Microstate geometries at a generic point in moduli space
aa r X i v : . [ h e p - t h ] S e p CPHT-RR023.052019IPhT-T19/019
Microstate geometriesat a generic point in moduli space
Guillaume Bossard a and Severin L¨ust b,a a Centre de Physique Th´eorique, CNRS, Institut Polytechnique de Paris,91128 Palaiseau Cedex, France b Institut de Physique Th´eorique, Universit´e Paris Saclay, CEA, CNRSOrme des Merisiers91191 Gif-sur-Yvette Cedex, France
Abstract
We systematically study all supersymmetric solutions of six-dimensional (2 ,
0) supergrav-ity with a null isometry. In particular, every such solution with at least four real supersym-metries is also a supersymmetric solution of a (1 ,
0) theory preserving the same absoluteamount of supersymmetry. This implies that no genuinely new solutions of this type canbe found in this framework. The microstate geometries associated to supersymmetric blackholes within Mathur’s proposal are generically supersymmetric solutions of six-dimensionalsupergravity. A direct consequence of our result is that supersymmetric microstate geome-tries of single centre supersymmetric black holes should carry only one compact 3-cycle. E-mail: [email protected] E-mail: [email protected] ontents (2 , supergravity 53 Supersymmetric solutions 84 Microstate geometries 245 Charge quantization 276 Conclusions 30A Conventions 32B Integrability conditions 33 One of the most fascinating problems in quantum gravity is the black hole information para-dox [1, 2]. In string theory the Bekenstein–Hawking entropy is understood to be associatedwith an exponentially large number of states which can be interpreted as string excitationswith D-brane boundary conditions in a weakly coupled regime [3]. This, by itself, does notyet provide a resolution of the paradox. Nevertheless, there are indications that in stringtheory the physics at the horizon scale may be sufficiently modified by quantum effects toresolve the paradox [4–9].Mathur proposed that the exponentially large number of accessible quantum states al-lows for non-negligible quantum effects, despite the fact that the Riemann tensor measuredin Planck units is very small in the vicinity of the horizon of a large black hole [2]. Hefurthermore proposes that one may be able to test this hypothesis semi-classically in someregime of superstring theory [4, 10].For a classical globally hyperbolic solution in supergravity, one may consider that thereexists a quantum gravity state that is well approximated on a Cauchy surface Σ by a Diracdistribution type wave functional, peaked on the pullback of this classical solution onto theCauchy surface. When quantum effects are negligible, the evolution of such a state shouldbe such that it is defined by the time evolution of the classical solution itself, so that inparticular, a stationary solution would correspond to a stationary state. This picture cannot1e directly applied to a black hole, which is not globally hyperbolic by essence. But a slightgeneralisation of the same picture is to consider instead a quantum state that is a linearsuperposition of microstates which would themselves be such Dirac distribution type wavefunctionals peaked on slices of globally hyperbolic solutions. From this point of view, ablack hole would be in a quantum superposition of a very large number of microstates whichcould individually be described semi-classically as peaked distributions on classical smoothglobally hyperbolic solutions. The microstate geometries associated to a given black hole areconstrained to have the same asymptotic charges as the black hole solution.If one could define the whole set of microstate geometries associated to a given blackhole, one could in principle compute any observable of the black hole as a quantum av-erage over some distribution of the same amplitude computed in the background of eachmicrostate separately. For example, one would expect that the gravitational attraction of amicrostate geometry does not necessarily capture a test particle for generic incoming bound-ary conditions [11]. The black hole, on the other hand, captures the test particle if theimpact parameter is small enough. The average of the evolution over a large enough setof microstate geometries should therefore reproduce the capture for a small enough impactparameter. Ideally, all classical observables computed in the microstate geometry shouldreproduce, after average, the observables computed in the black hole background to an ex-cellent approximation.However, it is not clear that one can construct a complete basis of microstates as suchpeaked distribution on microstates geometry. And even if this were possible, one may wonderif the typical solutions do not involve arbitrary small cycles such that one could not simplyuse them as classical backgrounds without neglecting quantum corrections. For this purposeit was suggested in [9] to distinguish between microstate geometries, that are differentiablemanifolds with an everywhere small Riemann tensor in Planck units, from microstate solu-tions, that possibly involve mild singularities which can be resolved in perturbative stringtheory, like orbifold singularities, and a bounded Riemann tensor, but not necessarily smallin Planck units. For microstate geometries one can use supergravity as an effective theory,while for microstate solutions one needs to take string theory corrections into account. Theyalso propose a third class, the fuzz-balls, that would be genuinely non-perturbative stringtheory quantum states for which there is no supergravity approximation. A large class ofthese geometries has been built explicitly. One can distinguish two main types of solutions.Firstly, the multi-bubble solutions that are regular in five dimensions and usually involveseveral cycles [5, 12–19], and secondly, the supertube solutions that usually only involveone cycle and are regular in six dimensions [20–25]. Both types can also be combined, butthis has not been done explicitly for the co-dimension one generic solutions [22–25], whichare called superstrata. One may also consider solutions that are only smooth locally, butfor which the patching between open sets would involve U-duality transformations [26–29].2he original supertube solutions are microstate geometries describing small black holes thathave a vanishing horizon in classical supergravity. It has been exhibited that their typicalmicrostates solutions involve arbitrarily small cycles for which one cannot trust the super-gravity approximation [30–34]. One may wonder if the same problem occurs for three-chargeblack hole that carry a microscopic horizon, but one may hope that arbitrarily small cyclesare only required when the horizon itself is small in string scale units.For a given black hole, a microstate geometry is defined as a globally hyperbolic smoothsolution with the same asymptotic charges as the black hole. One should in principle won-der about the uniqueness of the black hole solution. For a non-extremal asymptoticallyMinkowski black hole in four dimensions, the solution is uniquely determined by its totalmass, angular momentum and electromagnetic charges. But for BPS black holes one mustdistinguish a single centre black hole with a unique horizon from bound states of black holesthat carry the same electromagnetic charges and total energy [35]. The situation is evenmore complicated for black holes in five dimensions, because there the horizon can admitdifferent topologies [36, 37], and black objects can also be surrounded by topological cyclescarrying flux [38]. In four dimensions a BPS black hole has no angular momentum, so amicrostate geometry associated to the black hole should have no angular momentum either,or at least one should find that the angular momentum vanishes in average on the set of mi-crostate geometries. But there are also multi-centre black hole bound states without angularmomentum, and one must distinguish their microstate geometries. Such black hole boundstates have the property that they do not exist for arbitrary asymptotic values of the scalarfields. Therefore, the microstate geometries associated to the single-centre black hole can bedistinguished by the requirement that they exist everywhere in moduli space.It was indeed exhibited in [39] that multi-centre black hole solutions with at least twoblack holes with a large horizon do not exist at a generic point in moduli space. This propertyfollows from the fact that the most general asymptotically flat supersymmetric solution of(ungauged)
