BERTOTTI{ROBINSON GEOMETRY and SUPERSYMMETRY S. Ferrara*) ABSTRACT

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CERN-TH/97-10

hep-th/9701163

BERTOTTI{ROBINSON GEOMETRY AND SUPERSYMMETRY

)

S. Ferrara

Theoretical Physics Division, CERN

CH - 1211 Geneva 23

ABSTRACT

The role of Bertotti{Robinson geometry in the attractor mechanism of extremal black

holes is describ ed for the case of N =2 sup ersymmetry. Its implication for a mo del-

indep endent derivation of the Bekenstein{Hawking entropy formula is discussed.

Talk given at the 12th Italian Conference on

General Relativity and Gravitational Physics

Rome, September 1996

CERN-TH/97-10

January 1997

BERTOTTI{ROBINSON GEOMETRY AND SUPERSYMMETRY

Sergio FERRARA

Theory Division, CERN

1211 Geneva 23, Switzerland

E-mail: ferraras@vxcern.cern.ch

ABSTRACT

The role of Bertotti{Robinson geometry in the attractor mechanism of extremal

black holes is describ ed for the case of N = 2 sup ersymmetry. Its implication

for a mo del-indep endent derivation of the Bekenstein{Hawking entropy formula

is discussed.

1. Extremal Black Holes and Attractors

In this rep ort, I will discuss some recent work on the macroscopic determination

of the Bekenstein{Hawking entropy-area formula for extremal black holes, using du-

ality symmetries of e ective string theories enco ded in the sup ergravity low-energy

actions. A new dynamical principle emphasizes the role played by \ xed scalars"

and Bertotti{Robinson typ e geometries in this determination. Sup ersymmetry seems

to be related to dynamical systems with xed p oints describing the equilibrium and

stability. The particular prop erty of the long-range b ehavior of dynamical ows in

dissipative systems is the following: in approaching the attractors the orbits lose

practically all memory of their initial conditions, even though the dynamics is strictly

deterministic.

The rst known example of such attractor b ehavior in the sup ersymmetric system

1;2

was discovered in the context of N = 2 extremal black holes . The corresp onding

motion describ es the b ehavior of the mo duli elds as they approach the core of the

black hole. They evolve according to a damp ed geo desic equation (see eq. (20) in )

until they run into the xed p oint near the black hole horizon. The mo duli at xed

p oints were shown to b e given as ratios of charges in the pure magnetic case . It was

further shown that this phenomenon extends to the generic case when b oth electric

and magnetic charges are present . The inverse distance to the horizon plays the

role of the evolution parameter in the corresp onding attractor. By the time mo duli

reach the horizon they lose completely the information ab out the initial conditions,

i.e. ab out their values far away from the black hole, which corresp ond to the values

of various coupling constants, see Fig. 1. The recent result rep orted here is the

derivation of the universal prop erty of the stable xed p oint of the sup ersymmetric

A p oint x where the phase velo city v (x )isvanishing is named a xedpoint and represents the

x x

system in equilibrium, v (x )=0. The xed p oint is said to b e an attractor of some motion x(t)if

lim x(t)=x (t):

t!1 x 1

2 e

2 4 6 8 10 12 14

Figure 1: Evolution of the dilaton from various initial conditions at in nity to a

common xed p ointatr=0.

attractors: xed p oint is de ned by the new principle of a minimal central charge and

the area of the horizon is prop ortional to the square of the central charge, computed

at the p oint where it is extremized in the mo duli space. In N = 2, d = 4 theories,

which is the main ob ject of our discussion here, the extremization has to b e p erformed

in the mo duli space of the sp ecial geometry and is illustrated in Fig. 1. This results

in the following formula for the Bekenstein{Hawking entropy S , which is prop ortional

to the quarter of the area of the horizon:

S = = jZ j ; d =4 : (1)

This result allows generalization for higher dimensions, for example, in ve-dimensional

space-time one has

3=2

S = jZ j ; d =5 : (2)

There exists a b eautiful phenomenon in the black hole physics: according to the

no-hair theorem, there is a limited number of parameters which describ e space and

physical elds far away from the black hole. In application to the recently studied

black holes in string theory, these parameters include the mass, the electric and

magnetic charges, and the asymptotic values of the scalar elds.

It app ears that for sup ersymmetric black holes one can prove a new, stronger

version of the no-hair theorem: black holes lose all their scalar hair near the horizon.

We are assuming that the extremum is a minimum, as it can b e explicitly veri ed in some mo dels.

