ESI The Erwin S hr�odinger International Boltzmanngasse 9Institute for Mathemati al Physi s A-1090 Wien, AustriaSupersymmetri Bla k Holes are Extremaland Bald in 2D Dilaton SupergravityL. BergaminD. GrumillerW. Kummer

Vienna, Preprint ESI 1381 (2003) O tober 8, 2003Supported by the Austrian Federal Ministry of Edu ation, S ien e and CultureAvailable via http://www.esi.a .at

TUW{03{24Supersymmetri bla k holes are extremal andbald in 2D dilaton supergravityL. Bergamin�, D. Grumillery and W. KummerzInstitut f�ur Theoretis he PhysikTe hnis he Universit�at WienWiedner Hauptstr. 8{10, A-1040 WienAustriaAbstra tWe prove that solutions of 2D dilaton supergravity respe ting bothsupersymmetries have to belong to the rather trivial lass of onstantdilaton va ua. Then it is shown that solutions whi h retain at leasthalf of the supersymmetries are ground states with respe t to thebosoni Casimir fun tion { in physi al terms, the ADM mass has tovanish, whenever this notion is meaningful. Nevertheless, by tuningthe prepotential appropriately, bla k hole solutions may emerge insu h ground states with an arbitrary number of (Killing) horizons.Exploiting supersymmetri obstru tions for bosoni quantities it isproven that all horizons have to be extremal. In these derivations theknowledge of the general analyti solution of 2D dilaton supergravityplays an important role. Finally it is demonstrated that the in lusionof non-minimally oupled matter, a step whi h is already nontrivial byitself, does not hange these features in an essential way. As byprod-u ts we noti e the impossibility of a supersymmetri dS ground stateeven in the presen e of a dilaton, and the absen e of dilatino andgravitino hair for all supersymmetri bla k holes.�e-mail: bergamin�tph.tuwien.a .atye-mail: grumil�hep.itp.tuwien.a .atze-mail: wkummer�tph.tuwien.a .at

1 Introdu tionIn the mid 1990s, during and after the \se ond string revolution", BPS(Bogomolnyi-Prasad-Sommer�eld [1, 2℄) bla k holes (BHs) have attra tedmu h interest be ause in parti ular they allow to derive the BH entropyby ounting D-brane mi rostates exploiting string dualities (for reviews f.e.g. [3,4℄). We de�ne a BPS BH as a supergravity (SUGRA) solution respe t-ing half of the supersymmetries and exhibiting at least one (Killing) horizonin the bosoni line element.Typi ally BPS BHs were found to be extremal [3, 4℄. However, in lusionof non-geometri al �elds may hange this pi ture. For instan e, non-extremalBPS BHs were onstru ted in the presen e of Yang-Mills �elds [5℄. In this ontext the onsideration of two dimensional theories is espe ially useful,be ause of te hni al simpli� ations.For quite some time the key referen e for the properties of supersymmet-ri solutions in 2D dilaton SUGRA has been the work of Park and Stro-minger [6℄. As shown by these authors for the SUGRA version of the CGHSmodel [7℄, as well as for a SUGRA extended generalized 2D dilaton theory, a ertain va uum solution an be de�ned with vanishing fermions. A spe i� solution, whi h still retained one supersymmetry, was onstru ted for theCGHS-related model. In the generi ase the existen e of su h a solutionwas proven.Until quite re ently a systemati study of all supersymmetri solutionsin 2D dilaton SUGRA theories has not been possible. Re ently two of thepresent authors have shown [8, 9℄ that the super�eld formulation of [6℄ anbe identi�ed with the one of a ertain sub lass of graded Poisson-Sigmamodels (gPSMs). General gPSMs are fermioni extensions of bosoni PSMswhi h, in the present ase, are taken to be 2D dilaton theories of gravity [10℄.This sub lass of gPSMs has been dubbed minimal �eld SUGRA (MFS) inref. [9℄. The important onsequen e of this equivalen e is that the knownanalyti solution in the MFS formulation [9℄ represents the full solution fordilaton SUGRA [6℄, in luding all solutions with nonvanishing fermioni �eld omponents.This permits us to atta k in a systemati manner the problem of 2DSUGRA solutions whi h retain at least one supersymmetry, the main goalof our present work. It turns out that, although some information an beobtained from the symmetry relations, the knowledge of the full solution isa ne essary input. This is of parti ular importan e regarding the eventualexisten e of fermioni hair for BHs.In doing this we are also able to in orporate SUGRA invariant matter,where a spe ial previous model [11℄ is extended to generalized dilaton theories2

and also trans ribed into the more onvenient MFS formulation. Althoughno general analyti solution is possible when matter is in luded, the followingmain result an be proved also then:There are no nonextremal BH solutions of 2D SUGRA respe ting halfof the supersymmetries. The idea of the rather straightforward proof is asfollows: �rst, we show that the body of the Casimir fun tion (a quantity ingPSM theories related to the energy of the system) has to vanish and thenwe prove that these ground state solutions annot provide simple zeros ofthe Killing norm. En passant all solutions respe ting at least half of thesupersymmetries are lassi�ed in luding the ones of ref. [6℄.The paper is organized as follows: in se t. 2 all ground states with un-broken supersymmetry are dis ussed, whi h turn out to be onstant dilatonva ua. Su h va ua appear nontrivially in some, but not all dilaton theories.Se t. 3 is devoted to a lassi� ation of solutions respe ting half of the su-persymmetries. All of them imply a vanishing body for the bosoni Casimirfun tion. In se t. 4 it is demonstrated that for su h ground states the Killingnorm has to be positively semi-de�nite and thus only extremal horizons mayexist. Se t. 5 dis usses the oupling of MFS to matter degrees of freedomand se t. 6 generalizes the result of se t. 4 to the one in luding matter.The notation is explained in app. A. In app. B we summarize the ne es-sary formulas for MFS: the a tion as derived from a gPSM and the symmetryrelations. We also present the orresponding formulas for the equivalent dila-ton theory without auxiliary �elds (minimal �eld dilaton SUGRA, MFDS).Equations of motion and some aspe ts of the exa t solution are ontained inapp. C. The subje t of app. D are some key formulas [9℄ relating MFS tothe super�eld approa h to 2D dilaton SUGRA [6℄, whi h are needed for the onstru tion of the matter ouplings.2 Both Supersymmetries unbrokenFrom the MFS supersymmetry transformations in appendix B one an reado� di�erent onditions for solutions respe ting full supersymmetry. From(B.11) and (B.15) follows1 �+ = �� = + = � = 0. The two terms in(B.13) are linearly independent and thus X++ = X�� = u = 0 as well. Inaddition (B.16) implies that the transformation parameters must be ovari-antly onstant: (D")� = 0. For a solution where both supersymmetries areunbroken the Poisson tensor (B.3)-(B.6) vanishes identi ally.The equations of motion imply that the dilaton � has to be a onstant.It is restri ted to a solution of the equation u = 0. Su h onstant dilaton1Conventions and light one oordinates are summarized in app. A.3

