Conformal anomaly of dual QFT from higher dimensional dilatonic gravity Текст научной статьи по специальности «Физика»

НАУЧНЫЕ СТАТЬИ

Shinichi Nojiri", S.D. Odintsov"*, Sachiko Ogushi"** CONFORMAL ANOMALY OF DUAL QFT FROM HIGHER DIMENSIONAL DILATONIC GRAVITY

Department of Mathematics and Physics

"National Defence Academy, Hashirimizu Yokosuka 239, JAPAN "Tomsk State Pedagogical University, 634041 Tomsk, RUSSIA •"Department of Physics, Oehanomizu University Otsuka, Bunkyou-ku Tokyo 112, JAPAN

Five-dimensional gauged supergravity (SG) plays an important role in AdS/GFT correspondence [1]. It is known that different versions of d5 gauged SG (for example, N=8 d5 gauged SG [2] with forty-two scalars and non-trivial scalar potential) may appear as a result of truncation of dlO IIB SG. In particular, AdS5xS5 deformed truncation of IIB SG (with non-trivial scalars) corresponds to some specific solution of d5 gauged SG. Hence, it is often enough to study 5d gauged SG classical solutions in AdS/CFT set-up instead of the investigation of non-linear IIB SG solutions. Such (deformed) solutions describe RG flows in dual boundary field theory (for a very recent discussion of such flows, see [16, 3, 4] and refs. therein). It is very interesting that even 4d curvature or non-zero temperature effects may be taken into account in bulk description of such RG flows [5]. In consideration of extended d5 gauged SG solutions there are often more symmetric (special) RG flows where scalars lie in one-dimensional submanifold of complete scalars space. (Then such theory corresponds to d5 dilatonic gravity with non-trivial dilaton potential.) Such flows may also correspond to certain (D3)-brane distributions [6]. However, note that it is extremely difficult to make the explicit identification of deformed gauged SG solution with the corresponding non-conformal dual gauge theory.

The important characteristic of boundary (gauged) theory in AdS/CFT correspondence is the conformal anomaly which may be found from the bulk side (see paper by Witten in ref. [1]). The calculation of conformal anomaly in d5 gauged SG with single scalar and constant scalar potential (dilatonic gravity) on dilaton-gravitational background via AdS/CFT correspondence has been initiated in ref. [12]. It was shown that N=4 super YM theory covariantly coupled with N=4 conformal SG [7] is actual dual of d5 dilatonic gravity (see also derivation of anomaly in gravity-complex scalar background in refs. [8, 13]). From holographic SG description (see refs. [9, 10] for introduction) it is known that dilaton (or in more general case, scalars)

yflK 539.121.7

describe the coupling of dual (gauge) theory, say, masses, scalars or coupling constants. Hence, it is extremely interesting to get the conformal anomaly for gauged SG with non-trivial scalar potential. This may give much better understanding of RG flows in dual (non-conformal) boundary theory and also in the definition of analog of central charge (c-function) away of conformity. Even more, considering the conformal anomaly of dual general boundary theory with radiative corrections and comparing it with the one from bulk gauged SG may help in correct identification of dual boundary theory with correspondent bulk identification (which is currently noneasy task). Note also that conformal anomaly plays an important role in the construction of local surface counterterm for gauged SGs with non-constant scalar potential [14].

In the present letter we find the AdS/CFT conformal anomaly from d3 and d5 gauged SG with single scalar (dilaton) and arbitrary dilaton potential. This situation corresponds to special RG flow in dual description. The candidates for analogs of central charge (or more exactly, of c-function) away of conformity are presented. We believe that even for non-conformal theory this analogous c-function may be of importance as it measures the dilatonic deviation from conformity. The numerical study of c-function for two explicit choices of dilaton potential indicates to non-monotonic behaviour as expected.

We start with the bulk action of ¿/+1-dimensional dilatonic gravity with potential

1

16nC lMä+1 + Г(ф)Уф + Ф(ф) + 4Х2}.

(i)

Here M,it is d+1 dimensional manifold whose d+l

boundary is d dimensional manifold Md and we choose <|>(0) = 0. Such action corresponds to (bosonic sector) of gauged SG with single scalar (special RG flow). Note also that classical vacuum stability restricts the form of dilaton potential [15]. As well-

Вестник ТГПУ. 2000.. Выпуск 2 (18). Серия: ЕСТЕСТВЕННЫЕ НА УКИ (СПЕЦВЫПУСК)

known, we also need to add the surface terms [11] to the bulk action in order that the variational principle to be well-defined. We should only note here that the surface terms become irrelevant finally in the calculation for the Weyl anomaly given in this work. The equations of motion given by variation of (1) with respect to § and Guv are

0 = Ф'(Ф) - Г(ф) (^„фд„ф -

(2)

О = 1 3

12

Ф(Ф)+/2 (3)

Неге К(ф) = ЛХф)-ПФ).