N ≥ N = 2 truncation [40, 41]. This is very importantfor precise counting of black hole microstates in four dimensions, because only bound statesof two 1/2 BPS black holes contribute to the helicity supertrace [42,43], so that wall crossingcorrections are necessarily small and can accurately be separated from the single centre blackhole microstate counting [44].For a supersymmetric black hole, which type of microstate geometries preserving thesame supersymmetry exist at generic values of the moduli? This is the question we shallinvestigate in this paper. For this purpose we consider the effective supergravity descriptionfor type IIB string theory on T or K3, which is a supergravity theory in six spacetime di-mensions of type (2,2) or (2,0) respectively. In the maximal case, we shall only consider themaximal (2,0) truncation, disregarding the vector fields. For a four-dimensional supersym-3etric black hole, the microstate geometries are asymptotically four-dimensional Minkowskispace times two circles fibred over the Minkowski base. The electromagnetic charges of thefour-dimensional black hole originate geometrically from the momentum along the two cir-cles and their fibration over the base, as well as from the 3-form fluxes of the six-dimensional3-form field strengths. The 3-form fluxes are supported by 3-cycles of the Euclidean basespace of the microstate geometry, and in string theory each individual flux is constrainedto belong to an even-selfdual lattice. We shall find that the solutions only exist for genericvalues of the asymptotic scalar fields if all the fluxes associated to the different 3-cycle areproportional to each other, and therefore rational multiples of the total flux associated to theblack hole. This seems to mean that the superstratum microstate geometries found in [22–25]indeed describe black holes, while the multi-bubble type microstate geometries would rathercorrespond to bound states of black holes. Multi-bubble geometries always involve morethan one cycle with linearly independent fluxes and therefore do not exist everywhere inmoduli space.Supersymmetric solutions of supergravity theories have been characterized for the firsttime in [45] following results of [46] for the case of pure N = 2, d = 4 supergravity by assum-ing the existence of a Killing spinor and constructing bosonic objects from its bilinears. Insix dimensions this method was first applied in [47] on pure (1 ,
0) supergravity, not coupledto any matter multiplets. This work was later extended to Fayet-Iliopoulos-gauged super-gravity with vector multiplets and one tensor multiplet in [48], to ungauged supergravitywith an arbitrary number of tensor multiplets in [49] and finally to ungauged supergravitywith vector, tensor as well as hypermultiplets in [50]. It was discovered in [51] that theunderlying equations of these solutions exhibit a linear structure, and specific solutions havebeen constructed in [22,25,52–57]. For the six-dimensional (2 ,
0) theories, on the other hand,so far only maximally supersymmetric solutions have been classified [58], see also [59].It is one of the aims of this paper to fill this gap and to classify supersymmetric solutionsof six-dimensional (2 ,
0) supergravity coupled to an arbitrary number of tensor multiplets.Notice, that this theory does not allow for any gaugings or massive deformations [60]. Inparticular the former can easily be seen from the absence of any vector fields. Therefore, thisis already the most general (2 ,
0) theory. Supersymmetric solutions of supergravity theoriesadmit at least one isometry. The associated Killing vector field V can be obtained as abilinear of the Killing spinor ǫ , i.e. V µ = ¯ ǫγ µ ǫ . In the case at hand this isometry can beeither time-like or null, corresponding to V · V > V · V = 0. Black hole solutions in fiveor four dimensions uplift to supersymmetric solutions in six dimensions with a null isometry.Therefore, it is the second case of a null ( i.e. light-like) isometry that is relevant to microstategeometries, and hence this is the case we discuss in this paper. Here, we find that everysupersymmetric solution with four preserved supercharges (of the type allowing for a blackhole solution) is at the same time always also a supersymmetric solution of a (1 ,
0) theory,4reserving the same absolute amount of supersymmetry. Hence, it will not be possible tofind any genuinely new solutions with the same asymptotic metric as a supersymmetric blackhole.This paper is organized as follows. In Section 2 we review (2 ,
0) supergravity in sixdimensions. In Section 3 we study its supersymmetric solutions and show that all solutionswith a null isometry are equivalent to solutions of a (1 ,
0) theory. Section 4 discusses someimplications on the construction of microstate geometries. In Section 5 we finally show thatat a generic point in moduli space every solution has parallel fluxes. (2 , supergravity In this section we review the relevant properties of (2 ,
0) six-dimensional, i.e. chiral (un-gauged) half-maximal supergravity, which has been constructed in [61, 62].The field content of the theory includes one gravity multiplet coupled to n tensor multi-plets, which decompose as (cid:0) g µν , ψ Aµ , B Iµν , χ Ar , V aI (cid:1) , (2.1)where g µν is the metric and ψ Aµ , A = 1 , . . . ,
4, are symplectic Majorana–Weyl gravitini,transforming in the fundamental representation of the R-symmetry group
USp (4). The B Iµν , I = 1 , . . . , n , are chiral tensor fields and transform as a vector under the global symmetrygroup SO (5 , n ). Their field strengths G I = d B I satisfy a twisted selfduality equation thatwe shall display shortly. The spin-1/2 fermions χ Ar , r = 1 , . . . , n , are in the fundamentalrepresentation of USp (4) as well as of SO ( n ). All fermions in the theory are chiral, we have γ ψ Aµ = − ψ Aµ , γ χ Ar = χ Ar . (2.2)The tensor multiplets include 5 n scalar fields that parametrize the coset manifold M = SO (5 , n ) SO (5) × SO ( n ) , (2.3)through a coset representative in SO (5 , n ), V = ( V aI , V rI ) , I = 1 , . . . , n . (2.4)The coset representative satisfies η IJ = δ ab V aI V bJ − δ rs V rI V sJ , (2.5)where η IJ is a metric of signature (5 , n ). After introducing M IJ = δ ab V aI V bJ + δ rs V rI V sJ , (2.6)5he twisted selfduality equation for the tensor fields reads ⋆ G I = M IJ G J . (2.7)Thus G a = V aI G I is selfdual while G r = V rI G I is anti-selfdual.This supergravity theory describes the low energy effective theory of a type IIB super-string theory for n = 5 and n = 21. Type IIB string theory on a torus T , preserves allsupersymmetries and gives rise to (2 ,
2) supergravity in six dimensions. By removing thetwo gravitini multiplets which include the 16 vector fields one obtains a truncation to (2 , SO (5 ,
5) metric is then themetric of the even selfdual lattice II , with split signature η = ! . (2.8)The other possibility is type IIB string theory on K3, in which case the effective low energytheory is (2 ,
0) supergravity with 21 tensor multiplets and the SO (5 ,
21) metric is the one ofthe unique even selfdual lattice of signature (5 , η = − k , (2.9)with k the metric of the E ⊕ E root lattice, i.e. its Cartan matrix.Using the gamma matrices of Spin(5) ∼ = USp (4) one can express the components V aI asa symplectic traceless antisymmetric tensor of USp (4) with V ABI = V [ AB ] I , ω AB V ABI = 0 , (2.10)where ω AB is the symplectic matrix with the conventions displayed in (A.3). The relation(2.5) then becomes η IJ = V AB I V ABJ − δ rs V rI V sJ , (2.11)where the USp (4) indices are raised and lowered using the symplectic matrix ω AB accordingto (A.2). One decomposes the Maurer–Cartan form d VV − into its usp (4) component( Q µ ) AB = V AC I ∂ µ V BC I , (2.12)where we use η IJ to raise and lower global SO (5 , n ) indices, e.g. V ABI = ω AC ω BD η IJ V CDJ ,and its coset component P AB rµ = −V r I ∂ µ V ABI . (2.13)6 Q µ ) AB defines the USp (4) covariant derivative D µ = ∇ µ + Q . Its action on the cosetrepresentative reads D µ V ABI = ∂ µ V ABI + 2( Q µ ) C [ A V B ] C I , (2.14)and therefore D µ V ABI = P AB rµ V r I , D µ V rI = P AB rµ V AB I . (2.15)The field strength corresponding to the connection ( Q µ ) AB can be expressed in terms of P AB rµ , i.e. [ D µ , D ν ] X A = h ∂ [ µ (cid:0) Q ν ] (cid:1) B A − (cid:0) Q [ µ (cid:1) B C (cid:0) Q ν ] (cid:1) C A i X B = − P [ µ BC r P AC rν ] X B , (2.16)for any USp (4)-vector X A .The bosonic equations of motion of the theory are given by E µν = R µν − P AB rµ P rAB ν − M IJ G Iµκλ G Jν κλ = 0 , (2.17) E AB r = D µ P AB rµ − G ABµνρ G r µνρ = 0 , (2.18) E I = d (cid:0) ⋆M IJ G J (cid:1) = 0 . (2.19)The last equation is a direct consequence of the twisted selfduality equation and the Bianchiidentity for the tensor fields. In practice, we shall mostly use the dressed (anti) selfdualthree-form field strengths G AB = V ABI G I and G r = V rI G I which transform respectivelyas a vector under the compact R-symmetry group USp (4) and as a vector of the flavoursymmetry group SO ( n ). They satisfy the Bianchi identities DG AB = δ rs P AB r ∧ G s , DG r = P AB r ∧ G AB . (2.20)We finally need to give the supersymmetry transformations of the fermionic fields. Undera local supersymmetry transformation the gravitini vary as δψ Aµ = D µ ǫ A − G ABµνρ ω BC γ νρ ǫ C . (2.21)The supersymmetry variation of the tensorini reads δχ Ar = iP AB rµ ω BC γ µ ǫ C + i G rµνρ γ µνρ ǫ A . (2.22)Notice that ǫ A inherits the chirality of the gravitini, i.e. γ ǫ A = − ǫ A .7 Supersymmetric solutions