va ua (CDV) are, for instan e, en ountered [12℄ in the \kink" solution of thedimensionally redu ed gravitational Chern-Simons term [13℄.We re all in app. C that a key ingredient of the solution is the on-served Casimir fun tion. At this point its additive ambiguity an be �xed:supersymmetry ovarian e requires that solutions respe ting both supersym-metries have vanishing Casimir fun tion. This means that eventual additive onstants in (C.10),(C.11) are absent.Positivity of energy would imply u2 � 8Y be ause the Casimir fun tionCB is related to the negative ADM mass ( f. se tion 5 of ref. [14℄). If thisinequality is saturated the ground state is obtained. Eq. (C.10) in parti ularimplies that all CDV solutions with u = 0 have vanishing body of the Casimirfun tion.2As an illustration we onsider a two parameter family of models (theso- alled \ab-family") en ompassing most of the relevant ones [15℄. Amongother solutions, BHs immersed in Minkowski, Rindler or (A)dS spa e an bedes ribed. This family is de�ned by (B.8) withZ(�) = �a� ; u(�) = �� ; �; a; 2 R : (2.1)Supersymmetry restri ts the onstant B = 2(b+1)=4 in the potential (B.7)V (�) = �B2 �a+b ; � = a+ b+ 12 (2.2)to B > 0 if b > �1 and to B < 0 if b < �1. The urvature s alar of theground state geometry is given byR = b 22 (� � a)X2(��1) : (2.3)The Minkowski Ground State (MGS) ondition3 reads � = a, Rindler spa efollows for b = 0 and (A)dS means � = 1. In the latter ase supersymmetryrestri ts the urvature s alar R = 2(1 � a)2=2 to positive values, whi h inour notation implies AdS.For fully supersymmetri solutions of the ab-family the only possible val-ues for the dilaton are � = 0 or j�j = 1 (depending on the value of �),unless = 0, in whi h ase the prepotential u vanishes identi ally.2CDV solutions with u 6= 0 must obey u0=u = �12Z, whi h leads to CS = 0 whileCB 6= 0. Clearly, they annot respe t both supersymmetries.3It means simply that for vanishing bosoni Casimir fun tion the bosoni line elementis di�eomorphi to the one of Minkowski spa e.4

3 One Supersymmetry unbroken3.1 Casimir fun tion CB = 0The symmetry transformation Æ� = 0 of the dilaton from (B.11) implies�+"+ = ��"� : (3.1)The vanishing of (B.13) and (B.15) leads tou"� = �2p2X++"+ ; u"+ = �2p2X��"� ; (3.2)" a = �iZ8 �abeb(�") : (3.3)In (3.2) terms proportional to �2" have to vanish as a onsequen e of (3.1).Eqs. (3.2) require Y = 18u2 ; (3.4)whi h in turn implies that the body of the Casimir fun tion (C.10) vanishes.In this sense, BPS like states are always ground states. Noti e that eq. (3.4)remains valid in the ase u = 0, implying that at least one omponent of Xavanishes as well and vi e versa.3.2 Classi� ation of solutionsVanishing fermionsWe �rst onsider the ase of vanishing dilatino �� and gravitino �. Thenall three quantities u;Xa have to be nonvanishing or otherwise the solutionwith full supersymmetry of se t. 2 is re overed. With the onditions (3.2)and hen e also (3.4) the variations (B.11)-(B.15) vanish, while (B.16) equalzero represents two di�erential equations for "+, resp. "�. By inserting theexpli it solution (se t. 6 of ref. [9℄) it an be he ked that both are identi al.A straightforward al ulation, without using any further restri tions on thedi�erent variables involved, yieldsd"+ +�dX++2X++ � (u0u + 12Z) d�� "+ = 0 ; (3.5)possessing the general solution (Q is de�ned in (C.12))"+ = e 12Q upX++ ~" : (3.6)5

Here ~" is a spinorial integration onstant. "� is obtained via (3.2). Thus,all solutions without fermion �elds, exhibiting one supersymmetry, leave alinear ombination of the two supersymmetries unbroken, in agreement withthe dis ussion in ref. [6℄.There exists one spe ial ase of (3.5) where an even simpler solution exists:If Z = 0 (MFS0 in the parlan e of ref. [9℄) and if in addition u is a onstant,the di�erential equation (3.5) simply says that the symmetry parameter is ovariantly onstant. This model is generalized teleparallel dilaton gravity.As u is simply a osmologi al onstant, it drops out in the onstraint algebra,and therefore, has been referred to as rigid supersymmetry in ref. [10℄.Thus, all solutions without fermion �elds, exhibiting one supersymmetry,leave a linear ombination of the two supersymmetries unbroken, in agree-ment with the dis ussion in ref. [6℄.Nonvanishing fermions and No-Hair TheoremFurther supersymmetri solutions are possible for nonvanishing fermions, asituation not onsidered in ref. [6℄, but relevant for the question of fermioni hair4.Assuming that at least one omponent of �� is di�erent from zero, e.g.�+ 6= 0, eq. (3.1) an have the hiral solution "+ = �� = 0, while "� 6= 0provides the remaining supersymmetry, or it an relate "+ to "� by meansof �+ 6= 0 and �� 6= 0.Solutions of the �rst type are almost trivial, be ause they require u =X�� = 0. Sin e the dilaton must not be onstant (or else the CDV asewould be re overed) this implies that u must vanish identi ally, and not justat a ertain value of �. All fermions must be of one hirality, in parti ular"+ = �� = � = 0. The quantities X++; e++; !; �+ must be ovariantly onstant and may ontain soul ontributions. The dilaton has to ful�ll thelinear dilaton va uum ondition d� = onst: Only e�� and "� are slightlynontrivial and an be dedu ed from(De)�� = Z�e�� ; (D")� = Z2 �"� ; � = (X++e++ + 12�+ +) : (3.7)The bosoni line element is at and obviously the Casimir fun tion is iden-ti ally zero.For the remaining lass of solutions with both omponents of � and of "di�erent from zero the body of the Casimir fun tion still vanishes, but the4No-hair theorems are a re urrent theme in BH physi s (ref. [16℄ and refs. therein).6

