Instantons and Seven-Branes in Type Iib Superstring

Home , Cosmic string

DAMTP/R95-56

INSTANTONS AND SEVEN-BRANES

IN TYPE I IB SUPERSTRING THEORY.

Gary W. Gibb ons, Michael B. Green and Malcolm J. Perry

DAMTP, Silver Street, Cambridge CB3 9EW, UK.

Instanton and seven-brane solutions of typ e I IB sup ergravity carrying charges in

the Ramond-Ramond sector are constructed. The singular seven-brane has a quantized

R R `magnetic' charge whereas its dual is the instanton, which is non-singular in

the string frame and carries a global `electric' charge. The pro duct of these charges is

constrained by a Dirac quantization condition. The instanton has the form of a space-

time wormhole in the string frame, and is resp onsible for the violation of the conservation

of the `electric' charge.

? G.W.Gibb [email protected]; [email protected]; [email protected]

1. Intro duction

Recent developments in our understanding of sup erstring theory suggest that all

known theories may b e viewed as di erent p erturbative approximations of a single un-

derlying theory. Viewed from the standp ointofany given string theory the fundamental

strings of other theories are solitonic states that are not apparent in string p erturbation

theory. Other p-branes also arise as solitons, suggesting an imp ortantr^ole for ob jects

with all p ossible values of p in the theory.Intyp e I I theories the fundamental strings

carry the charges of the Neveu{Schwarz-Neveu{Schwarz (NS NS ) sector but do not

carry Ramond{Ramond (R R)charges. These latter charges are asso ciated with

(p + 1)-form p otentials or (p + 2)-form eld strengths, F , that are carried by some of

p+2

the known p-branes of the these theories. The solitons that carry the R Rcharge that

have b een constructed so far are the zero-brane (black hole), two-brane, four-brane and

six-brane solutions of the typ e I IA theory and the one-brane (string), self-dual three-

brane and ve-brane solitons of typ e I IB string theory (for a review see [1]). In fact,

there is an in nite-dimensional SL(2; Z)multiplet of b oth `dyonic' one-branes [2] and

ve-branes in the typ e I IB theory.

Recently, some p-brane solitons have b een asso ciated with sup erstring con gurations

known generically as D -branes (or D -instantons in the p = 1 case) [4] which are sources

of the R Rcharge. This asso ciation suggests that there should b e solitons for al l values

of p from p = 1top= 9. The case p = 9 is very sp ecial since the accompanying eld

strength vanishes identically, and is connected with the presence of chiral anomalies in

typ e I theories with any gauge group other than SO(32). The p = 8 soliton constructed

in [3] couples to a cosmological constant in the typ e I IA theory [4, 5] and is a solution

of `massive' typ e I IA sup ergravity [6] . This pap er considers the p = 1 (instanton) and

p = 7 (seven-brane) solutions of the typ e I IB theory whichhaveR Rcharges that are

related by a Dirac-like quantization condition.

Although the construction to b e describ ed b ears a resemblance to the construction

of the previously discovered p-branes there are fascinating new features. For example,

the `electric' R Rcharge carried by the instanton is a global charge. The instanton

solution is non-singular in the string frame { in fact it is a wormhole that leads to the

violation of the global R R No ether charge. Whereas other p-brane solutions can b e

thoughtofaswormholes with in nite throats the instanton is genuinely an Einstein-

Rosen wormhole [7] which connects two asymptotically euclidean regions of space-time.

The seven-brane solution, carrying the dual `magnetic' charge, is related to the stringy

cosmic string solution of [8].

The solutions of typ e I IB sup ergravitywe will consider are ones in which the two

scalar elds (the dilaton, , and the R R scalar, a) and the metric have non-trivial

b ehaviour while the other b osonic elds (the two third-rank eld strengths and the

self-dual fth-rank eld strength) vanish. The ten-dimensional lagrangian for the non-

vanishing elds is

2 2 2

1 1

L = R (@) e (@a) (1)

2 2

in the Einstein frame (where the signature is ( + + + + + + + + +)). De ning a 2

nine-form eld strength, F = e da, i.e.,

(9)

F = e @ a; (2)

:::

1 9

:::

1 9

the lagrangian can b e written in the equivalent form,

2 2 :::

1 9

(@) L = R e F F : (3)

:::

1 9

2(9!)

