8 May 1997 s . __ ‘Et3 li&d

ELSEVIER

PHYSICS LETTERS B

Physics Letters B 400 ( 1997) 32-36

Classical instanton and wormhole solutions of type III3 string theory

Jin Young Kima, H.W. Lee b, Y.S. Myungb a Division of Basic Science, Dongseo University, Pusan 616-010, South Korea

b Department of Physics, Inje University, Kimhae 621-749, South Korea

Received 23 January 1997; revised manuscript received 24 February 1997 Editor: M. Dine

Abstract

We study the p = -1 D-brane in type IIB superstring theory. In addition to the RR instanton, we obtain the RR charged wormhole solution in the Einstein frame. This corresponds to the ten-dimensional singular wormhole solution with infinite euclidean action. @ 1997 Elsevier Science B.V.

There has been much progress in nonperturbative

aspects of string theory [ 11. Recently some p-brane solitons appeared in the form of solitonic objects (Dirichlet (D) -branes) as well as D-instantons [ 21. The p D-branes were used mainly to understand the microscopic origin of the black hole entropy [ 31. In addition to solitonic objects, type IIB superstring theory possesses an instanton solution in ten dimen- sions [ 4,5]. This is the p = - 1 D-brane which carries the Ramond-Ramond (RR) charge. Such instantons are important in the theory. For example, they are responsible for point-like effects in fixed-angle scat- tering. We remind the reader that the possibility of inducing point-like structure is one of the original reasons for investigating string theories with Dirichlet boundary conditions [ 61.

On the other hand, there are many kinds of euclidean wormhole solutions. In four-dimensions the following matter fields which support the throat of the worm- hole were adopted: axion fields [ 71, scalar fields [ 81, SU( 2) Yang-Mills fields [ 91. Higher-dimensional wormhole solutions were obtained [ 10,111 and a

higher-derivative correction to the Einstein-Hilbert

action was considered [ 121. Especially, it is shown that there is no nonsingular wormhole solution for the axion in the four-dimensional stringy model which includes graviton, dilaton and axion [ 7,131. This is because the dilation is nontrivially coupled to the metric and the axion. The axion (pseudoscalar) is the dual of the NS-NS three-form H = dB and plays the role of the source for the wormhole.

In this paper we will study p = -1 D-brane of type IIB superstring in the Einstein frame. We ob- tain, in addition to the RR instanton (classical D- instanton), the new RR charged wormhole solution in ten-dimensional space with the euclidean signature (+ + . . .) . We are interested in the contribution of the RR instanton and RR charged wormhole configu- rations to the euclidean functional integral for the for- ward “flat space + flat space” amplitude [ 141. Worm- holes - solutions to the euclidean Einstein equations that connect two asymptotically flat regions - are con- sidered as saddle points of this integral and are very important for semiclassical calculations of the tran-

0370-2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00320-l

J.Y. Kim et al./ Physics Letters B 400 (1997) 32-36 33

sition probabilities of topological change in quantum gravity. If the action of the wormhole is given by ( S) , the transition probability of a topological change is proportional to e-s. For the wormhole of the metric- axion system [ 71, S is positive and large for large ax-

ionic charge flowing through the wormhole. And then the transition probability is small and the wormhole transition can be suppressed. On the other hand, for

the stringy wormhole of the dilaton-metric-axion the transition probability is almost zero because S is infi- nite [ 131.

We start with the ten-dimensional form of type II superstring action [ 15,161,

Ss = s

d”xe

x [

1 e-*%+4(V&*) - 2(P + 2) $+2 1 3 (1)

where FP+2 is an RR (p+2)-form field strength

C&+2 = dA,+l) and the subscript S indicates that the

string metric is being used. Here we observe that ki- netic term of RR field is not multiplied by the dilaton factor (e-*4) [ 171. This is because the special cou- pling of the RR fields to the dilaton lies in the struc- ture of local N = 2, D = 10 supersymmetry. In the Einstein frame with the signature ( - + . . .) , through the relation gs MN = .&I* gMN, one finds the action as

S= d”xfi J’

x [ R - i Web)* - 2(p y 2), e-(~-3W2F;+2].

(2)

The even (odd) p correspond to type IIA (type IIB) superstrings. The solution of (2) for all values from p = -1 to p = 9 was found in [ 15,161. In this paper we consider only the case of p = - 1. This is a special case because it requires a euclidean continuation. The

Ae-form is just an RR scalar (a) which is a source of the RR instanton and RR charged wormhole. This is essentially a ten-dimensional axion. First, we review

the RR instanton [4]. The corresponding action is given by

S,=_, =/d”&-[R- $(V$)* - ;e2d(Va)2].