soul an be non-vanishing as from (B.12):Z = �u0u CS = 132eQu0�2 (3.8)This is equivalent to the MGS ondition mentioned in se tion 2 below (2.3).Thus, only MGS models are allowed and sin e the solution has to be theground state, the geometry is trivially Minkowski spa e.Therefore, a solution with nonvanishing fermions must have a trivialbosoni ba kground (this feature is true also for the �rst type). Consequentlythere exist no BPS BHs with fermioni hair.There remains a te hni al subtlety about the non hiral solutions. As both omponents of " are non-zero, (3.2) implies that all interesting ases have u,X++ and X�� di�erent from zero. A tually, the solutions presented in [9℄ donot over this ase, be ause the one for C 6= 0 ((6.9)-(6.13) of [9℄), dependingon C�1, annot be used in the present ase, as the inverse of a pure soulis ill-de�ned. The solution for C = 0 ((6.17)-(6.21) of [9℄) depends on anarbitrary fermioni gauge potential ~A = �df , whi h is the gauge potentialasso iated with the additional fermioni Casimir fun tion (eq. (6.15) of [9℄)appearing in that ase. Analyzing the e.o.m.-s (app. C and f. also [10℄) forCB = 0 but CS 6= 0, this fermioni Casimir fun tion does no longer appear,but the solution remains valid if the gauge potential ~A = �df obeys the onstraint �2 ~A = 0.Finally (B.16) again leads to the di�erential equationD"+ �� Z16 u2X++ e�� � Z2X++e++� "+ = 0 : (3.9)Inserting the solution des ribed above all trilinear and higher spinorial termsare found to vanish. Thus, the solution of (3.9) redu es to (3.6).4 No nonextremal BPS bla k holesIt has been shown in se tion 3 that the body of the bosoni Casimir fun tionhas to vanish for all solutions respe ting at least half of the supersymmetries.We will fo us �rst on BPS solutions where either X++ 6= 0 or X�� 6= 0 (orboth).In the bosoni ase ground state solutions C = 0 imply for the line element( f. e.g. eq. (3.26), (3.27) of [14℄, denotes the symmetrized tensor produ t)(ds)2 = 2dF (dr � eQ(�)w(�)dF ) ; dr = eQ(�) d� : (4.1)7

Obviously, there exists always a Killing ve tor ���� = �F the norm of whi his given by K = �2eQ(�)w(�). By hoosing the fun tion w(�) appropriately,nonextremal Killing horizons are possible, e.g. for Q = 0;�2w = 1 � 2m=�a S hwarzs hild-like BH emerges.In SUGRA, however, (C.12) implies a negative (semi-)de�nite w andhen e the Killing norm K an only have zeros of even degree:K(�) = �2eQ(�)w(�) = �12u(�)eQ(�)�2 � 0 (4.2)Thus, if a Killing horizon exists it has to be an extremal one. This on ludesthe proof of our main statement. For instan e, with u = 2(p��M); Z(�) =�1=(2�) the line element reads (ds)2 = 2dF (dr + (1�M=r)2 dF ), whi his the two-dimensional part of an extremal Reissner-Nordstr�om BH.One an trivially generalize this result to all ground state solutions whi hare not CDV, i.e. non-supersymmetri solutions of 2D dilaton SUGRA withvanishing body of the Casimir fun tion and non- onstant dilaton, be ausethe key inequality (4.2) still holds.Finally, the simpler CDV ase X++ = 0 = X�� shall be addressed. Asshown in the previous two se tions CDV implies that both supersymmetriesare either broken or unbroken. The equations of motion imply a vanishingbody of torsion and a onstant body of urvature. Thus { as in the bosoni ase ( f. se t. 2.1 in [12℄) { only (A)dS, Rindler5 or Minkowski spa es are pos-sible. However, supersymmetry provides again an obstru tion: urvature isproportional to (u0)2 and thus again the dS ase, together with the possibilityof nonextremal Killing horizons, is ruled out by supersymmetry.65 Extension with onformal matterWe will prove in the next se tion that the on lusions of the se t. 4 do not hange when onformal matter is oupled to the dilaton SUGRA system. Asthese on lusions rely on the details of the symmetry transformations andthe onserved quantities, in a �rst step the extension to MFS with matter�elds is introdu ed in this se tion. To this end the lose relation between5If urvature is non-vanishing the Rindler term an always be absorbed by a linearrede�nition of the oordinates r ! r0 = �r + �, F ! F 0 = F=�, � 6= 0. For vanishing urvature the Minkowski term an always be absorbed by a similar rede�nition.6In fa t, there is one very trivial possibility that remains for a CDV solution withvanishing urvature whi h allows for the existen e of exa tly one nonextremal Killinghorizon: the line element (ds)2 = 2dF (dr + br dF ), b 6= 0 ontains a nonextremal(Rindler) horizon at r = 0. However, this is neither a BPS state nor a true BH solution.8

MFS and the models obtained from superspa e [9℄ is used. In superspa enon-minimally oupled onformal matter is des ribed by the LagrangianS(m) = 14 Z d2xd2� EP (�)D�MD�M : (5.1)Here P (�) is fun tion of the dilaton super�eld � ( f. (D.5)) and for the�-expansion of the matter multipletM we write7M = f � i�� + 12�2H : (5.2)Integrating out superspa e one arrives at ( f. app. D)S(m) = Z d2x e"P (�)�12(�mf�mf + i� m�m�+H2)+ i( n m n�)�mf + 14( n m n m)�2�+ 14P 0(�)�i(� ��)H � (� � m�)�mf � F�2�� 132P 00(�)�2�2# : (5.3)The a tion (5.3) depends on the auxiliary �elds H from the matter multipletand F from the dilaton multiplet. H an be integrated out without detailedknowledge of the geometri part of the a tion. To integrate out F , however,u(�) and Z(�) in (D.1) must be spe i�ed8.5.1 Matter Extension at �Z = 0A parti ulary simple situation is realized by hoosing �Z = 0 (following thenotation of [9℄, barred variables are used for this spe ial ase throughout).Then the a tion (D.1) is bilinear in �A and �F and the elimination ondition(D.12) is modi�ed a ording to �A = ��u0=2 + �P 0��2=4, �F = ��u=2. The part7Whenever the distin tion between super�eld omponents and MFS �elds is important,underlined symbols are used for the former ones. However, for simpli ity this is omittedin most formulas of this se tion, as the matter a tion is invariant under the rede�nition(D.18), while all other identi� ations are trivial.8As F appears in a term / P 0 the restri tion to �Z = 0 in the following is not ne essarywhen onsidering minimally oupled matter.9