However for the time b eing we cannot exclude situations with di erent extrema or even where the

equation D Z = 0 has no solutions.

This numb er can b e quite large, e.g. for N = 8 sup ersymmetry one can have56charges and 70

mo duli. 2

Black hole solutions near the horizon are characterized only by those discrete param-

eters which corresp ond to conserved charges asso ciated with gauge symmetries, but

not by the values of the scalar elds at in nity which maychange continuously.

A simple example of this attractor mechanism is given by the dilatonic black holes

4;5

of the heterotic string theory . The mo dulus of the central charge in question which

is equal to the ADM mass is given by the formula

0 0

M = jZ j = (e jpj +e jqj) : (3)

AD M

In application to this case the general theory, develop ed in this pap er gives the fol-

lowing recip e to get the area

i) Find the extremum of the mo dulus of the central charge as a function of a

2 2

dilaton e = g at xed charges

@ 1 1 1 @

jZ j(g ; p; q )= jpj + gjqj = jpj + jqj =0: (4)

@g 2@g g g

ii) Get the xed value of the mo duli

g = : (5)

iii) Insert the xed value into your central charge formula (3), get the xed value

of the central charge: the square of it is prop ortional to the area of the horizon and

de nes the Bekenstein{Hawking entropy

S = = jZ j = jpq j : (6)

5;6

This indeed coincides with the result obtained b efore by completely di erent metho ds .

In general sup ersymmetric N = 2 black holes have an ADM mass M dep ending

on charges (p; q )aswell as on mo duli z through the holomorphic symplectic sections

710

X (z );F (z) . The mo duli present the values of the scalar elds of the theory

far away from the black hole. The general formula for the mass of the state with

one half of unbroken sup ersymmetry of N = 2 sup ergravity interacting with vector

7 10

multiplets as well as with hyp ermultiplets is

2 2

M = jZ j ; (7)

where the central charge is

K(z;z)

Z (z; z; q; p)= e (X (z )q F (z ) p )= (L q M p ) ; (8)

so that

2 2 2

M = jZ j = M (z; z ; p; q ) : (9)

AD M AD M 3

The area, however, is only charge dep endent:

A = A(p; q ) : (10)

This happ ens since the values of the mo duli near the horizon are driven to the xed

p oint de ned by the ratios of the charges. This mechanism was explained b efore in

2 8

and on the basis of the conformal gauge formulation of N = 2 theory .

This attractor mechanism is by no means an exclusive prop erty of only N = 2

theory in four dimensions. Our analysis suggests that it may be a quite universal

phenomenon in any sup ersymmetric theory. It has in fact b een extended to all N > 2

theories in four dimensions and to all theories in ve dimensions . Further p ossible

extensions to higher dimensions and to higher extended ob jects (p-branes) have also

b een discussed in recent literature .

In this rep ort we will use the \co ordinate free" formulation of the sp ecial geo-

9;7;10

metry which will allow us to present a symplectic invariant description of the

system. We will b e able to show that the unbroken sup ersymmetry requires the xed

p oint of attraction to b e de ned by the solution of the duality symmetric equation

D Z =(@ + K )Z(z; z ; p; q )=0 ; (11)

i i i

which implies,

jZ j =0 (12)

Z = Z = Z L (p; q );M (p; q );p;q : (13)

Equation @ jZ j = 0 exhibits the minimal area principle in the sense that the area

is de ned by the extremum of the central charge in the mo duli space of the sp ecial

geometry, see Fig. 2 illustrating this p oint. Up on substitution of this extremal values

of the mo duli into the square of the central charge we get the Bekenstein{Hawking

entropy,

S = = jZ j : (14)

The area of the black hole horizon has also an interpretation as the mass of the

Bertotti{Robinson universe describing the near horizon geometry.

A=4 = M : (15)

This mass, as di erent from the ADM mass, dep ends only on charges since the mo duli

near the horizon are in their xed p oint equilibrium p ositions,

2 2 2

M = jZ j = M (p; q ) : (16)

BR BR 4

jZ j

i i

z (p; q ) z

Figure 2: Extremum of the central charge in the mo duli space.

