- Email: [email protected]
Supersymmetric black holes
Physics Letters B 282 (1992) 80-88 North-Holland
PHYSICS LETTERS B
Supersymmetric black holes R e n a t a K a l l o s h 1,2
Physics Department, Stanford University, Stanford CA 94305, USA Received 16 January 1992
The effective action of d = 4, N = 2 supergravity is shown to acquire no quantum corrections in background metrics admitting supercovariantly constant spinors. In particular, these metrics include the Robinson-Bertotti metric (product of two two-dimensional spaces of constant curvature) with all eight supersymmetries unbroken. Another example is a set of arbitrary number of extreme Reissner-Nordstr6m black holes. These black holes break four of eight supersymmetries, leaving the other four unbroken. We have found manifestly supersymmetric black holes, which are non-trivial solutions of the flatness condition 792 = 0 of the corresponding (shortened) superspace. Their bosonic part describes a set of extreme Reissner-Nordstr6m black holes. The super black hole solutions are exact even when all quantum supergravity corrections are taken into account.
1. Introduction Despite all successes in q u a n t u m gravity in dimensions d = 2, 3, development of this theory in d t> 4 is still a problem. One of the m a i n difficulties is the uncontrollable accumulation of divergent q u a n t u m corrections in each new order of perturbation theory. The purpose of this paper is to find some results in d = 4 q u a n t u m gravity, which remain valid with an account taken of all orders of perturbation theory. With this purpose we investigate some effective quantum actions of four-dimensional supergravity theories in very specific backgrounds, which admit supercovariantly constant spinors [ 1,2 ]. The main result of our investigation is rather surprising: The effective action of d = 4, N = 2 supergravity has no q u a n t u m corrections in the background of arbitrary n u m b e r of extreme Reissner-Nordstr6m (RN) black holes in neutral equilibrium. In some sense, to be defined later, the manifestly supersymmetric version of the extreme RN black holes provides an alternative to the trivial fiat superspace. We will start with a discussion of some earlier results on the vanishing of q u a n t u m corrections to the 1 On leave of absence from Lebedev Physical Institute, 117 924 Moscow, USSR. 2 Bitnet address: KALLOSH@SLACVM. 80
effective actions in supergravity theories. In N = l, 2, 3 supergravities the locally supersymmetric onshell effective action is given in terms of the following chiral superfields:
WAsc(X, Oi), WAB(x, Oi), W~(x, Oi),
WA,wc,(X, Oi), WA,w(X, Oi), WA,(X, Oi),
i = 1,
i= i=
1,2,
1,2,3.
(1)
Consider for example q u a n t u m corrections to N = 1 supergravity [ 3,4]. The superfields are
WAsc(x,O) = ~Asc(x) + CaBcDOD + .... _
-
- D
WA,s,c,(x,O) = ~A,B,c,(X) + CA,S,C,D,O + ....
t
(2)
where ~u, ~ are the gravitino field strength spinors and C, C' are the Weyl spinors of the space-time. Each locally supersymmetric term in the effective action depends both on W and W and their covariant derivatives #t. For example, the troublesome threeloop counterterm is [5 ]
#1 The only exceptions are the one-loop topological divergences proportional to W 2 or W2, related to the one-loop anomalies. Elsevier Science Publishers B.V.
Volume 282, number 1,2
PHYSICS LETTERS B
/" Sth . . . . loop = J
2. Absence of quantum corrections in Robinson-Bertotti background
d 4 x d40 det E
× WABcwABCwA,B,¢, W A'B'c' .
(3)
It is a supersymmetrized square of the Bell-Robinson tensor. This term, as well as any other term o f the effective quantum action, vanishes in the super-self, dual background [ 3 ]
WABC = O,
WA,B,C,~ O,
(4)
or in the super-anti-self-dual background
WASC ~ O,
WA'B'C' = 0.
(5)
Such non-trivial backgrounds exist only in space-time with euclidean signature. Indeed, in Minkowski space WABC ~
W real + i W im ,
IZVA,BtC , : ~ W real - i W im .
(6)
Therefore W and W cannot vanish separately, only together, in which case the background is trivial. With euclidean signature it is possible to have a vanishing right-handed spinor WABC(X, 0) and a non-vanishing left-handed spinor WA,B,C, (X, O) (or opposite). This half-fiat superspace, where the left-handed gravitino ~'A'8'C' (X) lives in the space with only left-handed curvature CA'B'C'D' (X), is the background where N = 1 supergravity effective action has no quantum corrections (up to the above-mentioned topological terms). In N = 2 supergravity we have
(7)
where FAB is the Maxwell field strength spinor. In the euclidean half-flat superspace, where
WAB = O,
WA,B, ~ O,
The special role of the Robinson-Bertotti metric in the context of the solitons in supergravity was explained in lectures by Gibbons [ 1 ]. His proposal was to consider the Robinson-Bertotti (RB) metric as an alternative, maximally supersymmetric, vacuum state. The extreme Reissner-Nordstr6m metric spatially interpolates between this vacuum and the trivial fiat one, as one expects from a soliton. In what follows we are going to prove a non-renormalization theorem for the effective action of d = 4, N = 2 supergravity in the RB background. The RB metric is known to be one particular example of a class of metrics, admitting supercovariantly constant spinors [ 1,2 ], which are called Israel-Wilson -Perjes (IWP) metrics [6]. It is also the special metric in this class, which does not break any of the eight supersymmetries of d = 4, N = 2 supergravity; all other IWP metrics break at least half of the supersymmetries. In general relativity the RB metric is known as the conformally fiat solution of the Einstein-Maxwell system with anisotropic electromagnetic field [6]. It describes the product of two two-dimensional spaces of constant curvature: ds 2 =
2 d( d (
2 du dv
( l "q- 0~(() 2
( l + O~U/)) 2 '
= const.
(9)
The metric can also be written in the form
WAB(X,O) = FAn(X) + ~]ec(X) Oic +CABcD oiCo~e ij + ....
21 May 1992
(8)
there are no quantum corrections to the effective action (up to topological terms). For N = 3 the half-flat superspace is given by Wa = O, WA, ~ O. To summarize, some examples of non-trivial background field configurations in supergravity, which receive no radiative corrections, have been known for more than 10 years [3,4]. They all require euclidean signature of space-time.
ds 2 = (1 + (1
+
-
2 y 2 ) d x 2 + (1
-
2y2)-ldy2
(1
+
222)d/2
2z2)-ld22
-
.
(10)
The corresponding Maxwell field Fab is constant (as well as the curvature tensor) and can be written as F~ B = v/-~sinfl, 2 = const.,
F~ B = v/~cosfl,
fl = const.
(11)
The property of the RB metric which is of crucial importance for our analysis is the conformal flatness of this metric, i.e., the vanishing of the Weyl tensor: RB
RB
RB
Cabcd = 0 ::~ C~BCD = C~I BI Ct DI -~- 0 .
(12) 81
Volume 282, number 1,2
PHYSICS LETTERS B
The curvature spinor, corresponding to the traceless Ricci tensor, satisfes Einstein's equation RB
RA'B'AB
~
~ R B ~,RB
(13)
"AB'A'B'.
The generic term in the effective q u a n t u m action o f N = 2 supergravity is given by
F ~ fd4xd4NO
detE
×A(WAs, WA,B,,Dcc, W, Dcc, W . . . . ),
- RB WA,B,
=
Fj,s,RB
Thus, we have proved that there are no quantum corrections to the effective action o f N = 2 supergravity in the R o b i n s o n - B e r t o t t i background. The basic difference with trivially flat superspace is the fact that the superfield, in terms o f which the on-shell quant u m corrections are expressed, is not zero, but is a constant superfield. However, all supersymmetric invariants vanish as in the case of a flat superspace with vanishing superficial.
