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Wilson Loops in 3d QFT from D-branes in AdS(4) x CP**3

We study the Wilson loops and defects in the three dimensional QFT from the D-branes in the AdS(4) x CP**3 geometry. We find out explicit D-brane configurations in the bulk which correspond to both straight and circular Wilson lines extended to the boundary of AdS(4). We analyze critically the role of boundary contributions to the D2-branes with various topology and to the fundamental string actions.


Introduction and Summary
Recently, Aharony, Bergman, Jafferis and Maldacena (ABJM) [1] have proposed a new class of gauge-string duality between N = 6 Chern-Simons theory and type IIA string theory on AdS 4 × CP 3 . More precisely, ABJM theory has been conjectured to be dual to M-theory on AdS 4 × S 7 /Z k with N units of four-form flux which for 1 << N << k 4 can be compactified to type IIA theory on AdS 4 × CP 3 , where k is the level of Chern-Simon theory with gauge group SU (N ). The ABJM theory is weakly coupled for λ << 1, where λ = N/k is the 't Hooft coupling.
In proving AdS 5 /CFT 4 duality, the integrability of both the string and the gauge theory side has played a key role. The semiclassical string states in the bulk side has been used to look for suitable gauge theory operators in the dual side, in establishing the correspondence. This important observation makes one to believe that perhaps a similar structure of integrability can be employed in the recently proposed AdS 4 /CFT 3 to understand it better. Indeed in [2,3,4] there has been attempts along this and it seems quite interesting to study the semiclassical rotating strings in particular sectors of the theory (see for example [5]- [16]). For example, the giant magnon solution and spike string solutions has been studied and they would certainly correspond to the trace operators in the three dimensional CFT 1 .
Further in the gauge theory, Wilson loop operators are non-local gauge invariant operators in gauge theory in which the theory can be formulated. In the absence of matter, the Wilson loops in Chern-Simons theory compute topological objects as knot invariants [38], and are somewhat less interesting than in four dimensions, where they can be used as an order parameter for confinement. For theories coupled to matter, on the other hand, we expect to find a similar structure to the four-dimensional case of N = 4 SYM, where the definition of these operators involves the scalar fields in a non-trivial way. In fact, one defines a Wilson loop as the trace in an arbitrary representation R of the gauge group G of the holonomy matrix associated with parallel transport along a closed curve C in spacetime. Since the beginning of the proposed AdS/CFT correspondence [39], it is known that Wilson loops in N = 4 SYM theory can be calculated in dual description using macroscopic strings [40,41]. This prescription is based on a picture of the fundamental string ending on the boundary of AdS 5 along the path C specified by the Wilson loop operator. The description of this Wilson loop in terms of a fundamental string is a well established part of the AdS/CFT dictionary. In a recent interesting paper [42] a class of new open string solutions in AdS 5 were found which end at the boundary on various Wilson lines. It was found that these configurations arising out of the solutions to the equations of motion corresponding to fundamental strings, they describe Wilson loops in the fundamental representations.
Motivated by the recent development of AdS 4 /CFT 3 duality, we would like to find out these Wilson line solutions in the AdS bulk, which will correspond to non-local gauge invariant operators in the CFT side. However, we will look at various D-branes in the AdS side and argue for the Wilson loop solutions. Some time back it was argued in very interesting paper [43], for type IIB theory, that Wilson loops have a gravitational dual description in terms of D5-branes or alternatively in terms of D3-branes in AdS 5 × S 5 background 2 . More precisely, in [57] it was argued that a Wilson loop with matter in the rank l symmetric representation is better described as a D3-brane embedded in AdS 5 with l units of electric flux. Further, it was argued in [58,59,66] that a Wilson loop with matter in the rank l antisymmetric representation is better described by a D5-brane whose world-volume is a minimal surface in the AdS part of the geometry times an S 4 inside the S 5 and that has a support from l-units of world-volume electric flux.
Hence, it seems that perhaps there are D-brane representations in type IIA theory which correspond to Wilson lines in the dual 3d CFT. In fact various particle like branes has been found out in [1] wrapping various cycles in CP 3 . There are D0-branes, D2-brane wrapped CP 1 ⊂ CP 3 , D4-branes wrapped on CP 2 ⊂ CP 3 and D6-brane wrapped on CP 3 and so on. We will however, find out solutions for the D-branes which correspond to Wilson lines in the 3d boundary theory. We study the D2-brane completely in the Euclidean AdS 4 in analogy with the recent paper [65]. Moreover, by adding a surface term, in accordance with [57] we find that this solution has similar form as D3-brane configuration in AdS 5 and it corresponds to line operator in N = 4 SYM theory. We also argue that the induced geometry on the world-volume of D2-brane is AdS 2 × S 1 . Moreover we also show that the action evaluated on the classical configuration vanishes and hence we can expect that the vacuum expectation value of dual line operator in 3d theory is equal to one. Then our results can be interpreted as a support for an existence of line operators in 3d theory. It would be really interesting to find their explicit form.
As the next step we propose another D2-brane configurations with topology AdS 2 × S 1 , corresponding to the straight and circular Wilson lines. These after adding the boundary terms to cancel the divergence of the action while evaluated on a straight surface have shown to behave like Wilson lines that end on the boundary of AdS.
The rest of the paper is organized as follows. In next section (2) we review basic facts about ABJM theory. Then in section (3) we review description of Wilson loops using fundamental string in dual geometry. Then in section (4) we study the D2-brane inside the Euclidean AdS 4 . We show that this configuration corresponds to the dual surface operator of dimension one. Then we study the D2-brane configuration which has AdS 2 × S 1 topology. We find the D2-branes which correspond to straight and circular Wilson line solutions. We study the boundary terms in both the cases. We conclude in section (5).

