CFT correspondence

30 December 1999

Physics Letters B 471 Ž1999. 155–161

Strong coupling limit of N s 2 SCFT free energy and higher derivative AdSrCFT correspondence Shin’ichi Nojiri a

a,1

, Sergei D. Odintsov

b,2

Department of Mathematics and Physics, National Defence Academy, Hashirimizu Yokosuka 239, Japan b Tomsk Pedagogical UniÕersity, 634041 Tomsk, Russia Received 9 August 1999; received in revised form 11 October 1999; accepted 26 November 1999 Editor: P.V. Landshoff

Abstract We study the role of higher derivative terms ŽRiemann curvature squared ones. in thermodynamics of SCFTs via AdSrCFT correspondence. Using IIB string effective action Žd5 AdS gravity. with such HD terms deduced from heterotic string via duality we calculate strong coupling limit of N s 2 SCFT free energy with the account of next to leading term in large N expansion. It is compared with perturbative result following from boundary QFT. Considering modification of such action where HD terms form Weyl squared tensor we found Žstrong coupling limit. free energy in such theory. It is interesting that leading and next to leading term of large N expanded free energy may differ only by factor 3r4 if compare with perturbative result. Considering HD gravity as bosonic sector of some Žcompactified. HD supergravity we suggest new version of AdSrCFT conjecture and successfully test it on the level of free energies for N s 2,4 SCFTs. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. AdSrCFT correspondence w1x Žfor an excellent review, see w2x. may provide new insights to strong coupling regions of SUSY QFTs. One example of this sort is given by strong coupling limit of N s 4 super YM theory free energy w3,4x which differs by factor 3r4 with perturbative result Žboundary QFT. in the leading order of 1rN expansion 3. It is quite interesting to understand what happens in the next order of 1rN expansion. Clearly that such analysis should be related with higher derivatives ŽHD. terms 1

E-mail: [email protected], [email protected] E-mail: [email protected] 3 Thermodynamics of N s 4 super YM theory in relation with AdSrCFT correspondence has been discussed in numerous works w5,6x. 2

on SG side. In more general framework HD terms may help in better understanding of AdSrCFT correspondence or even in formulation of new versions of bulkrboundary conjecture. That is the purpose of present work to study the role of HD terms in bulk action in AdSrCFT correspondence. In the next two sections we find strong coupling limit of N s 2 SCFT free energy and compare it with perturbative result up to the terms of next to leading order. HD terms are chosen in Riemann curvature squared form or in Weyl tensor squared form. The metric of effective five-dimensional gravity is BH in AdS background. In the last section we suggest new HD AdSrCFT conjecture. We show that it works well on the level of comparison of free energies for SCFTs Žor trace anomalies.

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 3 7 6 - 3

S. Nojiri, S.D. OdintsoÕr Physics Letters B 471 (1999) 155–161

156

if bulk sector is described by some HD Žsuper.gravity.

Here

2. The trace anomaly of d s 4, N s 2 and N s 4 SCFTs Žbulk side calculation. has been found up to next to leading order in the 1rN expansion in Refs. w7,8x. Even in next to leading order term it coincides with QFT result Žfor gravity side derivation of trace anomaly, mainly in N s 4 super YM case, see also Refs. w9,10x.. The N s 2 theory with the gauge group SpŽ N . arises as the low-energy theory on the world volume on N D3-branes sitting inside 8 D7branes at an O7-plane w11x. The string theory dual to this theory has been conjectured to be type IIB string theory on AdS5 = X 5 where X5 s S 5rZ2 w12x, whose low energy effective action is given by 4 Žsee the corresponding derivation in Ref. w7x..

2

5

H '

Ss d x g

½

N

6N q

24 P 16p 2

k

s

2

4p

Lsy

,

cs

k2

6N 24 P 16p 2

12 N 2

12 sy

4p 2

,

.

Ž 3.

We now treat the next-to-leading term of order N as perturbation of order N 2 terms. In the leading order, a solution is given by ds 2 s gmn dx mdx n 3

s ye 2 r dt 2 q ey2 r dr 2 q r 2

Ý Ž dx i .

Ž R q 12 .

,

1

Ž ym q r 4 . .

r2

Ž 4.

In the metric Ž4., we find

5

Rmnrs R mnrs .

Ž 1.