of the a tion independent of the matter �eld retains the form (D.13) with�Z = 0, while (5.3) after elimination of all auxiliary �elds be omes:�S(m) = Z d2x �e" �P�12(�m �f�m �f + i�� m�m��) + i( � n m n��)�m �f+ 14( � n m n � m)��2�+ �u8 �P 0��2 � 14 �P 0(�� � m��)�m �f� 132� �P 00 � 12 � �P 0�2�P ���2��2# (5.4)The symmetry transformations of the matter �elds �f and ��� after eliminationof �H read (it should be noted [9℄ that the symmetry parameters " and " aredi�erent in general)�Æ �f = i(�"��) ; �Æ��� = ��m �f + i( � m��)�( m")� � 14 P 0P (�� � ��)�"� ; (5.5)while the zweibein, the dilaton and the dilatino still transform a ording to(D.14), (D.16) and (D.17) resp. The transformation rule for the gravitino hanges, as it depends on the auxiliary �eld �A:Æ � m� = �( ~�D�")� + i4��u0 � 12 �P 0��2�(�" m)� (5.6)When working with the MFS formulation of the geometri part, it isadvantageous to formulate the matter a tion (5.4) in terms of di�erentialforms as well:�S(m) = ZM� �P�12 d �f ^ �d �f + i2 �� a�ea ^ �d�� + i � (�ea ^ �d �f)�eb ^ � � a b��+ 14 � (�eb ^ � � ) a b�ea ^ � � ��2�+ �u8 �P 0��2��� 14 �P 0(�� � a��)�ea ^ �d �f � 132� �P 00 � 12 � �P 0�2�P ���2��2��� (5.7)For the spe ial ase �Z = 0 dis ussed so far, the identi� ation (D.18)between the variables of MFS and of the super�eld formulation (D.1) in[6℄ be omes trivial. Thus, repla ing �" in (5.5) and (5.6) by �", the a tion(B.17) with �Z = 0 together with (5.4) is invariant under (B.18)-(B.20), (5.5)and (5.6). The spe ial ase of minimal oupling (P (��) = 1) of this matter10

extension of a gPSM based dilaton SUGRA model has already been obtainedin ref. [11℄ using Noether te hniques. But the above derivation using theequivalen e of this theory to a superspa e formulation has de�nite advantagesas it straightforwardly generalizes to more ompli ated matter a tions.As a �rst step we should derive from the result obtained so far the matterextension of MFS0 (in the following MFS0 indi ates MFS for �Z = 0). Thisstep is ne essary as only the �rst order formalism in terms of a gPSM allowsthe straightforward treatment of the model at the lassi al as well as at thequantum level. Also in the present ontext the MFS formulation is mu hsuperior. The MFS a tion is di�erent from (B.17), whi h was obtained afterelimination of Xa and (the torsion dependent part of) !. Now these auxiliaryvariables must be re-introdu ed together with the matter oupling. Se ond,we would like to extend the matter oupling to the most general MFS (B.8).So far, we arrived at a matter extension for the spe ial ase �Z = 0 only. Thegeneral matter oupling (Z 6= 0) will be obtained by the use of a ertaindilaton dependent onformal transformation ( f. [9, 10℄). It is argued in theend that the same result ould also have been derived in a di�erent way.The dis ussion of a onsistent matter extension of MFS0 onsiderablysimpli�es by observing that the a tion (5.4) or (5.7) does not hange whenthe independent variables �Xa and �! are re-introdu ed. This is trivial for �!, as(5.4) does not ontain the dependent spin onne tion ~�!. The independen eof �Xa is obvious as well [9, 10℄: The elimination ondition of �Xa dependson derivatives a ting onto the dilaton �eld and the matter a tion does not ontain su h terms. Thus the matter extension of MFS0 must be of the form9�Stot( �X; �A; �f; ��) = SMFS0( �X; �A) + �S(m)(�ea; � �; ��; �f; ��) : (5.8)Not ompletely trivial is the derivation of the orre t supersymmetry trans-formations. However, it is important to realize that (5.8) already is invariantup to equations of motion10 of �Xa and �!: As these e.o.m.-s are linear in�Xa and �!, the elimination of these �elds \ ommutes" with the symmetrytransformation. Therefore there exists a simple and systemati way to mod-ify the MFS0 symmetry laws (B.11)-(B.16) su h that (5.8) together with(5.5) is again invariant. In an abstra t notation the behavior of (5.8) under(B.11)-(B.16) and (5.5) may be written as�Æ �Stot( �X; �A; �f; ��) = NaX( �X; �A; �f; ��; �") � ( �Xa-e.o.m.)+ (�!-e.o.m.) ^N!( �X; �A; �f ; ��; �") : (5.9)9XI = (�;Xa; ��), AI = (!; ea; �)10We denote the equations of motion a ording to the �eld whi h has been varied. Thusthe Xa-e.o.m. refers to (C.7), while the !-e.o.m. to (C.3).11