Basic Facts About ABJM Theory
The purpose of this section is to outline the basic facts about the ABJM theory. It is a 3d superconformal Chern-Simons-matter theory with explicit N = 6 supersymmetry and it can be interpreted as a theory describing N coincident M2-branes at the singularity of the orbifold C 4 /Z k . This theory has following basic properties: • Gauge and global symmetries:  We also denote trace over U (N ) and U (N ) as Tr and Tr respectively.

Fundamental string as Wilson line
In this section we review the description of Wilson loop using fundamental string in the bulk of AdS 4 . We are interested in the configuration when string sigma model is embedded in AdS 4 only. Since AdS 4 can be trivially embedded in AdS 5 it is clear that many results that were derived for Wilson loops in N = 4 SYM are valid also for Wilson loops in N = 6 CS.
We begin with Nambu-Gotto action and where τ F = 1 2π in our units. In order to describe infinite Wilson line we consider an ansatz while remaining fields are constant. For this ansatz we have and hence the action (3.1) takes the form Further, the equation of motion for X takes the form that has clearly solution as X ′ = 0 ⇒ X = const. Then this ansatz above describes infinite long Wilson line. For this line the action takes the form where in order to derive finite result we imposed condition that Wilson line is extended in time interval (−T /2, T /2). Let us add boundary term to the action. Recall that we have firstly evaluate this term for arbitrary Y and X, then include the ansatz given above and finally evaluate it on the surface Y = σ = ǫ and we obtain familiar result that that shows that the expectation value of corresponding Wilson line is < W >= 1 so that this Wilson line preserves some fraction of supersymmetry. We discuss the number of unbroken supersymmetries below. It is important property of superconformal theories that straight Wilson loop can be conformally transformed to space-like circular Wilson loop that has following string theory description. To begin with let us consider the metric in the form (3.10) Then we consider an ansatz and hence the action (3.1) takes the form ¿From this action we easily determine the equation of motion for R (3.14) Then it can be easily shown that (3.14) can be solved with the ansatz (3.16) Further, we add to the action the boundary term in the form Then we obtain that the whole action is finite and equal to using the fact that τ F = 1 2π andR 2 = π √ 2λ. Let us now briefly review the explanation why the expectation values of the straight and circular Wilson lines are different [67], at least in case of CF T 4 /AdS 5 correspondence. As was argued there the origin in the difference is in the application of the conformal transformation that maps line to the circle. In fact, in order to map the line to the circle we have to add the point at infinity to the line. Then there is a small difference in the calculation of the perturbative theory when we have to add a total derivative to the propagator. Then it was shown in [67] that the perturbative calculation on SYM side agrees with the string theory description of this Wilson line.
Let us now briefly discuss the space-time symmetries that are preserved by these classical string solutions. It can be shown that each string configuration wraps an appropriate AdS 2 submanifold in AdS 4 so that it preserves SL(2, R) × SO(2) symmetry of isometry SO(2, 3) of AdS 4 . On the other hand the string is localized at CP 3 so that it breaks the original isometry U (4) of CP 3 to U (1) × U (3). Further, it can be shown that both these string configurations preserve 12 supercharges from 24 supercharges of original background.
With analogy with AdS 5 /CF T 4 correspondence we suggest that this string configuration is dual straight and circular Wilson line in dual theory. However we leave the analysis of the properties of these objects for further works.