The passage from (1) to (3) is a standard duality transformation that can b e achieved by

intro ducing a Lagrange multiplier constraint for (2). The eld equation for F coming

(9)

from (3) is

r e F =0; (4)

:::

1 8

which is equivalent to the Bianchi identity for a.

2. The instanton solution

The form of L remains unchanged after Wick rotation to euclidean signature, and

therefore the contribution of the dilaton and nine-form to the action,

I = L + b oundary terms; (5)

is p ositive de nite. The eld equations that arise from the euclidean version of (3) have

a form that could have b een obtained from

2 2 2

1 1

(@) + e (@ ) ; (6) L = R

2 2

which is (1) with the substitution a ! = ia. The actions (3) and (6) di er by surface

terms that can arise in replacing da by F . This action is invariant under a euclidean

(9)

version of N = 2 sup ersymmetry. The equations of motion that follow from (6) are

@ @ e @ @ =0; R

(7)

r e @ =0;

2 2 2

r e (@ ) =0:

Wenow turn to the conditions that need to b e satis ed for a solution of the euclidean

theory to preserve half the total sup ersymmetries which requires a brief discussion of the 3

euclidean version of the N = 2 sup ersymmetry of the typ e I IB theory. This could b e

expressed in terms of symplectic spinors but it is more succinct to express the euclidean

theory in a manner that parallels the usual discussion with lorentzian signature. Using

the conventions in [9] the lorentzian signature sup ersymmetry transformations of the

elds can b e written in a complex notation. They can then b e systematically adapted

to the case of euclidean signature by replacing the usual algebra C of complex numb ers,

generated over the reals R by 1 and i (where i = 1) by the so-called `double' or

hyp erb olic complex numb ers E, which are generated over the reals by 1 and e, where

e = +1. Conjugation of complex numb ers is an automorphism taking 1 to 1 and i to

i. Similarly conjugation of double numb ers takes 1 to 1 and e to e.Thus a general

2 2

double number u = a + eb (a; b 2 R) is conjugate to u = a eb and uu = a b .

As an algebra over R the double numb ers are reducible E = R R. The asso ciated

pro jectors P = (1 e) are lightlike, P P = 0, with resp ect to the inde nite inner

pro duct uu .

The sup ersymmetry transformations of the C-valued spin-1=2 and spin-3=2 elds,

and are given in [9] (in the Einstein frame). In a b osonic background in which only

= a + ie = + i and g are non-zero they are,

1 2

= (@ + i@ )( i ); (8)

1 2 1 2

+ i

1 1

= @ + ! i Q ( + i ); (9)

a 1 2

4 2

where the comp osite U (1) gauge p otential is de ned by

1 i + i

Q = (@ i@ ) (@ + i@ ) (10)

1 2 1 2

4 i + i

These expressions have b een transformed from the SU (1; 1)-invariant form in [9] to the

parameterization with manifest SL(2; R )-invariance.

For the application to the euclidean instanton solution, wechange i to e everywhere

. The ansatz in these transformations and make the identi cations = and = e

1 2

that leads to the preservation of half the sup ersymmetry is d = d and the metric

1 2

is at, i.e.

d = e d; g = : (11)

It follows from (8) that =0 if = .From hereon we will arbitrarily cho ose the

1 2

plus sign. Furthermore, to obtain =0we note that the spin connection vanishes

(! = 0) since the metric is at in the Einstein frame. Thus,

=(1+e)(@ + (@ ) ): (12)

1 2 1

4 (1 + 2 )

2 2

0 0

This can b e made to vanish by setting = f ( ) , where is an arbitrary constant

1 2

1 1

real spinor, and cho osing f ( ) appropriately.