ž

Dn Dm R m z n j s 0 ,

Rmnrs R mnrs s 40 q

1 2

The overall factor of the action is different by from that of the action which corresponds to N s 4 SUŽ N . gauge theory. The latter action is given by type IIB Žcompactified. string theory on AdS5 = S 5. The factor 12 comes from the fact that the volume of X 5 s S 5rZ 2 is half of S 5 due to Z2 . Note that Riemann curvature squared term in the above bulk action is deduced from heterotic string via heterotictype I duality w13x Ždilaton is assumed to be constant.. Then the equations of motion have the following form: c 1 0 s y gzj Rmnrs R mnrs q 2 R y L 2 k

ž

q cRzmnr R mnr j q

2

is1

e2 r s

2

4p 2

N2

1

1

k2

/ Ž 2.

r8

/

.

Ž 5. Then the metric is modified by ds 2 s gmn dx mdx n 3

s ye 2 r dt 2 q ey2 r dr 2 q r 2

Ý Ž dx i .

2

,

is1

e2 r s

1 r2

½

e ' ck 2 s

Rzj y 4 cDn Dm R m z n j .

72 m2

ym q Ž 1 q 23 e . r 4 q 2 e 1 16 N

m2 r4

.

5

,

Ž 6.

Then the radius r h of the horizon and the temperature T are given by r h ' m1r4 Ž 1 y 23 e . ,

m1r4 4

The conventions of curvatures are given by h Rs g mn Rmn , Rmn sy Gmll , n q Gmnl , l y Gmhl Gnhl q Gmn Glhl , h R l mnk s Gmkl , n y Gmnl , k q Gmk Gnhl y Gmnh Gkhl , h Gml s 12 g hn Ž gmn , l q g ln , m y gm l , n . .

Ts

p

m1r4

Ž1y2e . s

p

ž

1y

1 8N

/

.

Ž 7.

We now consider the thermodynamical quantities like free energy. After Wick-rotating the time variables by t ™ it , the free energy F can be obtained

S. Nojiri, S.D. OdintsoÕr Physics Letters B 471 (1999) 155–161

from the action S in Ž1. where the classical solution is substituted: Fs

1 T

S.

Ž 8.

Using Ž2., Ž3. and Ž6., we find N2

5

Ss Hd x'g 4p 2 s

N 2 V3 4p 2 T

`

Hr

½ ½

8y

drr 3 8 y

h

2e

ž ž

3 2e 3

40 q

40 q

72 m2 r8 72 m2 r8

/5 /5

.

Ž 9.

Here V3 is the volume of 3d flat space and we assume t has a period of 1rT. The expression of S contains the divergence coming from large r. In order to subtract the divergence, we regularize S in Ž9. by cutting off the integral at a large radius rmax and subtracting the solution with m s 0: Sreg s

N 2 V3 4p 2 T

`

žH ½ drr

3

8y

2e

ž

3

rh

ye r Ž rsr ma x .y r Ž rsr max ; m s0.

½

= 8y

80 e

5

3

40 q rmax

H0

72 m2 r8

drr 3

/5

/

.

Ž 10 .

The factor e r Ž rsr ma x .y r Ž rsr ma x ; m s0. is chosen so that the proper length of the circle which corresponds to the period 1rT in the Euclidean time at r s rmax coincides with each other in the two solutions. Then we find Fsy

N 2 V3 Ž p T . 4p

4

ž

2

1q

3 4N

/

.

Ž 11 .

The entropy S and the mass Ženergy. E are given by dF Ssy

s

N 2 V3 Ž p T .

dT

2

p T 2

EsFqT Ss

4

ž

3 N V3 Ž p T . 4p

2

1q

3 4N

4

ž

1q

/

,

3 4N

tiplet consists of two Weyl fermions, one complex scalar and one real vector what gives 4 bosonic Žfermionic. degrees of freedom on shell and hypermultiplet contains two complex scalars and two Weyl fermions, what also gives 4 bosonic Žfermionic. degrees of freedom on shell w14x. Therefore there appear 4 = Ž n V q n H . s 16 Ž N 2 q 2 N y 14 . bosonfermion pairs. In the limit which we consider, the interaction between the particles can be neglected. The contribution to the free energy from one bosonfermion pair in the space with the volume V3 can be easily estimated w3,4x. Each pair gives a contribution to the free energy of p 2 V3 T 4r48. Therefore the total free energy F should be Fsy

p 2 V3 N 2 T 4

.

Ž 12 .

