not all physical and independent. In order toevaluate them in terms of physical charges and asymptotic scalar values, we need to use the The field strengths F ( BLS ) in [18] are normalised differently from ours, with F ( BLS ) = 1 / (2 √ F . TheLagrangian (6.18) is written in terms of our convention for the field strength normalisation. Comparison ofthe expressions for the fields in the black hole solutions in [18] with our expressions leads to the relations ineqn (6.20). h I and h I : 0 = 2˜ h A h A − ˜ h I h I + (cid:16) d ABC ˜ h B ˜ h C + 2˜ h h A (cid:17) ¯ χ A , A = 1 , , .f A = d ABC ˜ h B ˜ h C + 2˜ h h A , A = 1 , , , (6.23)where ¯ χ A and f A ≡ e ¯ ϕ A are the asymptotic values of the scalars. These six equations, alongwith (6.8) and (6.9), enable us to solve for the eight constants, giving˜ h = a , ˜ h = − a ¯ χ + bf , ˜ h = − a ¯ χ + bf , ˜ h = − a ¯ χ + bf h = − a (cid:18) ¯ χ ¯ χ ¯ χ − ¯ χ f f − ¯ χ f f − ¯ χ f f (cid:19) − b (cid:18) f f f + ¯ χ ¯ χ f + ¯ χ ¯ χ f + ¯ χ ¯ χ f (cid:19) ,h = a (cid:18) f f + ¯ χ ¯ χ (cid:19) + b (cid:18) ¯ χ f + ¯ χ f (cid:19) ,h = a (cid:18) f f + ¯ χ ¯ χ (cid:19) + b (cid:18) ¯ χ f + ¯ χ f (cid:19) ,h = a (cid:18) f f + ¯ χ ¯ χ (cid:19) + b (cid:18) ¯ χ f + ¯ χ f (cid:19) , (6.24)where a and b are given by a = √ f f f √ D q q D q + D p , b = √ f f f √ D p q D q + D p (6.25)and D q = q + f f (cid:0) q − ¯ χ p − ¯ χ p − ¯ χ ¯ χ q (cid:1) + f f (cid:0) q − ¯ χ p − ¯ χ p − ¯ χ ¯ χ q (cid:1) + f f (cid:0) q − ¯ χ p − ¯ χ p − ¯ χ ¯ χ q (cid:1) ,D p = f f f (cid:0) p + ¯ χ q + ¯ χ q + ¯ χ q − ¯ χ ¯ χ p − ¯ χ ¯ χ p − ¯ χ ¯ χ p − ¯ χ ¯ χ ¯ χ q (cid:1) + f (cid:0) p + ¯ χ q (cid:1) + f (cid:0) p + ¯ χ q (cid:1) + f (cid:0) p + ¯ χ q (cid:1) . (6.26)Note that in the special case where the asymptotic values of the scalar fields ϕ i and χ i aretaken to vanish, the ˜ h I and h I constants become simply˜ h = ˜ h = ˜ h = ˜ h = P I P I √ p ( P J P J ) + ( P J ( Q J ) ,h = h = h = h = P I Q I √ p ( P J P J ) + ( P J ( Q J ) , (6.27)(after the change to the U-duality notation in (6.21)).Plugging the solutions (6.24) into the metric in (6.6), along with the mapping (6.20), onecan show that the metric is indeed equal to the static metric of the U-duality formulation44it is most easily done when the asymptotic scalars are set to zero). These voluminousbut symmetric equations enable one to find the eight constants ˜ h I and h I in terms ofthe eight charges and the asymptotic values of the six scalars. One can think of this asan “initial value” assignment for the eight ˜ h I and h I constants, prior to the inversion.One could then formulate the inversion problem entirely in terms of these constants andcharges, forgetting about the scalars. It is always possible, after inversion, to re-express:I) the transformed charges in terms of original charges and scalars using (6.11) and theaforementioned equations, and II) the transformed scalars in terms of the original chargesand asymptotic values of the original scalars using (6.6), after expressing (6.6) in terms of“hatted” quantities. The latter, as we have already shown, turn out to be equal to theoriginal set of scalars. Studies of BPS black holes in four-dimensional ungauged supergravity theories, such asthe extremal STU black holes of the N = 2 supergravity theory coupled to three vectorsupermultiplets, have been a subject of intense research ever since their discovery [5, 3]. Thestudy of their properties attracted extensive efforts over years, with recent ones focusingon their enhanced symmetries, such as Aretakis and Newman-Penrose charges [8, 19], andCouch-Torrence-type symmetries [9]. It is important to note that these features of extremalSTU black holes stem from and are