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48
UDC 530.1; 539.1
THE 1/N CORRECTION TO THE D3-BRANE DESCRIPTION OF CIRCULAR WILSON LOOPS
E. I. Buchbinder
School of Physics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia.
E-mail: evgenv.buchbinderQuwa.edu.au
We consider the one-loop correction to the probe D3-brane action in AdS5 x S5 expanded around the classical Drukker-Fiol solution ending on a circle at the boundary. It is given by the logarithm of the one-loop partition function of an Abelian N = 4 vector multiplet in AdS2 x S2 geometry. This one-loop correction is expected to describe the subleading 1/N term in the expectation value of circular Wilson loop in the totally symmetric rank k representation in N = 4, SU(N) supersymmetric Yang-Mills theory at strong coupling. We also discuss a comparison with the matrix model.
Keywords: The AdS/CFT correspondence, Wilson loops, D-branes, matrix models, the heat kernel technique.
1 Introduction
According to the AdS/CFT correspondence [1] N = 4, SU(N) supersymmetric Yang-Mills theory admits a dual description as string theory on the AdS5 x S5 background. The correspondence provides a string theory interpretation of various field theory phenomena and gives a way to perform computations in gauge theory in the regime of strong coupling where the standard field theoretic methods break down. However, since the AdS/CFT remains a conjecture finding its non-trivial checks is an important problem.
Supersymmetric circular Wilson loops represent a good lab for high precision tests of the AdS/CFT correspondence. The reason is that they admit a matrix model solution for any representation R as well as for any N and for any't Hooft coupling A [2,3]
(WR = Z J dM^Tr^e-^, (1)
where M is an Hermitian matrix. Recently, a general solution (though rather inexplicit and hard to use)
NA
any string calculation of (WR) can be compared with the matrix model result. Such a comparison is not based on any symmetry and represents a highly nontrivial test of the AdS/CFT correspondence.
In this paper, we will concentrate on circular Wilson loops in the symmetric representation. If in the large N limit the number of boxes k in the Young tableau is N
k
strings in AdS5 x S5. The case k = 1 corresponds to the Wilson loop in the fundamental representation □ and is described by a single string [5-7]. The situation
kN argued in [8-13] that as the number of boxes in the
N
is effectively described by D-branes rather than by-
strings. In particular, the Wilson loop in the symmetric representation is now described by a probe D3-brane in AdS5 x S5.
The classical solution corresponding to a probe D3-brane ending on a circle on the boundary was found by Drukker and Fiol in [8]. The investigation of the semiclassical quantization near the Drukker-Fiol solution was initiated in [14,15]. The action for the quadratic fluctuations was found to be that of an Abelian N = 4 multiplet in AdS2 x S2. In this paper we discuss the 1-loop effective action for these quadratic fluctuations following [16]. Essectially, we compute the 1-loop effective action on AdS2 x S2 using the heat kernel technique and the C-function regularization. At the end we comment on a comparison with the matrix model.
