hep-th/9709113CERN-TH/97-226September, 1997

Duality, Central Charges and Entropy ofExtremal BPS Black Holes

Laura Andrianopoli1, Riccardo D’Auria2 and Sergio Ferrara3

1 Dipartimento di Fisica, Universita di Genova, via Dodecaneso 33, I-16146 Genovaand Istituto Nazionale di Fisica Nucleare (INFN) - Sezione di Torino, Italy

2 Dipartimento di Fisica, Politecnico di Torino,Corso Duca degli Abruzzi 24, I-10129 Torino

and Istituto Nazionale di Fisica Nucleare (INFN) - Sezione di Torino, Italy

3 CERN Theoretical Division, CH 1211 Geneva 23, Switzerland

Talk given by S. Ferrara at the“STRINGS ’97” Conference,

16-21 June 1997, Amsterdam, The Netherlands

Abstract

We report on some general results on the physics of extremal BPS black holes in four and five dimensions. The

duality-invariant entropy-formula for all N > 2 extended supergravities is derived. Its relation with the fixed-

scalar condition for the black-hole “potential energy” wich extremizes the BPS mass is obtained. BPS black holes

preserving different fractions of supersymmetry are classified in a U-duality invariant set up. The latter deals

with different orbits of the fundamental representations of the exceptional groups E7(7) and E6(6). We comment

upon the interpretation of these results in a string and M-theory framework.

∗ Work supported in part by EEC under TMR contract ERBFMRX-CT96-0045 (LNF Frascati, Politecnico di Torino and

Univ. Genova) and by DOE grant DE-FGO3-91ER40662

1

Duality, Central Charges and Entropy of Extremal BPS Black Holes

L. Andrianopolia, R. D’Auriab and S. Ferrarac

aDipartimento di Fisica, Universita di Genova, via Dodecaneso 33, I-16146 Genovaand Istituto Nazionale di Fisica Nucleare (INFN) - Sezione di Torino, Italy

bDipartimento di Fisica, Politecnico di Torino,Corso Duca degli Abruzzi 24, I-10129 Torinoand Istituto Nazionale di Fisica Nucleare (INFN) - Sezione di Torino, Italy

cCERN Theoretical Division, CH-1211 Geneva 23, Switzerland

We report on some general results on the physics of extremal BPS black holes in four and five dimensions. Theduality-invariant entropy-formula for all N > 2 extended supergravities is derived. Its relation with the fixed-scalar condition for the black-hole “potential energy” wich extremizes the BPS mass is obtained. BPS black holespreserving different fractions of supersymmetry are classified in a U-duality invariant set up. The latter dealswith different orbits of the fundamental representations of the exceptional groups E7(7) and E6(6). We commentupon the interpretation of these results in a string and M-theory framework.

1. Introduction

In recent time, remarkable results have beenobtained in the study of general properties ofBPS states both in supersymmetric gauge theo-ries as well as in supersymmetric theories of grav-ity. The latter are described by string theory andM-theory [1] whose symmetry properties are en-coded in extended supergravity effective field the-ories.

Of particular interest are extremal black holesin four and five dimensions which correspond toBPS saturated states [2] and whose ADM massdepends, beyond the quantized values of electricand magnetic charges, on the asymptotic valueof scalars at infinity. The latter describe themoduli space of the theory Another physical rel-evant quantity, which depends only on quantizedelectric and magnetic charges, is the black holeentropy, which can be defined macroscopically,through the Bekenstein-Hawking area-entropy re-lation or microscopically, through D-branes tech-niques [3] by counting of microstates [4]. It hasbeen further realized that the scalar fields, in-dependently of their values at infinity, flow to-wards the black hole horizon to a fixed value ofpure topological nature given by a certain ratio

of electric and magnetic charges [5]. These “fixedscalars” correspond to the extrema of the ADMmass in moduli space while the black-hole entropyis the value of the squared ADM mass at thispoint in D = 4 [6] [7] and the power 3/2 of theADM mass in D = 5. In four dimensional the-ories with N > 2, extremal black-holes preserv-ing one supersymmetry have the further propertythat all central charge eigenvalues other than theone equal to the BPS mass flow to zero for “fixedscalars”. This is not true in D = 5 because thecharges transform in the antisymmetric represen-tation of Usp(N) instead of U(N) as in the fourdimensional cases.

The entropy formula turns out to be in all casesa U-duality invariant expression (homogeneous ofdegree two in D = 4 and of degree 3/2 in D = 5)built out of electric and magnetic charges andas such can be in fact also computed throughcertain (moduli-independent) topological quanti-ties which only depend on the nature of the U-duality groups and the appropriate representa-tions of electric and magnetic charges. For ex-ample, in the N = 8, D = 4 theory the entropywas shown to correspond to the unique quartic E7

invariant built with its 56 dimensional represen-tation [8]. In this report we intend to summarize

2

further progress in this subject by deriving, for allN > 2 theories in D = 4, 5, topological (moduli-independent) U-invariants constructed in termsof (moduli-dependent) central charges and mat-ter charges, and show that, as expected, they co-

incide with M2ADM or M

3/2ADM (in the case of the

four and five dimensional theories respectively),computed at “fixed scalars”.