(14)
where A can either be a local or non-local function in x o f the superfields IV, W and their covariant derivatives, the superfield W being given in eq. (7). F o r trivial flat superspace W = W = 0, since there are no Maxwell, gravitino or Weyl curvatures in the flat superspace. Therefore the path integral o f d = 4, N = 2 supergravity has no q u a n t u m corrections in the flat superspace. Consider now the superficial W, containing F RB o f the RB solution, given in eq. ( 11 ), and recall that there is no gravitino nor Weyl spinors in this background. The superfield W, W is a constant superfield but it does not vanish as would be the case for the trivial flat superspace. It has only the first c o m p o n e n t in the expansion in 0, i.e., it does not d e p e n d e n d on 0 at all: wffB = FRff,
3. Absence of quantum corrections in the extreme black hole background The p r o o f of the non-renormalization theorem for the RB background was almost trivial due to conformal flatness o f this metric and because the Maxwell f e l d is constant. These properties are not present for general metrics admitting supercovariantly constant spinors. In general relativity they are known as the conformal-stationary class of Einstein-Maxwell fields with conformally flat three-dimensional space. This class of metrics has been found by Neugebauer, Perjes, Israel and Wilson [6]: d s 2 ~-
(VV)(dt
d 4N 0 det E A ( WffBB , WA,8,) - RB •
(16)
If A is a local function o f W, W, this expression takes the form
.(
-..
= AfdV
=
f
2,
V-IVV),
V ~ O,
(18)
where V : is the fiat space laplacian in x. For real V this metric reduces to the M a j u m d a r - P a p a p e t r o u solutions [6], which, according to Hartle and Hawking [9], are the only regular black hole solutions in this class. They describe an arbitrary n u m b e r n o f extreme R N black holes with gravitational attraction balanced by electrostatic repulsion: s=,
Ms (19)
s=l
(17)
since the invariant volume o f the real superspace vanishes [7] in N = 2 supergravity. F o r non-local functions we end up with the integral over the volume o f the full superspace of certain functions of x. Those integrals are also equal to zero, according to ref. [7]. 82
(VV)-l(dx)
V - l = f/-i = 1 + - - ~ [ X - X s [ "
d4 x d4N O det E
O,
2 -
(15)
V 2 V -x = 0 ,
Wj, B, )
+ Adx)
V × A = -i/VF(V-tVV.
Now we only have to look for the terms in eq. (14) which depend on W, W but not on their covariant derivatives in bosonic or fermionic directions, since these derivatives are zero for the RB solution:
d4x
21 May 1992
It has been found by Gibbons and Hull [ 1], and in the most general form by T o d [2], that these metrics admit supercovariantly constant spinors o f N = 2 supergravity. We will reformulate here the results o f refs. [ 1,2 ] for the special case of pure N = 2 supergravity #2. For footnote see next page.
Volume 282, number 1,2
PHYSICS LETTERS B
We have found that in the treatment of supercovariantly constant spinors of N -- 2 supergravity it is very helpful to use the original Penrose notation [ 8 ]. We introduce a standard spinor basis, or dyad, o A, l A, and we define .3
21 May 1992
We use that 0 A = V - [ K A A ' l A' ,
lA' = - - V - I K A A ' O
A,
where Ka is the Killing vector (25)
KAA' = (l + n ) A A ' . ~ABOAI B = OA lA = V ,
(24)
Eqs. (23) can be rewritten as follows: (20)
~AIB~oAll Bt = OAtl AI = ~/'.
/~AA'OB ~ ~ A A ' O B -- V - I K A A ' F B c
OC = O,
Only when V = F = 1 the d y a d i s a spin frame. However, it is possible to work with a dyad which is not normalized to unity. Associated with any spinor basis o f the manifold is a null tetrad I a, n a, m a, m a defined by
The hatted derivatives have the standard meaning [8 ] o f derivatives in the conformally rescaled metric
l a = o A o A' '
~7 aAtO B = ~ AA,OB -- YBA,OA .
m a = oAl At ,
n a = lAl A' '
(21)
17na = lAo A' ,
and satisfying the following conditions: lala = Ylapla = m a m a
= mama
= 0,
gab = V Z g a b ,
(22)
Ta = V - I V a V ,
a(l -
=0,
n) = d ( / -
n) = 0,
V A A ' I B ' -- FA'B'OA = O.
= 0,
d A = d A - T V - I K = O,
a Y = d T = O,
c l w ab - w a C w c b - d w ab - w a C w c a = 0,
~7AA'OB + FABI A, = O,
(23)
F r o m now on we will limit ourselves to the case o f black holes only (real V), postponing the treatment of more general metrics to a future publication. In particular, basically the same techniques can be used to study eq. (20) with complex V, i.e., when V × A in eq. (18) in the definition o f the I W P metric is non-vanishing. Other interesting examples o f metrics admitting supercovariantly constant spinors are plane-wave s p a c e - t i m e s for which V = 0 in eq. (20). They also will be considered in a separate publication.
(27)
clrh-drh =0,
dK=dK-2VdA We will require that these spinors are supercovariantly constant [ 1,2 ]:
(26)
If the null tetrads are expressed according to eq. (21 ) through supercovariantly constant omicron and iota satisfying eqs. (23), one gets the following equations for differential forms ~4 : clm=dm
l a m a = lal~la = n a m a = n a m a = 0 , lana = - - m a f f t a = V V .
/~AAtlB t ~. ~ A A , IB t -- V - I K A A , F B , c t l C' = 0 .
(28)
where the null tetrad and Maxwell curvature forms are defined as m = dxama, l = d x a la,
#t = d X a t ~ l a , n = dx a na,
K=l+n,
F =dA,
(29)
and we have introduced the Lorentz connection form w ab and the Weyl connection form Y. These equations can be solved as follows: m = 2-1/2(dx + i d y ) , m = 2-UZ(dx - idy),
#2 Tod's parameter Q, related to dust density is equal to zero in our theory since there is no dust in pure supergravity. #3 Our spinorial indices take values 0, 1 and 0', 1'. The Greek letters chosen by Penrose for the basis, o, t (omicron and iota), visually resemble these numbers. The use of the equations eA = 0 A, ~ = tA and many others is particularly simple in this notation.
l-n
= v~dz,
A = v~Vdt,
K = v~V 2dt, T=
~I)AB = V - I ( I A d O B -
V-ldV, OAdlB "}- I 6 A B d V ) .
(30)
~4 The wedge product symbol is omitted for simplicity, when multiplying forms. 83
Volume 282, number 1,2
PHYSICS LETTERS B
This leads to the P a p a p e t r o u - M a j u m d a r metrics ds 2 =
V 2 dt 2 _ V -2 d x 2 '
F = v/2dV A dt,
(31)
where the flat-space laplacian in x, y, z of V - l is zero and F may still be subject to some dual rotation. These coordinates x, y, z are called comoving coordinates and t is defined a s K a V a = x / 2 0 / O t . To calculate the curvature of the manifold we act with V c c , on eqs. (23):
21 May 1992
bosonic background. To analyze the second component we first have to change variables. Instead of working with independent unconstrained eight fermionic coordinates 0/A,0 8'i of the real N = 2 superspace, for the black holes we need the following 16 coordinates, satisfying eight constraints: •1 -- r t ~AIJ 05i ~ UAi ~- 15AA CjiU , 0 Ati+ ~ 0 A'i 27 E A t A e i J O A j , Offi = -{-E AA,e jiO A' j ~ .
R=0,
Here E is a normalized Killing vector,
RA~BtAB = FABFAIB ~ ,
EA A' = V - I K A W '
CABCD = V A B ' F c o V - I K B ' B ,
(-ij• kj = t~ik .