We now compare the above results with those of field theory of N s 2 SpŽ N . gauge theory. N s 2 theory contains n V s 2 N 2 q N vector multiplet and n H s 2 N 2 q 7N y 1 hypermultiplet w7x. Vector mul-

ž

1q

2

1

/

. Ž 13 . 3 N 4N 2 Comparing Ž13. with Ž11., there is the difference of factor 43 in the leading order of 1rN as observed in w3,4x. Hence, we calculated strong coupling limit of free energy in N s 2 SCFT from SG side up to next to leading order term Žit was generated by Riemann curvature squared term.. Its weak coupling limit Ž13. obtained from QFT side cannot be presented as strong coupling limit free energy multiplied to some constant. This only holds for leading order terms where mismatch multiplier is 3r4. The next to leading term in Ž13. should be multiplied to 9r32 in order to produce the corresponding term in Ž11.. We have to pay attention once more that the main role in whole above analysis was played by Riemann curvature squared term. In Section 4 we try to understand the role of HD terms in AdSrCFT correspondence from the different point of view. y

3. We now consider the case where the action is given by the sum of Einstein term and the square of the Weyl tensor Cmnrs C mnrs 5 1 S s y d 5 x'y G R y L q cC ˜ mnrs C mnrs . k2 Ž 14 .

H

/

157

5

½

5

We thank A. Tseytlin for suggestion to write this section. His motivation was that the case with R 2 combination as C 2 does not make modification of original AdS 5 =S 5 solution as it already happened with C 4 correction w4x.

S. Nojiri, S.D. OdintsoÕr Physics Letters B 471 (1999) 155–161

158

In five dimensions, the square of the Weyl tensor is given by Cmnrs C mnrs s 16 R 2 y 43 Rmn R mn q Rmnrs R mnrs .

Ž 15 . The Weyl tensor term does not contribute to the radius L of AdS 5 . By using the equation of motion, we find 3 0sy R y 12 cC Ž 16 . ˜ mnrs C mnrs q 52 L . 2k 2 By using Ž16., we can delete R in Ž14. and obtain Ssy d

H

5

x'y G  23 L q 23 cC ˜ mnrs C mnrs 4 .

Ž 17 .

Cmnrs C mnrs s

r4

m

ž /

r2

r 04

r 04 ' y

y1 ,

H '

½

2 3

m2 L q 48c˜

5

r8

s

s

,

k 2L

.

12p

Sreg s

N V3 2

4p T

ž

`

Hr drr h

3

½

2 3

ye r Ž rsr ma x .y r Ž rsr max ; m s0.

s

V3 T

L

ž / y

12

y3

Žp T . k8

rmax

H0

Ž 23 .

y3

Žp T .

4

k8

12

y3

Žp T .

4

Ž 1 q 18 e . ,

Tk 8

12

y3

Žp T .

Ž 1 q 18 e . ,

4

Ž 1 q 18 e . .

Ž 20 .

Choosing k and L as in Ž3., above action may be considered as another modification of string action of the same sort as Ž1.. With it we obtain

5 drr 3 Ž 23 L .

q OŽ e 2 . .

E s 3V3 y

2

r8

4

k8

12

L

Fsy

Ss

m

/

18 e Ž p T .

ž /

ž / ž / ž /

Ž 1 y 52 e . ,

L q 48c˜

y3

L

r8

Ž 19 .

. Ž 21 . 12 Then by regularizing the action Ž20. as in Ž10., we find 2

y

ž

m2 72 c˜

Therefore we obtain the following thermodynamical quantities;

c˜ Lk 4

e'y

V3

3

0

L

12 m

k L r0

4p 2

T

2

Tsy

`

H drr T r

S s 4V3 y

The horizon radius and the temperature are given by r h ' r 0 Ž 1 y 12 e . ,

N 2 V3

F s yV3 y

we find S s d5 x g

H

Ž 18 .

r8 for the leading solution, which is given by

e2 r s

SW 2 s yc˜ d 5 x'y G Cmnrs C mnrs

L

Since the square of the Weyl tensor is given by 72 m2

evaluated by substituting the leading solution into the correction term, the square of the Weyl tensor, in the action Ž14.:

Es

k8

12

N 2 V3 Ž p T .