2 Circular Wilson loops in the fundamental representation
Let us review the string theory description of circular Wilson loops in the fundamental representation. In the large N limit, semiclassically they are described by a solution to the Nambu-Goto action in the AdS5 x S5 target space ending on a circle [5-7]. The solution sits in AdS3 and in conformai gauge it is of the form
cos a sin a
z = tanh t, xi =-, X2 =-. (2)
cosh t cosh t
Here (z, x1, x2) are the Poincare coordinates in AdS3 (z is the radial coordinate), t g (0, to), a G [0,2nj. The induced worldsheet is AdS2. To compute (Wn) semiclassically at large A we compute the Nambu-Goto action on this solution to get
ln(Wn) = A(AdS2) = VA, (3)
where A(AdS2) is the regularized volume of AdS2, A(AdS2) = — 2n. Expanding the Nambu-Goto action around this solution gives the following structure of the 1-loop correction [17]
(Wn) 1 —loop
Det8/2 [-V2 + 1 + R/4] Det3/2[-V2 + 2]Det3/2 [-V2] '
ln(Wn )i—ooP = -2 ln(2n) '
r- 12 ln(Wn) = v/Ä + 1 ln + ... ' 2 n
(6)
Comparing (6) with (3) and (5) we see that the leading terms %/A agree but the subleading terms do not
1 1 2
- -ln(2n) = -ln - ' 2 2 n
(7)
ds2 =
L2
sin2 n
[dn2 + cos2 nd^2 + dp2 + sinh2 p dsS2 ], (8)
where L is the radius of AdS and dsS2 is the metric on the unit 2-sphere. The circular Wilson loop is then given by the following solution of the Born-Infeld action
(4)
sin n
sinh p, F^p
îk\/a 2n sinh2 p
kVA
IN
That is to find the 1-loop correction we have to study the 1-loop effective action of the following fields propagating on AdS2: 5 scalars with m2 = 0, 3 scalars with m2 = 2 and 8 Majorana fermions with m2 = 1. The 1-loop effective action in (4) was computed using the methods of quantum field theory in curved space [16] and by the Gelfand-Yaglom method [18,19]. Both methods give
(9)
Note that the solution involves a non-trivial electric field propagating on the D3-brane worldvolume. Near the boundary n = 0 we also have p = 0 and, hence, from (8) we see that the solution at the boundary is parametrized by an angular variable ^ which means that the D3-brane ends on a circle. The induced metric gab on the D3-br^e is that of AdS2 x S2 where
(5) RAdS2 = W1 + K2 , Rs2 = lk '
(10)
Let us compare these results with the matrix model. The matrix integral (1) for the representation □ in the limit of large N and large A gives
Surprisingly, the resolution of this disagreement is currently unknown. It suggests that the string partition function (4) is missing a factor of 2 = eln2 which presumably should come from a proper division of the 1-loop determinant of the longitudinal modes by the ghost determinant.
Now let us briefly discuss the simplest higher representation k C N. At the level of the matrix integral (1) the modification is very simple: A ^ k2A. At the level of string theory we modify the classical solution (2) as t ^ kr, a ^ ka. Semiclassically, ln(Wnfc} = k\/A. However, finding the 1-loop effective action becomes a very difficult problem. The reason is that the induced metric now has a conical singularity at the origin with deficit angle 2n(1 — k) and we have to study quantum field theory on a singular space.
3 Wilson loops as D-branes
Now we will discuss the case when the number of kN of strings becomes large and Drukker and Fiol [8] proposed that the appropriate description is given in terms of D3-branes. Let us parametrize AdS5 as follows
The vev of the Wilson loop was found to be
ln (W} = 2N [k Vl + k2 + arcsinh«]. (11)
The above discussion shows that the description in terms of D3-branes is valid in the limit of large k, N and A but fixed k. Note that in the limit k C N, that is k C 1 we obtain ln(W} ^ k\/A which is the action k
Originally it was proposed in [8] that a probe D3-brane describes Wilson loops in the representation k
in [9] that D3-branes describe Wilson loops in the symmetric representation Symk. Indeed, let us take N D3-branes and intersect them with some other branes in a supersymmetric way so that the intersection is 1-dimensional. It was shown in [9] that there are only 2 such possibilities: a D3 — D3 system or a D3 — D5 system. The action describing the intersecting branes is given by Sn=4 + Sdefect, where Sn=4 is the action of N =4 SU(N) supersymmetric Yang-Mills theory supported on the stack of the original N D3-
Sdefect
the intersection. We can integrate the defect degrees of freedom which can be interpreted as an insertion of a Wilson loop in some representation into the path integral. In [9] it was shown that the D3 — D3-system corresponds to inserting a Wilson loop in the symmetric representation and the D3 — D5-system corresponds to inserting the Wilson loop in the antisymmetric representation. When we go to strong coupling limit we replace the stack of N D3-branes with the AdS5 x S5 geometry and the remaining brane becomes a probe. From this viewpoint, a probe D3-brane should describe Wilson loops in the symmetric representation. At the semiclassical limit the matrix integral (1) for both □^d Symk representations give the same answer (11) [8,10]. However, at the subleading
K
K
level we should expect to see the difference. Note that
N
hence, the higher order corrections will go as powers of 1/N.