D-branes in Euclidean AdS 4 × CP 3
We start by writing down the metric for AdS 4 ×CP 3 , which in a particular parametrization reads AdS 4 = − cosh 2 ρ dt 2 + dρ 2 + sinh 2 ρ dθ 2 + sin 2 θdφ 2 ds 2 CP 4 = dξ 2 + cos 2 ξ sin 2 ξ dψ + and The 't Hooft coupling constant is λ ≡ N/k where k is the level of the 3-dimensional N=6 ABJM model. The relation between the parameters of the string background and of the field theory are (for α ′ = 1)R At the same time the two-form field strength is given by and the four-form field where dΩ AdS 4 is unit volume form of the AdS 4 . There exist freedom in determination of the three-form C (3) and we choose following one The dynamics of Dp-brane in general background is governed by following Dirac-Born-Infeld type of action including the Wess-Zumino term: where τ p is Dp-brane tension, ξ α , α = 0, 1, . . . , p are the (p + 1) world-volume coordinates and where A α is gauge field living on the world-volume of Dp-brane. Note also that C in the last line in (4.8) means collection of Ramond-Ramond fields.

D2-brane
Our goal is to find D-brane description of Wilson lines in dual 3d theory. In order to do this we will consider Euclidean version of AdS 4 and also write it in the following form where the boundary of AdS 4 is at y = 0. Let us consider following D2-brane configuration where ξ, θ 1 , θ 2 , φ 1 , φ 2 are the coordinates of CP 3 . This configuration should correspond to a topological two dimensional operator in the dual CFT on the boundary of the AdS 4 . Further, for the ansatz (4.10) the matrix A takes the form where y ′ = dy dr ,φ 1 = dφ 1 dα . Then it is easy to see that the D2-brane action takes the form (1 + y ′2 ) r 2 + y 2 (cos 2 ξ sin 2 ξ cos 2 θ 1 + cos 2 ξ sin 2 θ 1 )φ 2 1 − r y 3 .

(4.12)
In order to simplify the analysis we note that the equation of motion for θ 1 for non-zero ξ has two solution θ 1 = 0 and θ 1 = π and we choose θ 1 = 0. Then the equation of motion for ξ has solutions for ξ = 0, π 2 or ξ = π 4 and in order to find non-trivial configuration we choose ξ = π 4 . Then the action (4.12) simplifies considerably  Similar half-BPS configuration was previously studied in [64] in the case of AdS 5 × S 5 background. The induced metric on the world-volume of D2-brane is equal to that clearly shows that this D2-brane configuration has a topology AdS 2 × S 1 . Let us now evaluate the action on this ansatz where T is regularized interval in x 0 direction and where we have introduced a regulator ǫ << 1. It is however important to stress that since we consider D2-brane that has a finite extend we should take the boundary terms into account. We will discuss the boundary contributions in more details bellow. Here we only stress that, following [57], that we should add to the action the boundary contribution in the form κǫ .