(32)
CA,e,c,o, = VA,BFc, o, V - I K B B ' •
Now we have enough information to investigate the effective q u a n t u m action in the black hole background ,5 with the properties: FAB(X) = V-2KAAtVAtB V,
= O,
C+BCD -- C a e c o + V A B , F c o V - I K S ' 8
~ O,
(33)
and the conjugate ones can be easily derived from eqs. (33). The basic on-shell superfield W in the real basis is given by i
W A B ( x , O , O ) = FAB(X) + gJj~c(x)O c + CaacoOicO~e ij + V c o , FaBoCo °'i
+ ....
(34)
The first component of this superfield is neither zero, as in flat superspace, nor a constant, as in the RB case. The second component is zero, we have just a ,5 The equations presented above for black holes can be derived also for the RB metric. There will be a second set of covariantly constant spinors, defined by eqs. (23) with opposite sign in front of F and F for the second set of omicron and iota. The second Killing vector will be built from the second set of these spinors. Both C + and C - are equal to zero for RB as a consequence of all those equations. 84
E A w E WB = dA B , (36)
In terms of these coordinates, whose supersymmetry variation is 4- ~ dO ~Ai = eAi -~- EAA, ~Ji ~A' j ~Ai Ati4- ~ dO Ati4- = e A'i -}- E A ' A e i J e A j ,
(37)
the supersymmetry breaking and the shortening of the unbroken superspace related to extreme RN black holes can be understood. The supersymmetric transformation of the gravitino field strength is
7"Asc ( X ) = O, CABCD =-- CABCD -- V A B ' F c D V - I K B ' B
(35)
d~JABCi = C+BCDef + + CABCDeD - .
(38)
It can be made zero under two conditions. The first is CffBCD -- CaBCD -- V A s ' F c D V - 1 K B ' B = 0,
(39)
which is satisfied for the black holes according to eq. (33). This condition is the property of extreme black holes that some combination of curvature and Maxwell fields vanish. It is an integrability condition for the existence of supercovariant spinors eA- (23). The second condition is e{ + = d0,A+ =
O,
(40)
and requires the breaking of four supersymmetries. It also indicates that after the change of coordinates, given by eq. (35), there are four independent combinations of fermionic coordinates. They are given by eight coordinates DAv i , v~At i - , constrained by four conditions 0 A- = --EAA,ejiO "¥j- .
(41)
Volume 282, number 1,2
PHYSICS LETTERS B
These combinations are still unbroken coordinates of the superspace, since e/A- in eq. (38) can take arbitrary values and the variation ofgravitino nevertheless vanishes. At this point it is appropriate to explain the difference between the Robinson-Bertotti solution and black holes from the point of view of supersymmetry. Both metrics belong to the general class of IsraelWilson-Perjes metrics, admitting supercovariantly constant spinors. For RB, both combinations, C + and C - , which define the supersymmetry transformation of the gravitino field strength, vanish, since the gravitational Weyl tensor and the derivative of a Maxwell tensor vanish separately. Therefore there are no restrictions on supersymmetry variations of all eight coordinates of the superspace, i.e., both e/A+ and e A- are arbitrary and nevertheless the supersymmetry variation of gravitino field strength is zero in the RB background. We have to analyze the structure of quantum corrections before using the properties of the black hole background. However, we will work with variables which are natural for this problem, like Weyl-Maxwell spinors C +, C ±, given in eq. (33) and fermionic coordinates of the superspace, given in eq. (35), considering the vector EAA' as some arbitrary one. When the quantum corrections are calculated in an arbitrary Lorentz-covariant background, there is no dependence on any such vector, of course. We have introduced this dependence through our choice of variables, and it should be absent in terms of the original variables after integration over fermionic variables. In the black hole background all terms which depends on C - or C - will vanish. The crucial question is: Are there terms which depend only on C + , C + and do not depend either on C - or on C - ? The answer is no, they do not exist. The point is that under the eft- transformations the non-vanishing combinations of curvature and Maxwell fields C +, C + do transform. However, the e/A- variation coming from any term containing fermions will have the combinations C-ei c-, according to eq. (38). Any term with C + or C + dependence but without C - or C - dependence will not satisfy the eA--supersymmetry requirements. To illustrate this general statement consider again the three-loop counterterm. The following combinations can be expected:
(C+)2 ( C + ) 2 ,
21 May 1992 (C-)2 (C-) 2,
(C+C-)(C+C-),
etc.
(42)
Only the first combination does not vanish in the black hole background. Let us show that it will not appear in the effective quantum action. The straightforward calculation is to check the dependence of each of these terms on the vector EAA' by substituting expressions for C + from eq. (33). The term (C + )2(C+ )2, which is forbidden by the above mentioned supersymmetry arguments, does depend on EAA', as opposed to the third term in (42), which is allowed by the e/A--supersymmetry and can be shown to be EAA,independent:
(C+C-)(C+C -) = [(CABcD -t- VAA'FcDEA'B)
× (C ABcD_ ~TA,FCDEB'B)] (C+C - ) = [(CABcD) 2 - (VAA,FcD) 2]
× [ (Cn,B,C,D,)2 _ (VAA, Fc,D, )2 ].
(43)
Thus, all terms in the on-shell effective quantum action of d = 4, N = 2 supergravity, which are locally supersymmetric and Lorentz invariant, vanish in the extreme multi black hole background.
4. B l a c k holes as a flat superspace
Our approach to the black hole superspace was inspired by the group manifold approach to N = 2 supergravity [ 10 ]. The superspace [ 10 ] is formulated in terms of the superspace one-forms E a, ~u~, ~UA,i, A associated to the supergravity physical fields and the spin connection w ab. We are interested only in onshell curvatures associated with d = 4, N = 2 superPoincar6 algebra with central charge. They are defined in terms of the following differential operator:
19 = d + AMTM,
(44)
where TM are the generators of the super-Poincar6 group [TM, TN} = fMLNTL,
(45)
and A M a r e connection forms E a, 7t~, ~A'i, Aii, W ab related to Pa, QA, QA'i, Mij, Mab generators of the superPoincar6 group (translation, eight supersymmetries, 85
Volume 282, number 1,2
PHYSICS LETTERS B
central charge and Lorentz generator). Thus, the operator 79 is 79 = d + E a e a + 7u+~aA + ~-lA,ia a'i
21 May 1992
where E~' is the normalized Killing vector, defined in eqs. (36), (25), (20). We choose the fermionic forms o f the black hole superspace to be ~_11 = OAV/I -{- lAY~2,
+AijM
~_12 ~-- OAV/2t -- l A y ~ I t ,
(46)
ij + w a b M a b .
~['/A'I = OA'V/It + IAtv/2t ,
The curvature o f this superspace is defined as ~'IA'2 =
792 = R M T M ,
R M = dA m + f~At'A
N.
(47)
The set o f curvatures includes R a = d E a _ W a b E b -- i ~ i y a ~ .
PAi=
-- D E a - i ~ i y a ~ i ,
=
- wab (
) A, =- D
F = d A + (eABeiJ~tAi~Sj
Rab = d~vab - w c wa
dv/it
=
dgt it = 0.
v/i = dO i '
+ h.c.),
(48)
The nilpotency o f the operator 79 is the requirement that all curvatures are equal to zero 792 = R g TM = O, or in detail PAl = O,
(54)
v/it = dO it .
Thus, our space contains four fermionic coordinates in addition to the four bosonic ones. The differential operator 79 for the black hole superspace is defined as follows: 79 = d + K P K + m P m + trnPm + (l - n ) P l - n
+ v/iQi + v/i,Q~ + AijMiJ R a = O,
+ ~13abMa b
p iA, = O,
(55)
+ TW , F =0,
R ab = 0 .