In this section we discuss necessary and sufficient conditions for the existence of a supersym-metric solution. Here we follow the approach of [47–50] on the classification of supersym-metric solutions of six-dimensional (1 ,
0) supergravity.Assuming that there is at least one spinor ǫ A solving δψ Aµ = δχ Ar = 0, one can constructthe following bi-linears ω AB V µ + V ABµ = ¯ ǫ A γ µ ǫ B , (3.1)Ω ABµνρ = ¯ ǫ A γ µνρ ǫ B . (3.2)Here, we have split ¯ ǫ A γ µ ǫ B into irreducible representations of USp (4), V ABµ = V [ AB ] µ , ω AB V ABµ = 0 , (3.3) i.e. it decomposes into a singlet and a vector of SO (5) ∼ = USp (4) / Z , whereasΩ AB = ⋆ Ω AB = Ω ( AB ) , (3.4)transforms in the adjoint representation. Notice that all combinations of even rank vanishidentically due to the chirality of ǫ A . Similarly to [47] one can use Fierz identities to show that¯ ǫ A γ µ ǫ B ¯ ǫ C γ µ ǫ D = − ε αβγδ ǫ Aα ǫ Bβ ǫ Cγ ǫ Dδ = − ω [ AB ω CD ] det[ ǫ ] , (3.5)so that V µ V AB µ = 0 , V
ABµ V CD µ = (cid:0) ω AB ω CD + 4 ω C [ A ω B ] D (cid:1) V µ V µ . (3.6)The reality condition on the spinor ǫ Aα ,( ǫ ∗ ) αA = ω αβ ω AB ǫ Bβ , (3.7)with the symplectic form ω αβ invariant under USp (4) ⊂ SU ∗ (4) ∼ = Spin(1 , ǫ as a non-zero four by four complex matrix must be of rank 2 or rank 4, because two of itseigenvalues are the complex conjugates of the two others. We are going to see that a rank 4 ǫ is associated to supersymmetric solutions with a time-like isometry whereas a rank 2 ǫ isassociated to supersymmetric solutions with a light-like isometry. Note that for a non-zero8pinor ǫ A , V µ and V µAB are both necessarily non-zero. From the supersymmetry variation ofthe gravitini (2.21) one obtains that ∇ µ V ν = − V σAB G ABµνσ = − V σa G aµνσ , (3.8)such that V µ is a Killing vector.This Killing vector is time-like if ǫ is of rank 4, in which case V and V a define anorthogonal frame with V · V > V a · V b = − δ ab V · V , with ǫ defining the identificationof the internal USp (4) with the spin group. On the other hand, if ǫ is of rank 2, (3.5) impliesthat V · V = 0. In this case V and V a are all orthogonal light-like vectors and are thereforeparallel. One can then introduce a spacetime scalar function v AB ( x µ ) such that V ABµ = v AB V µ . (3.9)In this paper we are interested in solutions generalizing already known solutions of the(1 ,
0) truncation in which the Killing vector is necessarily of null type. We shall thereforedisregard the possibility of a time-like Killing vector and will concentrate on the null typein the remainder.It will prove convenient to introduce u AB = 12 (cid:0) ω AB + v AB (cid:1) , (3.10)such that ¯ ǫ A γ µ ǫ B = 2 u AB V µ . Moreover, since ǫ is of rank 2, one has the additional Fierzidentity 3 ǫ [ C ¯ ǫ A γ µ ǫ B ] = 0 , (3.11)which implies after contraction with ω BC that ( ǫ A − u BA ǫ B ) V µ = 0 and hence u BA ǫ B = ǫ A . (3.12)Applying this result on the definition of u AB shows that u AC u C B = u AB , (3.13)which means that the matrix u AB is a rank 2 projector. In particular v AC v C B = δ BA , (3.14)so that v a is a norm 2 vector. One can also use the Fierz identity γ µ ǫ A ¯ ǫ B γ µ ǫ C = − γ µ ǫ C ¯ ǫ B γ µ ǫ A (3.15)9o show that γ µ ǫ A ¯ ǫ B γ µ ǫ C is antisymmetric in A, B, C so that for a rank 2 ǫγ + ǫ A ≡ V µ γ µ ǫ A = 0 . (3.16)The Fierz identity¯ ǫ A γ σ ǫ B ¯ ǫ C γ µνσ ǫ D = ¯ ǫ B γ σ ǫ C ¯ ǫ D γ µνσ ǫ D − ǫ A γ [ µ ǫ B ¯ ǫ C γ ν ] ǫ D − ǫ C γ [ µ ǫ B ¯ ǫ A γ ν ] ǫ D (3.17)implies in the rank 2 case that V σ (cid:0) u AB Ω CDµνσ − u BC Ω ADµνσ (cid:1) = 0 , (3.18)so that ι V Ω AB = 0 . (3.19)Moreover, from the rank 2 projection (3.12) it follows that u C A Ω CB = Ω AB , (3.20)so that the symmetric tensor Ω AB admits only three independent components along the twodimensional subspace defined by the projection u BA .Denoting the 1-form dual to V by e + , i.e. e + = V µ d x µ , one can always find a 1-form e − dual to V µ , satisfying e − ( V ) = 1, such that the space-time metric decomposes asd s = 2 e + e − − δ ij e i e j , (3.21)where e i , i = 1 , . . . ,
4, is an orthonormal frame of the four-dimensional space orthogonal to e + and e − . The orientation is fixed by ε + − ijkl = ε ijkl .Finally, we also want to express the three-forms Ω AB in this basis. Using the property(3.19) as well as their selfduality this decomposition readsΩ AB = e + ∧ I AB , (3.22)where I AB = I ABij e i ∧ e j are anti-selfdual two-forms with respect to the metric on thefour-dimensional base space. Using the Fierz identity¯ ǫ A γ µνλ ǫ B ¯ ǫ C γ σρλ ǫ D = 2¯ ǫ A ) γ [ σ ǫ ( C ¯ ǫ D ) γ ρ ] µν ǫ ( B − ǫ A ) γ [ µ ǫ ( C ¯ ǫ D ) γ ν ] σρ ǫ ( B + 4 η σ ][ µ ¯ ǫ A ) γ λ ǫ ( C ¯ ǫ D ) γ ν ] λ [ ρ ǫ ( B − η σ ][ µ ¯ ǫ A ) γ ν ] ǫ ( C ¯ ǫ D ) γ [ ρ ǫ ( B + 2 η µ [ σ η ρ ] ν ¯ ǫ A ) γ λ ǫ ( C ¯ ǫ D ) γ λ ǫ ( B (3.23)and contracting it with e + µ e iν e + σ e j ρ one obtains the identity I ABik I CDjk = 4 δ ij (cid:0) u AC u BD + u AD u BC (cid:1) + 4 u A )( C I D )( Bij . (3.24)10t follows that I AB define an almost hyper-complex structure, i.e. a triplet of almost complexstructures satisfying the quaternion algebra. In a basis in which u = 1 is the only non-zerocomponent, the triplet is defined as ( I − I ) , i ( I + I ) , − i I . In particular, the I AB define a complete basis of anti-selfdual forms and116 I ABij I ABkl = δ klij − ε ijkl . (3.25)There is another useful Fierz identity γ νσ ǫ C ¯ ǫ A γ µνσ ǫ B = − ǫ ( A ¯ ǫ B ) γ µ ǫ C + 3 γ µ γ ν ǫ ( A ¯ ǫ B ) γ ν ǫ C (3.26)that allows to show that for the rank 2 spinor I ABij γ ij ǫ C = 16 u C ( A ǫ B ) (3.27)so that the complex structures I AB rotate the spinor in the representation of SU (2) ⊂ USp (4).Using the relation { γ µν , γ σρ } = − δ σρµν + ε µν σρκλ γ κλ γ (3.28)with indices in the R basis one obtains { γ ij , γ kl } = − δ klij − ε ijkl (1 − γ − γ + ) γ , (3.29)which acting on ǫ A , gives { γ ij , γ kl } ǫ A = (cid:0) − δ klij + 2 ε ijkl (cid:1) ǫ A . (3.30)It follows that γ ij ǫ A = − ε ij kl γ kl ǫ A = − I ABij ǫ B . (3.31)To summarise this section, we note that the projection (3.12) halves the number of spinorsto those of a (1 ,
0) theory. The additional projection (3.16) further reduces the number ofsupersymmetries by a half, corresponding to 1/4 BPS in the (2 ,
0) theory ( i.e. ,
0) truncation). The last constraint (3.31), instead, is only satisfied for a single spinor.Therefore, we see already from the structure of the projectors that a consistent truncationto (1 ,
0) supergravity emerges naturally. However, this analysis is clearly not enough toguaranty that our solution is also a solution of (1 ,
0) supergravity. It will moreover benecessary that the projection matrix u AB is constant and also that all fields correspondingto (1 ,
0) gravitini multiplets and (non-factorised) hyper-multiplets, notably some componentsof G AB and P AB r , get consistently projected, too. We shall prove in the following that thisis indeed the case if the solution admits four Killing spinors satisfying the same conditions(3.12) and (3.16). 11 .2 Constraints from the Killing spinor equations
Gravitini variation
Let us first notice that the supersymmetry variation of the gravitini (2.21) implies D − ǫ A ≡ V µ D µ ǫ A = 12 G AB − ij ω BC γ ij ǫ C = 0 , (3.32)since G AB − ij = V µ G ABµij is selfdual as a two-form on the base space, see (A.13). Consequently,we also have ∇ − V µ = D − u AB = D − Ω ABµνρ = 0 . (3.33)More generally, we obtain from the gravitini variation thatD µ (cid:0) u AB V ν (cid:1) = 2 V κ G C [ Aκµν u CB ] − G C [ Aµκλ (cid:0) Ω C B ] (cid:1) κλν . (3.34)To proceed, we split G AB into its component parallel and orthogonal to v AB , G AB = v AB G + ˜ G AB , (3.35)such that v AB ˜ G AB = 0 and hence G = v AB G AB . Let us now also decompose (3.34) into itssymplectic trace and its components parallel and orthogonal to v AB . Both the trace as wellas the component parallel to v AB give ∇ µ V ν = − V κ G κµν , (3.36)which means that V µ is a Killing vector andd V = − ι V G . (3.37)Furthermore, if we denote the spin-connection by ω , (3.36) implies that ω µν − − G µν − = 0 . (3.38)On the other hand, the part of (3.34) orthogonal to v AB reads (cid:0) D µ u AB (cid:1) V ν = − V κ ˜ G ABκµν −