4

4p 2

4 N 2 V3 Ž p T .

4

4p 2 T 3 N 2 V3 Ž p T . 4p 2

4

ž ž ž

1q

1q

1q

18 P 4p 2 c˜ N2 18 P 4p 2 c˜ N2

18 P 4p 2 c˜ N2

/ / /

Therefore if we further choose c˜ by N c˜ s , 9 P 4p 2

Ž 24 .

,

,

.

Ž 25 .

6

Ž 26 .

the strong coupling limit free energy in Ž13. can be reproduced including the next-to-leading order term with the common overall factor 34 .

/

4

Ž 1 q 18 e q O Ž e 2 . . . 6

Ž 22 . Remarkably the above result coincides with that of the simplified procedure, where the correction is

Note that such choice of c˜ is not justified by string considerations. Actually, string theory indicates that c˜ should coincide with c from Eq. Ž1.. However, there are arguments that additional stringy corrections Žfor example, from antisymmetric tensor field. may appear.

S. Nojiri, S.D. OdintsoÕr Physics Letters B 471 (1999) 155–161

4. Instead of Ž1., we can consider more general action of R 2 gravity:

½

5

2

S s y d x'y G aR q bRmn R

H

1 q

k2

mn

q cRmnrs R

mnrs

5

ds s gmn dx dx

0s

80 a

144

n

3

s ye 2 r dt 2 q ey2 r dr 2 q r 2

e s

1 r2

ž

ym q

Ý Ž dx i .

2

k4

L

k 2 L2

yL .

Ž 31 .

y 16  20 a q 4 b 4 L G 0

r4 L2

/

.

Ž 28 .

5

Ž 32 .

which can been found from the determinant in Ž31.. Then we obtain 12 L2 s y

k

(

"

2

144

y 16  20 a q 4 b 4 L

k4

.

2L

Ž 33 .

The sign in front of the root in the above equation may be chosen to be positive, which corresponds to the Einstein gravity Ž a s b s 0.. With L in Ž33., the horizon radius r h and the temperature T are given by 1r4 1r2

rh ' m

L

m1r4 ,

Ts

Ž 34 .

p L3r2

and we can find the free energy F, the entropy S and the energy E by generalizing Ž11., Ž12.: Fsy

,

In this case, the curvature tensors become 20 4 R s y 2 , Rmn s y 2 gmn , Ž 29 . L L which tell that these curvatures are covariantly constant. Then in the equations of motion following from the action Ž27., the terms containing the covariant derivatives of the curvatures vanish and the equations have the following forms: 1 0 s y 12 gzj aR 2 q bRmn R mn q 2 R y L k 1 q 2 aRRzj q 2 bRmz R m j q 2 Rzj . Ž 30 . k

½

12 y

4

L

m V3 8

is1 2r

16 b q

4

Ž 27 .

One can imagine that this is bosonic sector of some HD multidimensional Žprobably compactified to AdS. SG. It was already shown w8x that such bulk theory may correctly reproduce trace anomaly of N s 4 super YM theory. We wish further check such HD AdSrCFT conjecture on the level of free energies. Note that such theory may represent Žyet unknown. resummation of string effective action or it may directly follow from strings or M-theory. It is known, for example, that HD quantum gravity has better UV properties than usual Einstein gravity Žsee w15x for a review.. If c s 0, the equations of motion given by Ž27. can be solved exactly. And as shown in w8x, the anomaly of N s 4 super Yang-Mills theory can be reproduced in case of c s 0. In the following, we consider only special case:c s 0. When c s 0, we can assume that the solution has the form: m

Then substituting Eqs. Ž29. into Ž30., we find

Eq. Ž31. can be solved with respect to L2 if

RyL .

2

159

Ss

Es

m V3 2T 3 m V3 8

ž

8

k

320 a

2

y

8

ž

k

ž

64 b y

L

320 a

2

y

2

2

320 a y

2

L

L2

64 b y

L

8

k

2

L2 64 b

y

L2

/

/

,

,

/

.

Ž 35 .

Here we delete L by using Ž33.. Note that we may also consider special case of no Einstein term in above equations since the above expressions are not perturbative but exact when c s 0. We can consider the case of N s 4 super YangMills theory. When the gauge group is UŽ N ., there are 8 N 2 set of the bosonic and fermionic degrees of freedom on-shell and when SUŽ N ., 8Ž N 2 y 1. since one multiplet corresponds to n V s n H s 1 in N s 2 theory and contains 8 boson Žfermion. degrees of

S. Nojiri, S.D. OdintsoÕr Physics Letters B 471 (1999) 155–161

160

freedom on shell. Then from the perturbative QFT its free energy is given by,

°y p F s~

¢y

2

V3 N 2 T 4

6 2 p V3 N 2 T 4 6

ž

1y

.