(49)
Eq. (49) has a trivial solution describing a flat d = 4, N = 2 superspace with four bosonic, eight fermionic coordinates and pure fermionic central charge form: E °
Oy a dO,
= dx a - i i
Y'tAi = d O A i ,
(53)
They have a simple solution:
'A, ,
cb .
Omicron and iota in eqs. (51) satisfy eqs. (23). O u r four independent Lorentz-invariant fermionic forms v/i, v/it satisfy the following equations: clv/i = dv/i = 0,
d~rlAi -- ~l)ab (¢Tab ~ii ) A -~ D~'tAi ,
(52)
On'V~2 - - I A t v / I .
~.t~, = dOi,,
A = 6.AB £ij OAi dOBj
-at- h.c.,
"Wab = O.
(50)
where PK, Pro, Pm, P t - , are the translation operators in the null tetrad basis. Qi, Q~t are the four supersymmetry generators, M ij = £iJM, Mab are generators of central charge and Lorentz symmetry and by W we have denoted the generator o f conformal transformations. The manifestly supersymmetric generalization o f the bosonic multi black hole can be obtained by solving the flatness condition 792 -- ( d A M + f N m L A N A L ) T M = 0
N o w we will build a non-trivial fiat superspace for the black holes. The variables which are natural for the black hole superspace are not Lorentz-covariant objects, like in eq. (49), but Lorentz-invariant ones, like in eqs. (28), (29), (21). The bosonic forms are m, rh, l - n, K = l + n and A. In addition we introduce eight fermionic forms satisfying four constraints, in accordance with the shortening o f a black hole superspace. These forms are required to be supercovariantly constant: VAA,~-'I~ + F A B £ i J ~ A , j = O, ~.1~ = _EAA,£ijtrI.IjA' ,
86
(51)
of the black hole superspace with the set o f curvatures R m required to vanish #6 : dm =0,
drh = 0 ,
d(/-n)
= 0,
d K - 2 V d A - i2v/2 vzv/2 = 0,
dv/i = 0, dA - TV-lK
dzo a b -
dv/it = 0, + i2x/2 VV/2 = 0,
ltlacZObc = 0,
dT = 0,
(56)
#6 Our notation here corresponds to the one in ref. [8]. The corresponding reference with supersymmetry, matching the notation of ref. [8], is ref. [11].
Volume 282, number 1,2
PHYSICS LETTERS B
vatures o f this superspace vanish, and there are no geometrical building blocks for quantum corrections. In this sense the supersymmetric black holes represent an alternative to the trivial flat superspace (46), (50), which also has no q u a n t u m corrections because all curvatures of the superspace vanish (49).
where ~2 = gli~it. These equations can be solved as follows: m = 2-1/2(dx + i d y ) ,
rh = 2 - 1 / 2 ( d x - i d y ) ,
l - n = v~ dz ,
K = x / - 2 V 2 ( d t - iOidO it + iOitdOi), gli = dOi,
~ll it =
A = v~V(dt"I-UAB =
V -1
dO i* ,
iOidO it + iOitdOi),
(IA dOB -- OA dls + ½£AB d V ) ,
T = V -~ d V ,
21 May 1992
Acknowledgement (57)
the flat-space laplacian in x, y, z o f V-~ being zero. In particular, V can be chosen in the form (19), and in this case eqs. (5 7) define a manifestly supersymmetric multi black hole: a flat superspace with four bosonic and four fermionic coordinates, the flatness condition being defined as the vanishing o f all curvatures (56) in this superspace. This superspace, when considered at 0i = 0 it = d0~ = dO ~t = 0, coincides with the set o f extreme R e i s s n e r - N o r d s t r 6 m black holes in the usual four-dimensional bosonic space. Thus, to find an extreme black hole we were solving not the classical Einstein-Maxwell equations but the flatness condition o f the superspace. The solution of the classical Einstein-Maxwell equations defines the Ricci tensor in terms o f a quadratic combination o f Maxwell tensors, the Riemann-Christoffel curvature tensor is non-vanishing. All on-shell quantum gravity corrections are expressed in terms o f the non-vanishing Riemann-Christoffel curvature tensor in the case o f non-extreme as well as extreme R N black holes. In both cases it is impossible to handle q u a n t u m corrections without additional information. The additional information indeed exists for the extreme R N black holes. It is possible to investigate the effective q u a n t u m action o f d = 4, N = 2 supergravity, taking into account the fundamental fact that the bosonic extreme R N black holes do not break four o f the original eight supersymmetries. The investigation shows that all on-shell q u a n t u m corrections, which are locally supersymmetric and Lorentz invariant, vanish in the extreme R N multi black hole background. Moreover, the four above mentioned supersymmetries can be m a d e manifest. A shortened superspace exists ( 5 5 ) , whose flatness condition (56) is solved by a supersymmetric black hole solution (57). All cur-
Our investigation was greatly stimulated by the recent activity in strings and black hole physics [12]. It is a pleasure to express my gratitude to S. Giddings, G. Gorowitz, M. Green, J. Hartle, A. Linde, T. Ortin, A. Strominger, L. Susskind and L. Thorlacius for interesting and fruitful discussions. This work was supported in part by N S F grant PHY-8612280 and by the John and Claire Radway Fellowship in the School o f Humanities and Sciences at Stanford University.
References [ l ] G.W. Gibbons and C.M. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in: Supersymmetry, supergravity and related topics, eds. F. del Aguila, J. de Azc~rraga and L. Ibfifiez (World Scientific, Singapore, 1985) p. 147. [2] K.P. Tod, Phys. Lett. B 121 (1983) 241. [3] R.E. Kallosh, JETP Lett. 29 (1979) 172; S. Christensen, S. Deser, M. Duff and M. Grisaru, Phys. Lett. B 84 (1979) 411; R.E. Kallosh, JETP Lett. 29 (1979) 449; R.E. Kallosh, Nucl. Phys. B 165 (1980) 119. [4] S. Gates, M. Grisaru, M. Ro~ek and W. Siegel, Superspace (Benjamin/Cummings, London, 1983) p. 451. [5] S. Deser, J.H. Kay and K.S. Stelle, Phys. Rev. Lett. 38 (1977) 527; R.E. Kallosh, Phys. Lett. B 99 (1981) 122; P. Howe and U. Lindstr6m, Nucl. Phys. B 181 (1981) 487. [6] D. Kramer, H. Stephani, M. MacCallum and E. Herlt, Exact solutions of Einstein's field equations (Cambridge U.P., Cambridge, 1980). [7] E. Sokatchev, Phys. Lett. B 100 (1981) 466. [8] R. Penrose and W. Rindler, Spinors and space-time (Cambridge U.P., Cambridge, 1988). 87
Volume 282, number 1,2
PHYSICS LETTERS B
[9] J.B. Hartle and S.W. Hawking, Commun. Math. Phys. 26 (1972) 87. [ 10 ] L. Castellani, R. D'Auria and P. Fr6, Supergravity and superstrings (World Scientific, Singapore, 1991 ); L. Castellani, R. D'Auria and S. Ferrara, Class. Quantum Gray. 7 (1990) 1767. I l l ] L. Brink and P. Howe, Phys. Lett. B 88 (1979) 268.
88
21 May 1992
[12] E. Witten, On black holes in string theory, lecture at Strings '91 (Stony Brook, June 1991); C.G. Callan Jr., J.A. Harvey and A. Strominger, Supersymmetric string solitons, lectures at Trieste 199 l Spring School on String theory and quantum gravity; T. Eguchi, Topological field theories and space-time singularity, preprint EFI 91-58 (1991).