12 ˜ G C [ Aµκλ (cid:0) Ω C B ] (cid:1) κλν . (3.39)This allows us to express the derivatives of the projector u AB in terms of ˜ G AB , in componentswe find D − u AB = ˜ G AB − ij = 0 ,D + u AB = −
12 ˜ G C [ A + ij (cid:0) I C B ] (cid:1) ij ,D i u AB = − ˜ G AB + − i − ˜ G C [ A + − j (cid:0) I C B ] (cid:1) i j , (3.40)12here the last equation is obtained by exploiting the (anti-)selfduality of ˜ G AB and I AB . Onthe other hand, it also follows from (3.39) that0 = (cid:0) D + u AB (cid:1) V i = ˜ G AB + − i + ˜ G C [ A + − j (cid:0) I C B ] (cid:1) i j , (3.41)and hence D i u AB = 0 . (3.42)Moreover, the selfduality of ˜ G AB and Ω AB implies that (cid:0) D [ µ u AB (cid:1) V ν ] = − V κ ˜ G ABκµν , (3.43)and therefore with the previous result we have˜ G AB + − i = 0 , (3.44)and by selfduality also ˜ G ABijk = 0.In a similar fashion we obtain from the Killing spinor equation that D µ Ω ABκλρ = 6 G µσ [ κ (cid:0) Ω AB (cid:1) σλρ ] − G C ( Aµσ [ κ (cid:0) Ω C B ) (cid:1) σλρ ] − G AC [ µκλ V ρ ] v C B − g µ [ κ ˜ G ACλρ ] σ V σ v C B = 6 G µσ [ κ (cid:0) Ω AB (cid:1) σλρ ] − G C ( Aµσ [ κ (cid:0) Ω C B ) (cid:1) σλρ ] , (3.45)which implies that D Ω AB = 4 V ∧ ˜ G AC v C B = 0 . (3.46)From (3.45) we also find D + I ABij = − G + k [ i (cid:0) I AB (cid:1) j ] k + 4 ˜ G C ( A + k [ i (cid:0) I C B ) (cid:1) j ] k ,D i I ABjk = − G il [ j (cid:0) I AB (cid:1) k ] l . (3.47)From the first equation in (3.47) we can compute (cid:0) D + u AB (cid:1) (cid:0) I BC (cid:1) ij = D + I ACij − u BA D + I BCij = − G AB + k [ i (cid:0) I BC (cid:1) j ] k . (3.48)We now use (3.24) to infer that (cid:0) D + u AB (cid:1) u BC = − (cid:0) D + u AB (cid:1) (cid:0) I BD (cid:1) ij (cid:0) I DC (cid:1) ij = 16 ˜ G AB + ij (cid:0) I BC (cid:1) ij , (3.49)which is in contradiction with (3.40) and hence D + u AB = 0 . (3.50)Moreover, using this result in (3.40) gives0 = ˜ G AB + kl (cid:0) I BC (cid:1) kl (cid:0) I C D (cid:1) ij = −
16 ˜ G AB + ij u BD − G AB + k [ i (cid:0) I BD (cid:1) j ] k (3.51)13nd since the second term vanishes due to (3.48) we obtain˜ G AB + ij = 0 . (3.52)In summary, we found that Du AB = ˜ G AB = 0 . (3.53)Note also that for a normalised SO (5) vector v AC v BC = δ BA there always exists a Λ( x µ ) BA ∈ USp (4) such that v AB ( x µ ) = Λ( x µ ) C A Λ( x µ ) DB v CD , (3.54)for v AB constant. We can now use the local USp (4) invariance of the theory to transform allfields by Λ − . Hence, it is always possible to choose a USp (4) frame in which ∂ µ u AB = 0 . (3.55)In particular, by means of (3.53) in this frame the USp (4) connection Q AB defined in (2.12)stabilizes v AB , i.e. Q C [ A v B ] C = 0 . (3.56)Let us finally give the general form of G . (3.38) fixes already all its components expectfor G + ij . To determine also these remaining components we write (3.47) as D µ I ABij = − G µl [ i (cid:0) I AB (cid:1) j ] l . (3.57)Using (3.24) and (3.25) this can be solved for G µij ,( ω µij − G µij ) − = 1256 I AB ij I C A kl ∂ µ I CBkl + 14 I ABij ( Q µ ) AB , (3.58)where the superscript − denotes the anti-selfdual part in the indices ij as a two-form on thebase-space.To write the solution more explicitly, it is convenient to introduce local coordinates( v, u, x m ), m = 1 , . . . , v the coordinate associate to the null Killing isometry V = ∂ v , u the conjugate coordinate and x m local coordinates on the four-dimensional base spacesuch that δ ij e i e j = f − γ mn d x m d x n . The general ansatz for the metric with V a null vectoris then [47] d s = 2 f (d u + β ) (cid:0) d v − H (d u + β ) + ω (cid:1) − f − γ mn d x m d x n . (3.59)Here, γ mn is the (possibly ambipolar) metric on the four-dimensional base-space, f and H aresome functions and β = β m d x m and ω = ω m d x m are one-forms on the base space. In general,14ll these objects depend on u and x m but not on v . Thus, the null vielbein introduced in(3.21) reads e + = f (d u + β ) , e − = d v − H (d u + β ) + ω , e i = f − / v i , (3.60)where v i = v im d x m is a vielbein of γ mn . With these definitions it is convenient to introducethe exterior derivative on the four-dimensional base space ˜ d = dx m ∂ m and the derivative D = ˜ d − β∂ u , as well as the Hodge star operator of the four-dimensional base space ˜ ⋆ ,defined with respect to the metric γ mn . Moreover, one can define a natural almost hyper-K¨ahler structure with respect to γ mn by introducing J AB = f I AB , (3.61)or in components J ABmn = I ABij v im v jm .In these coordinates one can now compute the spin-connection ω and with (3.38) and(3.58) one finds that G takes the form [47] G = 14 h − f − e + ∧ e − ∧ (cid:0) Df − f ˙ β (cid:1) − f e − ∧ Dβ + ˜ ⋆ (cid:0) Df − + f − ˙ β (cid:1) + e + ∧ (cid:0) ( Dω ) − − f − ψ (cid:1)i , (3.62)where ψ is an anti-selfdual two-form on the base, defined by ψ = (cid:18) J mnAC ∂ u J BC mn + 12 ( Q u ) AB (cid:19) J AB . (3.63)These results show for the gravitational multiplet sector that every supersymmetric con-figuration is at the same time also a supersymmetric configuration of a (1 ,
0) theory, inthe sense that the tensor fields of the gravitino multiplets vanish. While G corresponds tothe selfdual three-form in the (1 ,
0) gravity multiplet, the remaining four components ˜ G AB ,which would be part of a (1 ,
0) gravitino multiplet, are consistently projected out.
Tensorini variation
We now carry over with the analysis of the tensor multiplet sector. Contracting the spin-1/2variation (2.22) with ¯ ǫ B yields24 V µ P AC rµ u C B + (cid:0) Ω AB (cid:1) µνρ G rµνρ = 0 . (3.64)Both terms transform in different representations of USp (4) and therefore must vanish inde-pendently, P AB r − ≡ V µ P AB rµ = 0 . (3.65)15n the other hand, contracting (2.22) with ¯ ǫ B γ µν gives8 P AC r [ µ u CB V ν ] + 2 P AC rκ (cid:0) Ω C B (cid:1) µν κ + 2 G rκλ [ µ (cid:0) Ω AB (cid:1) ν ] κλ − u AB G rµνκ V κ = 0 . (3.66)As before for G AB , we split P AB r according to P AB = v AB P r + ˜ P AB r , (3.67)such that v AB ˜ P AB r = 0 and hence P r = v AB P AB r . The symplectic trace of (3.66) and itscomponent parallel to v AB read V ∧ P r = − ι V G r . (3.68)Therefore, G r takes the general form G r = (1 − ⋆ ) e + ∧ e − ∧ P r + e + ∧ F r (+) , (3.69)where F r (+) are arbitrary selfdual two-forms on the four-dimensional base space, see also(A.10). From the part of (3.66) which is orthogonal to v AB we infer that˜ P AC ri u C B = 12 ˜ P AC rj (cid:0) I C B (cid:1) ij . (3.70)Let us finally notice that there is an integrability condition on (3.53) which reads[ D µ , D ν ] u AB = − P AB r [ µ P rν ] = 0 . (3.71) In the previous section we have determined necessary conditions for the existence of a su-persymmetric configuration of (2 ,
0) supergravity with a null isometry. In particular, theyresemble very closely the conditions for a supersymmetric configuration of (1 ,
0) theories. Itremains to verify that these conditions are also sufficient. In the following we consider spinors ǫ A satisfying the conditions (3.12) and (3.16), each reducing the number of supersymmetriesby a factor of 1/2, such that only four of the original sixteen supercharges are preserved.The presence of hypermultiplet scalars ˜ P AB r will require another constraint on ǫ A .To determine if the previously determined conditions on P AB r and G r are sufficient forthe existence of a solution of δχ = 0 we insert (3.69) back into (2.22). Using (3.16) and(3.65) this gives δχ A r = iP AB ri ω BC γ i ǫ C + iG r + − i γ i ǫ A = i ˜ P AB ri ω BC γ i ǫ C . (3.72)For a single supersymmetry parameter, ǫ A satisfies γ i ǫ A = 16 (cid:0) I AB (cid:1) j i γ j ǫ B , (3.73)16o that the constraint ˜ P AB ri ω BC γ i ǫ C = 0 is automatically satisfied. Notice that (3.73) isequivalent to (3.31) and therefore always satisfied by the spinor from which I AB is defined.On the contrary, if one has four preserved supercharges, with the four linearly independentspinors ǫ A satisfying (3.12) and (3.16), then one gets ˜ P AB ri = 0. In principle there may existintermediary solutions with only two or three independent supersymmetries and the ˜ P AB ri are further constrained accordingly.After inserting (3.53) into (2.21) the gravitino variation δψ Aµ = 0 reduces to D µ ǫ A + 12 G µνρ γ νρ ǫ A = 0 . (3.74)Following [47, 48], this equation can be written as ∂ µ ǫ A −
