Five Dimensional Minimal Supergravities and Four Dimensional Complex Geometries

0.07 MB·English

by  Jai Grover

Five Dimensional Minimal Supergravities
and Four Dimensional Complex Geometries
Jai Grover , Jan B. Gutowski†, Carlos A.R. Herdeiro and Wafic Sabra‡
∗ ∗∗
DAMTP,CMS,UniversityofCambridge,WilberforceRoad,Cambridge,CB30WA,UK
9 ∗
0 †DepartmentofMathematics,King’sCollegeLondon,Strand,LondonWC2R2LS,UK
0 ∗∗DF-FCUPeCFP,UniversidadedoPorto,RuadoCampoAlegre,687,4169-007Porto,Portugal
2 ‡CentreforAdvancedMathematicalSciencesandPhysicsDepartment,AmericanUniversityof
Beirut,Lebanon
n
a
J
Abstract. We discuss the relation between solutionsadmitting Killing spinorsof minimalsuper-

6 gravitiesinfivedimensionsandfourdimensionalcomplexgeometries.Intheungaugedcase(van-
2 ishingcosmologicalconstantL =0)thesolutionsaredeterminedintermsofahyper-Kählerbase
space; in the gauged case (L <0) the complex geometry is Kähler; in the de Sitter case (L >0)
]
h the complex geometry is hyper-Kähler with torsion (HKT). In the latter case some details of the
t derivationaregiven.Themethodforconstructingexplicitsolutionsisdiscussedineachcase.
-
p
Keywords: Supersymmetry,Killingspinors,supersymmetricblackholes
e
PACS: 04.65.+e,04.70.Bw
h
[

1 INTRODUCTION
v
6
6 In the last few years it has been realised that there are some beautiful connections be-
0
tween solutions of minimal supergravities in five dimensions admitting Killing spinors
4
. andcomplexgeometries.Moreover,theseconnectionshavebeen usefulinfindingqual-
1
itatively new black hole solutions. The first case to be considered was the minimal un-
0
9 gaugedfivedimensionalsupergravity,i.ewithno cosmologicalconstant(L =0). It was
0 shown by Gauntlett et al [1] that the most general stationary solution admitting Killing
:
v spinorsisdefinedbyafourdimensionalhyper-Kähler basespaceandasetofconstraint
i
X equations. Using this construction, Elvang et al [2] obtained the supersymmetric black
r ring. The second case to be considered was the minimal gauged five dimensional su-
a pergravity, i.e with negative cosmological constant (L <0). It was shown by Gauntlett
and Gutowski [3] that the most general stationary solution admitting Killing spinors is
definedbyafourdimensionalKählerbasespaceandasetofconstraintequations.Using
this construction, Gutowski and Reall found the supersymmetric AdS black holes [4].
5
It is then natural to ask: i) Is there any interesting relation with complex geometry in
fivedimensionalminimaldeSittersupergravity(i.ewithpositivecosmologicalconstant
L > 0)? ii) Can we use the resulting structure to find interesting solutions? In the fol-
lowing we will discuss how indeed the answer to i) is yes and give some hints towards
answeringii).
MINIMAL UNGAUGED SUPERGRAVITY IN D = 4
Supersymmetric black holes are interesting gravitational objects. They are classically
andsemi-classicallystable;inmanycasesthereisano-forceconditionwhichallowsfor
a multi-object configuration. An early example is obtained in minimal N = 2, D = 4
supergravity,whosebosonicsectorhas action
1 F2
S = d4x√ g R ;
16p G − − 4
4
Z (cid:18) (cid:19)
theansatz
ds2 = H(x) 2dt2+H(x)2ds2 , A=H(x) 1dt ,
− − E3 −
reduces the full non-linear Einstein-Maxwell system to a single harmonic equation on
theEuclidean3-space E3
D H(x)=0.
E3
This is the well known Majumdar-Papapetrou solution, and taking H(x) to be a multi-
centred harmonic function yields a multi black hole solution. Physically, the exact lin-
earisationofthesupergravityequations,yieldingasuperpositionprinciple,isassociated
to the exact balance of electrostatic repulsion and gravitational attraction between any
pair of black holes. But it is also associated to supersymmetry. Indeed, Tod [5] showed
that the most general stationary solution of this theory admitting Killing spinors, i.e. a
non-trivialsolutionof
1
De F G abG e =0 ,
ab
−4
falls into the class of Israel-Wilson-Pérjes metrics (which have the modulus of charge
equaltomass).SincethemostgeneralblackholesolutioninthisclassistheMajumdar-
Papapetrou solution one concludes the latter is the most general static (indeed the most
general stationary)supersymmetricblack holesolutionofN =2,D=4supergravity.