PHYSICS LETTERS B
Supersymmetric black holes R e n a t a K a l l o s h 1,2
Physics Department, Stanford University, Stanford CA 94305, USA Received 16 January 1992
The effective action of d = 4, N = 2 supergravity is shown to acquire no quantum corrections in background metrics admitting supercovariantly constant spinors. In particular, these metrics include the Robinson-Bertotti metric (product of two two-dimensional spaces of constant curvature) with all eight supersymmetries unbroken. Another example is a set of arbitrary number of extreme Reissner-Nordstr6m black holes. These black holes break four of eight supersymmetries, leaving the other four unbroken. We have found manifestly supersymmetric black holes, which are non-trivial solutions of the flatness condition 792 = 0 of the corresponding (shortened) superspace. Their bosonic part describes a set of extreme Reissner-Nordstr6m black holes. The super black hole solutions are exact even when all quantum supergravity corrections are taken into account.
1. Introduction Despite all successes in q u a n t u m gravity in dimensions d = 2, 3, development of this theory in d t> 4 is still a problem. One of the m a i n difficulties is the uncontrollable accumulation of divergent q u a n t u m corrections in each new order of perturbation theory. The purpose of this paper is to find some results in d = 4 q u a n t u m gravity, which remain valid with an account taken of all orders of perturbation theory. With this purpose we investigate some effective quantum actions of four-dimensional supergravity theories in very specific backgrounds, which admit supercovariantly constant spinors [ 1,2 ]. The main result of our investigation is rather surprising: The effective action of d = 4, N = 2 supergravity has no q u a n t u m corrections in the background of arbitrary n u m b e r of extreme Reissner-Nordstr6m (RN) black holes in neutral equilibrium. In some sense, to be defined later, the manifestly supersymmetric version of the extreme RN black holes provides an alternative to the trivial fiat superspace. We will start with a discussion of some earlier results on the vanishing of q u a n t u m corrections to the 1 On leave of absence from Lebedev Physical Institute, 117 924 Moscow, USSR. 2 Bitnet address: KALLOSH@SLACVM. 80
effective actions in supergravity theories. In N = l, 2, 3 supergravities the locally supersymmetric onshell effective action is given in terms of the following chiral superfields:
WAsc(X, Oi), WAB(x, Oi), W~(x, Oi),
WA,wc,(X, Oi), WA,w(X, Oi), WA,(X, Oi),
i = 1,
i= i=
1,2,
1,2,3.
(1)
Consider for example q u a n t u m corrections to N = 1 supergravity [ 3,4]. The superfields are
WAsc(x,O) = ~Asc(x) + CaBcDOD + .... _
-
- D
WA,s,c,(x,O) = ~A,B,c,(X) + CA,S,C,D,O + ....
t
(2)
where ~u, ~ are the gravitino field strength spinors and C, C' are the Weyl spinors of the space-time. Each locally supersymmetric term in the effective action depends both on W and W and their covariant derivatives #t. For example, the troublesome threeloop counterterm is [5 ]
#1 The only exceptions are the one-loop topological divergences proportional to W 2 or W2, related to the one-loop anomalies. Elsevier Science Publishers B.V.
Volume 282, number 1,2
PHYSICS LETTERS B
/" Sth . . . . loop = J
2. Absence of quantum corrections in Robinson-Bertotti background
d 4 x d40 det E
× WABcwABCwA,B,¢, W A'B'c' .
(3)
It is a supersymmetrized square of the Bell-Robinson tensor. This term, as well as any other term o f the effective quantum action, vanishes in the super-self, dual background [ 3 ]
WABC = O,
WA,B,C,~ O,
(4)
or in the super-anti-self-dual background
WASC ~ O,
WA'B'C' = 0.
(5)
Such non-trivial backgrounds exist only in space-time with euclidean signature. Indeed, in Minkowski space WABC ~
W real + i W im ,
IZVA,BtC , : ~ W real - i W im .
(6)
Therefore W and W cannot vanish separately, only together, in which case the background is trivial. With euclidean signature it is possible to have a vanishing right-handed spinor WABC(X, 0) and a non-vanishing left-handed spinor WA,B,C, (X, O) (or opposite). This half-fiat superspace, where the left-handed gravitino ~'A'8'C' (X) lives in the space with only left-handed curvature CA'B'C'D' (X), is the background where N = 1 supergravity effective action has no quantum corrections (up to the above-mentioned topological terms). In N = 2 supergravity we have
(7)
where FAB is the Maxwell field strength spinor. In the euclidean half-flat superspace, where
WAB = O,
WA,B, ~ O,
The special role of the Robinson-Bertotti metric in the context of the solitons in supergravity was explained in lectures by Gibbons [ 1 ]. His proposal was to consider the Robinson-Bertotti (RB) metric as an alternative, maximally supersymmetric, vacuum state. The extreme Reissner-Nordstr6m metric spatially interpolates between this vacuum and the trivial fiat one, as one expects from a soliton. In what follows we are going to prove a non-renormalization theorem for the effective action of d = 4, N = 2 supergravity in the RB background. The RB metric is known to be one particular example of a class of metrics, admitting supercovariantly constant spinors [ 1,2 ], which are called Israel-Wilson -Perjes (IWP) metrics [6]. It is also the special metric in this class, which does not break any of the eight supersymmetries of d = 4, N = 2 supergravity; all other IWP metrics break at least half of the supersymmetries. In general relativity the RB metric is known as the conformally fiat solution of the Einstein-Maxwell system with anisotropic electromagnetic field [6]. It describes the product of two two-dimensional spaces of constant curvature: ds 2 =
2 d( d (
2 du dv
( l "q- 0~(() 2
( l + O~U/)) 2 '
= const.
(9)
The metric can also be written in the form
WAB(X,O) = FAn(X) + ~]ec(X) Oic +CABcD oiCo~e ij + ....
21 May 1992
(8)
there are no quantum corrections to the effective action (up to topological terms). For N = 3 the half-flat superspace is given by Wa = O, WA, ~ O. To summarize, some examples of non-trivial background field configurations in supergravity, which receive no radiative corrections, have been known for more than 10 years [3,4]. They all require euclidean signature of space-time.
ds 2 = (1 + (1
+
-
2 y 2 ) d x 2 + (1
-
2y2)-ldy2
(1
+
222)d/2
2z2)-ld22
-
.
(10)
The corresponding Maxwell field Fab is constant (as well as the curvature tensor) and can be written as F~ B = v/-~sinfl, 2 = const.,
F~ B = v/~cosfl,
fl = const.
(11)
The property of the RB metric which is of crucial importance for our analysis is the conformal flatness of this metric, i.e., the vanishing of the Weyl tensor: RB
RB
RB
Cabcd = 0 ::~ C~BCD = C~I BI Ct DI -~- 0 .
(12) 81
Volume 282, number 1,2
PHYSICS LETTERS B
The curvature spinor, corresponding to the traceless Ricci tensor, satisfes Einstein's equation RB
RA'B'AB
~
~ R B ~,RB
(13)
"AB'A'B'.
The generic term in the effective q u a n t u m action o f N = 2 supergravity is given by
F ~ fd4xd4NO
detE
×A(WAs, WA,B,,Dcc, W, Dcc, W . . . . ),
- RB WA,B,
=
Fj,s,RB
Thus, we have proved that there are no quantum corrections to the effective action o f N = 2 supergravity in the R o b i n s o n - B e r t o t t i background. The basic difference with trivially flat superspace is the fact that the superfield, in terms o f which the on-shell quant u m corrections are expressed, is not zero, but is a constant superfield. However, all supersymmetric invariants vanish as in the case of a flat superspace with vanishing superficial.