Of ourse, the two �eld-dependent quantities NaX and N! multiplying thee.o.m.-s must vanish in the absen e of matter �elds:NaX( �X; �A; �f = 0; �� = 0; �") = 0 ; N!( �X; �A; �f = 0; �� = 0; �") = 0 : (5.10)They are used to modify the symmetry transformations of �Xa and �! by�Æ �Xa = �ÆMFS0 �Xa �NaX ; �Æ�! = �ÆMFS0 �! �N! ; (5.11)where the transformations (B.11)-(B.16) with �Z = 0 have been renamed�ÆMFS0.The expli it al ulation of NaX and N! is straightforward. As �S(m) dependson the MFS0 �elds ��, �ea and � � only, the variations (B.11), (B.15) and (B.16)within (5.4) lead to potential non-invarian e. But (B.11) and (B.15) areequivalent to the supersymmetry transformations of these �elds within thesuper�eld formulation ( f. (D.6), (D.9)). There remains the supersymmetrytransformation of the gravitino. Again, most terms are equivalent to thesuperspa e formulation (remember that �Z = 0!), ex ept for the ovariantderivative (D")�. Here the dependent spin onne tion ~! has been repla edby the independent one. As the independent part of the spin onne tion iseliminated by means of the �Xa-e.o.m. (C.7) this leads to ontributions toNaX . A further sour e of non-invarian e is the modi� ation of the gravitinotransformation in (5.6). This yields another ontribution to NaX from the ovariant derivative a ting on � , but also to N! from the term / �Xa � a � .The latter ontributions are proportional to P 0(��) and vanish in the ase ofminimal oupling. Putting all terms together one �nds�Æ �Xa = �ÆMFS0 �Xa + i2P (�" m a ���)�m �f + 14P (�" m a � � m)��2� i16P 0(�� a��)��2 ; (5.12)�Æ�! = �ÆMFS0 �! � 14P 0(�� � � � ��� � � �)��2 : (5.13)With these new transformation laws the a tion (5.8) is �nally fully invariantunder supersymmetry, while lo al Lorentz invarian e and di�eomorphisminvarian e are manifest.5.2 Matter Extension at Z 6= 0To extend the matter ouplings to the general MFS (Z 6= 0 in (B.8)) we usethe onformal transformation (B.9) and (B.10) of appendix B. The matter12

a tion is invariant under those transformations of the �elds, when the newmatter �elds are de�ned asf = �f ; � = e� 14Q(�)�� : (5.14)After the ombined transformations (B.9), (B.10) and (5.14) an a tion withgeneral MFS as geometri al part oupled to onformal matter is obtained.�S(m) in (5.4) by onstru tion is invariant under the onformal transformationand therefore that equation, after dropping all bars, is the orre t matterextension of MFS. Of ourse, the new a tionStot(X;A; f; �) = SMFS(X;A) + S(m)(ea; �; �; f; �) (5.15)is invariant under the old �"-transformations that a t on the barred variables�ÆStot�X( �X); A( �A; �X); f( �f ); �(��; �X)� = 0 ; (5.16)but we have to be areful with the new transformations Æ (depending on "), asthe transformation parameters themselves hange under a onformal trans-formation as well. The importan e of this behavior for the understandingof gPSM based SUGRA has been pointed out in [9℄. Conformal transforma-tions represent a spe ial ase of target spa e di�eomorphisms in the (g)PSMformulation ( f. se t. 4.1 of [9℄). Under su h transformations the variablesand symmetry parameters hange as�Æ �XI = Æ �XI(X) ; (5.17)�Æ �AI = Æ �AI(A;X) + e.o.m.-s ; (5.18)�"I = �XJ� �XI "J ; (5.19)where the indi es are the ones used in the gPSM formulation (B.1). Eq. (5.19)together with (B.9) and (B.10) for a pure supersymmetry transformationyield ( f. se t. 5.2 in ref. [9℄, esp. eq. (5.8)):�" = (�"�; �"a; �"�) = (0; 0; �"�) onformal transformation������������! " = (Z4 (�"); 0; "�) (5.20)Thus for the general MFS the symmetries (5.5) are modi�ed by a lo alLorentz transformation with �eld-dependent parameter "� = (1=4)Z�". Thesymmetry law of f remains un hanged under both, the onformal transfor-mation (5.14) and the additional lo al Lorentz transformation (5.20), as this�eld is invariant under these symmetries. However, for � the lo al Lorentz13

transformation and the supersymmetry transformation of the onformal fa -tor in (5.14) add up to the new ontributions displayed in eq. (5.27) below.Still the a tion (5.15) is not invariant under (5.26), (5.27) and (B.11)-(B.16): First, the modi�ed laws of � (5.6), �Xa (5.12) and �! (5.13) shouldbe rewritten in terms of the MFS variables. But as none of these extensionsgenerates derivatives onto the onformal fa tors, this boils down to rewritethese transformation rules in terms of variables without bars. Se ond, thee.o.m.-s appearing on the r.h.s. of (5.18) may ne essitate further modi� a-tions of the MFS symmetries. As the onformal transformations (B.9) and(B.10) depend on the dilaton �eld only, under supersymmetry transforma-tions dis ussed so far the a tion (5.15) is invariant up to e.o.m.-s of !. Thesenew non-invariant terms originate from the variation of the gravitino ( f. eq.(4.8) of ref. [9℄ and omments below this equation). Indeed a straightforward al ulation shows that�Æ � = �d��� + 14Z d� ��� + : : : ; Æ � = �d�� + : : : ; (5.21)where the dots indi ate terms whi h do not ontain derivatives. The equationof motion of ! involved here drops out in the geometri part, but obviouslynot in the matter extension. However, from the above equation together withthe formulation of S(m) in (5.7), supersymmetry an be restored analogouslyto the arguments leading to (5.12). It an be read o� from (5.21) that thenew (matter-�eld dependent) pie e to the transformation of ! is obtained byrepla ing � in (5.7) by 1=4Z��:Æ! = (5.13)� 14ZP�i(" a b�)ema �mf(�eb)+ 12(" a b m)ema (�eb)�2� : (5.22)It is useful to summarize what we have obtained in se t. 5: The gPSMbased MFS models of eq. (B.8) an be extended by the oupling of matter�elds. The omplete a tion (5.15) is given by the sum of (B.8) and (5.4). Thesupersymmetry transformations (B.11), (B.13), (B.15) for �, �� and ea arenot hanged by the matter oupling. Eq. (B.16) for the gravitino re eives new ontributions from the elimination of the auxiliary �elds in superspa e, whileÆXa and Æ! are hanged by re-introdu ing the auxiliary �elds of the gPSMformulation. Thus, the omplete list of supersymmetry transformations isgiven by (B.11)-(B.16) plus new ontributions from the matter �elds,Æ(m)Xa = i2P (" m a ��)�mf+ 14P (" m a � m)�2 � i16P 0(� a�)�2 ; (5.23)14

