научная статья по теме ON COINCIDENCE OF ALDAY-MALDACENA-REGULARIZED σ-MODEL AND NAMBU-GOTO AREAS OF MINIMAL SURFACES Физика

Текст научной статьи на тему «ON COINCIDENCE OF ALDAY-MALDACENA-REGULARIZED σ-MODEL AND NAMBU-GOTO AREAS OF MINIMAL SURFACES»

Pis'ma v ZhETF, vol.86, iss.9, pp.643-645

© 2007 November 10

On coincidence of Alday-Maldacena-regularized cr-model and Nambu-Goto areas of minimal surfaces

A. Popolitov

Alikhanov Institute for Teoretical and Experimental Physics, 117218 Moscow, Russia Submitted 28 September 2007

For the cr-model and Nambu-Goto actions, values of the Alday-Maldacena-regularized actions are calculated on solutions of the equations of motion with constant non-regularized Lagrangian. It turns out that these values coincide up to a factor, independent of boundary conditions.

PACS: 11.25.Tq

1. Introduction. The idea of string/gauge theory duality is far from being new (see [1, 2] for intoduction to the subject). A particular incarnation of such a duality, namely the AdS/CFT correspondence, first stated in [3] and then reformulated in [4] and [5], proved to be very helpful in obtaining results in strongly coupled gauge theories2^.

Alday and Maldacena [8] have recently calculated the four-point gluon scattering amplitude in N = 4 SYM. The point is, that at the string theory side of the duality at the zeroth order of semiclassical expansion the scattering amplitude is equal to etScl. Here, Sci stands for the action of a string, evaluated on some solution of the classical equations of motion with certain boundary conditions. Alday and Maldacena chose the cr-model action, and found some solution to the equations of motion. However, they realized that the Lagrangian is constant on this solution, so that Sci diverges and seems to be independent of boundary conditions. They proposed a regularization procedure in order to make answers finite. After regularization was applied, the dependence on boundary conditions was recovered.

Later, Mironov, Morozov and Tomaras in [9] and [10] found a whole family of solutions with a constant Lagrangian for the cr-model and Nambu-Goto actions, which are of the form z = "EazaekaU, v = "EavaehaU, where certain conditions are imposed on za,va and ka (here, z and v are convinient coordinates on the AdS$ space, za,va and ka are, respectively, constant scalars, 4-vectors and 2-vectors). It is known that solutions of the non-regularized NG equations of motion are solutions of the non-regularized cr-model equations of motion too, and these solutions correspond to Alday-

e-mail: popolitov0itep.ru 2^For breef historical review on the AdS/CFT correspondence, see [6]. Quite intuitition-developing view on the topic can be found in [7].

Maldacena's choise of za. It seems that the regularization can break this simple relation between these two actions, hence the regualrized areas, a priori, can be different. In this paper, we calculate these areas explicitly and it turns out that they coincide up to a factor, dependent only on regularization parameter e and angle 4> between ka. This is quite surprising and seemingly indicates presense of a more subtle relationship between the cr-model and NG actions.

Throughout this paper we freely use the notations and results of [9] and [10].

2. 0--modeI case. The action under consideration is of the form

= J GijS'Wu, (1)

where Gy = (dizdjz + (zdiV — vdiZ, zdjv — vdjz))z2 -the metric induced by mapping R2 —t AdS5. Following the regularization procedure given in [8], we substitute z —t z( 1 + e/2)-1/2 and add a factor of zc. Further, we use the fact that Gy = const on solutions, in order to exclude v

M'+r^H+H^))^-

(2)

As soon as we are interested only in terms which do not go to zero while e —y 0 , in what follows all the equalities should be understood as right up to terms infinitesimal in the limit of e 0. Solutions of the undeformed equations of motion are of the form: z = "EazaehaU, v = T,avaehaU, where k2 = trG. The integral fzed2u was evaluated in [9] and is equal to

IlHCbMa b ?K3T<J> tom 86 Bbin.9-10 2007

643

644

A. Popolitov

/

z€d2u =

\[h,k2}\

x -9 + ~ ln(ziz2z3z4) - — -\ez с 3

i (ln2(ziz2z3z4) - In2 8 V \г2г4

(3)

Thus, one immediately obtains ' dizdiz

I

tr

aub.

c(c — 1) а,Ь,г 1

£ kfkfZaZbg

z£d2u =

92

дг Г

agzb J

z€d2u =

(ka,kb) S ,rr, r.'.ZaZb X

dz

c(c — 1) а,Ь Itfc1,^2]!

d2 Г

adzb J

z42u

(4)

fc2 = l,(fc\fc2)=0

Writing down the matrices (ka,kb) and

it is easy to see

ZaZbgßgjr J zed2u

that

(4)=ПГ

|sin(0)|(l-e)e;

fc2 = l,(fc\fc2)=0

-2 (l+l ln(ziz2z3z4)) • (5)

Since (l + e/2)-1-^2 = l-c/2, substituting (5) and (3) into (2) gives

(l-§) /4 1, .