MINIMAL UNGAUGED SUPERGRAVITY IN D = 5
It follows from the above that Einstein-Maxwell theory (seen as the bosonic sector of
supergravity) does not admit any supersymmetric rotating, asymptotically flat, black
holes. And for some time it was even doubtful that such an object could exist; indeed
supersymmetry is a statement of stability and rotation is normally associated to the
instabilities which arise from the existence of an ergo-region (Penrose process and
superradience). However one such solution was found in five dimensional minimal
ungaugedsupergravity,whosebosonicsectorhas action
1 2
S = d5x √ g R F2 A F F ;
16p G5 − − −3√3 ∧ ∧
Z (cid:20) (cid:21)
(cid:0) (cid:1)
theansatz
√3
ds2 = H(x) 2(dt+w )2+H(x)ds2 , A= H(x) 1(dt+w ) ,
− − E4 2 −
reduces thesupergravityequationsto theconstraints[6]
D H(x)=0, dw = ⋆(4)dw ,
E4
−
where ⋆(4) is theHodgedual on theEuclidean 4-spaceE4. This is knownas theBMPV
solution [7]. Taking H(x) to be a multi-centred harmonic function, and an appropriate
choiceforw ,yieldsasolutionwithmultipleblackholesinanasymptoticallyflatspace-
time.Physically,theexactlinearisationofthesupergravityequationsisagainassociated
to the exact balance between electromagneticand gravitational forces between any pair
of black holes. But now, besides electric we will have magnetic effects (spin-spin and
magnetic dipole-dipole forces). And again it is also associated to supersymmetry. In-
deed, Gauntlett et al [1] showed that the most general stationary solution of this theory
admittingKillingspinors,i.e. anon-trivialsolutionof
Da + 1 G abg 4d ab G g Fbg e a =0,
4√3 −
(cid:20) (cid:21)
(cid:16) (cid:17)
isoftheform
√3 G+
ds2 = f2(dt+w )2+ f 1ds2 , F = d(f[dt+w ]) ,
− M
− 2 −√3
andthat themethodtoconstruct explicitexamplesisthefollowing:
1) ChooseM tobea4 dimensionalhyper-Kählermanifold;
2) Decompose fdw =G++G ;
−
3) Solve
2
dG+ =0, D f 1 = (G+)2 .
−
9
Taking G+ = 0 and M = E4, we find the BMPV (multi) black hole solution. But this
choice also includes other, qualitatively different solutions, namely maximally super-
symmetricGödeltypeuniversesand black holesinGödel typeuniverses[1, 8].
Taking G+ = 0 one can find supersymmetric black rings. But note that in this case
6
one does not find a harmonic equation any longer, but rather a Poisson type equation,
to which G+ is the source. Thus, the superposition principle is, in general, lost. And in
fact thereisnosolutionwithmultipleblack ringswherethesecan beplaced atarbitrary
positions, as for the multiple black holes seen above. Nevertheless, writing M as a
Gibbons-Hawkingspaceonecan constructmultipleconcentricblack rings[9].
MINIMAL GAUGED SUPERGRAVITY IN D = 5
A negative cosmological constant is introduced by moving to minimal gauged super-
gravity in five dimensions. This theory is particularly relevant because it is related by
the AdS/CFT duality to the well understood N = 4, D = 4 Super Yang Mills theory.
Hence one mightexpect to givea microscopicinterpretation to theblack hole solutions
inthistheoryusingtheduality.A genericanalysisoftheKillingspinorequation
Da + 1 G abg 4d ab G g Fbg e a ge ab G a √3Aa e b =0,
4√3 − − 2 −
(cid:20) (cid:21) (cid:18) (cid:19)
(cid:16) (cid:17)
showsthatall susysolutionswithatimelikeKillingvectorfield are oftheform [3]:
√3 G+
ds2 = f2(dt+w )2+ f 1ds2 , F = d(f[dt+w ]) +√3gf 1J ,
− M −
− 2 −√3
andthat themethodtoconstruct explicitsolutionsisthefollowing:
1) ChooseM tobea4 dimensionalKählermanifold;
2) Compute
24g2 1 R
f = , G+ = R+ J ,
− R −2g 4
(cid:20) (cid:21)
which determines completely f and G+ in terms of the properties of the Kähler
space: itsRicci scalar, R, itsRicci form, R,and itsKählerform,J; solvealso
2
D f 1 = (G+)mn(G+) gf 1(G )mnJ 8g2f 2 ,
− mn − − mn −
9 − −
which determines the components of G that have a non trivial contraction with
−
theKählerform;
3) Solvetheconstraint
fdw =G++G ,
−
whichdeterminestheremainingcomponentsofG .