14 ( ω µνρ − G µνρ ) γ νρ ǫ A + ( Q µ ) B A ǫ B = 0 , (3.75)and using (3.12) and (3.38) it becomes ∂ µ ǫ A −
14 ( ω µij − G µij ) γ ij ǫ A + ( Q µ ) B A ǫ B = 0 . (3.76)To proceed, we use (3.58) and obtain ∂ µ ǫ A − (cid:20) I BC ij I DB kl ∂ µ I DCkl + 14 I BCij ( Q µ ) BC (cid:21) γ ij ǫ A + ( Q µ ) B A ǫ B = 0 . (3.77)As for the (1 ,
0) theory [47], the solution to (3.24) can always be chosen up to an SU (2) ⊂ SO (4) local transformation on the frame e i such that the I ABij are canonical constant coeffi-cients, so that ∂ µ I ABij = 0, and one gets ∂ µ ǫ A − I BCij ( Q µ ) BC γ ij ǫ A + ( Q µ ) B A ǫ B = 0 . (3.78)Once again this is automatically integrable in the two extreme cases discussed above. Eitherone has only one supercharge satisfying (3.27), and after choosing a USp (4) gauge suchthat (3.55) one obtains that this equation reduces to ∂ µ ǫ A = 0. If we have instead fourindependent supercharges, then ˜ P ABi = 0 and thus (2.16) and (3.71) imply dQ BA − Q BC ∧ Q C B = 0 . (3.79)Consequently, we can find a USp (4) gauge such that Q BA = 0 and obtain ∂ µ ǫ A = 0 again.For the intermediary case with two or three Killing spinors one will get further constraintson Q BA . 17 .4 Equations of motion Let us finally discuss the equations of motion and inspect under which conditions a supersym-metric configuration, such that the supersymmetry variations of the fermionic fields vanish,is also a solution of the equations of motion. It was found in [47–50] that a supersymmetricconfiguration of a (1 ,
0) theory is automatically also a solution of its equations of motion ifmoreover the three-form Bianchi identities are satisfied, as well as the ++ component of theEinstein equations. All remaining equations of motion are already implied by the Killingspinor equations. As we show in Appendix B the same holds true for the (2 ,
0) theories.Under the previously determined conditions, the Einstein equations (2.17) reduce to R µν − G µκλ G νκλ − G rµκλ G νκλr − P rµ P ν r − ˜ P AB rµ ˜ P ν AB r = 0 , (3.80)while the scalar equations of motion (2.18) split into D µ P rµ = 23 G µνρ G r µνρ , D µ ˜ P AB rµ = 0 , (3.81)and the Bianchi identities (2.20) give dG = P r ∧ G r , DG r = 4 P r ∧ G , (3.82)as well as ˜ P AB r ∧ G r = 0 . (3.83)These equations resemble closely the corresponding equations of (1 ,
0) supergravity with anadditional constraint (3.83) that would trivially be satisfied in (1 ,
0) supergravity. How-ever, P r and P AB r are still momenta of the full coset space SO (5 , n ) / ( SO (5) × SO ( n )),and we must explicitly decompose this coset space into a tensor multiplet moduli space SO (1 , n T ) /SO ( n T ) and a quaternionic K¨ahler coset space SO (4 , n H ) / ( SO (4) × SO ( n H )) inorder to understand the solution in (1 ,
0) supergravity.To proceed, we make use of the fact one can always write the coset representative V as V = V T V H , with V T ∈ SO (1 , n ) and V H ∈ SO (4 , n ) . (3.84)Here, V H carries the right rigid SO (5 , n ) index of V and a left local SO (5) × SO ( n ) vectorindex, i.e. it takes the form V H = ( V H ABI , V H rI ) T , (3.85)while V T has only SO (5) × SO ( n ) indices, so its components are given by V T = V T ABCD V T ABs V T rCD V T rs ! . (3.86)18he underlined indices are associated to the ambiguity in the split of V , which we fix partiallyby imposing the following conditions on V T and V H . According to the previously introducednotation we take the USp (4) frame in which v AB is constant and decompose V T as V T ABCD = v AB V T CD + ˜ V T ABCD , (3.87)with V T AB = v AB V T , ˜ V T ABCD = δ [ AC δ B ] D − ω AB ω CD − v AB v CD , (3.88)where the components of ω AB and v AB are the same as those of ω AB and v AB , and we onlykeep the underlined indices to recall that they are associated to the spurious SO (5) × SO ( n )that we have introduced in the splitting (3.84). The remaining components of V T satisfy V T ABr = v AB V T r , V T rAB = v AB V T r , (3.89)while V T rs is unconstrained. Notice, that V T ∈ SO (1 , n ) implies4 V T V T − V T r V T r = 1 , V T V T r − V T r V T rr = 0 , V T r V T s − V T rr V T rs = − δ rs . (3.90)Similarly, we decompose V H according to V H ABI = v AB v I + ˜ V H ABI , v AB ˜ V H ABI = 0 , (3.91)such that v I is a constant SO (5 , n ) vector of norm 2. In particular, this decompositionimplies v I v J + ˜ V AB H I ˜ V H AB J − V H rI V H r J = η IJ . (3.92)Notice, that V H and V T are defined only up to an arbitrary local SO ( n ) transformation actingon the underlined indices.Following this decomposition, we can now compute2 Q C [ A v B ] C = − δ AC δ BD P CD r H V T r , (3.93)where P AB r H denotes the respective component of the Maurer–Cartan form of V H . Hence, Dv AB = 0 implies V T r P AB r H = 0 . (3.94)Using this result we determine the various other components of the Maurer-Cartan form.For the USp (4)-part of the composite connection (2.12) we find Q AB = δ AA δ BB Q H AB , (3.95)which we shall abbreviate as Q H AB , while the SO ( n ) part of the connection reads Q rs = Q rs T + V T rr Q rs H V T ss . (3.96)19or P AB r , on the other hand, we find P r = P r T − V T rs Q st H V T t , (3.97)and ˜ P AB r = V T rr δ AC δ BD P CD r H . (3.98)One can show that (3.97) and (3.98) together with (3.94) satisfy (3.71). Moreover, we find D µ ˜ P AB rν = V T rr δ AC δ BD D H µ P CD r H ν , (3.99)where D H denotes the covariant derivative with respect to only Q H . The second equation in(3.81) is thus equivalent to D µ H P AB r H µ = 0 , (3.100)and indeed describes (1 ,
0) hypermultiplet scalar fields parametrising the quaternionic K¨ahlermanifold SO (4 , n ) / ( SO (4) × SO ( n )). However, due to the mixing with Q H in (3.97), P r does not directly correspond to scalars on a coset space of the form SO (1 , n ) /SO ( n ). Inparticular the first equation of (3.81) depends non-trivially on both V T and V H , which is notthe case in a genuine (1 ,
0) theory. Instead, the scalar geometry is described by a fibrationof SO (1 , n ) /SO ( n ) over SO (4 , n ) / ( SO (4) × SO ( n )).To continue the discussion of the tensor multiplet sector, let us furthermore rewrite theconstraint ˜ G AB = 0 as ˜ G AB = ˜ V T ABCD V H CDI G I = δ AC δ BD ˜ V H CDI G I = 0 . (3.101)It follows using (3.92) that − V H r I V H r J G J = ( δ IJ − v I v J ) G J . (3.102)Therefore, −V H r I V H r J acts on G I as the constant projection onto the subspace orthogonalto the constant vector v I . Moreover, because V H rI V H r J is by construction a positive defi-nite matrix, one can define this projection as a manifestly positive constant matrix v rI v rJ .Moreover, one checks that V H rI v sI acts as an SO ( n ) rotation on V H r J G J , so that one can usethe local SO ( n ) invariance to choose V H r J such that V H rJ G J = v rJ G J . The set of matrices( v I , v rI ) satisfies by construction (cid:0) v I v J − v rI v r J (cid:1) G J = η IJ G J , (3.103)and one chooses the constant matrices v rI of lowest possible rank n T such that this identityholds for all G I at all points in spacetime. By definition, n T is the dimension of the span of V H rI G I in R n . The set of matrices ( v I , v rI ) then define a constant metric of signature (1 , n T ) η T IJ ≡ v I v J − v rI v r J , (3.104)20hich acts trivially on G I , i.e. η T IJ G J = η IJ G J . (3.105) η T I J = η T IK η JK then defines a projector onto a subspace of dimension 1 + n T . One can verifythat v I and v rI are orthogonal in this subspace, so v rI v I = 0 , v rI v s I = − δ rs T , (3.106)where δ rs T is a rank n T projector that satisfies G r = δ rs T G s . Therefore, we can define V T I ≡ V T v I + V T r v rI , V T rI ≡ V T r v I + V T rs v sI , (3.107)satisfying 4 V T I V T J − V T rI V T r J = η T IJ . (3.108)By construction we have G = V T I G I , G r = V T rI G I . (3.109)and hence we would like to identify ( V T I , V T rI ) with the tensor multiplet coset representative.However, this would require P r T = − η IJ T V T rI d V T I which would only be valid if δ rs T d V T s = d V T r ,but this does not need to be true in general.This will be true if we suppose the additional constraint V T r Q rs H = 0 , (3.110)which is together with (3.94) equivalent to V T r ∂ µ V H rI = 0 . (3.111)This conditions implies that the tensor and the hypermultiplets are locally defined in or-thogonal subspaces. In particular, one obtains then that they decouple and P r = P r T as wellas D µ P rν = D T µ P r T ν . Hence, the first equation in (3.81) becomes D µ T P r T µ = 23 G µνρ G r µνρ . (3.112)Moreover, it now follows from ( δ rs − δ rs T ) G s = 0 in combination with the supersymmetryconditions (3.62) and (3.69) that ( δ rs − δ rs T ) D V T s ∝ ( δ rs − δ rs T ) V T s . The only solution to thisrelation is ( δ rs − δ rs T ) V T s = 0 and hence P r T = − V T r d V T + δ st T V T rs d V T t = − η IJ T V T rI d V T J . (3.113)Therefore, under the assumption that (3.110) is satisfied, we can indeed identify ( V T I , V T rI )with the tensor multiplet coset representative. It is then convenient to introduce M T IJ = 4 V T I V T J + V T rI V T rJ , (3.114)21o ⋆ G I = M T IJ G J , (3.115)and the Einstein equations take the form R µν − δ rs P r T µ P s T ν − M T IJ G I µκλ G J ν κλ − δ rs P AB r H µ P H rAB ν = 0 . (3.116)We have thus shown that (3.110) is a sufficient condition for a supersymmetric solution of(2 ,