4 The 1-loop correction to the D3-brane effective action
Expansion of the D3-action around the Drukker-Fiol solution was performed in [14,15]. The result reads
Sb = J dVVM[1 G0bd0dfc*1 + 4 fabfab] , Sf = J dVVM©A(iGabraVb)©A , (12)
where SB and SF are the actions for the bosonic and fermionic fluctuations respectively. In (12) are the
fab
Abelian gauge field and ©A are the 4 Majorana spinors. Except for the %/M the action is defined using the background metric Gab
Gab = gab + (2W)2gcdFacFbd, Gabdaadab = L2K2(dsAdS2 + ds|2 ), RAdSs = Rs2 = lk . Finally, the matrix M is given by
Mab = gab +(2W)2Fab .
VM = cVG, C =
V1 + K2
prefactor c and discuss the effective action on just AdS2 x S2 where both AdS2 and S2 have the same radius a. We can study the effective action using the heat kernel K(s):
K(s) = | d4a £ e-*/(a)/n(a)
Vol • ^ e-A"
/(a/a),
(19)
where {fn(^)} is a full set of orthonormal functions with eigenvalues An, Vol is the regularized volume of AdS2 x S2 and in the last equality we used the fact that on a homogeneous space the integrand does not depend on the point. The 1-loop effective action is then given by
-K(s),
e/a2 s
(20)
(13)
(14)
where e is a UV cut-off. Note that since AdS2 x S2 is conformally flat the dependence on the radius a appears only via the combination a2/e. Let us expand the heat kernel for small s which corresponds to concentrating on the UV divergences. In 4 dimensions the expansion is of the form
K(s) = Vol • [^ + ^ + 64 + O(s)].
(21)
Gab
it is related to M as follows Gab = M(ab). Furthermore, we have a relation
(15)
In the present case the coefficient 60 = 0 because we have equal number of bosonic and fermionic degrees of freedom. The coefficient 62 =0 because we the total curvature of AdS2 and S2 vanishes. The coefficient
b4
review)
To summarize, we find that Sb + S F = cSNb=e4ian (AdS2 x S2),
ri =Tv + 6rs +4r;, ln(W )i_ioop = -ri,
b4
(360(4n)2
[2GabDaDbR
(16)
+ 5r2 - 2rlb + 2rlbcd],
(22)
that is the action of the quadratic fluctuations is, up to the prefactor c, the action of an Abelian N =4 multiplet on AdS2 x S2 where both AdS2 and S2 have the same radius Lk. The prefactor c, however, is important as it depends on k. It also means that the fluctuations have a non-trivial norm given by
||$||2 = y d4aVM$J= c J d4aVG$J(17)
and similarly for the vector and spinor fluctuations.
Now let us explicitly compute the 1-loop effective action
where b equals 1 for a scalar, 11/2 for a Majorana fermion and 62 for a vector. In the present case we obtain
Vol • 64 = -1.
(23)
This means that the 1-loop effective action is given by
22
ri = -Vol • 64ln- +T/m 2 e '
1 a2 2 ln T + r/i" :
(24)
where rfin is finite ^d independent of the radius a.
The term rfin ^^ ^e ^^^^ted using the Ç-function regularization [16]
(18)
Z (s) =
1
where we have contributions from a vector, 6 scalars and 4 Majorana fermions. Let us first ignore the
r(s)
dt ts-iK(t) :
1 a2 1
ri = -1 z(0) ln ar - 1Z'(0).
