A Semiclassical String Description of Wilson Loop with Local Operators

We discuss a semiclassical string description to circular Wilson loops without/with local operator insertions. By considering a semiclassical approximation of type IIB string theory on AdS_5 X S^5 around the corresponding classical solutions, quadratic actions with respect to fluctuations are computed. Then the dual corresponding operators describing the fluctuations are discussed from the point of view of a small deformation of the Wilson loops. The result gives new evidence for AdS/CFT correspondence.


Introduction
Almost a decade has passed from a discovery of AdS/CFT correspondence [1,2]. Now it is firmly supported by enough evidences, but there is no proof of it now. Hence it is still important to continue to seek further, new confirmation to support it.
One of the difficulties is to analyze type IIB string on AdS 5 ×S 5 . The action is constructed in [3] and its classical integrability is shown in [4]. However, it still seems difficult to quantize the theory manifestly, simply because the action is quite non-linear. A sensible way is to find a solvable subsector such as the BMN sector [5]. The BMN sector is pulled out by taking a Penrose limit [6]. Then the simplified string theory is exactly solvable [7,8], and hence one can test the duality at stringy level though the argument is restricted to a certain region.
It is pointed out in [9] that a non-relativistic limit of type IIB string on AdS 5 ×S 5 gives a new arena to test the AdS/CFT. It is shown in [10] that the limit is regarded as a semiclassical approximation around a static AdS 2 solution [11] like as the Penrose limit is around a BPS particle [12]. This equivalence holds even for AdS-branes [10,13]. With this semiclassical interpretation it has been shown that the corresponding operator in the gauge theory is nothing but a small deformation of straight Wilson line [10].
The purpose of this paper is to generalize the result for the straight line to circular Wilson loops without/with local operator insertions. An AdS 2 solution corresponds to a 1/2 BPS circular Wilson loop without the insertions [14]. A semiclassical approximation around the solution has already been studied in [15]. We newly compute a quadratic action around the solution corresponding to a circular Wilson loop with the local operators, Z J and its complex conjugate. Here Z is a complex scalar composed of the two real scalar fields in N =4 SYM like Z ≡ φ 1 + iφ 2 . The resulting quadratic fluctuations describe the action in [15] around σ = 0 while those behave as a pp-wave string at σ = ∞ .
Then we clarify that the fluctuations correspond to a small deformation of the circular Wilson loops without/with the insertions. In particular, the dictionary of impurity insertion is derived. With no local operator the dictionary is the same as in the case of the straight line [10]. This result is not so surprising since the difference between a straight Wilson line and a circular Wilson loop is the behavior at infinity and it only gives an anomalous contribution to the expectation value (and the value of classical action of the corresponding string solution). However the local behavior around a finite point should not be different. With the local operators, it is the same as the case without them apart from the insertion points while it is nothing but the BMN dictionary [5] on the inserted local operators. This result nicely agrees with the behavior of the semiclassical action. This paper is organized as follows. In section 2 we reproduce a semiclassical action around an AdS 2 solution whose boundary is a circular Wilson loop with no local operator.
Then, in section 3, we discuss the corresponding operators in the gauge theory from a small deformation of the circular Wilson loop. In section 4, as a further generalization, we consider a semiclassical action around the Miwa-Yoneya solution [16], which is a generalization of the solution constructed by Drukker-Kawamoto [17] in the Lorentzian case.
This solution corresponds to a circular Wilson loop with local operator insertions. The resulting action interpolates the pp-wave string action and the semiclassical action around the AdS 2 as expected. In section 5 we consider a small deformation of the Wilson loop corresponding to the semiclassical action obtained in section 4. The configuration of the Wilson loop is more involved. Section 6 is devoted to a summary and discussions.

Semiclassical limit around a circular solution
In this section, as a warming up, let us consider a classical string solution whose boundary describes a circular Wilson loop [14] without local operator insertions. Note that the quadratic string action with respect to the fluctuations has already been computed by Drukker-Gross-Tseytlin [15]. To make the present paper self-contained, however, we shall rederive the result of [15] here. Then we show the agreement between the fluctuations around the classical solution in the string side and those around the circular Wilson loop in the gauge theory.

Classical solution for a circular Wilson loop
First let us discuss a classical solution describing a circular Wilson loop. We begin with the string action in the Polyakov formulation and the bosonic part is given by where γ ij is an auxiliary world-sheet metric and we work in conformal gauge, The spacetime metric G M N describes AdS 5 ×S 5 and it is given by Hereafter we will work in Euclidean signature and Poincare coordinates. See Appendix A for the detail expressions of vielbeins and spin connections.
The equation of motion reads The Virasoro constraints to be imposed are It is easy to see that Here it is valuable to comment on the relation between a circular loop and a straight line. First we move to Cartesian coordinates, where the radius of the circle R has been recovered. Then let us rescale τ and σ as τ → τ R , σ → σ R and take the large R limit. As a result, (2.4) is reduced to a static AdS 2 solution At the boundary σ = 0 this solution describes a straight Wilson line.