(4)

We chose the metric on Md +, and the metric

g on Md in the following form

/2

ds2 = Gilvdxildxv = p"2dp# + 4

d . . + TJg,idx,dx', g„=p gy.

»=I

(5)

T of the action renormalized by the subtraction of the terms which diverge when e->0 (d = 4)

J d4xf^T.

(8)

First we consider the case of d = 2, i.e. three-dimensional gauged SG. Then anomaly term is found as follows:

(9)

T = Ù i й(0) + ЛГ(Ф(0)Х^Ф(0))2 +НФ(0))АФ(0) +

+ 1 |ЗФ'(Ф(0))

+ 21.......12

Ф"(Ф(0)) Ф(Ф(0)) +

3

(Ф'(Ф(0))2)-1 -ф(ф(0))

X (R( 0) + *ЧФ<0) )g(0)di (Ф(0) )dj (Ф(0) ))х

Ф(Ф(0))+/у

ЗФ'(Ф(0))

Ф*(ф(0)) Ф(Ф(0,)+ 2 -Ф'(ф(0))2

V ' )

Here I is related with A,2 by 4 A2 = d(d • 1) I2. If g=r).., the boundary of AdS lies at g . Note that we follow the method of calculation in ref. [12,13] where dilatonic gravity with constant dilaton potential has been considered.

The action (1) diverges in general since the action contains the infinite volume integration on Md+]. The action is regularized by introducing the infrared cutoff e and replacing

(6)

We also expand gSj and <j) with respect to p:

gy = g(0)ij + pg(r)i/+p2g(2)y+-,

§ = <|>(0) + p (|>(1) + p2<|>(2) + • ■ ■. (7)

Then the action is also expanded as a power series on p. The subtraction of the terms proportional to the inverse power of e does not break the invariance under the scale transformation 5g)iv = 2§ag|iv and 8e = 28c7e. When d is even, however, the term proportional to In s appears. This term is not invariant under the scale transformation and the subtraction of the In e term breaks the invariance. The variation of the In e term under the scale transformation is finite when e-»0 and should be canceled by the variation of the finite term (which does not depend on e) in the action since the original action (1) is invariant under the scale transformation. Therefore the In e term Sin gives the Weyl anomaly

ПФ(о))4)0ДФ(0))5ДФ(о)) +

+ 2 3,(V-«(0)*<VAo)>

V~£< 0)

For F(<|>)=0 case, the central charge of the conformai field theory is given by the coefficient of R. Then it might be natural to introduce the analog of central charge c, i.e. c-function for the case when the conformai symmetry is broken by the deformation as follows:

2G

, 1 Г ЗФ'(Ф(0)) t

/ + —J -------- ' x

2 Î1

Ф'(Ф(0)) Ф(Ф(0))+/2

v '

\

-I

Ф'(ф(0)) Ф(Ф«»П Ф(Ф{0)) +

I2

-1

(10)

Comparing this with radiatively-corrected central charge of boundary QFT may help in correct bulk description of such theory.

For the case oi d = 4, we obtain the following expression for the anomaly:

1

"ïnG

(И)

h = [3{(24 - 10Ф)Ф'6 + (62208+22464Ф + + 2196Ф2 + 72Ф3 + Ф4 )Ф'(Ф* + 8V)2 + '

2<D'4 {(108 +162<D + 7<D2)0>" + 72(-8 + + ] 4$ + 4>2 )V} - 20'2 {(6912 + 27364) + + 1923>2 +4>3)4)''2 +4(11232 + 6156<D + + 5520>2 +13<I>3)0"K+32(-2592 + 4680 + + 96<t>2 + 5<J>3)F2} -3(—24 + <I>) (6 + <i>)2 x x cD'3 (<D '' + 8F')} ], [16(6 + ®)2 {-20)'2 +] + (24 + 0)0>"} {-2d>'2 + (18 + 4>)(0' + 8F)}2

-3

{(12-5$)$'2 + (288 + 72® + i>2)i>"}

{8(6+ <»)2{-2<i>'2+(24+ <£)<!>"}} ' (12)

Here ... expresses the terms containing the derivative with respect to x., whose explicit forms is given in [22]. In case of the dilaton gravity in [12] corresponding to <&=0 (or more generally in case that the axion is included as in [13]), we have the following expression:

l6nG

Jd4xJ- g(0) x

RmjRso) '

1 R2 +

24

+i R»0)d,<№djm - ~ RioygyMWjm+

-AiFg^skd MOA

(13)

G = R-4R::Rij+RijkiR

ijkl

F = _R2 -2RjjR'j +RijURiikl.