N

/

2

SU Ž N . case

Ž 36 . If we assume k 2 , L, a, and b are given by the powers of N, we find the above free energy F can be reproduced if

k2s

6p 2 N2

Lsy

,

°0 1 ¢24p

2N2

p2

SU Ž N . case

2

.

Ž 37 .

As a special case, we can consider the theory without Einstein term, i.e., 1rk 2 s 0. Then we find L2 in Ž33. is given by

(

L2 s "2

20 a q 4 b yL

k2

y 40 aL y 8 bL s

.

Ž 38 .

Since the y sign in Ž38. does not correspond to AdS but to de Sitter space, we only consider the case of q sign. Then the free energy F in Ž35. has the following form:

1

k2

scs0 ,

Lsy

3 2r3 2 16r3

10 a q 2 b s y

P

sy

L

°N Ž 10 a q 2 b .

2

.

Ž 39 .

L

2

24p s~

U Ž N . case

2

N 2y1

¢ 24p

2

2

.

Ž 41 .

2

,

31r3 2 8r3

P

N2

Ž 4p .

2

,

Ž 42 .

5. In summary, we calculated strong coupling limit for free energy of N s 2 SCFT from AdSrCFT correspondence in the next to leading order of 1rN expansion. As bulk side we used AdS Einstein gravity with Riemann curvature Žor Weyl tensor. squared term. For general five-dimensional HD gravity considered as bosonic sector of some HD Žprobably compactified. SG we formulated new HD AdSrCFT conjecture which works on the level of free energies for N s 4 super YM. Note also that it works for N s 2 SCFT. Indeed, in this case the analog of Eqs. Ž37. is given by

k2s

3p 2 N2

,

Lsy

10 a q 2 b s y

In the " sign in the first line, q corresponds to negative L and y to positive one since 20 a q 4 b ) 0 if L - 0 and 20 a q 4 b - 0 if L ) 0 from Ž32.. Therefore we can obtain the free energy in Ž36. if 2

N2

Ž 4p .

(

4

Ž 4p .

It is remarkable that Ž41. can be compatible with Ž40. if

F s "mV3 y L Ž 20 a q 4 b . 4V3 Ž p T . Ž 10 a q 2 b .

2N2

in the leading order of 1rN expansion, or UŽ N . case.

,

U Ž N . case

10 a q 2 b s~

L3

cs0 ,

U Ž N . case 1

Yang-Mills theory can be reproduced by above R 2 gravity if

.

Ž 40 .

SU Ž N . case

Note that L should be positive. In w8x, it has been shown that the conformal anomaly of N s 4 super

N

4N 2

p2

,

q O Ž 1. . Ž 43 . 9p Notice finally that it would be really interesting to consider the role of above HD terms to thermodynamics of AdSrCFT correspondence in the case of non-constant dilaton. Without HD terms the corresponding Žapproximate. AdS BH solution has been found in Ref. w6x. It could be of interest also to investigate the role of spatial curvature to above analysis. Without HD terms such investigation has been presented in Refs. w16x, which is much related to the study of finite gauge theories, including N s 4 super YM theory in curved spacetime ŽQFT side. where curvature squared and scalar-gravitational divergences appear, see Refs. w17x.

S. Nojiri, S.D. OdintsoÕr Physics Letters B 471 (1999) 155–161

Acknowledgements We are very grateful to A.A. Tseytlin for participation at early stages of this work and many helpful discussions.