Semiclassical limit
Next we consider a semiclassical approximation of the full type IIB string on AdS 5 ×S 5 around the classical solution (2.4).
Let us expand the string action (2.1) about the classical solution (2.4) z = tanh σ +z , r = 1 cosh σ +r , θ = τ +θ , where quantum fluctuations are denoted as the symbols with tilde likeX . Hereafter the overall factor of the action (2.1), √ λ is absorbed into the definition of fluctuations by rescaling the variables asX → λ −1/4X . Then the value of λ should be taken to be large in order for the semiclassical (quadratic) approximation to be valid.
An additional redefinition of the variables is performed as r =r coth σ ,x =x coth σ ,z =z coth σ ,θ =θ 1 sinh σ , and the following quadratic action is obtained, Although one should impose the Virasoro constraints to eliminate the longitudinal modes, it is not an easy task for small fluctuations. Thus we shall take another course following [15] instead. Note that the action (2.5) can be rewritten with conformal gauge as follows: Here the following quantities have been introduced: where E A and Ω A B are vielbein and spin connection of AdS 5 ×S 5 evaluated with the classical solution (2.4). In the present case we obtain Here the following abbreviations have been introduced: Then the mass term for ζ 1 and ζ 4 can be expressed as and it is diagonalizable by taking the following linear combination The mass eigenvalues are given bỹ The new linear combination has been introduced and then the covariant derivative accordingly turn to be where we have defined Substituting the above quantities into the action, the resulting action is Let us write it as the action on the two-dimensional induced metric Imposing Virasoro constraints is equivalent to removing the longitudinal modes by adding the ghost action We choose g ij as the induced metric (2.8). The covariant derivative is defined as where ω is the two-dimensional spin connection: ω 0 1 = − c s dτ , then The ghost action is the same as that of ζ 0 andζ 1 . Thus these modes may be eliminated by the constraints. The final gauge-fixed action is Finally, let us comment on the fermionic fluctuations. The quadratic action is where E and Ω are evaluated with the classical solution. After rotating the spinor basis so that a two-dimensional spinor covariant derivative is manifest, and fixing κ-symmetry appropriately, the mass squared for the fermions is m 2 = 1 and the mass term is propor- is not broken by the fermions.

Small deformations of circular Wilson loop
From now on let us discuss the corresponding gauge-theory operator describing the fluctuations obtained in the previous section by following [10,18].
Let us consider a Wilson loop gives the following expression Thus the Wilson loop (3.1) is invariant under supersymmetry transformation ifẎ M Γ M ǫ = 0. The locally supersymmetry condition is derived as the integrability We are interested in a circular Wilson loop configuration C 0 Hence the Wilson loop W (C 0 ) is which satisfies the locally supersymmetric condition. We identify W (C 0 ) as the vacuum operator.
Let us consider a small deformation of C 0 : The Wilson loop can be expanded as where ellipsis implies higher order fluctuations. Note that where a = r, 2, 3 .
The mass dimensions of fermionic fluctuations ∆ = 1 2 (1 + 2|m|) is since m 2 = 1 . Thus we expect that the eight fermionic impurities with conformal dimension 3 2 are inserted in the Wilson loop as well as bosonic impurities (3.4). This expectation is correct as we show in Appendix B.

Semiclassical limit around Miwa-Yoneya solution
As the second issue we consider a rotating classical solution in which the boundary is a circular Wilson loop with local operator insertions. The classical solution was constructed in [16]. This solution is a generalization of [17], which was constructed for a straight Wilson line in Lorentzian signature. We will examine a correspondence between the fluctuation about the classical solution and small deformation of the circular Wilson loop with local operator insertions.

Classical solution for a circular Wilson loop with local operator insertions
First of all, let see the classical solution found in [16]. We work in Euclidean AdS 5 ×S 5 with the Poincare coordinates (See appendix A for vielbeins and spin connections.) The solution corresponding to a circular Wilson loop with the local operator insertions is given by [16] for the AdS 5 part, and ψ = τ , cos θ = tanh σ (4.3) for the S 5 part. The parameter α parametrizes the radius of the loop R like Note that we have performed a Wick rotation as ψ → −iψ following [16,19]. That is why we can define a sensible angular momentum even in Euclidean signature. But we should keep it in mind that the signature of dψ 2 in the metric (4.1) is flipped as −dψ 2 due to the Wick rotation of ψ .

Semiclassical limit
Next lets us consider a semiclassical action around the solution (4.2) and (4.3) .
The quadratic action around the classical solution (4.2) and (4.3) is basically given by (2.6) , where the covariant derivatives and mass matrices are replaced by 3,7,8,9) , respectively. Here we should note that an additional term should be added to the action because of the presence of a conserved charge associated with ψ . The only effect of adding the term is to change the sign of the kinetic term of ψ , and that is why we arrive at the same action even after the Wick rotation of ψ .
The corresponding eigenvectors are found to be Therefore we find (1) , v (4) ) .