Æ(m)! = �14P 0(� � � �� � �)�2� 14ZP�i(" a b�)ema �mf(�eb) + 12(" a b m)ema (�eb)�2� ; (5.24)Æ(m) � = iP 08 �2( b")�eb ; (5.25)together with the transformations of the matter �eldsÆf = i("�) ; (5.26)Æ�� = ��mf + i( m�)�( m")�� 18Z�(� �")�� + (�")( ��)��� P 04P (� ��)"� : (5.27)Of ourse, we ould have used the relation to the general Park-Stromingermodel SFDS of eq. (D.13) to the MFS ( f. (D.18) and also ref. [9℄) insteadof the onformal transformation of MFS0 to derive the matter oupling atZ 6= 0. Thus we may eliminate Xa and ! in (5.15) and by this pro edurearrive at the matter a tion (5.4) oupled to MFDS. Using the te hniquesdeveloped in [9℄ this equivalen e follows almost trivially. Considering thesymmetry transformations we note that we �nd again ( f. (D.19))��� = �14Z(�")( ��)� (5.28)in agreement with the result derived in [9℄.One might wonder whether, on a di�erent route, it is possible to derivea di�erent matter extension of gPSM based SUGRA, where modi� ations ofthe transformation laws of XI and AI do not o ur. The answer is negative,as long as this extension shall preserve both, lo al Lorentz invarian e andsupersymmetry. Indeed, the ommutator of two lo al supersymmetry trans-formations is a lo al Lorentz transformation Æ� plus a \lo al translation" Æa.Invarian e under stri t gPSM symmetry transformations would imply thatthe matter a tion is invariant under Æa, whi h, ex ept for rigid supersymme-try, annot be ful�lled.6 Supersymmetri Ground States with Mat-terThe matter extension of MFS derived in the previous se tion allows thedis ussion of supersymmetri ground states in luding matter �elds. The15

fully supersymmetri states are trivial: The geometri variables obey thesame onstraints as derived already in se tion 2, the matter �elds must obeyf = onst. and � = 0.More involved are the states with one supersymmetry: Eq. (5.26) leadsto �+"+ = ���"� (6.1)whi h, in analogy to (3.1), implies �2" � 0. Furthermore, as (B.11), (B.13)and (B.15) did not re eive new matter-�eld dependent ontributions, therelations (3.1), (3.2) and (3.3) still hold. Of ourse, this still implies (3.4),but the geometri part of the Casimir fun tion is no longer onserved (seedis ussion below).As a onsequen e of (6.1) and (3.1)-(3.3) the vanishing of Æ�� in (5.27)redu es to Æ�� = ( m")��mf = 0 : (6.2)It is straightforward to he k that all matter-�eld dependent modi� ationsin (5.23)-(5.25) vanish due to (6.1) and (6.2). Thus the matter ouplings donot hange the lassi� ation of the solutions as given in se tion 3 as well asthe results of se tion 4.In order to understand the ondition (6.2) on the matter �eld on�gura-tions it is advantagous to reformulate it asf++"+ = 0 ; f��"� = 0 ; (6.3)where f�� = �(e�� ^df) are (anti-)selfdual �eld on�gurations of the s alar�eld. Thus the hiral solution of (3.1) and (6.1) with �� = �� = "+ = 0admits selfdual s alar �elds while the anti- hiral one allows anti-selfdual f .The third solution with "+ 6= 0 and "� 6= 0 is ompatible with stati f , only.Even in the presen e of matter a onserved quantity an be onstru ted.Its physi al relevan e is displayed in the lose relationship to energy de�ni-tions well-known from General Relativity, su h as ADM-, Bondi- and quasi-lo al mass (for details we refer to se t. 5 of [14℄ and referen es therein).The onservation law dC = 0 in the presen e of matter �elds is modi�edby analogy to the pure bosoni ase ( f. [17℄) a ording toe�Q dC +X��W++ +X++W�� + 18(u0 + 12uZ)(��W� � �+W+) = 0(6.4)W�� = ÆÆe��S(m) W� = ÆÆ �S(m) (6.5)16

In the presen e of matter (�W��) appear on the r.h.s. of the e.o.m-s (C.4),(�W�) on the r.h.s. of (C.5). Eq. (6.4) results from a suitable linear om-bination of the e.o.m-s (C.3)-(C.5), C now is only part of a total onservedquantity, whi h also ontains a matter ontribution e�Q dC(m) from the Wterms.A straightforward al ulation from eq. (5.4) yieldsW�� = �e���P�f++f�� + i � (e++ �)f�� + i � (e�� �)f+++12 � ( e��)� � ( e++)�(��)� + u8P 0�2 + 132�P 00 � 12 P 02P ��2�2�+ dfhP �f�� + i � (e�� �)�� i2p2P 0����i+ Phi( �)f�� � 12 � ( e��)(��) � 1p2�� d��i ; (6.6)W� = P��i��(e++f�� + e��f++ � df)� 12 �(��)� 12�e++ � ( �e��) + e�� � ( �e++)�(��)� : (6.7)Now also the question may be posed about the meaning of the restri tion(3.4) within that generalized onservation law. The body of dC in (6.4)vanishes trivially due to that equation. The restri tion to (anti-)selfdual orstati f as derived from (6.3) ensures that the body of (6.4) vanishes withoutimposing further onstraints on the �elds. Thus the result of se t. 4 for thematterless ase ontinuous to hold if non-minimally oupled onformal matteris in luded.7 Con lusionsIn our present work we present the omplete lassi� ation of all BPS BHsin 2D dilaton SUGRA oupled to onformal matter. The use of a �rst orderformulation as suggested from the graded Poisson-Sigma model approa h forthe geometri part of the a tion plays a ru ial role in the al ulations. As nomatter extension thereof had been onsidered in the literature, its derivationis an important result on its own. For future referen e we ompile the MFSa tion non-minimally oupled to onformal matter (with oupling fun tion17