2 4

The conditions (11) combined with the equations of motion (7) lead to e =

2 2

and @ = (@) , so that

@ (e )=0: (13)

This has a spherically symmetric solution describing a single instanton,

e = e + ; (14)

where is the value of the dilaton eld at r = 1 and c is a constant that is arbitrary

at this stage, and will b e shown b elow to b e prop ortional to the instanton charge.

This single instanton soluton is evidently singular at r = 0 in the Einstein frame.

However, it is natural to transform from the Einstein frame (in which the metric is simply

2 2

ds = dx ) to the string frame in which the metric is given by,

1=2

2 2 2 2 =2 2

dr + r d ; (15) + ds = e ds = e

2 n

where d is the SO(n)-invariant line elementonS . This metric is manifestly invariant

under the inversion transformation,

1=4

) r ! (ce ; (16)

which shows that the region r ! 0 is another asymptotically Euclidean region identical

to that near r = 1. In fact, the solution in this frame is a worm-hole in which there are

twoaymptotically euclidean regions connected by a neck. The space-time is geo desically

complete so in this sense it is non-singular.

The euclidean action of the instanton is given (in the Einstein frame) by substituting

the solution into (5),

Z Z

2 2 :::

1 9

(@) + e F I = F R + 2 [TrK ]; (17)

::: inst

1 9

2(9!)

M @M

where [TrK ] is the di erence b etween the trace of the extrinsic curvature on the b oundary

and the value it would have if the b oundary were lled in with at space. For our

instanton solution R = 0 and the b oundary contributions also vanish. The contribution

, but from in nity clearly vanishes. That from r = 0 also vanishes since TrK =

when integrated over a small nine-sphere of radius r , it will vanish. Using the fact that 5

2 2 :::

1 9

(@) = e F F =9!, one obtains

:::

1 9

Z Z Z Z

2 2

@ = @ d + @ d : (18) (@) = I =

inst

10 10

r =1

r=0

R R

Using the explicit form of in (14) the contribution from r = 0 is seen to vanish whereas

the contribution from r = 1 gives the total action,

5=2

I =8c Vol (S )= c; (19)

inst

9 5=2

where Vol(S )=2 =(5) is the volume of the unit nine-sphere.

(1)

The electric charge Q asso ciated with the one-form F = d is conserved by

(1)

virtue of the eld equation

2 2

r (e F )= r (e @ )=0 (20)

(1)

The electric charge Q carried by the instanton coincides with the No ether charge for

the translation symmetry ! + constant. Thus

Z Z Z

(1) 2 N oether

Q = e F d = J d = F (21)

(9)

9 9 9

S S S

N oether 2

where J = e @ . Conservation of the No ether current is equivalent to the eld

equation for F ,

(1)

N oether 2

r J = r (e F )=0 (22)

(1)

For our solution

Z Z

2 9

Q = e @ d = e @ d = 8cg V ol (S ) (23)

(1)

so that

(1)

jQ j

; (24) I =

inst

where g = e is the string coupling constant. This shows the exp ected dep endence

of the instanton action on the coupling constant. The energy of the p-brane solitons

of the R R sector have a similar 1=g dep endence [10,11]. Such e ects are exp ected

from general considerations in string theory [12] based on estimates of the divergence of

closed-string p erturbation theory and matrix mo del calculations. Furthermore, we will

(1)

show later that the constant c is determined by a quantization condition that xes Q

to b e 2n, where n is an integer and the action (24) is that of a D-instanton [4]. 6

Multi-instanton solutions, with k + 1 asymptotically Euclidean regions lo cated at

x are obtained by taking the solution of (13),

i=k

e = e ; (25) +

(x x )

i=1

(1)

where c are p ositive constants that de ne the charges, jQ j =8cg Vol (S ). These

i i

instantons are simply connected. One may also construct more complicated non-simply

connected wormhole typ e solutions using in nitely many mass p oints.