(14)

h

3x 62208$ "(8F)2 16x62 x24xl8<D'(8K)2

1

24'

3x2884>"

1

8x6 x24«l>" 8

and we can find that the standard result (conformal anomaly of N = 4 super YM theory) is reproduced. In order that the region near the boundary at p = 0 is asymptotically AdS, we need to require <D->0 and <E>'-»0 when p->0. We can also confirm that A~> !/24 and in the limit of <X>->0 and <J>'-»0 even if

3>"#0 and d)""* 0. In the AdS/CFT correspondence, k and h should be related with the central charge c of the conformal field theory (or its analog for nonconformal theory). Since we have two functions h and k, there are two natural ways to define the c-function when the conformal field theory is deformed:

24k h .......G

Cl

8nk

G

(16)

2 ' Ifweput F® = 4k* +<m , we have / = (l2/F(0))l.

We should note that we have chosen I = 1 in the

expressions in (16). We can restore /-dependence by

changing h -> l3h and k l3k and <&' W, P<&"and

<E>'" ^/iO'"in (11). Then in the limit of <D-»0, we obtain

12

V(0)

(17).

which agrees with the definition used in the works [17, 18] in above limit. The ct- or c2-functions give the new candidate for c-function away of conformity.

We now consider some examples. In [6] and [16], the following dilaton potentials appeared:

Here <p can be regarded as dilaton. When i> is not trivial, of course, there appear extra terms which are denoted by ... in (11). When <& is not trivial, for example, the coefficient of (g'^clfpCOjo.cpiQ)2 becomes dilaton dependent. And there would be appear the terms like R^, 0^(0)0^(0) and R«!0)âq>(0)acp(Û) and their dilaton dependent coefficients are quite complicated.

We should also note that the expression (11) cannot be rewritten as a sum of the Gauss-Bonnet invariant and the square of the Weyl tensor F, which are

PGPW

(+) = 4

exp

+ 2 exp

'24^

L V

41

4).2+<t>cppz(§) :

3 +

cosh

S

+ 4 cosh

24

S

,(18)

(19)

This is the signal that the conformal symmetry is broken. In the limit of <E>->0, we obtain

(15)

In both cases V is a constant and V—-2. In the classical solutions for the both cases, $ is the monotoni-

cally decreasing function of the energy scale (~p 4) and § =0 at the UV limit corresponding to the boundary and <j)—in the UV limit. Then in order to know the energy scale dependences of c, and c2, we only need to investigate the f dependences of h and k in (11). By the numerical calculations, their behaviors are given in [21]. In any case, h and k are increasing functions when <j> is small as expected but the monotonities are broken when <|> is not small. That proves that such bulk regime corresponds to non-conformal boundary gauge theory. Furthermore there appear singularities coming from 0 —20/J + (24+<£>)<I>" which are included in the denominators in h and k. In h for <J>FGpw, there also appears a singularities coming from 0 —2<I>'2 + (18+<S)(i>"+8 K), which is also included in the denominator of h. Hence, our candidate c-func-

Вестник ТГПУ. 2000. Выпуск 2 (18). Серия: ЕСТЕСТВЕННЫЕ НАУКИ (СПЕЦВЫПУСК)

tions may be seriously considered as realistic ones only in the region with small dilaton.

In summary, we found the conformal anomaly from d3 and d5 gauged supergravity with single scalar and arbitrary scalar potential on the scaiar-gravitational background. It corresponds to the conformal anomaly of dual boundary theory, The attempt to define c-function away of conformity is also presented. Our work may be extended for d5 gauged SG with bigger number of scalars (say N-% gauged SG) and arbitrary scalar potential. The final result appears in really complicated and lengthy form as it will be shown in another place. This opens the possibility of explicit check if the results on RG flows

in dual gauge theory (deformed N=4 super Yang-Mills) presented in ref. [4, 16] from bulk side indeed describe 4d gauge Yang-Mills theory with lesser supersymmetry and the correspondent identification is correct. From another side, our conformal anomaly in the spirit of ref. [19] may be used to calculate the Casimir energy in dilatonic gravity. As the final remark let us note that dilaton-dependent conformal anomaly found in this work may be used for calculation of anomaly induced effective action of non-conformal boundary QFT in the presence of scalars (see ref. [20] for related example of dilaton dependent induced effective action in SUSY Yang-Mills theory).

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