The situation of black-hole backgrounds pre-serving more than 1/8 of the original supersym-metry (32 charges) is further explored.

Sections 2 and 3 deal with the four and five di-mensional cases respectively, section 4 describesthe absolute duality invariants, section 5 de-scribes BPS states preserving more than one su-persymmetry and section 6 gives a further classi-fication of BPS states in terms of duality orbits.

2. Central charges, U-invariants and en-tropy in D = 4

In D = 4, extremal black-holes preserving onesupersymmetry correspond to N -extended multi-plets with

MADM = |Z1| > |Z2| · · · > |Z[N/2]| (1)

where Zα, α = 1, · · · , [N/2], are the proper val-ues of the central charge antisymmetric matrixwritten in normal form [11]. The central chargesZAB = −ZBA, A,B = 1, · · · , N , and mattercharges ZI , I = 1, · · · , n are those (moduli-dependent) symplectic invariant combinations offield strenghts and their duals (integrated overa large two-sphere) which appear in the grav-itino and gaugino supersymmetry variations re-spectively [12], [13], [14]. Note that the totalnumber of vector fields is nv = N(N − 1)/2 + n(with the exception of N = 6 in which case thereis an extra singlet graviphoton)[15].It was shown in ref. [7] that at the attractor point,where MADM is extremized, supersymmetry re-quires that Zα, α > 1, vanish together with thematter charges ZI , I = 1, · · · , n (n is the num-ber of matter multiplets, which can exist only forN = 3, 4)

This result can be used to show that for “fixedscalars”, corresponding to the attractor point, the

scalar “potential” of the geodesic action [9][10]

V = −1

2P tM(N )P (2)

is extremized in moduli space. Here P is the sym-plectic vector P = (pΛ, qΛ) of quantized electricand magnetic charges and M(N ) is a symplec-tic 2nv × 2nv matrix whose nv × nv blocks aregiven in terms of the nv × nv vector kinetic ma-trix NΛΣ (−ImN , ReN are the normalizations ofthe kinetic F 2 and the topological F ∗F terms re-spectively) and

M(N ) =

(A BC D

)(3)

with:

A = ImN +ReN ImN−1ReN

B = −ReN ImN−1

C = −ImN−1ReN

D = ImN−1 (4)

The above assertion comes from the importantidentity, shown in ref. [13], [14] to be valid in allN ≥ 2 theories:

−1

2P tM(N )P =

1

2ZABZ

AB+ ZIZ

I(5)

Indeed, let us consider the differential relationssatisfied by the charges [14]:

∇ZAB =1

2PABCDZ

CD+ PABIZ

I

∇ZI =1

2PABIZ

AB+ PIJZ

J(6)

where the matrices PABCD, PABI , PIJ are thesubblocks of the vielbein of G/H embedded inUSp(n, n) [14]:

P ≡ L−1∇L =

(PABCD PABJPICD PIJ

)(7)

written in terms of the indices of H = HAut ×Hmatter . By computing the extremum of (2) andusing equations (5),(6) we obtain

PABCDZABZCD = 0; ZI = 0 (8)

PABCD being the vielbein of the scalar manifold,completely antisymmetric in its SU(N) indices.

3

It is easy to see that in the normal frame theseequations imply:

MADM |fix ≡ |Z1| 6= 0 (9)

|Zi| = 0 (i = 2, · · · , N/2) (10)

The main purpose of this section is to provideparticular expressions which give the entropy for-mula as a moduli–independent quantity in theentire moduli space and not just at the criti-cal points. Namely, we are looking for quanti-

ties S(ZAB(φ), Z

AB(φ), ZI(φ), Z

I(φ))

such that∂∂φi

S = 0, φi being the moduli coordinates.These formulae generalize the quartic E7(7) in-

variant of N = 8 supergravity [8] to all othercases.

Let us first consider the theories N = 3, 4,where matter can be present [16], [17].

The U-duality groups are, in these cases,SU(3, n) and SU(1, 1) × SO(6, n) respectively(Here we denote by U-duality group the isom-etry group G acting on the scalars, althoughonly a restriction of it to integers is the properU-duality group [18]). The central and mattercharges ZAB, ZI transform in an obvious way un-der the isotropy groups

H = SU(3)× SU(n)× U(1) (N = 3) (11)

H = SU(4)×O(n)× U(1) (N = 4) (12)

Under the action of the elements of G/H thecharges get mixed with their complex conjugate.For N = 3:

PABCD = PIJ = 0 , PABI ≡ εABCPCI

ZAB ≡ εABCZC (13)

Then the variations are:

δZA = ξAI ZI

(14)

δZI = ξAI ZA (15)

where ξAI are infinitesimal parameters of K =G/H. Indeed, once the covariant derivatives areknown, the variations are obtained by the substi-tution ∇→ δ, P → ξ.