3. The seven-brane solution

The seven-brane is a soliton with an eight-dimensional world-volume characterized

by non-trivial b ehaviour of the scalar elds and the metric in the two transverse euclidean

dimensions. De ning = a and = e (where = + i spans the upp er-half

1 2 1 2

plane) the lagrangian (1) is identi ed with the SL(2; R)-invariant lagrangian considered

in [8],

Z Z

2 2

(@ ) +(@ )

1 2

I = R + 2[TrK ]; (26)

The global SL(2; R ) symmetry of this action is exp ected to b e broken in string theory to

a residual SL(2; Z ) that is interpreted as a lo cal symmetry, so that should b e restricted

to the fundamental domain of the mo dular group, j j1, 1=2 1=2. Thus,

the target space of the sigma mo del is a non-compact orbifold of nite volume with two

, and a cusp at = 1. Many arguments have b een orbifold p oints at = i and = e

advanced that p oint to this breaking of the continuous global symmetry. This restriction

of the domain of the scalar elds is an essential ingredient in the following. Moreover

it app ears that unless one restricts the domain in this way, there are probably no nite

energy solutions.

The ansatz for the seven-brane solution is one in which the elds are trivial in the

eight dimensions of the world-volume of the brane and non-trivial in the transverse space

(r , ) so that the metric in the Einstein frame takes the form,

2 1 2 7 2 2 2 2 2 2

= dt +(dx ) + :::(dx ) + (dr + r d ); (27) ds

where = (r;). Intro ducing the complex co-ordinate z = re , the complex scalar

eld is taken to satisfy the holomorphic (or antiholomorphic) ansatz, @ = 0 (or

@ = 0), used in [8] to give stringy cosmic string solutions. The Einstein equation for 7

2@ @

@ @ ln = =2@@ln : (28)

( )

For a single seven-brane the solution of (28) is

j ( (z )) = z; (29)

where j is the elliptic mo dular function. Then b ecause r as r !1, the space

transverse to the seven-brane is asymptoically conical with de cit angle = .We can

easily see that this is consistent with sup ersymmetry by considering (8) and (9). We will

now use with complex numb ers to describ e sup ersymmetry, rather than double numb ers,

1 2

b ecause the signature is Lorentzian. Demanding that ( + i )=i( +i )

1 2 1 2

enforces the condition = 0 since ! = Q for this ansatz. In other words the

spin connection and the comp osite gauge connection cancel. Asymptotically conical

spacetimes usually do not admit covariantly constant spinors, but in this case, the non-

trivial gravitational holonomy is neutralized by that of the U (1) gauge eld. The fact

that = 0 follows from (8) making use of the holomorphicity of the eld .

A solution of the equations of motion that follows from (26) has R = ((@ ) +

2 2

(@ ) )= so that the energy of the solution comes entirely from the b oundary con-

tribution. The energy p er unit seven-volume of the seven-brane with these b oundary

conditions satis es a Bogomol'nyi b ound

; (30) E =2

with equality if and only if is either holomorphic or antiholomorphic corresp onding to

the sup ersymmetric case [13].

(7)

The magnetic charge P of the seven-brane is

I Z

(7)

P = F d = d = 1 (31)

where the line integral is taken around a closed lo op at in nity, and wehave used the

fact that as r !1;a . This charge is lo calized on the inverse images of i and

e , the two orbifold p oints of the target space. Thus this seven-brane b ehaves as one

would exp ect of a singular Coulomb-like source for the electric eld.

One can straightforwardly extend the analysis of [8] for multi-cosmic strings to nd

multi-seven-brane solutions by replacing (29) by

P (z )

j ( (z )) = (32)

Q(z )

where P (z ) and Q(z ) are p olynomials in z of order m and n resp ectively with no common

factors. If m>n;we obtain a solution with k = m seven-branes. If m n; the 8

solution rep esents k = n seven-branes. Provided that k<12, the transverse space is

asymptotically conical. If k = 12, the transverse space is asymptotically cylindrical, and

if k>12 the transverse space has nite volume; in general it is singular apart from

the exceptional case of k = 24. There app ears to b e an interesting di erence b etween

multi-seven-branes and multi-instantons. Two seven-branes may continuously approach

one another until they coincide, the resultant seven-brane having a charge equal to the

sum of the individual charges. By contrast, for our instantons the analogous pro cess

do es not seem to b e p ossible.