−
TakingM tobeBergamann space, which can bewrittenas [10]
sinh2gs dx2
ds2 =ds 2+ +H(x)dy 2+cosh2gs (df +xdy )2 ,
M
4g2 H(x)
(cid:18) (cid:19)
with H(x)=1 x2, one finds AdS . The Gutowski-Reall black hole [4] and the Chong
5
−
etal.blackhole[11]arefoundbytakingmoregeneralquadraticand cubicpolynomials
[10]. Note that for arbitrary H(x) the above metric is Kähler; but the supersymmetry
constraintsimpose(H2H ) =0,whereprimesdenotexderivatives,whichshowsthat
′′′′ ′′
agivenbasespacemightnot giverisetoa fivedimensionalsolution.Also,a givenbase
spacemightgiverisetoafamilyofsolutionswithaninfinitenumberofparameters[10].
ItisworthnotingthatthesupersymmetricAdS blackholesfoundusingthisconstruc-
5
tionmustrotate,whichissimilartowhathappensinthree[12]andfour[13]dimensions.
MINIMAL DE SITTER SUPERGRAVITY IN D = 5
It is well known that de Sitter superalgebras have only non-trivial representations in a
positive-definite Hilbert space in two dimensions [14, 15]. Nevertheless one can take
the perspective of fake supersymmetry, in analogy to the recently explored Domain
Wall/Cosmology correspondence [16]: that there is a special class of solutions in a
gravitational theory with a positive cosmological constant admitting “pseudo-Killing
spinors”. Thus, fake supersymmetry becomes a solution generating technique, as we
shall explain. Note that, nevertheless, a relation to fundamental theory still exists via
compactificationsoftheIIB* theory [17,18].
Wenowfollowclosely[19].TheactionforminimaldeSittersupergravityinD=5is
1 1 1 2
S = (5R c 2)⋆1 F ⋆F F F A ,
4p G5Z (cid:18)4 − −2 ∧ −3√3 ∧ ∧ (cid:19)
andtheKillingspinorequationis
1 i 3i
¶ + W N1N2G FN1N2G G + F NG
M 4 M, N1N2−4√3 M N1N2 2√3 M N
(cid:20)
i 1
+c ( G A ) e =0.
M M
4√3 −2
(cid:21)
The sense in which fake supersymmetrycan be used as a solutiongenerating technique
is the following. If a non-trivial solution of the (pseudo) Killing spinor equation exists
and the gauge field equations are satisfied, the integrability conditions of the former
place constraints on the Ricci tensor. For the solutions we consider here, for which the
Killing spinor generates a timelike vector field, these constraints are equivalent to the
Einstein equations. Note this would not be so for the null case, for which the Killing
spinorgenerates anullvectorfield.
Let us now give some details of the derivation of the (fake) supersymmetry con-
straints. The basic principle is to assume the existence of, at least, one non-trivial
(pseudo)Killingspinor.Thisputsconstraintson thespinconnectionand gaugefield.In
practice we use spinorial geometry techniques [20]. That is, we take the space of Dirac
spinors to be the space of complexified forms on E2, which is spanned over the space
of complex numbers by 1,e ,e ,e where e =e e . The action of complexified
1 2 12 12 1 2
G -matriceson thesespino{rsisgivenb}y ∧
G a =√2ea , G a¯ =√2iea ,
∧
fora =1,2, andG satisfies
0
G 1= i1, G e12 = ie12, G ej =iej , j =1,2,
0 0 0
− −
wherewework withan oscillatorbasisin whichthespacetimemetricis
ds2 = (e0)2+2d ea eb¯ .
a b¯
−
The Killing spinor can be put in a canonical form using the Spin(4,1) transformations.
Thecanonicalformise =h1,wherehisafunction,ifitoriginatesatimelikevectorand
e =1+e ifitoriginatesanullvector.Wewillbeinterestedintheformercase.
1
Defininga1-formV =e0 andintroducingat coordinatesuchthatthedualvectorfield
isV = ¶ /¶ t, acomputationshowsthattheframes taketheform
−
e0 =dt+2√3P+e√c t3Q , ea =e−2√c 3teˆa ,
c
where
L eˆa =0 , L Q =0 , L P =0.
V V V
Werefer tothe4-manifoldwitht-independentmetric
ds2 =2d eˆa eˆb¯ ,
M a b¯
as the “base space” M. It follows that part of the geometrical constraints imposed by
theKillingspinorequationare equivalenttodJi = 2P Ji fori=1,2,3where
− ∧
J1 =eˆ1 eˆ2+eˆ1¯ eˆ2¯ , J2 =ieˆ1 eˆ1¯ +ieˆ2 eˆ2¯ , J3 = ieˆ1 eˆ2+ieˆ1¯ eˆ2¯ ,
∧ ∧ ∧ ∧ − ∧ ∧
defines a triplet of anti-self-dual almost complex structures on M which satisfy the
algebraoftheimaginaryunitquaternions.Thus,M ishyper-Kählerwithtorsion,HKT;
inotherwords,thealmostcomplexstructuresarepreservedbyaconnectionwithtorsion:
(cid:209) +Ji =0 , G (+)i = i +Hi ,
jk jk jk
{ }
wherethetorsionisH =⋆ P, and ⋆ istheHodgedualon M.