P (�)) at this pla e:S = ZM"�d!+XaDea +��D � + ��V + Y Z � 12�2�V Z + V 02u + 2V 2u3 ��+ Z4 Xa(� a beb � ) + iVu (� aea ) + iXa( a )� 12�u+ Z8 �2�( � )+ P�12 df ^ �df + i2� aea ^ �d� + i � (ea ^ �df)eb ^ � a b�+ 14 � (eb ^ � ) a bea ^ � �2�+ u8P 0�2�� 14P 0(� � a�)ea ^ �df � 132�P 00 � 12 �P 0�2P ��2�2�# (7.1)This a tion is invariant under the supersymmetry transformations (B.11)-(B.16) supplemented by (5.23)-(5.27) and (5.26),(5.27).Starting from this a tion it has been shown that all BPS like states havevanishing body of the Casimir fun tion and thus are ground states. Solutionswith vanishing fermions allow a non-trivial bosoni geometry, but all Killinghorizons were found to be extremal. On the other hand, the geometry ofsolutions with non-vanishing fermions must be Minkowski spa e and onse-quently there exist no supersymmetri BHs with dilatino or gravitino hair.The impossibility of supersymmetri dS ground states has been reprodu edfor our lass of models and the absolute onservation law {the modi� ationof the Casimir fun tion in presen e of matter �elds{ has been al ulatedexpli itely.A knowledgementThis work has been supported by proje ts P-14650-TPH and P-16030-N08of the Austrian S ien e Foundation (FWF). We thank P. van Nieuwenhuizenand Th. Mohaupt for helpful dis ussions on dS va ua and BPS like bla kholes, resp. We are grateful to D. Vassilevi h and V. Frolov for asking veryrelevant questions. This work has been ompleted in the hospital atmosphereof the International Erwin S hr�odinger Institute.A Notations and ConventionsThese onventions are identi al to [10,18℄, where additional explanations anbe found. 18

Indi es hosen from the Latin alphabet are ommuting (lower ase) orgeneri (upper ase), Greek indi es are anti- ommuting. Holonomi oor-dinates are labeled by M , N , O et ., anholonomi ones by A, B, C et .,whereas I, J , K et . are general indi es of the gPSM. The index � is used toindi ate the dilaton omponent of the gPSM �elds:X� = � A� = ! (A.1)The summation onvention is always NW ! SE, e.g. for a fermion �:�2 = ����. Our onventions are arranged in su h a way that almost everybosoni expression is transformed trivially to the graded ase when usingthis summation onvention and repla ing ommuting indi es by general ones.This is possible together with exterior derivatives a ting from the right, only.Thus the graded Leibniz rule is given byd (AB) = AdB + (�1)B (dA)B : (A.2)In terms of anholonomi indi es the metri and the symple ti 2�2 tensorare de�ned as�ab = � 1 00 �1 � ; �ab = ��ab = � 0 1�1 0 � ; ��� = ��� = � 0 1�1 0 � :(A.3)The metri in terms of holonomi indi es is obtained by gmn = ebneam�ab andfor the determinant the standard expression e = det eam = p�det gmn isused. The volume form reads � = 12�abeb ^ ea; by de�nition �� = 1.The -matri es are used in a hiral representation: 0�� = � 0 11 0 � 1�� = � 0 1�1 0 � ��� = ( 1 0)�� = � 1 00 �1 �(A.4)Covariant derivatives of anholonomi indi es with respe t to the geometri variables ea = dxmeam and � = dxm �m in lude the two-dimensional spin- onne tion one form !ab = !�ab. When a ting on lower indi es the expli itexpressions read (12 � is the generator of Lorentz transformations in spinorspa e): (De)a = dea + !�abeb (D )� = d � � 12! ��� � (A.5)Light- one omponents are very onvenient. As we work with spinors ina hiral representation we an use�� = (�+; ��) ; �� = ��+��� : (A.6)19

For Majorana spinors upper and lower hiral omponents are related by �+ =��, �� = ��+, �2 = ���� = 2���+. Ve tors in light- one oordinates aregiven by v++ = ip2(v0 + v1) ; v�� = �ip2(v0 � v1) : (A.7)The additional fa tor i in (A.7) permits a dire t identi� ation of the light- one omponents with the omponents of the spin-tensor v�� = ip2v �� .This implies that �++j�� = 1 and ���j++ = ��++j�� = 1.B Minimal Field SUGRA (MFS)The a tion of a generi graded Poisson sigma model (gPSM) [10,19{24℄ withtarget spa e variables XI , gauge �elds AI and Poisson tensor P IJSgPSM = ZM dXI ^AI + 12P IJAJ ^AI (B.1)is invariant under the symmetry transformationsÆXI = P IJ"J ; ÆAI = �d"I � ��IP JK� "K AJ : (B.2)Not every gPSM may be used to des ribe 2D SUGRA. A sub lass of ap-propriate models ( alled minimal �eld SUGRA, MFS) has been identi�edin [8, 9℄ for N = (1; 1) SUGRA. It ontains a dilatino �� and a gravitino �, both of whi h are Majorana spinors (XI = (�;Xa; ��), AI = (!; ea; �),Y = X++X��): P a� = Xb�ba P ��= �12�� ��� (B.3)P ab = �V + Y Z � 12�2�V Z + V 02u + 2V 2u3 ���ab (B.4)P �b = Z4 Xa(� a b �)� + iVu (� b)� (B.5)P �� = �2iX �� + �u+ Z8 �2� ��� (B.6)V , Z and the prepotential u are fun tions of the dilaton �eld � and obey therelation V = �18 �(u2)0 + u2Z� : (B.7)20

This Poisson tensor leads to the a tionSMFS = ZM��d!+XaDea+��D �+��V +Y Z� 12�2�V Z + V 02u +2V 2u3 ��+ Z4 Xa(� a beb � ) + iVu (� aea )+ iXa( a )� 12�u+ Z8 �2�( � )� : (B.8)An important lass of simpli�ed models is des ribed by the spe ial hoi e�Z = 0. Following the nomen lature of [9℄ it is alled MFS0 and barredvariables are used. As an be veri�ed easily the a tion (B.8) and SMFS0 arerelated by a onformal transformation of the �elds (Q0(�) = Z(�))� = �� ; Xa = e� 12Q(�) �Xa ; �� = e� 14Q(�)��� ; (B.9)! = �! + Z2 � �Xb�eb + 12 ��� � �� ; ea = e 12Q(�)�ea ; � = e 14Q(�) � � : (B.10)The symmetry transformations (B.2) with (B.3)-(B.6) be ome:Æ� = 12(� �") (B.11)ÆXa = �Z4Xb(� b a �")� iVu (� a") (B.12)Æ�� = 2iX (" )� � �u+ Z8 �2�(" �)� (B.13)Æ! = Z 04 Xb(� b a �")ea + i�Vu �0(� a")ea + �u0 + Z 08 �2�(" � ) (B.14)Æea = Z4 (� a b �")eb � 2i(" a ) (B.15)Æ � = �(D")� + Z4Xa( a b �")�eb + iVu ( b")�eb + Z4 ��(" � ) (B.16)By eliminating Xa and the torsion dependent part of the spin onne tion anew a tion (MFDS) in terms of dilaton, dilatino, zweibein and gravitino is21