References w1x J.M. Maldacena, Adv. Theor. Math. Phys. 2 Ž1998. 253; E. Witten, Adv. Theor. Math. Phys. 2 Ž1998. 253; S. Gubser, I.R. Klebanov, A.M. Polyakov, Phys. Lett. B 428 Ž1998. 105. w2x O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, Y. Oz, hep-thr9905111. w3x S. Gubser, I. Klebanov, A. Peet, Phys. Rev. D 54 Ž1996. 3915. w4x S. Gubser, I. Klebanov, A. Tseytlin, Nucl. Phys. B 534 Ž1998. 202, hep-thr9805156. w5x E. Witten, Adv. Theor. Math. 2 Ž1998. 505, hep-thr9803131; A.A. Tseytlin, S. Yankielowicz, Nucl. Phys. B 541 Ž1999. 145, hep-thr9809032; N. Itzhaki, J.M. Maldacena, J. Sonnenschein, S. Yankielowicz, Phys. Rev. D 58 Ž1998. 046004, hep-thr9802042; S.-J. Rey, S. Theisen, J.-T. Yee, Nucl. Phys. B 527 Ž1998. 171, hep-thr9803135; A. Brandhuber, N. Itzhaki, J. Sonnenschein, S. Yankielowicz, JHEP 06 Ž1998. 001, hep-thr9803263; A. Chamblin, R. Emparan, C.V. Johnson, R.C. Myers, Phys. Rev. D 59 Ž1999. 064010, hep-thr9808177; S.W. Hawking, C.J. Hunter, D.N. Page, Phys. Rev. D 59 Ž1999. 044033, hep-thr9809035; J.L.F. Barbon, I.I. Kogan, E. Rabinovici, Nucl. Phys. B 544 Ž1999. 104, hep-thr9809033; A. Hashimoto, Y. Oz, Nucl. Phys. B 548 Ž1999. 167, hep-thr9809106; S. Kalyana Rama, B. Sathiapalan, Mod. Phys. Lett. A 13 Ž1998. 3137, hepthr9810069; A.W. Peet, S.F. Ross, JHEP 12 Ž1998. 020, hep-thr9810200; Y. Kinar, E. Schreiber, J. Sonnenschein,

w6x w7x w8x w9x w10x

w11x

w12x

w13x w14x w15x

w16x

w17x

161

hep-thr9811192; G. Ferreti, D. Martelli, hep-thr9811208; R.-G. Cai, K.-S. Soh, hep-thr9812121; JHEP 05 Ž1999. 025; J. Greensite, P. Olesen, JHEP 04 Ž1999. 001, hepthr9901057; K. Landsteiner, Mod. Phys. Lett. A 14 Ž1999. 379, hep-thr9901143; J. Ellis, A. Ghosh, N.E. Mavromatos, Phys. Lett. B 454 Ž1999. 193, hep-thr9902190; M. Cvetic an S.S. Gubser, JHEP 04 Ž1999. 024, hep-thr9902195; E. Kiritsis, T.R. Taylor, hep-thr9906048; D.S. Berman, M.K. Parikh, hep-thr9907003. S. Nojiri, S.D. Odintsov, hep-thr9906216. M. Blau, K.S. Narain, E. Gava, hep-thr9904179. S. Nojiri, S.D. Odintsov, hep-thr9903033. S. Nojiri, S.D. Odintsov, Phys. Lett. B 444 Ž1998. 92, hep-thr9810008. M. Nishimura, Y. Tanii, hep-thr9904010; V. Balasubramanian, P. Kraus, hep-thr9902121; W. Mueck, K.S. Viswanathan, hep-thr9905048; P. Mansfield, D. Nolland, hep-thr9906054. A. Sen, Nucl. Phys. B 475 Ž1996. 562, hep-thr9605150; T. Banks, M.R. Douglas, N. Seiberg, hep-thr9605199; O. Aharony, C. Sonnenstein, S. Yankielowicz, S. Theisen, hepthr9611222; M.R. Douglas, D.A. Lowe, J.H. Schwarz, hepthr9612062. A. Fayyazuddin, M. Spalinski, Nucl. Phys. B 535 Ž1998. 219, hep-thr9805096; O. Aharony, A. Fayyazuddin, J.M. Maldacena, JHEP 9807 Ž1998. 013, hep-thr9806159. A.A. Tseytlin, Nucl. Phys. B 467 Ž1996. 383. N. Seiberg, E. Witten, Nucl. Phys. B 426 Ž1994. 19, hepthr9407087. I.L. Buchbinder, S.D. Odintsov, I.L. Shapiro, Effective Action in Quantum Gravity, IOP Publishing, BristolrPhiladelphia, 1992. C. Burgess, N. Constable, R.C. Myers, hep-thr9907188; B. Sundborg, hep-thr9908001; K. Landsteiner, E. Lopez, hepthr9908010. I.L. Buchbinder, I. Lichtzier, S.D. Odintsov, Class. Quant. Grav. 6 Ž1988. 605; S.D. Odintsov, Fortschr. Phys. 39 Ž1991. 621.