closely related to those of extremal Reissner-Nordstr¨omand Kerr-Newman black holes of Maxwell-Einstein gravity, these are a special case of STUblack holes. The STU BPS black holes are fourteen-parameter solutions, specified by fourelectric and four magnetic charges, and by the asymptotic values of the six scalar fields.In the past, most analyses focused on the BPS black holes where the asymptotic valuesof scalar fields were taken to be their canonical zero values. The explicit form of such blackholes was constructed in [3], in the heterotic formulation, and the analogous process wasemployed recently in [9] to construct the eight-charge solutions in the U-duality formulationof STU supergravity.Surprisingly, until now the explicit form of the eight-charge solutions together with ar-bitrary asymptotic values of the scalar fields has not been studied. The main purpose ofthe present paper is to rectify this omission: we have systematically derived the full explicit In [3] explicit results for the mass and the horizon area of these black holes in the heterotic frame wereobtained. This analysis showed that the horizon area, and thus the Bekenstein entropy, is independent ofthe asymptotic values of the scalar fields. N = 8 and N = 4 supergravity theory of toroidallycompactified effective Type II and heterotic superstring theory, respectively. In particular,we expect that the BPS solution of N = 4 supergravity with O (6 , 22) T-duality symmetryand SL (2 , R ) S-duality symmetry (at the level of the equations of motion) can be obtainedin a straightforward way by acting on the STU black hole with a subset of the global sym-metry generators in the heterotic frame. The final solution will then be parameterised by28 electric and 28 magnetic charges, and the asymptotic values of the 134 scalar fields in46he coset of O (6 , / [ O (6) × O (22)] × SL (2 , R ) /U (1). It is expected that the final resultshould represent a straightforward generalisation of the resulting section 4, with chargevectors ~α and ~β characterising the 28 electric and 28 magnetic charges, respectively, thematrix L now denoting the O (6 , 22) invariant matrix, and the matrix µ + parameterising thecorresponding asymptotic values of 132 scalar fields in the O (6 , / [ O (6) × O (22)] coset.(Note again, that in this case the mass and the horizon area were obtained in [3].) Onthe other hand, N = 8 supergravity has an E , U-duality symmetry, and the general BPSblack hole solution can in principle be generated by acting on the STU black hole solutionwith a subset of E , generators. It would be of great interest to obtain the explicit formof the full BPS black hole solution in terms of manifestly E , -covariant coefficients.Another important application is to focus on black hole properties as a function of theasymptotic values of the scalar fields, which in effective string theory specify the moduliof the corresponding string compactification. The explicit expressions obtained in thispaper would allow us to employ these black holes for studies of various so-called swamplandconjectures [20, 21, 22]. In particular, we plan to explore the implications of these black holesfor the swampland distance conjecture [21], and its connection to the weak gravity conjecture[22]. (The swampland distance conjecture argues that as one moves toward the boundaryof moduli space, there appears an infinite tower of states with masses approaching zeroexponentially as a function of the traversed distance. The weak gravity conjecture arguesthat in any consistent quantum gravity there must exist a particle whose charge-to-massratio equals or exceeds the extremality bound for black hole solutions of that theory.) Forrecent