0) supergravity to satisfy the equations of motions of (1 ,
0) supergravity.We shall see in the next section that in the special case of ˜ P AB ri = 0, i.e. when thehypermultiplets only depend on the coordinate u , one can choose a gauge in which (3.110)is indeed satisfied. This is also the case if the the moduli space factorises completely, i.e. if δ rs T d V T s = d V T r , δ rs T d V H sI = 0 , (3.117)in which case one can simply take v rI = δ rs T V H sI which is constant. We found in Section 3.3 that a generic solution with a light-like isometry preserves onlyone supersymmetry. In the following, let us however focus on solutions preserving fourindependent supersymmetries, i.e. P AB ri = 0 , (3.118)and the only component of ˜ P AB r which can be possibly non-vanishing is ˜ P AB r + . This com-ponent of P AB r is projected out of (2.22) by (3.16) and is therefore unconstraint by thesupersymmetry variations. Notice that (3.118) is equivalent to P AB r H i = 0 according to(3.98). Moreover, as discussed at the end of Section 3.3, (3.118) implies that Q AB is locallypure gauge and hence can be chosen to vanish.However, there is an integrability condition on (3.118) that can be used to further con-strain ˜ P AB r + . For this purpose we use the explicit split V ABI = v AB V I + ˜ V ABI to rewrite(3.65) and (3.118) as D − V ABI = D i ˜ V ABI = 0 , (3.119)which follows from (2.15) and (3.53). Again, D + ˜ V ABI is a priori unconstraint. The Frobeniusintegrability condition for (3.119) reads[ e i + Q i , e j + Q j ] ˜ V ABI = P r [ i CD P rj ] D [ A ˜ V B ] C I + [ e i , e j ] µ D µ ˜ V ABI , (3.120)22here e i + Q i is the vector field e i acting as a Lie derivative plus the USp (4) connection along e i . Firstly, we notice that (3.118) implies P r [ i AC P r CBj ] = − P [ i P j ] δ BA = 0 . (3.121)Moreover, the commutator between two frame vector fields e iµ can be expressed in terms ofthe spin-connection ω , so (3.120) reduces to[ e i + Q i , e j + Q j ] ˜ V ABI = 2 ω ij + D + ˜ V ABI . (3.122)Hence, ˜ P AB r + can only be non-trivial if ω ij + = 0.For the general supersymmetric metric (3.59) this component of the spin-connectionreads [47] ω ij − = − f ( Dβ ) ij , (3.123)where ( Dβ ) ij was introduced below (3.59). f − γ ij is a regular metric on the base space,and the function f can only vanish on measure zero surfaces interpreted as evanescent er-gosurfaces [63]. The condition ω ij + = 0 therefore requires Dβ = 0 by continuity. One caninterpret β ( x, u ) as a connection over the four-dimensional base of a Virasoro group actingon the circle parametrized by u . Dβ is then its field strength and for Dβ = 0, β is a flatconnexion which is locally pure gauge and can be written as β ( x, u ) = ∂ m α ( x, u )1 + ∂ u α ( x, u ) dx m . (3.124)Thus, such a flat connection β can always be reabsorbed by a change of coordinate u → u − α ( x, u ) and a redefinition of the function f .To summarize, we can distinguish two branches of solutions. The first is characterisedby Dβ = 0 which in turn enforces ˜ P AB r + = 0 , (3.125)so there are no (1 ,
0) hypermultiplets. On the second branch we have β = 0 and ˜ P AB r + isunconstrained. The corresponding equation of motion (3.81) reads D µ ˜ P AB rµ = D − ˜ P AB r + = 0 , (3.126)and is identically satisfied.Let us finally discuss the implications of (3.118) on the split of the coset representative(3.84). In particular, since V T rs is an invertible matrix, we find from (3.98) that P AB r H i = 0and therefore P AB r H ∧ P CD s H = 0 . (3.127)23onsequently, the curvature of Q H vanishes and we can choose a gauge such that Q H = 0 , (3.128)and in which V H is a function of u only. This condition implies (3.110), therefore, accordingto the discussion of the preceding section, every solution with four supercharges is a solutionof a (1 ,
0) theory, where the hypermultiplet scalars are arbitrary functions of u , satisfyingpointwise the algebraic constraints (3.94) and (3.101). Let us now come back to the discussion of microstate geometries and their dependence onthe moduli parametrizing SO (5 , n ) / ( SO (5) × SO ( n )), for n = 5 or 21. It is convenient forthe microstate geometry interpretation to write the metric (3.59) asd s = − f H (cid:0) du + β − H − ( dv + ω ) (cid:1) + fH ( dv + ω ) − f − γ mn d x m d x n . (4.1)such that one can identify u as the coordinate of a circle fibered over a five-dimensionalpseudo-Riemannian spacetime. The microstate geometry generically depends non-triviallyon the circle coordinate u , but asymptotically the leading contributions to the metric areconstant on the circle, and the metric approaches the one of a black hole solution for which ∂ u is an isometry. The metric field (4.1) of the black hole solution can therefore be decomposedinto an additional dilaton R y = f H , a vector field A = β − H − ( dv + ω ), and the Einsteinframe metric of the five-dimensional spacetimed s = (cid:18) f H (cid:19) ( dv + ω ) − (cid:18) Hf (cid:19) γ mn d x m d x n . (4.2)For a five-dimensional black hole one takes the asymptotic value of f H to be one. Theisometry coordinate v can then be interpreted as the asymptotic time coordinate t of theblack hole solution. The typical example is the D1-D5-P BMPV black hole [64] with γ mn = δ mn , H = 1 + Q | x | , f = 1 √ Z Z , Z I = 1 + Q I | x | , β = 0 , (4.3)and ω a harmonic 1-form on R with anti-selfdual exterior derivative, ω = J + mn x m dx n | x | , (4.4)which carries the selfdual angular momentum J + mn . The microstate geometries associatedto such a five-dimensional black hole admit the same asymptotic fall-off for the gauge fieldsand the metric, and in particular γ mn is asymptotically Euclidean.24he most important example of a microstate geometry is probably the superstratumsolution [21,22,25,65]. This class of solutions is a deformation of a supertube solution with acircular profile [10]. The supertube solution is defined for a general closed parametric curve f m ( s ) in R such that γ mn = δ mn , H = 1 , f = 1 √ Z Z ,Z = 1 + Q Z π ds π | x − f ( s ) | , Z = 1 + Q Z π ds π | f ′ ( s ) | | x − f ( s ) | ,ω m = Q Z π ds π f m ( s ) | x − f ( s ) | , dβ = (1 + ˜ ⋆ ) dω , (4.5)with the identification of the coordinates as v = t and u = y + t , which defines t as a timecoordinate in six dimensions. The identification of either u or y as the periodic coordinatealong which both the D1 and the D5 branes are wrapped is consistent since a v -independentfunction that is periodic in u → u +2 πR y is also periodic in y → y +2 πR y . The superstratumsolutions are obtained by solving the system iteratively when the functions Z I are deformedby functions of u and x m , starting from a set of selfdual forms Θ I over R depending peri-odically on u . The function H then becomes non-trivial and reproduces asymptotically thefall-off of the BMPV black hole solution H ∼ Q | x | + O ( | x | − ) for some Q determined bythe original deformation. The superstratum solutions generally inherit from the supertubethe property that both five-dimensional angular momenta do not vanish, and that thereis a magnetic dipole associated to β . It has been shown in [25] that the selfdual angularmomentum can be pushed below the regularity bound for the black hole solution, whereasthe anti-selfdual component is non-zero and therefore necessarily remains over the regularitybound since it must vanish for the black hole solution.For a four-dimensional black hole one must take the metric γ mn to be asymptoticallyTaub-NUT, with an additional compact S fibered over R . The equivalent of the super-stratum solution has not been constructed explicitly in this case.For a globally hyperbolic metric, one requires moreover that the isometry coordinate v defines a null foliation of spacetime over a Riemannian base space, such that the 1-form field ω is globally defined over the base space and the pullback metric on a leaf B of the foliationdefines the Riemannian metricd s B = f H (cid:16) du + β − ωH (cid:17) + f − γ mn d x m d x n − fH ω . (4.6)In particular f H > f − γ mn − fH ω m ω n > B . Note that this is The conventional choice is rather to take v = t − y , u = y + t and H = 0 [21,22,25,65], which is equivalent,but we prefer to keep this definition because this change of variables is not a well defined diffeomorphism,as it shifts the time coordinate by a periodic variable. B after the change ofvariable u = y + t .As we have seen in the preceding section, supersymmetric solutions of the (2 ,