Note that in the Einstein{Maxwell system without scalar elds the ADM mass of the

extreme sup ersymmetric black hole simply coincides with the Bertotti{Robinson one,

b oth b eing functions of charges:

2 2

M (p; q )= M (p; q ) : (17)

AD M BR

We will describ e b elow a near horizon black holes of N = 2 sup ergravity inter-

acting with vector and hyp er multiplets. The basic di erence from the pure N = 2

sup ergravity solutions comes from the following: the metric near the horizon is of the

Bertotti{Robinson typ e, as b efore. However, the requirementofunbroken sup ersym-

metry and duality symmetry forces the mo duli to b ecome functions of the ratios of

charges, i.e. take the xed p oint values. We will describ e these con gurations, show

that they provide the restoration of full unbroken N = 2 sup ersymmetry near the

horizon. We will call them N = 2 attractors, see Sec. 2. We will brie y rep ort on

the extension of the attractor mechanism to more general (higher N and higher D )

theories in Section 3.

2. Bertotti{Robinson Geometry and xed scalars

The sp ecial role of the Bertotti{Robinson metric in the context of the solitons in

sup ergravity was explained by Gibb ons . He suggested to consider the Bertotti{

Robinson (BR) metric as an alternative, maximally sup ersymmetric, vacuum state.

The extreme Reissner{Nordstrom metric spatially interp olates between this vacuum

and the trivial at one, as one exp ects from a soliton. 5

Near the horizon all N = 2 extremal black holes with one half of unbroken sup er-

symmetry restore the complete N =2unbroken sup ersymmetry. This phenomenon of

the doubling of the sup ersymmetry near the horizon was discovered in the Einstein{

14 15

Maxwell system in . It was explained in that the manifestation of this doubling of

unbroken sup ersymmetry is the app earance of a covariantly constant on shell sup er-

eld of N = 2 sup ergravity. In presence of a dilaton this mechanism was studied in .

In the context of exact four-dimensional black holes, string theory and conformal the-

ory on the world-sheet the BR space-time was studied in . In more general setting

the idea of vacuum interp olation in sup ergravity via sup er p-branes was develop ed

in .

We will show here using the most general sup ersymmetric system of N = 2 sup er-

gravity interacting with vector multiplets and hyp ermultiplets how this doubling of

sup ersymmetry o ccurs and what is the role of attractors in this picture. The sup er-

symmetry transformation for the gravitino, for the gaugino and for the hyp erino are

9;10 d

given in the manifestly symplectic covariant formalism in the absence of fermions

and in absence of gauging as follows:

= D + T ;

A A AB

i AB iA i A

F ; = i @ z +

B u A

= i U @ q C ; (18)

iA A

where , are the chiral gaugino and gravitino elds, isahyp erino, ; are

A A

the chiral and antichiral sup ersymmetry parameters resp ectively, is the SO(2)

Ricci tensor. The mo duli dep endent duality invariant combinations of eld strength

i B 19

T ; F are de ned by eqs. (31), U is the quaternionic vielb ein .

Our goal is to nd solutions with unbroken N = 2 sup ersymmetry. The rst one

is a standard at vacuum: the metric is at, there are no vector elds, and all scalar

elds in the vector multiplets as well as in the hyp ermultiplets take arbitrary constant

values:

2 i i i u u

ds = dx dx ; T = F =0 ; z =z ; q = q : (19)

0 0

This solves the Killing conditions = = = 0 with constant unconstrained

values of the sup ersymmetry parameter . The unbroken sup ersymmetry manifests

itself in the fact that each non-vanishing scalar eld represents the rst comp onent

of a covariantly constant N = 2 sup er eld for the vector and/or hyp er multiplet, but

the sup ergravity sup er eld vanishes.

The second solution with unbroken sup ersymmetry is much more sophisticated.

First, let us solve the equations for the gaugino and hyp erino by using only a part of

the previous ansatz:

i i u

F =0 ; @ z =0 ; @ q =0 : (20)

d 10

The notation is given in . 6

The Killing equation for the gravitino is not gauge invariant. We may therefore

15 16

consider the variation of the gravitino eld strength the wayitwas done in , . Our

ansatz for the metric will b e to use the geometry with the vanishing scalar curvature

and Weyl tensor and covariantly constant graviphoton eld strength T :

R =0 ; C =0 ; D (T )= 0 : (21)

15 16

It was explained in , that such con guration corresp onds to a covariantly constant

sup er eld of N = 2 sup ergravity W (x; ), whose rst comp onentisgiven byatwo-

comp onent graviphoton eld strength T . The doubling of sup ersymmetries near

the horizon happ ens by the following reason. The algebraic condition for the choice

of broken versus unbroken sup ersymmetry is given in terms of the combination of the