obtained11SMFDS = Z d2x e�12 ~R�+ (�~�) + V � 14u�2�V Z + V 0 + 4V 2u2 �� 12Z��m��m�+ 12(� � m)�m�+ 12�mn�n�(� m)�� iVu �mn(� n m) + u2�mn( n � m)� : (B.17)Its supersymmetry transformations readÆ� = 12(� �") ; (B.18)Æ�� = �2i�mn��n�+ 12(� � n)�(" m)� � �u+ Z8 �2�(" �)� ; (B.19)Æema = Z4 (� a b �")emb � 2i(" a m) ; (B.20)Æ m� = �( ~D")� + iVu ( m")� + Z4 ��n�( m n")� + 12( m n�)( n �")��(B.21)C E.o.m.-s and Conserved QuantityThe equations of motion for a generi gPSM aredXI + P IJAJ = 0 ; (C.1)dAI + 12(�IP JK)AKAJ = 0 : (C.2)Consequently, the ones for the MFS (B.8) a tion be ome:d��Xb�baea + 12(� � ) = 0 (C.3)DXa + �abeb�V + Y Z � 12�2�V Z + V 02u + 2V 2u3 ���Z4Xb(� b a � )� iVu (� a ) = 0 (C.4)11Quantities with a tilde refer to the dependent spin onne tion ~!a = �mn�nema �i�mn( n a m), i.e. ~�� = �( ~D )� and ~R = 2 � d~!.22

D�� + Z4Xa(� a b �)�eb + iVu (� a)�ea+2iXa( a)� � �u+ Z8 �2�( �)� = 0 (C.5)d! + ��V 0 + Y Z 0 � 12�2��V Z + V 02u �0 + �2V 2u3 �0��+Z 04 Xb(� b aea � ) + i�Vu �0(� aea )� 12�u0 + Z 08 �2�( � ) = 0 (C.6)Dea + �ab�XbZ + Z4 (� a beb � ) + i( a ) = 0 (C.7)D � � ����V Z + V 02u + 2V 2u3 �+ Z4 Xa( a beb � )�+iVu ( aea )� � Z8 ��( � ) = 0 (C.8)We re-emphasize that V;Z and the prepotential u are related by (B.7).The full analyti solution of MFS has been given in se t. 6 of [9℄. Ea hsolution is hara terized by a ertain value of the Casimir fun tion, a quantity onserved in spa e and time. It onsists of a bosoni part (body) and afermioni one (soul):C = CB + CS (C.9)CB = eQ(�)Y + w(�) = eQ(�)�Y � 18u2(�)� (C.10)CS = 116eQ�2(u0 + 12uZ) (C.11)In this equation the (logarithm of the) integrating fa tor and the onformallyinvariant ombination of the bosoni potentialQ(�) := Z � Z(�0)d�0 ; w(�) := Z � eQ(�0)V (�0)d�0 = �18eQ(�)u2(�) � 0(C.12)have been introdu ed. In [9℄ the solutions for C 6= 0 (eqs. (6.9)-(6.13)) andC = 0 (eqs. (6.17)-(6.21)) have been given whi h are not reprodu ed here.23

D Dilaton SUGRA in Superspa eThe a tion for a general dilaton SUGRA in superspa e [6℄ may be writtenas12 SSFDS = Z d2xd2� E��S � 14Z(�)D��D�� + 12u(�)� ; (D.1)with13 [25℄E = e�1 + i� a a + 12�2(A+ �ab b � a) ; (D.2)S = A+ 2� �~� � iA� a a+ 12�2��mn�n~!m �A(A+ �ab b � a)� 2i a a �~�� ; (D.3)D� = �� + i( a�)��a : (D.4)Quantities with a tilde are de�ned in analogy to footnote 11. � is the dilatonsuper�eld with omponent expansion� = �+ 12� ��+ 12�2F : (D.5)The supersymmetry transformations of the omponent �elds of the superde-terminant are given byÆema = �2i(" a m) ; Æema = 2i(" m a) ; (D.6)Æ m� = ��( ~D")� + i2A(" m)�� ; (D.7)ÆA = �2�(" �~�)� i2Aema(" a m)� ; (D.8)while the ones of the dilaton super�eld read:Æ� = �12" �� (D.9)Æ�� = �2( �")�F + i( � b")�( b ��)� 2i( � m")��m� (D.10)ÆF = i(" a a)F � i2�" m �( ~Dm�)�+ ("�m)�( m ��)� 2�m�� (D.11)12In ref. [6℄ the �rst term in the bra kets was hosen as EJ(�)S. If a global re-parametrization J(�) ! � is not possible, these models are not equivalent globally toMFS [9℄.13Ex ept for the zweibein, omponents of super�elds are denoted by underlined variablesto distinguish them from the �elds in the gPSM approa h.24

Integrating out superspa e and eliminating the auxiliary �elds A and F usingtheir equations of motionF = �u2 ; A = �12(u0 + uZ) + 18Z 0�2 : (D.12)one arrives at the a tionSSFDS = Z d2x e�12 ~R�+ (�~�)� 12Z��m��m�� i4� m�m�� ( n m n ��)�m��� 18�(u2)0 + u2Z�+ u2 �mn( n � m)+ i2u0(� ��) + 18�u00 + 14uZ 0 + 12Z( n m n m)�(��)� ;(D.13)while the symmetry transformations of the remaining �elds take the formÆema = �2i(" a m) ; Æema = 2i(" m a) ; (D.14)Æ m� = �( ~D")� + i4�u0 + uZ � 18Z 0(��)�(" m)� ; (D.15)Æ� = �12" �� ; (D.16)Æ�� = u( �")� + i( � b")�( b ��)� 2i( � m")��m� : (D.17)In ref. [9℄ it has been shown that this a tion is equivalent to the a tion (B.17)of MFDS if the identi� ations �m = �m + i8Z(�)eam�ab(� b)� ; � = � ; � = � ; (D.18)are made. The supersymmetry transformations are equivalent up to a lo alLorentz transformation with �eld dependent parameter:" = " � = ÆMFDS � ÆSFDS = Z2 �" Æ� (D.19)Referen es[1℄ E. B. Bogomolny, \Stability of lassi al solutions," Sov. J. Nu l. Phys.24 (1976) 449. 25

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