(14)
where A can either be a local or non-local function in x o f the superfields IV, W and their covariant derivatives, the superfield W being given in eq. (7). F o r trivial flat superspace W = W = 0, since there are no Maxwell, gravitino or Weyl curvatures in the flat superspace. Therefore the path integral o f d = 4, N = 2 supergravity has no q u a n t u m corrections in the flat superspace. Consider now the superficial W, containing F RB o f the RB solution, given in eq. ( 11 ), and recall that there is no gravitino nor Weyl spinors in this background. The superfield W, W is a constant superfield but it does not vanish as would be the case for the trivial flat superspace. It has only the first c o m p o n e n t in the expansion in 0, i.e., it does not d e p e n d e n d on 0 at all: wffB = FRff,
3. Absence of quantum corrections in the extreme black hole background The p r o o f of the non-renormalization theorem for the RB background was almost trivial due to conformal flatness o f this metric and because the Maxwell f e l d is constant. These properties are not present for general metrics admitting supercovariantly constant spinors. In general relativity they are known as the conformal-stationary class of Einstein-Maxwell fields with conformally flat three-dimensional space. This class of metrics has been found by Neugebauer, Perjes, Israel and Wilson [6]: d s 2 ~-
(VV)(dt
d 4N 0 det E A ( WffBB , WA,8,) - RB •
(16)
If A is a local function o f W, W, this expression takes the form
.(
-..
= AfdV
=
f
2,
V-IVV),
V ~ O,
(18)
where V : is the fiat space laplacian in x. For real V this metric reduces to the M a j u m d a r - P a p a p e t r o u solutions [6], which, according to Hartle and Hawking [9], are the only regular black hole solutions in this class. They describe an arbitrary n u m b e r n o f extreme R N black holes with gravitational attraction balanced by electrostatic repulsion: s=,
Ms (19)
s=l
(17)
since the invariant volume o f the real superspace vanishes [7] in N = 2 supergravity. F o r non-local functions we end up with the integral over the volume o f the full superspace of certain functions of x. Those integrals are also equal to zero, according to ref. [7]. 82
(VV)-l(dx)
V - l = f/-i = 1 + - - ~ [ X - X s [ "
d4 x d4N O det E
O,
2 -
(15)
V 2 V -x = 0 ,
Wj, B, )
+ Adx)
V × A = -i/VF(V-tVV.
Now we only have to look for the terms in eq. (14) which depend on W, W but not on their covariant derivatives in bosonic or fermionic directions, since these derivatives are zero for the RB solution:
d4x
21 May 1992
It has been found by Gibbons and Hull [ 1], and in the most general form by T o d [2], that these metrics admit supercovariantly constant spinors o f N = 2 supergravity. We will reformulate here the results o f refs. [ 1,2 ] for the special case of pure N = 2 supergravity #2. For footnote see next page.
Volume 282, number 1,2
PHYSICS LETTERS B
We have found that in the treatment of supercovariantly constant spinors of N -- 2 supergravity it is very helpful to use the original Penrose notation [ 8 ]. We introduce a standard spinor basis, or dyad, o A, l A, and we define .3
21 May 1992
We use that 0 A = V - [ K A A ' l A' ,
lA' = - - V - I K A A ' O
A,
where Ka is the Killing vector (25)
KAA' = (l + n ) A A ' . ~ABOAI B = OA lA = V ,
(24)
Eqs. (23) can be rewritten as follows: (20)
~AIB~oAll Bt = OAtl AI = ~/'.
/~AA'OB ~ ~ A A ' O B -- V - I K A A ' F B c
OC = O,
Only when V = F = 1 the d y a d i s a spin frame. However, it is possible to work with a dyad which is not normalized to unity. Associated with any spinor basis o f the manifold is a null tetrad I a, n a, m a, m a defined by
The hatted derivatives have the standard meaning [8 ] o f derivatives in the conformally rescaled metric
l a = o A o A' '
~7 aAtO B = ~ AA,OB -- YBA,OA .
m a = oAl At ,
n a = lAl A' '
(21)
17na = lAo A' ,
and satisfying the following conditions: lala = Ylapla = m a m a
= mama
= 0,
gab = V Z g a b ,
(22)
Ta = V - I V a V ,
a(l -
=0,
n) = d ( / -
n) = 0,
V A A ' I B ' -- FA'B'OA = O.
= 0,
d A = d A - T V - I K = O,
a Y = d T = O,
c l w ab - w a C w c b - d w ab - w a C w c a = 0,
~7AA'OB + FABI A, = O,
(23)
F r o m now on we will limit ourselves to the case o f black holes only (real V), postponing the treatment of more general metrics to a future publication. In particular, basically the same techniques can be used to study eq. (20) with complex V, i.e., when V × A in eq. (18) in the definition o f the I W P metric is non-vanishing. Other interesting examples o f metrics admitting supercovariantly constant spinors are plane-wave s p a c e - t i m e s for which V = 0 in eq. (20). They also will be considered in a separate publication.
(27)
clrh-drh =0,
dK=dK-2VdA We will require that these spinors are supercovariantly constant [ 1,2 ]:
(26)
If the null tetrads are expressed according to eq. (21 ) through supercovariantly constant omicron and iota satisfying eqs. (23), one gets the following equations for differential forms ~4 : clm=dm
l a m a = lal~la = n a m a = n a m a = 0 , lana = - - m a f f t a = V V .
/~AAtlB t ~. ~ A A , IB t -- V - I K A A , F B , c t l C' = 0 .
(28)
where the null tetrad and Maxwell curvature forms are defined as m = dxama, l = d x a la,
#t = d X a t ~ l a , n = dx a na,
K=l+n,
F =dA,
(29)
and we have introduced the Lorentz connection form w ab and the Weyl connection form Y. These equations can be solved as follows: m = 2-1/2(dx + i d y ) , m = 2-UZ(dx - idy),
#2 Tod's parameter Q, related to dust density is equal to zero in our theory since there is no dust in pure supergravity. #3 Our spinorial indices take values 0, 1 and 0', 1'. The Greek letters chosen by Penrose for the basis, o, t (omicron and iota), visually resemble these numbers. The use of the equations eA = 0 A, ~ = tA and many others is particularly simple in this notation.
l-n
= v~dz,
A = v~Vdt,
K = v~V 2dt, T=
~I)AB = V - I ( I A d O B -
V-ldV, OAdlB "}- I 6 A B d V ) .
(30)
~4 The wedge product symbol is omitted for simplicity, when multiplying forms. 83
Volume 282, number 1,2
PHYSICS LETTERS B
This leads to the P a p a p e t r o u - M a j u m d a r metrics ds 2 =
V 2 dt 2 _ V -2 d x 2 '
F = v/2dV A dt,
(31)
where the flat-space laplacian in x, y, z of V - l is zero and F may still be subject to some dual rotation. These coordinates x, y, z are called comoving coordinates and t is defined a s K a V a = x / 2 0 / O t . To calculate the curvature of the manifold we act with V c c , on eqs. (23):
21 May 1992
bosonic background. To analyze the second component we first have to change variables. Instead of working with independent unconstrained eight fermionic coordinates 0/A,0 8'i of the real N = 2 superspace, for the black holes we need the following 16 coordinates, satisfying eight constraints: •1 -- r t ~AIJ 05i ~ UAi ~- 15AA CjiU , 0 Ati+ ~ 0 A'i 27 E A t A e i J O A j , Offi = -{-E AA,e jiO A' j ~ .
R=0,
Here E is a normalized Killing vector,
RA~BtAB = FABFAIB ~ ,
EA A' = V - I K A W '
CABCD = V A B ' F c o V - I K B ' B ,
(-ij• kj = t~ik .