With a simple calculation, the U-invariant ex-pression is:

S = ZAZA − ZIZI

(16)

In other words, ∇iS = ∂iS = 0, where the covari-ant derivative is defined in ref. [14].

Note that at the attractor point (ZI = 0) it co-incides with the moduli-dependent potential (2)computed at its extremum.For N = 4

PABCD = εABCDP, PIJ = ηIJP

PABI =1

2ηIJεABCDP

CDJ(17)

and the transformations of K = SU(1,1)U(1) ×

O(6,n)O(6)×O(n) are:

δZAB =1

2ξεABCDZ

CD+ ξABIZ

I(18)

δZI = ξηIJZJ

+1

2ξABIZ

AB(19)

with ξABI

= 12ηIJεABCDξCDJ .

There are three O(6, n) invariants given by I1,I2, I2 where:

I1 =1

2ZABZAB − ZIZ

I(20)

I2 =1

4εABCDZABZCD − ZIZ

I(21)

and the unique SU(1, 1) × O(6, n) invariant S,∇S = 0, is given by:

S =√

(I1)2 − |I2|2 (22)

At the attractor point ZI = 0and εABCDZABZCD = 0, so that S reduces tothe square of the BPS mass.

For N = 5, 6, 8 the U-duality invariant expres-sion S is the square root of a unique invariant un-der the corresponding U-duality groups SU(5, 1),O∗(12) and E7(7). The strategy is to find aquartic expression S2 in terms of ZAB such that∇S = 0, i.e. S is moduli-independent.

As before, this quantity is a particular combi-nation of the H quartic invariants.

For SU(5, 1) there are only two U(5) quartic in-

variants. In terms of the matrix A BA = ZACZ

CB

they are: (TrA)2, Tr(A2), where

TrA = ZABZBA

(23)

Tr(A2) = ZABZBC

ZCDZDA

(24)

4

As before, the relative coefficient is fixed by the

transformation properties of ZAB under SU(5,1)U(5)

elements of infinitesimal parameter ξC :

δZAB =1

2ξCεCABPQZ

PQ(25)

It then follows that the required invariant is:

S =1

2

√4Tr(A2)− (TrA)2 (26)

For N = 8 the SU(8) invariants are:

I1 = (TrA)2 (27)

I2 = Tr(A2) (28)

I3 = Pf Z

=1

244!εABCDEFGHZABZCDZEFZGH (29)

TheE7(7)

SU(8) transformations are:

δZAB =1

2ξABCDZ

CD(30)

where ξABCD satisfies the reality constraint:

ξABCD =1

24εABCDEFGHξ

EFGH(31)

One finds the following E7(7) invariant [8]:

S =1

2

√4Tr(A2)− (TrA)2 + 32Re(Pf Z) (32)

The N = 6 case is the more complicated becauseunder U(6) the left-handed spinor of O∗(12) splitsinto:

32L → (15, 1) + (15,−1) + (1,−3) + (1, 3) (33)

The transformations of O∗(12)U(6) are:

δZAB =1

4εABCDEF ξ

CDZEF

+ ξABX

δX =1

2ξABZ

AB(34)

where we denote by X the SU(6) singlet. Thequartic U(6) invariants are:

I1 = (TrA)2 (35)

I2 = Tr(A2) (36)

I3 = Re(Pf ZX)

=1

233!Re(εABCDEFZABZCDZEFX) (37)

I4 = (TrA)XX (38)

I5 = X2X2

(39)

The unique O∗(12) invariant is:

S =1

2

√4I2 − I1 + 32I3 + 4I4 + 4I5 (40)

∇S = 0 (41)

Note that at the attractor point Pf Z = 0, X = 0and S reduces to the square of the BPS mass.

We note that in the normal frame the transfor-mations of the coset which preserve the normalform of the ZAB matrix belong to O(1, 1)3 bothfor N = 6 and N = 8 theories. The relevantO(1, 1)3 transformations can be read out fromeqs. (30), (34) going to the normal frame. Theensuing transformations correspond to commut-ing matrices which are proper, non compact, Car-tan elements of the coset algebra of N = 8, N = 6respectively[20].

3. The attractor point condition in D = 5

In this section we extend the previous analysisto the D = 5 dimensional case, for theories withN > 2 supersymmetries. Theories with N = 2 atD = 5 have been considered earlier [6] and fixedscalars recently analyzed in great detail [19]. Atechnical important difference in this case is thatalthough matter charges vanish for fixed scalars,preserving one supersymmetry requires that theeigenvalues of the central charges which are notthe BPS mass do not generally vanish at the hori-zon, but are all equal and fixed in terms of theentropy.