(3)

Defining a nine-form field strength as F9 = e*& * da, the action (3) can be rewritten in the equivalent dual- form,

Sp=7 = /d”x&[R - $(V$)* - &eC2dFi].

(4)

The equations of motion for the euclidean version can be derived from (3) with the substitution a + LY = ia. These are

RMN - ~(VMC$VN# - e*‘V,@J!VNa) = 0,

VM( e2’dMa) = 0,

V24 + e*$(Va)* = 0. (5)

We consider the conditions that need to be satisfied

for a solution of the euclidean theory to preserve half of the N = 2 supersymmetry of the type IIB theory.

These are

da = e@dqb, ds* = dp* + p2df2;, (6)

where p is the radius of the nine-sphere (p) and its line element (da;). Substitution of these into (5) leads to C?*C#J = -(&$)* and thus a*(e&) = 0. This allows us a spherically symmetric solution which de- scribes a single RR instanton,

e@ = e+m + $, (7)

where & is the value of the dilaton field at p = cc and c is a constant that will be shown later to be proportional to the instanton charge. This instanton solution is evidently singular at p = 0 in the Einstein frame. However, in the string frame, one finds the metric

dsg = e@/*&* = J---J

e+= + c(dp* + p*d@). (8)

The above metric is invariant under the inversion trans- formation

(9)

This shows that the region of p -+ 0 is another asymp- totically euclidean region which is identical to that of p --f cc. The solution in the string frame is thus a

34 J.Y. Kim et d/Physics Letters B 400 (1997) 32-36

wormhole in which there exist two asymptotically eu- clidean regions connected by a neck.

Now we wish to find the RR charged wormhole solution in the Einstein frame. The euclidean action

for this purpose is given by

s worm = J [-It+ $(vqb>* + p(va)*] M -2 [TrK],

s aM

(10)

where M(dM = 9) is a ten-dimensional euclidean space (its boundary) and [TX] is the boundary con-

tribution. However, the boundary contribution is not relevant here. We note here that the RR scalar (a) can be considered as the source for the RR instanton and RR charged wormhole [ 131. One thus takes the Noether current JM = e24JMa and requires its conser-

vation

%4(&J”) = 0 (11)

which is equivalent to the field equation in (5). There- fore we have to perform the functional integration over conserved current densities. Let us introduce the gen- eral 0( 10) -symmetric euclidean metric as

ds* = N*(p)dp* + R*(p)d,n; (12)

with two scale factors (N, R). The 0( lO)-symmetric

current density has one non-zero component ( Jc ( p) ) and its conservation in ( 11) means that dJc is a constant. This constant is related to the global charge Q of the RR instanton (Q/vOZ( 9) ) . Thus one finds

12Q 1 -- + NR9’

Then the 0( lo)-symmetric action is given by

S,,,, = -62 .I [

dp ?&,,R)* + NR’

(13)

(14)

From the above action, we find three equations

= 0, (15)

R7d R 7$d,R)* - 2&(--g-) i- 7NR6

(17)

For simplicity we choose the N = 1 gauge. We then

obtain the exact solution of ( 15) -( 17))

NJ = 1, (RI)* = p*,

,Q = ,& + se’ 2& #3. (18)

Comparing ( 18) with (6) and (7)) one immediately recovers the RR instanton solution with c = 3Q/2&.

With different boundary conditions, we can also find the new wormhole (RR charged wormhole) solution. First, let us consider the region of p = 0. From ( 16) one obtains the first integral equation for the dilaton

(19)

Here the integration constant has been fixed so that #I has vanishing derivative at p = 0. This point will be the

RR charged wormhole neck with the radius Rw( p = 0) = Ro. At this point one requires apRWj,=o = 0 and

RA6 = (Q/&)*ee2fi. Eq. (19) is derived actually by integration of ( 16) from 0 to p. Substituting ( 19) into

the ( pp) -metric equation ( 15) leads to the important equation

(t$,R)* = 1 - $. (20)

Further, using ( 19) and (20), it turns out that the angular-component metric equation ( 17) is redundant and leads to (20). In order to study the behavior of wormhole dilaton, let us solve (19) to obtain

e*6 - - &$cos* [$cos-’ (2) *I- (21) 0

When Rw(pc,) = 2@Ro for finite p = per, the right- hand side is zero and thus one has +w(p,,.) = --00. Substituting this into ( 14)) the last term (action den- sity) gives rise to an infinity at p = per. It suppresses the transition probability ( ePSwom ) compIetely. When a dilaton is coupled to metric and axion as in (3)) one

J.Y. Kim et al./Physics Letters B 400 (1997) 32-36 35

can always find a singular solution with infinite eu- clidean action. This shows there is no nonsingular RR charged wormhole solution for the p = -1 D-brane of type IIB theory.