(4.19)
Consequently we obtain the action after adding the boundary contribution We see that the action vanishes and hence the vacuum expectation value of the dual line operator corresponding to the D2-brane explained above, is equal to one. This is a strong indication that the dual line operator is stable and preserves some fractions of supersymmetry. It would be really interesting to study these operators from the point of view of 3d QFT, following [60,61].

D2-brane with topology AdS 2 × S 1
In this section we present another example of D2-brane solution that has a topology AdS 2 × S 1 . Once again we start with the metric on AdS 4 as This D2-brane can be described by the following parametrization: The DBI part of the D2-brane action is written as Further, there is a WZ term in the following form where the non-zero one form R-R field is The fact that the WZ term in Euclidean signature contains factor i implies that we have to consider imaginary electric flux so that we can write cos 2 ξ sin 2 ξ cos 2 θ 1 + cos 2 ξ sin 2 θ 1 ) , (4.26) where X ′ = ∂X/∂σ. Hence we obtain Let us now solve the equations of motion coming from the above DBI action including the WZ term. First of all the variation of the action with respect to θ 1 implies 3 cos 4 ξ sin θ 1 cos θ 1 sin 2 ξ cos 2 ξ cos 2 θ 1 + cos 2 ξ sin 2 θ 1 When cos ξ = 0, the equation above is solved with θ 1 = 0, π and we choose for our convenience θ 1 = 0. Further the equation of motion for ξ implies Now let us consider the equations of motion for A 0 , A 1 that again imply the existence of a conserved electric flux Π: Simplifying the above equations one gets: Inserting the above result into the equation of motion for ξ we obtain . (4.32) Note that this equation is obeyed for any ξ. Notice also that in deriving the above, we have used the fact that Finally we consider equation of motion for X that takes the following simple form It is clear that this equation has natural solution X ′ = 0 ⇒ X = const.. Then the metric induced on the world-volume of D2-brane takes the form that clearly has a form AdS 2 × S 1 . Let us again compute the current J 01 which takes the form Note that it is again proportional to the number of fundamental strings k.
Let us now evaluate the action on the solution given above (4.36) Where deriving above we have used the following identities Notice that this action derived above is proportional to the world volume electric flux Π.

Boundary terms
The D2-brane solution that we found corresponds to D2-brane that extends all the way to the boundary of AdS 4 and ends there along a one-dimensional curve. Since this D2-brane action has finite extent we have to discuss the possibility of adding boundary terms to the action. These boundary terms should not change the equations of motion and hence the solution will still be the same, but the value of the action when evaluated at this solution will in general depend on the boundary terms.
As it is well known that when we calculate the Wilson loop using string surfaces the bulk action is divergent but this divergence can be canceled by boundary term as we will review in the Appendix. Explicitly, the string that corresponds to Wilson loop has to satisfy three Dirichlet boundary conditions on the three directions parallel to the boundary of AdS 4 , and the seven Neumann ones that combine the radial coordinate of AdS 4 and coordinates along CP 3 and hence it is the appropriate action assuming that we have Dirichlet boundary conditions. So we have to add appropriate boundary terms that change the boundary conditions.
In case of the fundamental string we have the coordinate Y with Neumann boundary condition (before imposing the static gauge). Then it is natural to define p y as the momentum conjugate to it where n is normal derivative to the boundary. The new action including the term that changes the boundary conditions isS where the integral is over the boundary at a cutoff Y = Y 0 . In fact, the variation of the original action is where we used the fact that the field Y obeys the equation of motion in the bulk. The boundary term above shows (since it is proportional to δY ) that on-shell action is functional of Y . Then including the boundary term we obtain and hence the new action is functional of p Y as it should be. Taking a lesson from the review of the boundary term for the fundamental string mentioned above we now return to D2-brane action. In case of D2-brane action there is another subtlety. The DBI action is a functional of the gauge field, but the Wilson loop observable should depend on the variable Π. Then in order to find correct form of the action we have to add to it following boundary term where δL δ∂σ A 0 = −Π = const. Now if we collect all these terms together we obtain S + δS = 0 (4.43) We find the straight Wilson line has vanishing action after adding the boundary term which is the same result as in case of AdS 5 /CF T 4 correspondence.