Weyl tensor plus or minus a covariant derivative of the graviphoton eld strength,

dep ending on the sign of the charge. However, near the horizon b oth the Weyl cur-

vature and the vector part vanish. Therefore b oth sup ersymmetries are restored and

we simply have a covariantly constant sup er eld W (x; ). The new feature of the

generic con gurations which include vector and hyp er multiplets is that in addition to

acovariantly constant sup er eld of sup ergravity W (x; ), we have covariantly con-

stant sup er elds, whose rst comp onent is given by the scalars of the corresp onding

multiplets. However, now as di erent from the trivial at vacuum, which admits any

values of the scalars, we have to satisfy the consistency conditions for our solution,

which requires that the Ricci tensor is de ned by the pro duct of graviphoton eld

strengths,

0 0

R = T T ; (22)

0 0

and that the vector multiplet vector eld strength vanishes

F =0 : (23)

Before analysing these two consistency conditions in terms of symplectic structures

of the theory, let us describ e the black hole metric near the horizon.

The explicit form of the metric is taken as a limit near the horizon r = j~xj! 0of

the black hole metric

2 2U 2 2U 2

ds = e dt + e d~x ; (24)

where

e =0 : (25)

We cho ose

A M

e = = ; (26)

2 2

4 j~xj r

where the Bertotti{Robinson mass is de ned by the black hole area of the horizon

M = : (27)

4 7

We may show that this metric, which is the Bertotti{Robinson metric

2 2

j~xj M

2 2 2

ds = dt + d~x ; (28)

M j~xj

is conformally at in the prop erly chosen co ordinate system. In spherically symmetric

co ordinate system

2 2

M r

2 2 2 2

dt + (dr + r d ) : (29) ds =

M r

2 2

After the change of variables r = M = and j~xj = M =j~yj the metric b ecomes

BR BR

obviously conformally at

2 2 2

M M M

BR BR BR

2 2 2 2 2 2

ds = dt + (d + d ) = (dt + d~y ) ; (30)

2 2 2

j~yj

which is in agreement with the vanishing of the Weyl tensor.

Now we are ready to describ e our solution in terms of symplectic structures, as

de ned in . The symplectic structure of the equations of motion comes by de ning

the Sp(2n +2) symplectic (antiselfdual) vector eld strength (F ; G ).

Two symplectic invariant combinations of the symplectic eld strength vectors

are:

T = M F L G

i ij

F = G (D M F D L G ) : (31)

j j

The central charge as well as the covariant derivative of the central charge are de ned

as follows:

T ; (32) Z =

and

Z D Z = F G : (33)

i i

The central charge, as well as its derivative, are functions of mo duli and electric

and magnetic charges. The ob jects de ned by eqs. (31) have the physical meaning

of b eing the (mo duli-dep endent) vector combinations which app ear in the gravitino

and gaugino sup ersymmetry transformations resp ectively. In the generic p oint of the

mo duli space there are two symplectic invariants homogeneous of degree 2 in electric

and magnetic charges :

2 2

I = jZ j + jD Z j ;

1 i

2 2

I = jZ j jD Zj : (34)

2 i 8

Note that

I = I (p; q; z; z)= P M(N)P ;

1 1

I = I (p; q; z; z)= P M(F)P : (35)

2 2

Here P =(p; q ) and M(N ) is the real symplectic 2n +2 2n+ 2 matrix

A B

(36)

C D

where

1 1

A = ImN +ReNImN ReN ; B = ReN ImN

1 1

C = ImN ReN ; D =ImN : (37)

7 10

The vector kinetic matrix N was de ned in Refs. . The same typ e of matrix

app ears in (35) with N ! F = F . Both N ; F are Kahler invariant functions,

710

which means that they dep end only on ratios of sections, i.e. only on t ;f .

The unbroken sup ersymmetry of the near horizon black hole requires the consis-

tency condition (23), which is also a statement ab out the xed p oint for the scalars

z (r )as functions of the distance from the horizon r

z (r) =0 =) DZ =0 : (38)

Thus the xed p oint is de ned due to sup ersymmetry by the vanishing of the covariant

derivative of the central charge. At this p oint the critical values of mo duli b ecome

functions of charges, and two symplectic invariants b ecome equal to each other:

2 2

I = I =(jZj ) jZ j : (39)

1 x 2 x DZ=0 x

The way to explicitly compute the ab ove is by solving in a gauge-invariant fashion

eq. (38), yielding:

p = i(ZL ZL ) ; q = i(ZM ZM ) : (40)

From the ab ove equations it is evident that (p; q ) determine the sections up to a

0 K=2

(Kahler) gauge transformation (which can b e xed setting L = e ). Vice versa the

xed p oint t can only dep end on ratios of charges since the equations are homoge-

neous in p,q.