The five dimensional case exhibits analogiesand differencies with respect to the four dimen-sional one. Exactly like in the four dimen-sional theories, the entropy is given, through theBekenstein-Hawking relation, by an invariant ofthe U-duality group over the entire moduli spaceand its value is given in terms of the moduli-dependent scalar potential of the geodesic ac-tion[9] at the attractor point [5], [6], [7]:

S =A

4=π2

12M

3/2extr =

π2

12

[√3

2V (φfix, g)

]3/4

(42)

where we have used the relation Vextr = 43M

2extr

which is valid for any N in D = 5, as we will showin the following. Note that, while in the four di-mensional case the U-invariant is quartic in the

5

charges, in D = 5 it turns out to be cubic. Fur-thermore, in five dimensions the automorphismgroup under which the central charges transformis USp(N) instead of SU(N) × U(1) as in thefour dimensional theories [14], [15]. As it is ap-parent from the dilatino susy transformation law,this implies that, at the minimum of the ADMmass, the central charges different from the max-imal one do not vanish, contrary to what hap-pens in D = 4. Indeed in D = 5 we may performagain the extremization of the geodesic potentialas in D = 4 but, due to the traceless conditionof the antisymmetric symplectic representationsof the vielbein PABCD and of the central chargesZAB,the analogous equations:

PABCDZABZCD = 0; ZI = 0 (43)

do not imply anymore the vanishing of the centralcharges different from the mass. Here A,B, · · ·are USp(N) indices and the antisymmetric ma-trix ZAB satisfies the reality condition:

ZAB

= CACCBDZCD (44)

CAB being the symplectic invariant antisymmet-ric matrix satisfying C = −CT , C2 = −11. Letus now consider more explicitly the various fivedimensional theories. In N = 4 matter coupledsupergravity [14], the scalar manifold is given by

G/H = O(5,n)O(5)×O(n) × O(1, 1). The black-hole po-

tential is given by:

V (φ, q) =1

2ZABZ

AB+ 2X2 + ZIZ

I

= qΛ(N−1)ΛΣqΣ (45)

where X is the central charge associated to thesinglet photon of the N = 4 theory, qΛ ≡∫S3NΛΣ

?FΣ and NΛΣ are the electric chargesand vector kinetic matrix respectively. The cen-tral charge ZAB can be decomposed in its C-

traceless part◦ZAB and trace part X according

to:

ZAB =◦ZAB −CABX (46)

This decomposition corresponds to the combi-nation of the five graviphotons and the singletphoton appearing in the gravitino transformation

law [14]. The matter charges ZI are instead inthe vector representation of O(n). Note that inthe dilatino transformation law a different com-bination of the five graviphotons and the singletphoton appears corresponding to the integratedcharge:

Z(χ)AB ≡

1

2(ZAB + 3CABX)

=1

2(◦ZAB + 2CABX) (47)

The differential relations satisfied by the centraland matter charges are:

∇ZAB = PIABZI − 2Z

(χ)ABdσ

→ ∇◦ZAB = PIABZ

I −◦ZABdσ (48)

∇X = 2Xdσ (49)

∇ZI =1

4(ZABP

AB

I

+ ZABPABI)− ZIdσ (50)

To minimize the potential, it is convenient to go

to the normal frame where◦ZAB has proper values

e1, e2 = −e1. In this frame, the potential and thedifferential relations become:

V (φ, q) = e21 + e2

2 + 4X2 (51)

∇e1 = −e1dσ + PIZI (52)

∇X = 2Xdσ (53)

where PI ≡ PI12 is the only independent com-ponent of the traceless vielbein one-form PIABin the normal frame. We then get immediatelythat, in the normal frame, the fixed scalar condi-tion ∂V

∂φ= 0 requires:

ZI = 0 (54)

e1 = −e2 = −2X (55)

where ei (i = 1, 2) are the proper values of◦ZAB.

It follows that Mextr = |Z12 extr| =32e1 so that

Vextr = 3e21 =

4

3M2extr (56)

In the N = 6 theory, the scalar manifold isG/H = SU?(6)/Sp(6) [14]. The central charge

6

ZAB can again be decomposed in a C-traceless

part◦ZAB and a trace part Z according to:

ZAB =◦ZAB +

1

3CABX (57)

The traceless and trace parts satisfy the differen-tial relations:

∇◦ZAB =

◦ZC[APB]DCCD

+1

6CAB

◦ZLMP

LM +2

3XPAB (58)

∇X =1

4

◦ZABP

AB (59)

where PAB is the C traceless vielbein of G/H.The geodesic potential has the form:

V (φ, q) =1

2ZABZ

AB +4

3X2

= qΛΣ(N−1)ΛΣ,Γ∆qΓ∆ (60)

where Λ,Σ, · · · = 1, · · · , 6 are indices in the fun-damental representation of SU?(6), the couple ofindices ΛΣ in the elctric charges qΛΣ are anti-symmetric and (N )ΛΣ,Γ∆ is the kinetic matrixof the vector field-strengths FΛΣ To perform theminimization of the potential we proceed as inthe N = 4 case going to the normal frame where◦ZAB has proper values e1, e2, e3 = −e1− e2. Thepotential becomes:

V (φ, q) = e21 + e2

2 + (e1 + e2)2 +4

3X2 (61)

Moreover, the differential relations (58) take theform:

∇e1 =1

3(−e1 + e2 + 2X)P1

+1

3(e1 + 2e2)P2 (62)

∇e2 =1

3(2e1 + e2)P1

+1

3(e1 − e2 + 2X)P2 (63)

∇X =1

2(2e1 + e2)P1 +

1

2(e1 + 2e2)P2 (64)

where P1, P2, P3 = −P1 − P2 are the proper val-ues of the vielbein one-form PAB in the normalframe. Imposing the attractor-point constraint

∂V∂φ

= 0 on the potential we get the following re-lations among the charges at the extremum:

e2 = e3 = −1

2e1 = −

4

3X

Vextr =27

16e2

1 =4

3M2extr. (65)

Note that, using eqs. ( 62)-(64), (65), the masse1 + 1

3X satisfies ∂∂φi (e1 + 1

3X) = 0 at the ex-

tremum, with value Mextr = 98e1.

In the N = 8 supergravity the scalar manifoldis G/H = E6(−6)/Sp(8) and the central chargessit in the twice antisymmetric, C-traceless, repre-sentation of USp(8) [14]. The scalar “potential”of the geodesic action is given by:

V (φ, g) =1

2ZABZ

AB

= qΛΣ(N−1)ΛΣ,Γ∆(φ)qΓ∆ (66)

where qΛΣ ≡∫NΛΣ,Γ∆F

Γ∆ are the electriccharges and NΛΣ,Γ∆ the vector kinetic matrix.The extremum of V can be found by using thedifferential relation [14]:

∇ZAB =1

2PABCDZ

CD(67)

where PABCD is the four-fold antisummetric viel-bein one-form of G/H. One obtains:

PABCDZABZCD = 0 (68)

To find the values of the charges at theextremum we use the traceless conditionsCABPABCD = 0, CABZAB = 0. In thenormal frame, the proper values of ZAB aree1, e2, e3, e4 = −e1 − e2 − e3 and we take, asindependent components of the vielbein, P1 =P1234 = P5678, P2 = P1256 = P3478 (P3456 =P1278 = −P1 − P2). In this way, the covariantderivatives of the charges become:

∇e1 = (e1 + 2e2 + e3)P1

+ (e1 + e2 + 2e3)P2 (69)

∇e2 = (e1 − e3)P1 + (−e1 − e2 − 2e3)P2 (70)

∇e3 = (−e1 − 2e2 − e3)P1 + (e1 − e2)P2 (71)

7

Using these relations, the extremum condition ofV implies:

e2 = e3 = e4 = −1

3e1

Vextr =4

3e2

1 =4

3M2extr (72)

4. Topological invariants

In this section we determine the U-invariantexpression in terms of which the entropy can beevaluated over the entire moduli space. Our pro-cedure is the same as in ref. [20], namely we com-pute cubic H-invariants and determine the appro-priate linear combination of them which turns outto be U-invariant. The invariant expression of theentropy for N = 4 and N = 8 at D = 5 in termsof the quantized charges was given in [7]. Let usbegin with the N = 4 theory, controlled by the

coset O(5,n)O(5)×O(n) × O(1, 1). In this case there are

three possible cubic H-invariants, namely:

I1 =1

2

◦ZAB

◦

Z

AB

X (73)

I2 = ZIZIX (74)

I3 = X3 (75)

In order to determine the U ≡ G = O(5, n) ×O(1, 1)-invariant, we set I = I1+αI2+βI3 and us-ing the differential relations (48) one easily findsthat ∂I = 0 implies α = 1, β = 0. Therefore:

I = I1 − I2 =

(1

2

◦ZAB

◦

Z

AB

− ZIZI

)X (76)

is the cubic (O(5, n)×O(1, 1))-invariant indepen-

dent of the moduli and the entropy S = π12M

32 is

then given as:

S ∼ I1/2 =

√(1

2ZABZ

AB− ZIZI)Z (77)

In the N = 6 theory, where the coset mani-fold is SU∗(6)/Sp(6), the possible cubic Sp(6)-invariants are:

I1 = Tr(ZC)3 (78)

I2 = Tr(ZC)2X (79)

I3 = X3 (80)

Setting as before:

I = I1 + αI2 + βI3 (81)

the covariant derivative ∂I is computed using thedifferential relations (58) and the parametrer αand β are then determined by imposing ∂I = 0.Actually, the best way to perform the computa-tion is to go to the normal frame. Using the dif-ferential relations (62)-(64) and the expression forthe invariants in the normal frame:

Tr(ZC)3 = −6(e21e2 + e1e

22) (82)

Tr(ZC)2 = 4(e21 + e2

2 + e1e2) (83)

the vanishing of ∂I fixes the coefficients α and β.The final result is:

I = −1

6Tr(ZC)3 −

1

6Tr(ZC)2X +

8

27X3 (84)

and the entropy is:

S ∼

√−

1

6Tr(ZC)3 −

1

6Tr(ZC)2X +

8

27X3 (85)

Finally, for the N = 8 theory, described by thecoset E6(6)/Sp(8), it is well known that there isa unique E6 cubic invariant, namely:

I3(27) = Tr(ZC)3 = Z BA Z C

B Z AC (86)

where the Sp(8) indices are raised and loweredby the antisymmetric matrix CAB. Curiously,the E6-invariant corresponds to a single cubicUSp(8)-invariant. Again, the invariance of I canbe best computed in the normal frame where theinvariant (85) takes the form:

I3(27) = Tr(ZC)3 = −2(e31 + e3

2 + e33 + e3

4)

= 6(e22e3 + e2e

23 + e2

1e2 + e1e22

+ e21e3 + e1e

23 + 2e1e2e3) (87)

One finds indeed ∂I = 0 and therefore

S ∼ I1/2 =√Tr(ZC)3 (88)

As in D = 4 the transformations of the cosetwhich preserve the normal form of the ZAB ma-trix belong to the non compact Cartan subalge-bra of SU∗(6) and E6(6) for N = 6 and N = 8respectively, which in both cases turns out to beSO(1, 1)2.

The relevant O(1, 1)2 transformations can beread out from eqs. (62)-(64), (69) written in thenormal frame[21].

8

5. BPS Conditions for Enhanced Super-symmetry

In this section we will describe U-duality in-variant constraints on the multiplets of quantizedcharges in the case of BPS black holes whosebackground preserves more than one supersym-metry [22].

We will still restrict our analysis to four and fivedimensional cases for which three possible casesexist i.e. solutions preserving 1/8, 1/4 and 1/2 ofthe original supersymmetry (32 charges).

The invariants may only be non zero on solu-tions preserving 1/8 supersymmetry. In dimen-sions 6 ≤ D ≤ 9 black holes may only preserve1/4 or 1/2 supersymmetry, and no associated in-variants exist in these cases.

The description which follows also make con-tact with the D-brane microscopic calculation, asit will appear obvious from the formulae givenbelow. We will first consider the five dimensionalcase.

In this case, BPS states preserving 1/4 of su-persymmetry correspond to the invariant con-straint I3(27) = 0 where I3 was defined in eq.86. This corresponds to the E6 invariant state-ment that the 27 is a null vector with respectto the cubic norm. As we will show in a mo-ment, when this condition is fulfilled it may beshown that two of the central charge eigenvaluesare equal in modulus. The generic configurationhas 26 independent charges.

Black holes corresponding to 1/2 BPS statescorrespond to null vectors which are critical,namely

∂I(27) = 0 (89)

In this case the three central charge eigenvaluesare equal in modulus and a generic charge vectorhas 17 independent components.

To prove the above statements, it is useful tocompute the cubic invariant in the normal frame,given by:

I3(27) = Tr(ZC)3

= 6(e1 + e2)(e1 + e3)(e2 + e3)

= 6s1s2s3 (90)

where:

e1 =1

2(s1 + s2 − s3)

e2 =1

2(s1 − s2 + s3)

e3 =1

2(−s1 + s2 + s3) (91)

are the eigenvalues of the traceless antisymmetric8 × 8 matrix. We then see that if s1 = 0 then|e1| = |e2|, and if s1 = s2 = 0 then |e1| = |e2| =|e3|. To count the independent charges we mustadd to the eigenvalues the angles given by USp(8)rotations. The subgroup of USp(8) leaving twoeigenvalues invariant is USp(2)4, which is twelvedimensional. The subgroup of USp(8) leaving in-variant one eigenvalue is USp(4)×USp(4), whichis twenty dimensional. The angles are therefore36−12 = 24 in the first case, and 36−20 = 16 inthe second case. This gives rise to configurationswith 26 and 17 charges respectively, as promised.

Taking the case of Type II on T 5 we can chooses1 to correspond to a solitonic five-brane charge,s2 to a fundamental string winding charge alongsome direction and s3 to Kaluza-Klein momen-tum along the same direction.

The basis chosen in the above example is S-dualto the D-brane basis usually chosen for describ-ing black holes in Type IIB on T5. All other basesare related by U-duality to this particular choice.We also observe that the above analysis relatesthe cubic invariant to the picture of intersectingbranes since a three-charge 1/8 BPS configura-tion with non vanishing entropy can be thoughtas obtained by intersecting three single charge 1/2BPS configurations [31], [32], [33]

By using the S–T-duality decomposition wesee that the cubic invariant reduces to I3(27) =10−210−214 + 16116110−2. The 16 correspond toD-brane charges, the 10 correspond to the 5 KKdirections and winding of wrapped fundamentalstrings, the 1 correspond to the N-S five-branecharge.