Circular Wilson line
In order to find this solution we consider following form of AdS 4 metric ds 2 AdS 4 = 1 y 2 (dy 2 + dr 2 + r 2 dφ 2 + d(x 0 ) 2 ) (4. 44) and consider the Wilson line that is siting at x 0 = 0. Our goal is to find Wilson line that for y = 0 takes the form of circle with r 2 = R 2 . In order to find such a configuration we consider an ansatz Then we easily obtain A 00 =R 2 r 2 y 2 , A 11 =R 2 1 y 2 (1 + r ′2 ) , A 01 = i(2πα ′ )F , A 10 = −i(2πα ′ )F , A 22 =R 2 (cos 2 ξ sin 2 ξ cos 2 θ 1 + cos 2 ξ sin 2 θ 1 ) , (4.46) where r ′ = ∂ σ r. Then we obtain so that DBI part of the action takes the form Further, there is a coupling to C (1) field in the form As in the case of straight Wilson line we start with the equation of motion for θ 1 and we found that it is solved for θ 1 = 0, or θ 1 = π and we choose for our convenience θ 1 = 0. Further the equation of motion for ξ implies Further, the equations of motion for A 0 , A 1 imply a conserved electric flux Π that allows us to find (4.52) Inserting the above results into the equation of motion for ξ we again obtain Note that in deriving the above, we have used the fact that Finally we determine the equation of motion for r 1 Now we will argue that the ansatz r 2 = R 2 − y 2 solves the equation above. Indeed, using the fact that r ′ = − y r we one can check that the above equation is identically zero. Let us now evaluate the action for the D2-brane configuration on this solution (4.55) Further, we have the first boundary contribution at σ = ǫ. We again proceed as in previous section for the boundary contributions. Namely, we evaluate the contribution to the action for general y, then insert the ansatz y = σ and finally evaluated the action at y = ǫ δS Y = − dφ 2 dφ δL δ∂ σ y y = −4π 2 τ 2 e Φ 0R k 2 8 R 2 − ǫ 2 ǫR . (4.56) In the same way the boundary contribution from the gauge fields takes the form (4.57) Collecting all these terms together we obtain that the divergent terms cancel as in the case of straight Wilson line. On the other hand we find finite contribution to the action in the form where we used the convention that τ 2 = 1 4π 2 and where S F S is the fundamental string action evaluated on circular Wilson line. Borrowing the interpretation of Wilson lines in AdS 5 /CF T correspondence using D3-branes we can argue that our solution describes Wilson line in symmetric k 4 representation where k is level of CS action. It would be certainly very interesting to study the problem whether there exists D-brane description of Wilson loops in arbitrary representations. We hope to return to these problems in future.

Conclusions
We have studied in this paper various D-brane configurations corresponding to Wilson loops and lines in the boundary of AdS 4 . We have taken D2-brane as an example and have shown various configurations of this in the Euclidean AdS 4 × CP 3 correspond to straight and circular Wilson line solutions. We analyzed the D2-brane which has a topology of AdS 2 × S 1 and corresponds to straight Wilson line gives vanishing action after adding the boundary term. We have also studied the circular Wilson line solution and show that the action gives a non-zero contribution after adding the boundary term. One can study the D4-brane which correspond to both straight and circular Wilson line in the boundary AdS 4 , however we mention that the action will take a similar form like that of D2-brane and hence the analysis is very similar to the one performed in the present paper.
Note added: While we were finalizing to submit our paper, we came across [62] and [63] which have some overlap with our present work.