(32)
CA,e,c,o, = VA,BFc, o, V - I K B B ' •
Now we have enough information to investigate the effective q u a n t u m action in the black hole background ,5 with the properties: FAB(X) = V-2KAAtVAtB V,
= O,
C+BCD -- C a e c o + V A B , F c o V - I K S ' 8
~ O,
(33)
and the conjugate ones can be easily derived from eqs. (33). The basic on-shell superfield W in the real basis is given by i
W A B ( x , O , O ) = FAB(X) + gJj~c(x)O c + CaacoOicO~e ij + V c o , FaBoCo °'i
+ ....
(34)
The first component of this superfield is neither zero, as in flat superspace, nor a constant, as in the RB case. The second component is zero, we have just a ,5 The equations presented above for black holes can be derived also for the RB metric. There will be a second set of covariantly constant spinors, defined by eqs. (23) with opposite sign in front of F and F for the second set of omicron and iota. The second Killing vector will be built from the second set of these spinors. Both C + and C - are equal to zero for RB as a consequence of all those equations. 84
E A w E WB = dA B , (36)
In terms of these coordinates, whose supersymmetry variation is 4- ~ dO ~Ai = eAi -~- EAA, ~Ji ~A' j ~Ai Ati4- ~ dO Ati4- = e A'i -}- E A ' A e i J e A j ,
(37)
the supersymmetry breaking and the shortening of the unbroken superspace related to extreme RN black holes can be understood. The supersymmetric transformation of the gravitino field strength is
7"Asc ( X ) = O, CABCD =-- CABCD -- V A B ' F c D V - I K B ' B
(35)
d~JABCi = C+BCDef + + CABCDeD - .
(38)
It can be made zero under two conditions. The first is CffBCD -- CaBCD -- V A s ' F c D V - 1 K B ' B = 0,
(39)
which is satisfied for the black holes according to eq. (33). This condition is the property of extreme black holes that some combination of curvature and Maxwell fields vanish. It is an integrability condition for the existence of supercovariant spinors eA- (23). The second condition is e{ + = d0,A+ =
O,
(40)
and requires the breaking of four supersymmetries. It also indicates that after the change of coordinates, given by eq. (35), there are four independent combinations of fermionic coordinates. They are given by eight coordinates DAv i , v~At i - , constrained by four conditions 0 A- = --EAA,ejiO "¥j- .
(41)
Volume 282, number 1,2
PHYSICS LETTERS B
These combinations are still unbroken coordinates of the superspace, since e/A- in eq. (38) can take arbitrary values and the variation ofgravitino nevertheless vanishes. At this point it is appropriate to explain the difference between the Robinson-Bertotti solution and black holes from the point of view of supersymmetry. Both metrics belong to the general class of IsraelWilson-Perjes metrics, admitting supercovariantly constant spinors. For RB, both combinations, C + and C - , which define the supersymmetry transformation of the gravitino field strength, vanish, since the gravitational Weyl tensor and the derivative of a Maxwell tensor vanish separately. Therefore there are no restrictions on supersymmetry variations of all eight coordinates of the superspace, i.e., both e/A+ and e A- are arbitrary and nevertheless the supersymmetry variation of gravitino field strength is zero in the RB background. We have to analyze the structure of quantum corrections before using the properties of the black hole background. However, we will work with variables which are natural for this problem, like Weyl-Maxwell spinors C +, C ±, given in eq. (33) and fermionic coordinates of the superspace, given in eq. (35), considering the vector EAA' as some arbitrary one. When the quantum corrections are calculated in an arbitrary Lorentz-covariant background, there is no dependence on any such vector, of course. We have introduced this dependence through our choice of variables, and it should be absent in terms of the original variables after integration over fermionic variables. In the black hole background all terms which depends on C - or C - will vanish. The crucial question is: Are there terms which depend only on C + , C + and do not depend either on C - or on C - ? The answer is no, they do not exist. The point is that under the eft- transformations the non-vanishing combinations of curvature and Maxwell fields C +, C + do transform. However, the e/A- variation coming from any term containing fermions will have the combinations C-ei c-, according to eq. (38). Any term with C + or C + dependence but without C - or C - dependence will not satisfy the eA--supersymmetry requirements. To illustrate this general statement consider again the three-loop counterterm. The following combinations can be expected:
(C+)2 ( C + ) 2 ,
21 May 1992 (C-)2 (C-) 2,
(C+C-)(C+C-),
etc.
(42)
Only the first combination does not vanish in the black hole background. Let us show that it will not appear in the effective quantum action. The straightforward calculation is to check the dependence of each of these terms on the vector EAA' by substituting expressions for C + from eq. (33). The term (C + )2(C+ )2, which is forbidden by the above mentioned supersymmetry arguments, does depend on EAA', as opposed to the third term in (42), which is allowed by the e/A--supersymmetry and can be shown to be EAA,independent:
(C+C-)(C+C -) = [(CABcD -t- VAA'FcDEA'B)
× (C ABcD_ ~TA,FCDEB'B)] (C+C - ) = [(CABcD) 2 - (VAA,FcD) 2]
× [ (Cn,B,C,D,)2 _ (VAA, Fc,D, )2 ].
(43)
Thus, all terms in the on-shell effective quantum action of d = 4, N = 2 supergravity, which are locally supersymmetric and Lorentz invariant, vanish in the extreme multi black hole background.
4. B l a c k holes as a flat superspace
Our approach to the black hole superspace was inspired by the group manifold approach to N = 2 supergravity [ 10 ]. The superspace [ 10 ] is formulated in terms of the superspace one-forms E a, ~u~, ~UA,i, A associated to the supergravity physical fields and the spin connection w ab. We are interested only in onshell curvatures associated with d = 4, N = 2 superPoincar6 algebra with central charge. They are defined in terms of the following differential operator:
19 = d + AMTM,
(44)
where TM are the generators of the super-Poincar6 group [TM, TN} = fMLNTL,
(45)
and A M a r e connection forms E a, 7t~, ~A'i, Aii, W ab related to Pa, QA, QA'i, Mij, Mab generators of the superPoincar6 group (translation, eight supersymmetries, 85
Volume 282, number 1,2
PHYSICS LETTERS B
central charge and Lorentz generator). Thus, the operator 79 is 79 = d + E a e a + 7u+~aA + ~-lA,ia a'i
21 May 1992
where E~' is the normalized Killing vector, defined in eqs. (36), (25), (20). We choose the fermionic forms o f the black hole superspace to be ~_11 = OAV/I -{- lAY~2,
+AijM
~_12 ~-- OAV/2t -- l A y ~ I t ,
(46)
ij + w a b M a b .
~['/A'I = OA'V/It + IAtv/2t ,
The curvature o f this superspace is defined as ~'IA'2 =
792 = R M T M ,
R M = dA m + f~At'A
N.
(47)
The set o f curvatures includes R a = d E a _ W a b E b -- i ~ i y a ~ .
PAi=
-- D E a - i ~ i y a ~ i ,
=
- wab (
) A, =- D
F = d A + (eABeiJ~tAi~Sj
Rab = d~vab - w c wa
dv/it
=
dgt it = 0.
v/i = dO i '
+ h.c.),
(48)
The nilpotency o f the operator 79 is the requirement that all curvatures are equal to zero 792 = R g TM = O, or in detail PAl = O,
(54)
v/it = dO it .
Thus, our space contains four fermionic coordinates in addition to the four bosonic ones. The differential operator 79 for the black hole superspace is defined as follows: 79 = d + K P K + m P m + trnPm + (l - n ) P l - n
+ v/iQi + v/i,Q~ + AijMiJ R a = O,
+ ~13abMa b
p iA, = O,
(55)
+ TW , F =0,
R ab = 0 .