00
k
00
0
With the proper zero mode treatment one can show that Z(0) = Vol • b4 = —1. Explicit calculations presented in [16] give
_ N 45 - 158ln2 - 160lnn , „
C(o) = -120-+ 4ln A
- 30ZR(-3)Ji + J2'
(26)
Here A is the Glaisher's constant ln A = y^ - ZR(-1) and Jy,2 are given by
f ~ dvv
= 32Jo Hi,2(v),
. r(-iv + 1/2) Hl = iV r(iv + 1/2)
+ 2)(-iv + 1/2) + 2)(iv + 1/2), H2 = iv^ + 2)(-iv) + 2)(iv),
(27)
(TrA1)res = K (s)
0
= Vol • b4 ,
(30)
ri = 1ln (^i21 ) + f ■
Recalling that a = Lk, c
T, 1, L2 1 K3
ri = - ln--+ - ln .
i 2 e +2
^/i+k2 we obtain
+ r
fin ■
(31)
(32)
Note that the result is not UV finite and, hence, is not well-defined. This is not surprising since the Born-Infeld action is only the effective action for massless
open string modes. However, we observe that the second term in (32) is cut-off independent. Thus, we
propose that it is robust and survives the proper
k
independent constant the 1 /N correction to the Wilson loop reads
1 K3 ln(W)i — loop = ln H , 2
2 1 + K2
(33)
Let us compare it with the matrix model results. First, lets us compare with the matrix integral for the representation □k. The matrix model result in this case is [221
ln(Wnfc ) = - -ln[KV 1 + k2]
(34)
where is the poly-gamma function. We were not
able to evaluate the integrals Ji,2 analytically but it is straightforward to evaluate them numerically.
Finally, let us take into account the contribution from the prefactor c in (16). Let us consider
Det(c • A) = Det(c) • Det(A) = eTrln c+TrlnA . (28)
We are interested in the first factor
Trln c = ln c • TrA1 =ln c • Vol • ^ f* (a )/„(a), (29)
n
where we represented the trace of the unit operator in terms of a complete set of eigenfunctions of the corresponding differential operator. Comparing with (19) we notice that
which is not the same as (33). Hence, we showed explicitly that D3-branes do not describe Wilson loops
k
as was originally proposed by Drukker and Fiol in [8].
Note, however, that (33) and (34) agree in the limit k
dimensions of the representations □^d Symk,
dak = Nk , dS
(N + k - 1)!
Symk k!(N — 1)! ' N k/N
ln dak = ln dSymk = k ln N. Due to this equality it is natural to expect that In(Wnfc} =ln(WSymfc}
(35)
(36)
(37)
where we recalled that 6o = b2 =0 after we sum over the bosonic and fermionic degrees of freedom. Thus, the full 1-loop contribution is
in this limit.
Finally, let us compare (33) with the matrix integral for the representation Symk. The matrix model calculation of the subleading correction in the limit
N, A k
recently performed in [23] and gave exactly the same result as (33). This can be viewed cUS cl confirmation that D3-branes indeed describe Wilson loops in the symmetric representation and clS cl highly non-trivial check of the AdS/CFT correspondence.
Acknowledgement
The work was supported by the ARC Future Fellowship FT120100466 and in part by the ARC Discovery project DP140103925. The author would like to thank A. A. Tseytlin for collaborations on [16].
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Received, 05.11.20Ц
Е. И. Вухбиндер
ПОПРАВКА К БЗ-БРАННОМУ ОПИСАНИЮ КРУГОВЫХ ПЕТЕЛЬ ВИЛЬСОНА
Мы рассматриваем однопетлевую поправку к действию пробной БЗ-браны в Л<1Я5 х Я5, разложенному около классического решения Друккера-Фиола, заканчивающегося на окружности на границе. Она дается логарифмом одно-петлевой статистической суммы Абелева N = 4 векторного мультиплета в геометрии Лё,Я2 х Я2. Эта однопетлевая поправка описывает 1 /М слагаемое в вакуумном среднем петли Вильсона, имеющей форму окружности, в симметричном представлении в N = 4, Яи(М) теории Янга-Миллса в пределе сильной связи. Мы также обсуждаем сравнение с матричной моделью.
Ключевые слова: А4Я/СРТ соответствие, петли Вильсона, Д-браны, матричные модели, ядро теплопроводности.
Вухбиндер Е. И., доктор. Университет Западной Австралии.
35 Stirling Highway, Crawley WA 6009, Австралия. E-mail: [email protected]