The rst invariant provides an elegant expression of jZ j which only involves the

charges and the vector kinetic matrix at the xed p oint N = N t ; t ;f ; f .

x x x

x x

2 2 t 2

(I ) =(jZj +jD Zj ) = P M(N )P =(jZ j ) : (41)

1 x i x x x

2 9

Indeed eq. (41) can be explicitly veri ed by using eq. (42). For magnetic solutions

the area formula was derived in . This formula presents the area as the function of

the zero comp onent of the magnetic charge and of the Kahler p otential at the xed

p oint.

0 2 K

A = (p ) e : (42)

In the symplectic invariant formalism wemaycheck that the area formula (42) which

is valid for the magnetic solutions (or for generic solutions but in a sp eci c gauge

only) indeed can be broughtto the symplectic invariant form :

0 2 K 2 2 2

A = (p ) e =4(jZj +jDZj ) =4(jZ j )=2p ImF p : (43)

i x x

One can also check the rst consistency condition of unbroken sup ersymmetry

(22), which relates the Ricci tensor to the graviphoton. Using the de nition of the

central charge in the xed p ointwe are lead to the formula for the area of the horizon

(which is de ned via the mass of the Bertotti{Robinson geometry) in the following

form

A A

2 2 2

M = =(jZj ) ; S = = M : (44)

DZ=0

BR BR

4 4

The new area formula (44) has various advantages following from manifest sym-

plectic symmetry. It also implies the principle of the minimal mass of the Bertotti{

Robinson universe, whichisgiven by the extremum in the mo duli space of the sp ecial

geometry.

@ M =0 : (45)

i BR

3. Attractors in more general theories

The previous analysis admits an extension to higher N theories at D =4 and to

N 2 theories at D =5. The general condition for getting a non-vanishing entropy-

area formula is that extremal black-holes preserve only one sup ersymmetry (1/4 for

N = 4 and 1/8 for N = 8). This condition is again obtained by extremizing the

ADM mass in the mo duli space. In this case the ADM mass is given, at D =4, by

the highest eigenvalue of the central charge matrix and sup ersymmetry implies that

when the ADM mass is extremized, the other eigenvalues vanish. The vanishing of

the other eigenvalues imply that the variation of the spin-1/2 partners of the gravitino

vanishes on the residual unbroken (Killing) sup ersymmetry .

The entropy-area formula is always given byaU-dualityinvariant expression built

20 ; 21

out of the electric and magnetic charges . This is a consequence of the fact that

b oth for D = 4 and 5 the area is given by an appropriate p ower of the extremized ADM

e 1

In this pap er we have a normalization of charges which is di erent from due to the use of the

10 8

conventions of and not . 10

mass whichisaU-dualityinvariant expression (the central charge is an expression of

3 ; 11

rst degree in terms of electric and magnetic charges) .

Similar arguments show that the attractor mechanism gives a vanishing (or con-

stant) result for D > 5 since a U -duality invariant expression do es not exist in that

case. Similar considerations can b e extended to higher p-extended ob jects in any D .

Finally, we note that the mechanism of the doubling of sup ersymmetry at the

attractor p oint is still op erating in ve dimensions . The analogue of the Bertotti{

Robinson geometry (which is AdS S ) is in this case the Tangherlini extremal

23 3

D = 5 black hole (with top ology of AdS S ) which indeed admits two Killing

spinors. This is the xed mo duli geometry of the D = 5 attractors.

Recently many applications of these ideas have b een worked out, esp ecially in

the case of string theory compacti ed on three-dimensional Calabi{Yau complex

manifolds . Determination of the top ological entropy formula by counting micro-

cospic states in string theory,by means of D -brane techniques, has also b een p erformed

and shown to give results, whenever obtainable, in agreement with the mo del-indep endent

determination which uses the attractor mechanism.

Finally it should b e mentioned that several prop erties of \ xed scalars" have b een

investigated . In particular, it has b een shown that the attractor mechanism is also

relevant in the discussion of black hole thermo dynamics out of extremality .

4. Acknowledgements

It was a pleasure to give this talk on the o ccasion of the 65th birthday of Prof.

Bruno Bertotti. The material covered in this talk comes mainly from work done

jointly with Renata Kallosh and Andrew Strominger, b oth of which I would like to

thank for very pleasant and fruitful collab oration.

This work was supp orted in part by EEC under TMR contract ERBFMRX-CT96-

0045, Frascati, and by DOE grant DE-FG03-91ER40662. 11

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