4. Quantization condition

The R Rcharges carried by the seven-brane and the instanton are related by the

same quantization condition that applies to magnetic and electric charges of the other

pairs of branes with p andp ~ [14,15,16] where in the case of sup erstrings p +~p=6,

(~p) (p)

P Q =2n; n 2 Z: (33)

In our case, p = 1 andp ~ = 7, so that substituting the fact that the electric charge

(7)

of the elementary seven-brane solution has a quantized charge, P = m (where jmj =

0; 1;:::;11; 12; 24), gives a quantization condition on the charge carried by the instanton,

(1)

P =2n: (34)

Using (23) this determines c in the single instanton solution to b e

3jnj

c = : (35)

3=2

With this value the action for the instanton in (24) agrees with that of the D -instanton

[4], where it is determined in an altogether di erent pro cedure as a functional integral

over a string world-sheet with the top ology of a disk. 9

5. Discussion

In this pap er wehave argued that the D-Instanton of typ e I IB string theory maybe

identi ed with a BPS spacetime wormhole solution of the euclidean ten-dimensional typ e

I IB sup ergravity theory carrying R R electric charge. Wehave also argued that the dual

magnetically charged seven-brane may b e identi ed with a solution constructed from the

stringy cosmic string of [8]. Consistent with our interpretation, the instanton solution

(1) (1)

has nite euclidean action I = jQ j, where jQ j is the electric charge. The

inst

instanton is geo desically complete in the string frame and the dilaton eld is everywhere

nite though it diverges at in nity in one asymptotically at region. It is also consistent

that the seven-brane has nite energy p er unit seven-volume, saturates a Bogomol'nyi

b ound and is geo desically complete in the Einstein frame, but not in string frame. The

(7)

magnetic charge resides entirely at two p oint sources. The total magnetic charge P is

quantized and satis es a Dirac-Teitelb oim-Nep omechie quantization condition.

Our R R electrically charged D-instanton or 1-brane has a numb er of features

which distinguish it from other R R pbranes. Firstly it has a nite throat and two

asymptotically at regions, while in other cases any throat is necessarily in nitely long.

This may b e partly understo o d using sup ersymmetry. The Lorentzian solutions have

Killing spinors and the asso ciated Killing vector can never b ecome spacelike.

Thus if these solutions have regular horizons they must b e of extreme typ e which means

that the surfaces of constant time resemble those in the extreme Reissner-Nordstrom

solution and have the form of in nitely long throats with an internal in nity rather than

the nite Einstein-Rosen throats encountered on the surfaces of constant time of non-

extreme black holes. For our euclidean instanton one cannot construct a timelike Killing

vector from the Killing spinor. In fact in the string frame the metric of our instanton

coincides exactly with the constant time surfaces of the 11-dimensional vacuum black

hole solution of the vacuum Einstein equations but we susp ect that this a coincidence

with no deep er meaning.

It is also interesting to contrast our ten-dimensional R R instanton with the

closely related four dimensional axionic instanton of the NS NS sector [17,18]. This

may also b e thought of as the four dimensional transverse space of the neutral ve-

brane [19]. In string frame, the solution is the pro duct of at 6-dimensional Minkowski

1 5 2

spacetime (t; y ; :::y ) with a curved four dimensional transverse space of the form ds =

2 2 2 2

)(dr + r d ). In the Einstein frame the transverse space is at. If t is (e +

imaginary, the solution maybeinterpreted as an instanton with an in nite throat. In the

at four-dimensional Einstein frame the dilaton and NS NS 3-form eld strength F

(3)

satisify the self-duality condition which guarantees sup ersymmetry: d = e F

(3)

which is similar to the condition for our ten-dimensional instanton. The main di erence

is that due to the NS NS three-form F coupling with a di erentpower of e compared

(3)

to the R R elds, it is the square of e which is harmonic. This is why the throat is

in nite rather than nite. Moreover, for the same reason, the Euclidean action of this

1 1

NS NS instanton is prop ortional to not as it is for our ten-dimensional R R

instanton.