0) theorypreserving the same supersymmetry as the BMPV black hole are necessarily solutions in a(1 ,
0) theory. If Dβ = 0, the hypermultiplet scalar fields must be constant, while if β = 0 theycan be arbitrary functions of the coordinate u which do not depend on the four-dimensionalbase space coordinates x m . However, one may anticipate that this kind of solution withnon-trivial hypermultiplet profile cannot lead to a regular microstate geometry since thehyper-multiplet scalar fields are not constant at asymptotic infinity, but oscillate insteadalong the circle parametrized by the coordinate u . In the absence of hypermultiplets, thesystem of equations can be solved as in [21, 22, 56]. Here we shall discuss the case β = 0,and for simplicity we shall assume that γ mn does not depend on the coordinate u . Usingthe property that one can choose a gauge such that the coefficients I AB ij are constants, oneconcludes directly that the 4-dimensional manifold of metric γ mn is hyper-K¨ahler.One can always parametrize the tensor multiplet scalar fields by projective coordinatessuch that V T I = Z I p ( Z, Z ) , where ( Z, Z ) = η IJ T Z I Z J , (4.7)and therefore M T IJ = V T I V T J + δ rs V r T I V s T J = 2 Z I Z J ( Z, Z ) − η T IJ , (4.8)where η T IJ is the restriction of the even-selfdual metric to the sublattice of signature (1 , n T )with n T ≤ n , introduced in (3.104).Exploiting the remaining freedom in the definition of Z I , one can always define the scalingfactor f such that f = 2( Z, Z ) . (4.9)Using these definitions, one computes that the coset momentum satisfies δ rs V r T I P T µs = ∂ µ Z I p ( Z, Z ) = −√ f − δ rs V r T I V s T J η JK T ∂ µ Z K ( Z, Z ) . (4.10)Using (3.62) and (3.69), one concludes that in an appropriate gauge, the 2-form fields canbe written as B I = 12 √ (cid:16) η IJ Z J ( Z, Z ) du ∧ ( dv + ω ) + A I ∧ du + b I (cid:17) , (4.11)where A I are 1-forms and b I G I are then G I = 12 √ (cid:20) η IJ ˜ d Z J ( Z, Z ) ∧ du ∧ ( dv + ω ) + (cid:18) ˜ dA I + ˙ b I − η IJ Z J ( Z, Z ) ˜ dω (cid:19) ∧ du + ˜ db I (cid:21) , (4.12)26here ˙ b I = ∂ u b I . From the selfduality of Z I G I and the anti-selfduality of ( Z I Z J ( Z,Z ) − η IJ ) G J ,one obtains that db I = ˜ ⋆dZ I , (1 + ˜ ⋆ )(2 ˜ dω − Z I Θ I ) = 0 , (cid:18) Z I Z J ( Z, Z ) − η IJ (cid:19) (1 − ˜ ⋆ )Θ J = 0 , (4.13)where one defines for convenience Θ I = ˜ dA I + ˙ b I . (4.14)As in [56], we further assume that all the Θ I are selfdual, such that one gets˜ dω + ˜ ⋆ ˜ dω = Z I Θ I , (1 − ˜ ⋆ )Θ J = 0 . (4.15)Then the last equation that remains is the Einstein equation along the null coordinate u R ++ − M IJ G I + ij G J + ij − δ rs P + r P + s − δ rs ˜ P AB r + ˜ P + ABs (4.16)= 12 ˜ ⋆ ˜ d ˜ ⋆ ( ˜ dH + 2 ˙ ω ) + 14 η IJ Θ I ∧ Θ J − ( Z, ¨ Z ) − ( ˙ Z, ˙ Z ) + ( Z, Z )2 ˙ V ABI H ˙ V H ABI , where V H ABI only depends on the u coordinate.We therefore retrieve the same system of equation as in [21, 22] at β = 0, with additionalarbitrary functions V H ABI that further source the Laplace equation for the function H . Thesystem can be solved in steps starting from a given hyper-K¨ahler metric γ mn . One firstfinds u dependent harmonic functions Z I on the four-dimensional base. Then, one can solve db I = ˜ ⋆ tdZ I for the 2-forms b I and determines the vector fields A I such that the Θ I areselfdual, up to arbitrary harmonic vectors of selfdual field strength. The 1-form ω can besolved modulo a harmonic form of anti-selfdual field strength. Finally, one needs to solvethe Laplace equation with source for the function H .However, one can easily convince oneself that there is no regular solution of this kindwhich is asymptotically R , × S or R , × T . In the asymptotic region ˙ ω and η IJ Θ I ∧ Θ J must fall off rapidly, so that ˜∆ H is sourced by a non-zero positive function of the coordinate u which is constant in x m . It follows directly that H is singular and that the solution is nota smooth geometry. The five-dimensional base space metric (4.6) generically does not admit any isometry. As aRiemannian space, it admits a third homology group H ( Z ) of compact cycles, and the fluxquantization imposes that for any homology cycle Σ ∈ H ( Z ) one has18 √ π Z Σ G I = Q I Σ ∈ Λ ,n , (5.1)27here Λ ,n is the even-selfdual lattice of integral vectors of SO (5 , n ). For n = 5 one hasΛ , = II , , the standard Lorentzian lattice, while Λ , = II , ⊕ E ⊕ E for n = 21.For a given choice of primitive cycles Σ A ∈ H ( Z ) such that any Σ = n A (Σ)Σ A for someintegers n A (Σ), the solution admits the corresponding set of primitive fluxes18 √ π Z Σ A G I = Q IA ∈ Λ ,n , (5.2)which must individually be quantized in Λ ,n . There is always at least one cycle at infinityΣ ∞ that defines an S (or more generally a Lens space for a four-dimensional black hole)embedded in the four-dimensional base space parametrized by the coordinates x m . For afive-dimensional black hole solution, this asymptotic S is homotopic to the horizon of theblack hole, and 18 √ π Z Σ ∞ G I = Q I ∈ Λ ,n (5.3)are the NS and RR charges of the black hole in Λ ,n . The typical example is the D1-D5system, in which Q = ( Q n , Q n ) for a primitive vector n ∈ Z that can be chosen to be n = (1 , , , ,
0) in the appropriate basis. The so-called large black hole, with a macroscopichorizon area, must also include a momentum Q along the circle parametrized by u , whichcan be interpreted as an electric charge for the vector potential A = β − H − ( dv + ω ) infive dimensions.In this paper we consider supersymmetric solutions with a null isometry ∂ v . We willnow show that the quantization condition implies that, at a generic point in moduli space,all charges Q IA must be proportional to the total charge Q I . For this we use the fact thatthe equations are invariant under SO (5 , n ). The stabilizer of the charge Q of a super-symmetric black hole ( Q Q >
0) is SO (4 , n ). We have seen in the previous section thatall solutions with the same supersymmetry as the five-dimensional black hole (respectivelyfour-dimensional) can be obtained from solution of a (1 ,
0) theory with no hypermultiplets.If one starts from a given embedding of the (1 ,
0) theory in the (2 ,
0) or (2 ,
2) theory inwhich the scalar fields parametrize SO (1 , n ), one can obtain any values of the asymptoticscalar fields using the property that any element of V ∈ SO (5 , n ) can be written as V g − with V ∈ SO (1 , n ) the tensor multiplet coset representative and g ∈ SO (4 , n ) a constantgroup element. All the fields of the theory are then defined from the (1 ,
0) solution and theconstant group element g . In the notations introduced in Section 3.4, one can define thisembedding from the projection η T I J such that for a specific choice of metric η IJ in the (1 , η T IJ = g I K g J L η KL . (5.4)In particular, the 3-form field strengths are G I = g I J G J , (5.5)28o that the charges associated to the basis of primitive cycles Σ A are Q IA = 18 √ π Z Σ A G I = 18 √ π Z Σ A g I J G J = g I J Q JA (5.6)where Q A is valued in the vector space Λ ,n ⊗ R . Notice, that supersymmetry indeed requiresthe solution to be defined in the (1 ,
0) truncation. The original solution, however, has nophysical significance so one should only quantize Q A and not the original charges Q A . Tounderstand the set of charges that is allowed, one must find the intersection of g (Λ ,n ⊗ R ) ∩ Λ ,n . (5.7)We shall find that for a generic g this intersection is the one-dimensional lattice of chargevectors proportional to Q .One considers a solution with total charge Q = ( Q n , Q n ) for a primitive vector n ∈ Z that could be chosen to be n = (1 , , , , SO (5 ,
5) element g ( u ) for areal vector u ∈ R orthogonal to n , g ( u ) = + u × n ⊺ − Q Q (cid:0) u × n ⊺ − ( n + | n | u ) × u ⊺ (cid:1) − n × u ⊺ ! , (5.8)such that it stabilizes the charge Q , g ( u ) Q n Q n ! = Q n Q n ! . (5.9)The charges Q A of the (1 ,