£i£3 \

г2г4 /

I sin(0)| \e2 с у + i (ln2(ziz2z3z4) - In2 2 1

, ln(ziz2z3z4) + 2 = . x

с 2 I sm(^)|e2

x U + e2 il^T^l (l+Un(Zlz2z3zA)

4 12// V 4

(ln2(ziz2z3z4) - In2 ( о/ \ \г2г4

1 TT2

1 + ez T - x

| sin(^)|e2 V ' " V4 12,

Following [8], to get a regularized area, this expression should be multiplied by

VhiCD _ VWi^y

V

1 + e ~

(7)

2irae 2irae V 12

Thus, the regularized area for the cr-model is equal to3)

3^We still reproduce the original Alday-Maldacena result, cf.

[Ill

Areaff =

x (1+ —(1— ln2)+

( [W7

(Vho7

1

7Гб2 | sin)

2 /

fc 1 VI

8 I' 3

JXß2e ■ -

V

-2 In 2+In 2 I I x

■ln^

3. Nambu-Goto case. In this case the action has the form

Sng =

N

£ik£3lGijGhid2u.

(9)

Solutions with constant Gy look very similar to those of cr-model: they have the same form, but different constraints are imposed on za,va,ka (see [10]). Expanding in powers of epsilon, one obtains for the regularized action

/ £n-1-C/2

Sng = (1 + 2 j x

[ V det G ( 1

tG

-lkl

dkzdiz

r2

(_G-ikiG-iij dkzdizdizdjz\ £ 32 z4

(10)

It is easy to see that the action is invariant under coordinate transformations of the worldsheet u = f(u). Hence, without loss of generality, we put fci = (1,0), fc2 = (0,l), Gw = diag(l/2, 1/2)

-l-e/2

/

l+|tr

1 2

dizdjz

г'

2 2 (dizdjz

с

~tr

z£d2u. (11)

Difference with the a - model case is in overall factor of 1/2 and in the presense of a term contributing to the finite (i.e. e-independent) part of the action. This term is equal to

/

tr

2 I dizdjz) z42u =

4 1

12

e

-- x

e(l — e)(2 — e)(3 — e)e2 4

Xa4z^d^!ln(ziz2z3z4) = i-

(12)

Substituting (12), (5) and (3) into (11) and manipulating with the expression just as in the cr-model case, one obtains

Письма в ЖЭТФ том 86 вып. 9-10 2007

г

г

г

On coincidence of Alday-Maldacena-regularized a-model

645

Thus, the answer for the regularized area in the NG case is

Areajmg =

27re2

x

x (l+i(l_ln2) + y f~ ^21n2+ln22^| | x

I [Tip rn&

Who7 V^)7

: In

(?) • (")

4. Conclusions. Thus, we found that the difference between the two areas reduces to a factor, which is equal to

Area„

Areajmg | sin <j)\

(15)

2 in the numerator is due to S& instead of | S&, which is actually related to Sng> was used. For orthogonal ka, and for Alday-Maldacena's solution particularly sin <j> = 1, and substitution of one action by another in the scattering amplitude results only in constant phase shift, which is equal to VX/8 7r. So, physics of the process is left unaffected by this substitution. This gives us a hope, that in further investigations and applications one action can be replaced with the other one. However, it still remains unknown whether this simple relation between amplitudes would survive if one takes into account further terms of the semiclassical expansion.

We would like to thank A. Mironov and A. Morozov for discussions. This work is partly supported by Federal Agency of Atomic Energy of Russia, by grant for support of scientific schools # NSh-8004.2006.2 and by grant RFBR # 07-02-00878.

1. V. Schomerus, arXiv:0706.1209.

2. J. Baez, arXiv:hep-th/9309067.

3. J. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998); Int. J. Theor. Phys. 38, 1113 (1999); arXiv:hep-th/9711200.

4. S. Gubser, I. Klebanov, and A. Polyakov, Phys. Lett. B 428, 105 (1998); arXiv:hep-th/9802109.

5. E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998); arXiv:hep-th/9802150.

6. I. Klebanov, Int. J. Mod. Phys. A 21, 1831 (2006); arXiv:hep-ph/0509087.

7. A. Morozov, arXiv:hep-th/9810031.

8. L. Alday and J. Maldacena, arXiv:0705.0303.

9. A. Mironov, A. Morozov, and T. Tomaras, arXiv:0708.1625.

10. A. Mironov, A. Morozov, and T. Tomaras, Some Properties of Alday-Maldacena Minimum in the Moduli Space of Minimal Areas, to appear, 2007.

11. S. Naculich and H. Schnitzer, arXiv:0708.3069.

ÜHCbMa b ?K3T<J> tom 86 Bbin.9-10 2007

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