We see that to have a non-vanishing area weneed a configuration with three non-vanishing N-S charges or two D-brane charges and one N-Scharge.

Unlike the 4-D case, it is impossible to have

9

a non-vanishing entropy for a configuration onlycarrying D-brane charges.

We now turn to the four dimensional case.In this case the situation is more subtle because

the condition for the 56 to be a null vector (withrespect to the quartic norm defined through eq.32) is not sufficient to enhance the supersymme-try. This can be easily seen by going to the nor-mal frame where it can be shown that for a nullvector there are not, in general, coinciding eigen-values. The condition for 1/4 supersymmetry isthat the gradient of the quartic invariant vanish.

The invariant condition for 1/2 supersymmetryis that the second derivative projected into theadjoint representation of E7 vanish. This meansthat, in the symmetric quadratic polynomials ofsecond derivatives, only terms in the 1463 of E7

are non-zero. Indeed, it can be shown, going tothe normal frame for the 56 written as a skew 8×8matrix, that the above conditions imply two andfour eigenvalues being equal respectively.

The independent charges of 1/4 and 1/2 pre-serving supersymmetry are 45 and 28 respec-tively.

To prove the latter assertion, it is sufficient tosee that the two charges normal-form matrix isleft invariant by USp(4)×USp(4), while the onecharge matrix is left invariant by USp(8) so theSU(8) angles are 63 − 20 = 43 and 63− 36 = 27respectively.

The generic 1/8 supersymmetry preservingconfiguration of the 56 of E7 with non vanish-ing entropy has five independent parameters inthe normal frame and 51 = 63−12 SU(8) angles.This is because the compact little group of thenormal frame is SU(2)4. The five parameters de-scribe the four eigenvalues and an overall phaseof an 8× 8 skew diagonal matrix.

If we allow the phase to vanish, the 56 quar-tic norm just simplifies as in the five dimensionalcase:

I4(56) = s1s2s3s4

= (e1 + e2 + e3 + e4)(e1 + e2 − e3 − e4)

×(e1 − e2 − e3 + e4)(e1 − e2 + e3 − e4) (92)

where ei (i = 1, · · · , 4) are the four eigenvalues.1/4 BPS states correspond to s3 = s4 = 0 while

1/2 BPS states correspond to s2 = s3 = s4 = 0.An example of this would be a set of four D-

branes oriented along 456, 678, 894, 579 (wherethe order of the three numbers indicates the ori-entation of the brane). Note that in choosing thebasis the sign of the D-3-brane charges is impor-tant; here they are chosen such that taken to-gether with positive coefficients they form a BPSobject. The first two possibilities (I4 6= 0 andI4 = 0, ∂I4

∂qi 6=0 ) preserve 1/8 of the supersymme-

tries, the third (∂I4∂qi = 0, ∂2I4∂qi∂qj |Adj E7 6= 0) 1/4

and the last ( ∂2I4∂qi∂qj |Adj E7 = 0) 1/2.

It is interesting that there are two types of 1/8BPS solutions. In the supergravity description,the difference between them is that the first casehas non-zero horizon area. If I4 < 0 the solutionis not BPS. This case corresponds, for example,to changing the sign of one of the three-branecharges discussed above. By U-duality transfor-mations we can relate this to configurations ofbranes at angles such as in [23]

Going from four to five dimensions it is naturalto decompose the E7 → E6×O(1, 1) where E6 isthe duality group in five dimensions and O(1, 1)is the extra T-duality that appears when we com-pactify from five to four dimensions. Accordingto this decomposition, the representation breaksas: 56→ 271 + 1−3 + 27′−1 + 13 and the quarticinvariant becomes:

564 = (271)31−3 + (27′−1)313 + 13131−31−3

+ 27127127′−127′−1 + 27127′−1131−3(93)

The 27 comes from point-like charges in fivedimensions and the 27′ comes from string-likecharges.

Decomposing the U-duality group into T- andS-duality groups, E7 → SL(2, IR) × O(6, 6), wefind 56 → (2,12) + (1,32) where the first termcorresponds to N-S charges and the second termto D-brane charges. Under this decompositionthe quartic invariant (92) becomes 564 → 324 +(12.12′)2 + 322.12.12′. This means that we canhave configurations with a non-zero area thatcarry only D-brane charges, or only N-S chargesor both D-brane and N-S charges.

It is remarkable that E7(7)-duality gives addi-tional restrictions on the BPS states other than

10

the ones merely implied by the supersymmetryalgebra. The analysis of double extremal blackholes implies that I4 be semi-definite positive forBPS states. From this fact it follows that config-urations preserving 1/4 of supersymmetry musthave eigenvalues equal in pairs, while configu-ratons with three coinciding eigenvalues are notBPS.