One of the most intriguing asp ects of our work is the relation to the breaking of

continous global symmetries. On the one hand sup erstring theory is b elieved to have 10

no continous symmetries. On the other hand, in low energy quantum gravity the thesis

that black holes and p ossibly wormholes should lead to a violation of the conservation of

charges asso ciated with continuous and p ossibly discrete symmetries has b een strenously

argued [20,21], and equally strenously rebutted [22,23]. The connection b etween real

black holes, virtual black holes and instantons in this context remains, however, obscure

[24,25].

The relevant continous symmetries are contained in SL(2; R). From the p ointof

view of string theory it is b elieved on quite general grounds that this must break down to

the mo dular subgroup SL(2; Z). Now SL(2; R) is certainly a symmetry of the classical

equations of motion of typ e I IB sup ergravity theory written in terms of the dilaton and

pseudoscalar eld a. In particular the equation of motion for a may b e thoughtofas

No ether 2

the conservation of a No ether current J = e @ a arising from the translation

(1)

subgroup: a ! a + constant . The asso ciated charge is Q .Itmay b e argued that

quantum mechanically spacetime wormholes will lead to the violation of the conservation

No ether 2

= e @ a may owdown the of any No ether charge b ecause some of the current J

throat. It may also b e argued that black holes or black branes should lead to a violation

of the conservation of No ether charges. We shall comment on this later.

The discussion so far has assumed that the global translational symmetry a !

a+constantiswell-de ned. This would b e true of our sigma mo del if the target space were

the entire upp er half plane SL(2; R)=S O (2). But it is not. Rather it is the fundamental

domain of the mo dular group SL(2; Z)nSL(2; R)=S O (2). Thus the translations do not

act globally on the target space of our sigma mo del. In other words the continuous

global symmetry is not well de ned. There are two apparently unrelated reasons for

quotienting out by the mo dular group. One was string inspired. The other is that unless

we do so, we cannot obtain a seven-brane with nite total energy.

Thus b oth for stringy reasons and by virtue of wormhole e ects we do not exp ect

(1) (7)

Q to b e conserved. What ab out the magnetic charge P of the seven-brane and

what ab out black holes? The usual physical arguments for the violation of global charges

by black holes rely on two main planks. Firstly, there should b e a no-hair theorem for

the charge, and secondly there should b e a lower b ound to the mass of any state carrying

the relevantcharge. In our case, it is straightforward to show that if the target space

is the entire upp er half plane then there are no black hole solutions with regular event

horizons and non-constant scalar elds. If the target space is the fundamental domain

of the mo dular group, the argument is not quite so straightforward but as far as we

can tell the result seems to hold. Moreover, the same no-hair results seem to apply to

other p-branes. Thus wehave the rst plank. The only states carrying R Rcharge are

(7)

p-branes, in particular the only states we know of that carry P are the seven-branes

of this pap er, which satisfy a Bogomol'nyi b ound on the energy p er unit seven-volume.

Therefore ther are no light states in the theory that carry R Rcharges. Thus it seems

(7)

plausible that the conservation of P could b e violated by dropping seven-branes into

a black hole or a black p-brane and letting it evap orate

Finally we conclude with the observation that the construction of the D-instanton

wormhole solutions in this pap er op ens up the prosp ect of studying a variety of non-

p erturabative e ects of great interest in quantum gravity within a controlled computa-

tional scheme. These include the breaking of sup ersymmetry, the nature of a p ossible 11

non-p erturbative dilaton p otential and the problem of the cosmological constant.

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