In order to obtain the large p-behavior (p > p,,), one has to integrate ( 16) from 00 to p. In this case, one also finds the same forms as in ( 19) -( 21) but with the substitutions:

G-t(bZ + (l/2) ln2;

RA6 --f R, 16 = 3Q/7ij)“e-24w-_

Eqs. (19)-(21) are now given by

(cT,R)~ = 1 - g

e-24 e-24: -_ R’8 -17 2R18

(22)

(23)

e24 _ -( Q -)2cos2 [; cos-qp8].

VSR& (24)

Now we consider the asymptotically flat region. From (23) the resulting solution has an asymptotic behav-

ior RW (p) -+ fp for p + foe. On the other hand, it

has a minimum at the wormhole neck (Ro) . This cor- responds to the ten-dimensional RR charged worm- hole solution, which is based on the RR sector. Here we wish to point out the distinction between the RR instanton wormhole and the RR charged wormhole. This is related to the different behavior of the dila-

tion at the boundary (p = 0). In the case of RR in- Stanton, one has 4’(O) 4 00 and #( 00) -+ &,= constant. On the other hand, although the asymptotic structure of the RR charged wormhole is the same as that of the RR instanton, we have +w(0) = 40” (= exp{ (Q/6)2R,‘6}) with d&w IP+ = 0. That is, we have the singular solution of the dilaton at p = 0 for the RR instanton, while the nonsingular solution at p = 0 is required for the RR charged wormhole.

For an explicit calculation, we first have to solve the differential equations (20) and (23) by numerical analysis. We introduce the resealings

(P/PO, RIRo, +I+,“) with pc = Ro for (20), (21)

and (P/PO, R/R,,4/4&) with PO = R, for W),

(24). The resulting solution is shown in Fig. 1 for RO = R, and C# = CJ~E. Far from the RR charged wormhole throat (p/R0 > 1)) one can ignore the

095

t :

i

Fig. 1. R/R0 as a function of p/Ro. The solid, dashed and dotted

lines correspond to R/R0 = 2’.lz, the RR charged wormhole

scale factor ( Rw/Ro) and the RR instanton scale factor (RI/R()), respectively. The singular critical point (pC,.) is determined from

the solution of RW (per) /Ro = 2’.lz5.

1

Fig. c#J/& as a function p/Ro. The solid and

correspond to the charged wormhole dilaton (qSw/dr)

the instanton dilaton ($‘/4:),

W N p). Here one can find the wormhole neck

;fRw=O) near p = 0. Also one can get the point p = per: as a solution to the relation Rw(p,,) = 2°.125Ra. Now let us substitute the result of RW ( p) / Ro in Fig. 1 into (21) and (24). Then one obtains the wormhole dilaton ( $w ( p) )-behavior. As is shown in Fig. 2, dw is obviously singular at p = per N 1.08Ra. For p < per, one can observe the feature of the RR charged

36 J.Y. Kim et al./Physics Letters B 400 (1997) 32-36

wormhole throat. Clearly +w is different from 4’. The instanton dilation (4’) is singular at p = 0, while

the wormhole dilaton ( +w) is nonsingular at p = 0. On the other hand, for p > per, +w approaches 4’.

In conclusion, we find the RR charged wormhole solution in p = -1 D-brane of type III3 superstring theory. Unfortunately, this is a singular solution with an infinite euclidean action. This is because the ki- netic term of RR field is not multiplied by the dilaton

factor (e-*4) in the string frame. Also we clarify the distinction between the known RR instanton and RR charged wormhole solutions. This is mainly due to the different dilaton configurations. Finally we point out that the RR instanton is supersymmetric, while the RR

charged wormhole is not supersymmetric. This is be- cause half of the N = 2 supersymmetry is preserved when the metric is flat as in the case of the RR instan- ton ( 18). For this case the solution is easily found and thus the numerical integration is not necessary. It is obvious that the RR charged wormhole does not pre- serve any supersymmetry because its metric is not flat

as in (20).

This work was supported in part by the Basic Sci- ence Research Institute Program, Ministry of Educa- tion, Project Nos. BSRI-96-2413,BSRI96-2441 and

by Inje Research and Scholarship Foundation.

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