It follows that
Thus we have obtained the diagonalized mass-matrix Here we have used the notation (2.7) .
Next let us examine the kinetic terms for ζ 0 , ζ 1 and ζ 4 . Then we see that where the following quantities have been introduced These expressions lead us to Thus we have obtained We have derived the eigenvalues of the mass matrix in the bosonic sector so far. It seems to be difficult to add the ghost action so that it should delete the unphysical longitudinal modes in the matrix. Actually, we have not completed this step and we will leave it as a future problem. Instead of trying to delete the unphysical modes from the full action, let us consider the Lagrangian density in the two special regions, 1) σ = 0 and 2) σ = ∞ . For the former case we expect that the Lagrangian density should behave as that in section 2. For the latter case the Lagrangian density is expected to be the one for the pp-wave string action.
The fluctuations near σ = 0 The ghost action (2.9) is introduced as before. Let us consider the two-dimensional metric which is AdS 2 and corresponds to the induced metric of AdS 5 part only. The covariant derivative is defined by where ω is the two-dimensional spin connection: ω 0 1 = − 1 s dτ , then Thus we see that the covariant derivative defined in (4.5) is nothing but two-dimensional covariant derivative.
The mass dimension of a fluctuation is derived through its behavior near the boundary.
Near σ = 0 Here we note that the contribution of fluctuationsζ 0 andζ 1 is the same form as the ghost action near σ = 0, so that it is canceled out by the ghost contribution. In addition we should note that g ij D iζ a D jζ a ≈ g ij ∂ iζ a ∂ jζ a (a = 5, 6).

The fluctuations near σ = ∞
In the region near σ = ∞ we should introduce the ghost action (2.9) with flat twodimensional metric Then we can see that the longitudinal modes are canceled out with the ghost action as follows.
First let us consider the fluctuations in the neighbor of σ = ∞ . Then by using (4.7) L 2B can be rewritten as where D iζ ≈ ∂ iζ . The contribution of fluctuationsζ 0 andζ 5 is the same form as the ghost action, hence it is canceled out by the ghost contribution. Thus we are left with eight massive bosons with m 2 = 1 propagating in the two-dimensional flat space. The fermionic fluctuations with m 2 = 1 break SO (8) to SO(4) × SO(4) (see for example [20]). This is nothing but the Lagrangian density of a pp-wave string. solution on the boundary is described by and the circle is represented by That is, the radius of the loop is R and its center is located at (x 0 , x 1 ) = (0, −αR) . It is also convenient to introduce radial coordinates r and θ Then the circle lies along (r, θ) = (R, s) .
We may consider three kinds of impurity insertions as depicted in Fig. 1. Let us consider each of the cases below.
This is nothing but the dictionary for the gauge-theory operators corresponding to the fluctuations around the circular solution (non-relativistic string) [10].
A small deformation at s = s 1 , s 2 Next let us expand W (C) around C 0 at s = s 1 . This corresponds to the cases (ii) and (iii) in Fig. 1. The argument here basically follows [18].
In this case the Wilson loop can be expanded as By requiring that the small fluctuations should satisfy the locally supersymmetry condition, we obtain the following condition, In addition a reparametrization invariance allows us to fix the fluctuations as Thus we can impose the condition δẏ 1 = δẏ 2 = 0 .
The impurities respect SO(4) × SO(4) symmetry. This dictionary is nothing but the BMN one.
In summary, the resulting dictionary says that the impurity insertion at the local operators should follow the BMN-dictionary and other than the position of local operators it should follow the dictionary we obtained in [10].

Summary and Discussion
We have discussed a semiclassical approximation around the classical solutions corresponding to circular Wilson loops without/with local operator insertions Z J andZ J . We For the case of the circular Wilson loop with local operator insertions, it remains as a problem to be solved to find a ghost terms to remove the unphysical longitudinal modes for an arbitrary σ . It would be interesting to compute a semiclassical partition function after solving these problems.
One may consider a spin chain description for the circular Wilson loop case. The gauge-theory analysis is discussed in [17]. The remaining work is to construct circular Wilson loop solutions rotating with two or more spins and compare them with the gaugetheory results. A part of this issue has already been discussed in [21].
It is also nice to consider a semiclassical analysis to a dual giant Wilson loop [22].
A rotating dual giant Wilson loop solution has been already constructed in [23]. By following this paper, one can discuss the semiclassical limit around the solution and it is possible to deduce the corresponding gauge-theory side.
We believe that our approach would give a new window to test the AdS/CFT duality.

B Fermionic fluctuations of Wilson loop
A supersymmetrized Wilson loop, which is proposed in [24], is given by Here the loop includes a superpartner of (x µ (s), y i (s)), which couples to the fermion Ψ .
The supersymmetry transformation in N =4 SYM is given by and the following relation may also be included: A small deformation for the fermionic variables may be considered. By setting that ζ =ζ = 0 on C 0 , the only contribution is evaluated as Here note that the matrix defined as is a projection operator. The original fermionic variable Ψ has 16 components but it projects out half of it. As a result, the physical eight components of the fermionic variables remain.
Thus the operator insertion for the fermionic fluctuations are described by the eight fermionic variables. We have discussed the circular case so far, but the argument for the straight line is the same. It is also the same even for the BMN case, where the eight fermions iΓ 1 h + Ψ (h + = 1 2 (1 + iΓ 12 )) are inserted.