(49)
Eq. (49) has a trivial solution describing a flat d = 4, N = 2 superspace with four bosonic, eight fermionic coordinates and pure fermionic central charge form: E °
Oy a dO,
= dx a - i i
Y'tAi = d O A i ,
(53)
They have a simple solution:
'A, ,
cb .
Omicron and iota in eqs. (51) satisfy eqs. (23). O u r four independent Lorentz-invariant fermionic forms v/i, v/it satisfy the following equations: clv/i = dv/i = 0,
d~rlAi -- ~l)ab (¢Tab ~ii ) A -~ D~'tAi ,
(52)
On'V~2 - - I A t v / I .
~.t~, = dOi,,
A = 6.AB £ij OAi dOBj
-at- h.c.,
"Wab = O.
(50)
where PK, Pro, Pm, P t - , are the translation operators in the null tetrad basis. Qi, Q~t are the four supersymmetry generators, M ij = £iJM, Mab are generators of central charge and Lorentz symmetry and by W we have denoted the generator o f conformal transformations. The manifestly supersymmetric generalization o f the bosonic multi black hole can be obtained by solving the flatness condition 792 -- ( d A M + f N m L A N A L ) T M = 0
N o w we will build a non-trivial fiat superspace for the black holes. The variables which are natural for the black hole superspace are not Lorentz-covariant objects, like in eq. (49), but Lorentz-invariant ones, like in eqs. (28), (29), (21). The bosonic forms are m, rh, l - n, K = l + n and A. In addition we introduce eight fermionic forms satisfying four constraints, in accordance with the shortening o f a black hole superspace. These forms are required to be supercovariantly constant: VAA,~-'I~ + F A B £ i J ~ A , j = O, ~.1~ = _EAA,£ijtrI.IjA' ,
86
(51)
of the black hole superspace with the set o f curvatures R m required to vanish #6 : dm =0,
drh = 0 ,
d(/-n)
= 0,
d K - 2 V d A - i2v/2 vzv/2 = 0,
dv/i = 0, dA - TV-lK
dzo a b -
dv/it = 0, + i2x/2 VV/2 = 0,
ltlacZObc = 0,
dT = 0,
(56)
#6 Our notation here corresponds to the one in ref. [8]. The corresponding reference with supersymmetry, matching the notation of ref. [8], is ref. [11].
Volume 282, number 1,2
PHYSICS LETTERS B
vatures o f this superspace vanish, and there are no geometrical building blocks for quantum corrections. In this sense the supersymmetric black holes represent an alternative to the trivial flat superspace (46), (50), which also has no q u a n t u m corrections because all curvatures of the superspace vanish (49).
where ~2 = gli~it. These equations can be solved as follows: m = 2-1/2(dx + i d y ) ,
rh = 2 - 1 / 2 ( d x - i d y ) ,
l - n = v~ dz ,
K = x / - 2 V 2 ( d t - iOidO it + iOitdOi), gli = dOi,
~ll it =
A = v~V(dt"I-UAB =
V -1
dO i* ,
iOidO it + iOitdOi),
(IA dOB -- OA dls + ½£AB d V ) ,
T = V -~ d V ,
21 May 1992
Acknowledgement (57)
the flat-space laplacian in x, y, z o f V-~ being zero. In particular, V can be chosen in the form (19), and in this case eqs. (5 7) define a manifestly supersymmetric multi black hole: a flat superspace with four bosonic and four fermionic coordinates, the flatness condition being defined as the vanishing o f all curvatures (56) in this superspace. This superspace, when considered at 0i = 0 it = d0~ = dO ~t = 0, coincides with the set o f extreme R e i s s n e r - N o r d s t r 6 m black holes in the usual four-dimensional bosonic space. Thus, to find an extreme black hole we were solving not the classical Einstein-Maxwell equations but the flatness condition o f the superspace. The solution of the classical Einstein-Maxwell equations defines the Ricci tensor in terms o f a quadratic combination o f Maxwell tensors, the Riemann-Christoffel curvature tensor is non-vanishing. All on-shell quantum gravity corrections are expressed in terms o f the non-vanishing Riemann-Christoffel curvature tensor in the case o f non-extreme as well as extreme R N black holes. In both cases it is impossible to handle q u a n t u m corrections without additional information. The additional information indeed exists for the extreme R N black holes. It is possible to investigate the effective q u a n t u m action o f d = 4, N = 2 supergravity, taking into account the fundamental fact that the bosonic extreme R N black holes do not break four o f the original eight supersymmetries. The investigation shows that all on-shell q u a n t u m corrections, which are locally supersymmetric and Lorentz invariant, vanish in the extreme R N multi black hole background. Moreover, the four above mentioned supersymmetries can be m a d e manifest. A shortened superspace exists ( 5 5 ) , whose flatness condition (56) is solved by a supersymmetric black hole solution (57). All cur-
Our investigation was greatly stimulated by the recent activity in strings and black hole physics [12]. It is a pleasure to express my gratitude to S. Giddings, G. Gorowitz, M. Green, J. Hartle, A. Linde, T. Ortin, A. Strominger, L. Susskind and L. Thorlacius for interesting and fruitful discussions. This work was supported in part by N S F grant PHY-8612280 and by the John and Claire Radway Fellowship in the School o f Humanities and Sciences at Stanford University.
References [ l ] G.W. Gibbons and C.M. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in: Supersymmetry, supergravity and related topics, eds. F. del Aguila, J. de Azc~rraga and L. Ibfifiez (World Scientific, Singapore, 1985) p. 147. [2] K.P. Tod, Phys. Lett. B 121 (1983) 241. [3] R.E. Kallosh, JETP Lett. 29 (1979) 172; S. Christensen, S. Deser, M. Duff and M. Grisaru, Phys. Lett. B 84 (1979) 411; R.E. Kallosh, JETP Lett. 29 (1979) 449; R.E. Kallosh, Nucl. Phys. B 165 (1980) 119. [4] S. Gates, M. Grisaru, M. Ro~ek and W. Siegel, Superspace (Benjamin/Cummings, London, 1983) p. 451. [5] S. Deser, J.H. Kay and K.S. Stelle, Phys. Rev. Lett. 38 (1977) 527; R.E. Kallosh, Phys. Lett. B 99 (1981) 122; P. Howe and U. Lindstr6m, Nucl. Phys. B 181 (1981) 487. [6] D. Kramer, H. Stephani, M. MacCallum and E. Herlt, Exact solutions of Einstein's field equations (Cambridge U.P., Cambridge, 1980). [7] E. Sokatchev, Phys. Lett. B 100 (1981) 466. [8] R. Penrose and W. Rindler, Spinors and space-time (Cambridge U.P., Cambridge, 1988). 87
Volume 282, number 1,2
PHYSICS LETTERS B
[9] J.B. Hartle and S.W. Hawking, Commun. Math. Phys. 26 (1972) 87. [ 10 ] L. Castellani, R. D'Auria and P. Fr6, Supergravity and superstrings (World Scientific, Singapore, 1991 ); L. Castellani, R. D'Auria and S. Ferrara, Class. Quantum Gray. 7 (1990) 1767. I l l ] L. Brink and P. Howe, Phys. Lett. B 88 (1979) 268.
88
21 May 1992
[12] E. Witten, On black holes in string theory, lecture at Strings '91 (Stony Brook, June 1991); C.G. Callan Jr., J.A. Harvey and A. Strominger, Supersymmetric string solitons, lectures at Trieste 199 l Spring School on String theory and quantum gravity; T. Eguchi, Topological field theories and space-time singularity, preprint EFI 91-58 (1991).