work, tying together the two conjectures via special examples of BPS black holes, see[23] and references therein. We would also like to emphasise that since the BPS black holesolutions presented in this paper account for both electric and magnetic charges, they wouldallow for probing non-perturbative effects in the moduli space of string compactifications,such as the appearance of light/massless dyonic BPS states in the middle of moduli space,where they could signify, for example, the appearance of enhanced gauge symmetry and/orsupersymmetry (c.f., [24]). Acknowledgments The work of M.C. is supported in part by the DOE (HEP) Award DE-SC0013528, the FayR. and Eugene L. Langberg Endowed Chair (M.C.) and the Slovenian Research Agency(ARRS No. P1-0306). The work of C.N.P. is supported in part by DOE grant DE-FG02-473ER42020. A Kaluza-Klein T Reduction From Six Dimensions Using the notation and conventions of [25, 26], the T reduction of the six-dimensionalLagrangian (2.1) is accomplished by means of the reduction ans¨atze d ˆ s = e 12 ( ϕ + ϕ ) ds + e − 12 ( ϕ + ϕ ) h e − ˜ ϕ ( h ) + e ˜ ϕ ( h ) i , (A.1)ˆ B (2) = B (2) + A (1) ∧ dz + A (1) ∧ dz − A (0) dz ∧ dz , (A.2)ˆ φ = 1 √ ϕ − ϕ ) , (A.3)where h = dz + A (1) + A (0) dz , h = dz + A (1) , (A.4)and ( z , z ) are the coordinates on the internal 2-torus. The six-dimensional 3-form fieldstrength is then given byˆ H (3) = d ˆ B (2) = H (3) + F (2) ∧ h + F (2) ∧ h − F (1) h ∧ h , (A.5)with the four-dimensional field strengths H (3) , F (2) , F (2) and F (1) being read off by sub-stituting (A.2) into (A.5). It turns out to be advantageous to make redefinitions of certainof the potentials in order to obtain the four-dimensional theory in a parameterisation inwhich the axionic scalars A (0) from the metric and A (0) from the ˆ B (2) field occur with-out derivatives in their couplings to the vector fields. Thus we define primed potentials asfollows A (1) = A ′ (1) + A (0) , A (1) = A ′ (1) + A (0) A (1) , A (1) = A (1) ′ + A (0) A (1) . (A.6)We also define a primed potential B ′ (2) for the four-dimensional 3-form H (3) via B (2) = B ′ (2) + A (0) A ′ ∧ A (1) . (A.7)The four-dimensional dressed field strengths are now given by F (2) = d A (1) ′ + A (0) d A (1) , F (2) = d A (1) ,F (2) = dA ′ (1) − A (0) d A (1) ,F (2) = dA ′ (1) + A (0) d A (1) ′ − A (0) dA ′ (1) + A (0) A (0) d A (1) ,H (3) = dB ′ (2) − dA ′ (1) ∧ A (1) ′ − dA ′ (1) ∧ A (1) . (A.8)48n this form, after then dualising the 2-form potential B (2) to an axion, the bosonic STUsupergravity Lagrangian was presented in [27].In this paper we shall denote the four gauge potentials by ( e A , e A , A , A ), with e A = A (1) ′ , e A = A ′ (1) , A = A (1) , A = A ′ (1) . (A.9)We also define the axions e χ and χ by e χ = −A (0) , χ = A (0) . (A.10)(The tildes are placed on the potentials e A and e A to signify that these are the fields thatwe shall dualise when passing to the U-duality formulation of the STU supergravity. Thetildes on e χ , and on ˜ ϕ earlier, are used because we are reserving the symbols χ and ϕ for redefined fields we shall be using later.) The four-dimensional Lagrangian is then givenby L = R ∗ − ∗ dϕ ∧ dϕ − e ϕ ∗ dχ ∧ dχ − ∗ d ˜ ϕ ∧ d ˜ ϕ − e ϕ ∗ d e χ ∧ d e χ − ∗ dϕ ∧ dϕ − e − ϕ ∗ dH (3) ∧ dH (3) − e − ϕ h e ˜ ϕ − ϕ ∗ e F ∧ e F + e − ˜ ϕ + ϕ ∗ e F ∧ e F + e − ˜ ϕ − ϕ ∗ F ∧ F + e ˜ ϕ + ϕ ∗ F ∧ F i , (A.11)where e F = d e A , e F = d e A , F = dA and F = dA are the “raw” field strengths, e F = e F − e χ F , e F = e F − χ F ,F = F , F = F + χ e F + χ e F − e χ χ F , (A.12)are the “dressed” field strengths appearing in the kinetic terms in (A.11), and H (3) = dB ′ (2) − e A ∧ d e A − A ∧ dA . 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