0) truncation belong to the real extension of the lattice Λ , = II , ⊕ A , where the first factor is parametrized by ( q n , q n ) ∈ II , , while one canparametrize the A component by a vector ( q , − q ) with q ∈ Z orthogonal to n . Weshall consider q , q and q to be real since they do not define the quantized physical charges.The physical charge is defined after the action of g ( u ) as g ( u ) q n + q q n − q ! = ( q + Q Q u · q ) n + q + u | n | (cid:0) q − Q Q ( q − u · q ) (cid:1) ( q + u · q ) n − q ! . (5.10)For the resulting charge to be in the lattice Λ , , one needs that q ∈ Z mod n , q + u · q = q ′ ∈ Z , q + Q Q u · q = q ′ ∈ Z , (5.11)and therefore also u | n | (cid:16) q ′ − Q Q q ′ + Q Q u · q (cid:17) ∈ Z . (5.12)29f u is generic in R , there is no vector x ∈ Q such that x · u ∈ Q , except x = 0. Thus,if one component of u (cid:0) q ′ − Q Q q ′ + Q Q u · q (cid:1) is an integer, then the others cannot be integerunless they all vanish. One finds therefore that the only solution is the trivial one for which q = 0 and Q q = Q q . This means that the lattice of allowed charges is generated by Q/ gcd( Q , Q ).One can similarly find an SO (5 ,
21) group element g ( u , v ) stabilizing Q with u ∈ R such that u · n = 0 and v ∈ ( E ⊕ E ) ⊗ R , so that g ′ ( u , v ) q n + q q n − qp = q n + q + u (cid:0) k ( v , p ) − k ( v , v ) u · q (cid:1) q n − qp − v ( u · q ) = q n + q + u (cid:0) k ( v , p ′ ) + k ( v , v ) u · q (cid:1) q n − qp ′ . (5.13)Here k denotes the Killing Cartan form on the root lattice E ⊕ E . One obtains the similarcondition that u (cid:0) k ( v , p ′ ) + k ( v , v ) u · q (cid:1) ∈ Z , (5.14)with p ′ ∈ E ⊕ E and q ∈ Z , so that for generic u and v the only allowed solutionis q = p = 0. Combining two elements of the form g ( u ) g ′ ( u ′ , v ) as defined above, onearrives at the conclusion that the only allowed charges are proportional to the total charge Q/ gcd( Q , Q ). In this paper we studied supersymmetric solutions of (2 ,
0) supergravity in six dimensions.The spinors are defined in a real R × vector space inside the C × tensor product repre-sentation of SU ∗ (4) and USp (4). A Killing spinor is at least rank 2 as such a four by fourmatrix, and for any given rank 2 spinor one can decompose R × into four R × orthogonalcomponents using (3.12) and (3.16). Supersymmetric black hole solutions in R , × S and R , × T with four Killing spinors have their four Killing spinors in the same R × subspace.We proved that all supersymmetric solutions with four Killing spinors of this type are alsosolutions of a (1 ,
0) theory with the same preserved supersymmetries.This result has a direct consequence for the search of microstate geometries withinMathur’s proposal. We exhibit that smooth solutions with the same four supersymmetriesas BPS black holes are also solutions of a (1 ,
0) theory with hypermultiplet scalar fields that30nly depend on the coordinate parametrising the circle at infinity. It follows therefore fromthe asymptotic behaviour of these solutions that smooth microstate geometries only existif the hypermultiplet scalar fields are constant. We conclude that all such solutions can beobtained from solutions of standard (1 ,
0) supergravity involving tensor multiplets only.The same proof applies to maximal supergravity if one disregards the vector fields. Inprinciple it is possible that solutions involving the vector fields would allow for more generalsolutions than the ones that can be constructed in a (1 ,
0) theory. But one does not expect theblack hole microstate realisation of supersymmetric D1-D5-P black holes to be qualitativelydifferent on T and on K3, so it is difficult to believe that including the vector fields of (2 , ,
0) supergravity mod-uli space unless all the three-form fluxes are proportional to the total flux of the solution.Multi-cycle solutions that define bound states, in the sense that one cannot move the vari-ous cycles apart without changing the moduli or the flux they carry, only exist when theirfluxes are linearly independent [5,12–19]. Supersymmetric solutions where the various cyclescan freely be moved in spacetime are understood to describe multi-centre black holes of thePapapetrou–Majumdar type [66,67], and are therefore not relevant for microstate geometriesof a single black hole solution. Furthermore, single-centre black holes exist everywhere inmoduli space [68], the same must hence be true for their microstate geometries. We concludetherefore that all microstate geometries which are relevant for the description of single-centresupersymmetric black holes can only carry one single cycle.This therefore rules out the possibility that multi-bubble solutions are admissible super-symmetric black hole microstate geometries. Instead, it suggests that one must concentrateon supertube-like solutions as the superstratum. Our findings are also consistent with previ-ous results on the D1-D5 orbifold conformal field theory, since only supertube-like solutionsseem to admit a holographic description within this framework [69–72].
Acknowledgements
We would like to thank Iosif Bena, Massimo Bianchi, Jose F. Morales, Andrea Puhm, AshokeSen and Nick Warner for useful discussions. This work was partially supported by the ANRgrant Black-dS-String (ANR-16-CE31-0004). The work of S.L. is also supported by the ERCStarting Grant 679278 Emergent-BH. 31
Conventions
The
USp (4) invariant tensor ω AB satisfies ω AC ω BC = δ BA . (A.1)It is used to raise and lower indices according to X A = ω AB X B , X A = X B ω BA . (A.2) USp (4) is locally isomorphic to SO (5). In particular, its antisymmetric traceless representa-tion agrees with the vector representation of SO (5). Explicitly, every vector A a , a = 1 , . . . SO (5) can be transformed into an antisymmetric symplectic traceless USp (4) tensor A AB with the help of the Γ-matrices of SO (5), i.e. A AB = 12 A a (Γ a ) AB . (A.3)Therefore, as a direct consequence of the anticommutation relation of the Γ a , any two anti-symmetric tensors A AB and B AB of USp (4) satisfy the identity A AC B BC + B AC A BC = A CD B CD δ BA . (A.4)We use a ‘mostly negative’ spacetime signature, η αβ = diag(+1 , − , − , − , − , − . (A.5)The chirality projector γ is given by γ = γ , (A.6)and the supersymmetry parameter ǫ A is anti-chiral, γ ǫ A = − ǫ A . (A.7)Hodge duality acts on the γ -matrices as γ µ ...µ n = ( − ⌊ n ⌋ (6 − n )! ε µ ...µ n ν ...ν p γ ν ...ν p γ . (A.8)Moreover, for a selfdual three-form G + = ⋆G + it follows that G + µνρ γ µνρ ǫ A = 0 . (A.9)32he general expansion of an (anti-)selfdual three-form G ± = ± ⋆ G ± with respect to thenull frame (3.21) reads G ± = e + ∧ A ∓ + e − ∧ B ± + (1 ± ⋆ ) e + ∧ e − ∧ C , (A.10)where A ± = A ± ij e i ∧ e j and B ± = B ± ij e i ∧ e j are (anti-)selfdual two-forms with respect to − δ ij , i.e. ⋆ A ± = ± A ± and C = C i e i .In the null frame (3.21) we introduce the four-dimensional chirality matrix: γ ∗ = γ . (A.11)A relation analogous to (A.8) holds for the four-dimensional γ -matrices. Moreover, theprojection (3.16) is equivalent to γ ∗ ǫ A = ǫ A . (A.12)Thus it follows that A + ij γ ij ǫ A = 0 , (A.13)for every selfdual two-form A + . B Integrability conditions
The gravitini Killing spinor equation (2.21) implies the integrability condition (cid:16) R µνκλ δ AB γ κλ + 8 P [ µ BC r P AC rν ] + 4 D [ µ G ACν ] κλ ω CB γ κλ − G ACκλ [ µ G DEν ] γδ ω CD ω EB γ κλ γ γδ (cid:17) ǫ B = 0 . (B.1)After contracting this with γ µ we obtain (cid:20) ω AB E µν γ ν + 23 V ABI E Iµνκλ γ νκλ + 4 V ABI ( ⋆E I ) µν γ ν (cid:21) ω BC ǫ C + i (cid:20) P AC rµ ω CB + 13 G rνκλ γ νκλ γ µ δ AB (cid:21) δχ B r = 0 , (B.2)where E µν and E I are defined in (2.17) and (2.19) and represent the Einstein equations andthe equations of motion of the two-form fields. Therefore, if the Bianchi identity (2.20) for G AB is satisfied, the integrability condition reduces to E µν γ ν ǫ A = 0 , (B.3)and by multiplying with ¯ ǫ B γ κλ from the left we find E µ [ κ V λ ] = 0 , (B.4)33r equivalently E µ − = E µi = 0 . (B.5)So all but the ++ component of the Einstein equations follow from the Killing spinor equa-tion.There is another integrability condition for the tensorini variation (2.22) which reads − iγ µ D µ δχ A r = (cid:20) E AB r ω BC + 12 V rI ( ⋆E I ) µν γ µν δ AC (cid:21) ǫ C + i G ABµνρ ω BC γ µνρ δχ C r = 0 , (B.6)with E AB r and E I from (2.18) and (2.19). Consequently, if the Bianchi identity (2.20) for G r holds, also the scalar equation of motion is implied by the Killing spinor equations.34 eferences [1] S. W. Hawking, “Breakdown of Predictability in Gravitational Collapse,” Phys. Rev.
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