4 4
ThebottomlineisthatthemostgeneralsolutionoffivedimensionalminimaldeSitter
supergravity admitting a (pseudo) Killing spinor, from which a timelike vector field is
constructed,isoftheform:
2
2√3 c c dt P √3 c
ds2 =− dt+ c P+e√3tQ! +e−√3tds2M , A= 2√3+ c + 2 e√3tQ ,
where all t dependence is explicit, and the method to construct explicit solutions is the
following:
1) Take the base space M to be a four dimensional HKT geometry with metric ds2
M
and torsiontensorH;
2) The1-formP isgivenbyP = ⋆ H;
4
−
3) Choosea1-formP obeyingtheconstraints
16
dQ 2P Q + =0, d⋆ Q+ dP dP =0 ,
− ∧ 4 √3c 3 ∧
(cid:0) (cid:1)
where + denotes the self-dual projection on the base-manifold M, with positive
orientation fixed with respect to the volume form eˆ1 eˆ1¯ eˆ2 eˆ2¯. Note that one
∧ ∧ ∧
can always solvethesecondconstraint;thegeneral solutionisgivenby
16
Q = ⋆ (P dP)+⋆ dF ,
√3c 3 4 ∧ 4
where F is a 2-form on M. On substituting this expression back into the first
constraintequation,onefinds an equationconstrainingF , which mustbesolved.
TheRicci scalarofthesolutionis
R = 5c 2+e√2c3t (dP)2 c 2e√c3tQ2+3e√2c3t(dQ 2P Q)2 ,
3 3 c 2 − 2 4 − ∧
(cid:20) (cid:21)
where the norms are computed with respect to the t-independent base space metric.
Therefore, the t-dependence of the Ricci scalar can be read directly from the above
expression. Thus, in particular, for the solution to be regular at both t = ¥ we must
requireQ=0anddP =0.Inparticular,thisimpliesthatthebasespaceis±conformally
hyper-Kähler.
If one assumes that M is a conformally hyper-Kähler, then dP = 0. After some
coordinatetransformations,thesolutioncan becast intheform
√3
ds2 = f2(dt+w )2+ f 1ds2 , F = d f(dt+w ) ,
− − HK 2
(cid:18) (cid:19)
where
c
f 1 =H t ,
−
−√3
and
D H =0, (dw )+ =0 .
HK
dS isobtainedby takingthehyper-KählerspacetobeE4, H =const.andw =0.
5
This form of the solution is exactly the form of the solutions of minimal ungauged
supergravity with G+ =0, except for the linear term int which arises in f 1 due to the
−
cosmologicalconstant.Thusweareled tothefollowingtheorem:
Any solution of D = 5 minimal de Sitter supergravity with a pseudo Killing spinor
and a base space which is conformal to a hyper-Kähler manifold can be obtained from
a “seed” solution of minimal ungauged supergravity simply by adding a linear time
dependencetotheharmonicfunction.
ThisresultgeneralisesanearlierresultbyBehrndtandCvetic[21].Moreoveritmakes
clear why we can superimpose certain solutions (like the BMPV black hole [22] or
Gödel type universes [23]) with a positive cosmological constant. But it also suggests
thatsolutionswithG+ =0 donot generaliseeasilyto deSitterspace. Mostnotablythis
includestheblack ring.
Hence, in contrast to the AdS theory, the de Sitter theory admits multi-black hole
solutions (like the multi-BMPV de Sitter), in which one finds multiple black holes co-
moving with the expansion of the universe. This latter solution can be considered as a
fivedimensionalrotatinggeneralisation of theMajumdar-Papapetrou de Sittersolution,
foundbyKastorand Traschen[24].
As a second class of examples one can take M to be an HKT manifold admitting
a tri-holomorphic Killing vector field X. This means that both the base space metric
and the almost complex structures are preserved by Lie dragging along the integral
lines of X. Such HKT manifolds have been classified [25, 26] and their structure is
completely specified in terms of a constrained 3-dimensional Einstein-Weyl geometry.
Explicit solutions can then be obtained, the simplest of which takes the Einstein-Weyl
geometry to be a round 3-sphere [19]. But all these solutions are singular, as expected
from the above analysis of the Ricci scalar. The outstanding question is if, for any of
these solutions, these singularities have interesting interpretations in terms of either
blackholeorbigbang/bigcrunch singularities.That remainsto beseen.
ACKNOWLEDGMENTS
C.H. and W.S. would like to thank the hospitality of DAMTP Cambridge. CFP is
partially funded by FCT through the POCI programme. The work of W.S. is supported
inpart bytheNationalScience FoundationundergrantnumberPHY-0703017.
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