To see this, it is sufficient to write the quarticinvariant in the normal frame basis. A genericskew diagonal 8× 8 matrix depends on four com-plex eigenvalues zi. These eight real parameterscan be understood using the decomposition [34]:

56 → (8v,2,1,1) + (8s,1,2,1) + (8c,1,1,2)

+ (1,2,2,2) (94)

under

E7(7) → O(4, 4)× SL(2, IR)3 (95)

Here O(4, 4) is the little group of the normalform and the (2,2,2) are the four complex skew-diagonal elements. We can further use U(1)3 ⊂SL(2, IR)3 to further remove three relative phasesso we get the five parameters zi = ρie

iφ/4 (i =1, · · · , 4).

The quartic invariant, which is also the uniqueSL(2, IR)3 invariant built with the (2,2,2), be-comes [22]:

I4 =∑i

|zi|4 − 2

∑i<j

|zi|2|zj |

2

+4(z1z2z3z4 + z1z2z3z4)

= (ρ1 + ρ2 + ρ3 + ρ4)(ρ1 + ρ2 − ρ3 − ρ4)

× (ρ1 − ρ2 + ρ3 + ρ4)(ρ1 − ρ2 − ρ3 + ρ4)

+8ρ1ρ2ρ3ρ4(cosφ− 1) (96)

The last term is semi-definite negative. Thefirst term, for ρ1 = ρ2 = ρ becomes:

−[4ρ2 − (ρ3 + ρ4)2](ρ3 − ρ4)2 (97)

which is negative unless ρ3 = ρ4. So 1/4 BPSstates must have

ρ1 = ρ2 > ρ3 = ρ4 , cosφ = 1 (98)

For ρ1 = ρ2 = ρ3 = ρ, the first term in I4 be-comes:

−(3ρ+ ρ4)(ρ− ρ4)3 (99)

so we must also have ρ4 = ρ , cosφ = 1 whichis the 1/2 BPS condition.

An interesting case, where I4 is negative, cor-responds to a configuration carrying electric andmagnetic charges under the same gauge group,for example a 0-brane plus 6-brane configura-tion which is dual to a K–K-monopole plus K–K-momentum [24], [25]. This case correspondsto zi = ρeiφ/4 and the phase is tanφ/4 = e/gwhere e is the electric charge and g is the mag-netic charge. Using (92) we find that I4 < 0 un-less the solution is purely electric or purely mag-netic. In [26] it was suggested that 0 + 6 does notform a supersymmetric state. Actually, it wasshown in [27] that a 0 + 6 configuration can beT-dualized into a non-BPS configuration of fourintersecting D-3-branes. Of course, I4 is negativefor both configurations. Notice that even thoughthese two charges are Dirac dual (and U-dual)they are not S-dual in the sense of filling out anSL(2,ZZ) multiplet. In fact, the K–K-monopoleforms an SL(2,ZZ) multiplet with a fundamentalstring winding charge under S-duality [28]

6. Duality Orbits fo BPS States PreservingDifferent Numbers of Supersymmetries

In this section we give an invariant classifica-tion of BPS black holes preserving different num-bers of supersymmetries in terms of orbits of the27 and the 56 fundamental representations of theduality groups E6(6) and E7(7) resperctively [29],[30].

In five dimensions the generic orbits preserv-ing 1/8 supersymmetry correspond to the 26 di-mensional orbits E6(6)/F4(4) so we may think thegeneric 27 vector of E6 parametrized by a pointin this orbit and its cubic norm (which actuallyequals the square of the black-hole entropy).

The light-like orbit, preserving 1/4 supersym-metry, is the 26 dimensional coset E6(6)/O(5, 4)�T16 where � denotes the semidirect product.

The critical orbit, preserving maximal 1/2 su-persymmetry (this correspond to ∂I4

∂qi6= 0) corre-

spond to the 17 dimensional space

E6(6)

O(5, 5)� T16(100)

11

In the four dimensional case, we have two inequiv-alent 55 dimensional orbits corresponding to thecosets E7(7)/E6(2) and E7(7)/E6(6) depending onwhether I4(56) > 0 or I4(56) < 0. The firstcase corresponds to 1/8 BPS states whith non-vanishing entropy, while the latter corresponds tonon BPS states.

There is an additional 55 dimensional light-like orbit (I4 = 0) preserving 1/8 supersymmetry

given byE7(7)

F4(4)�T26.

The critical light-like orbit, preserving 1/4supersymmetry, is the 45 dimesnsional cosetE7(7)/O(6, 5)� (T32 ⊕ T1)

The critical orbit corresponding to maximal1/2 supersymmetry is described by the 28 dimen-sional quotient space

E7(7)

E6(6) � T27(101)

We actually see that the counting of parametersin terms of invariant orbits reproduces the count-ing previously made in terms of normal frame pa-rameters and angles. The above analysis makesa close parallel between BPS states preservingdifferent numbers of supersymmetries with time-like, space-like and light-like vectors in Minkowskispace.

Acknowledgements

The results of sections 5 and 6 have been ob-tained by one of the authors (S. F.) in collabora-tions with J.M. Maldacena and M. Gunaydin.

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