Wilson Loops in 3d QFT from D-branes in AdS(4) x CP**3
aa r X i v : . [ h e p - t h ] M a r Preprint typeset in JHEP style - HYPER VERSION hep-th/
Wilson Loops in d QF T from D-branes in AdS × CP J. Klusoˇn
Department of Theoretical Physics and AstrophysicsFaculty of Science, Masaryk UniversityKotl´aˇrsk´a 2, 611 37, BrnoCzech RepublicE-mail: [email protected]
Kamal L. Panigrahi
Department of PhysicsIndian Institute of Technology Guwahati,Guwahati , 781 039, INDIAE-mail: [email protected]
Abstract:
We study the Wilson loops in the three dimensional QFT from the D-branesin the AdS × CP geometry. We find out explicit D-brane configurations in the bulk whichcorrespond to both straight and circular Wilson lines extended to the boundary of AdS .We analyze critically the role of boundary contributions to the D2-branes with varioustopology and fundamental string actions. Keywords:
D-branes, AdS-CFT Correspondence. ontents
1. Introduction and Summary 12. Basic Facts About ABJM Theory 33. Fundamental string as Wilson line 44. D-branes in Euclidean AdS × CP × S
5. Conclusions 15
1. Introduction and Summary
Recently, Aharony, Bergman, Jafferis and Maldacena (ABJM) [1] have proposed a newclass of gauge-string duality between N = 6 Chern-Simons theory and type IIA stringtheory on AdS × CP . More precisely, ABJM theory has been conjectured to be dual toM-theory on AdS × S /Z k with N units of four-form flux which for 1 << N << k canbe compactified to type IIA theory on AdS × CP , where k is the level of Chern-Simontheory 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 /CFT duality, the integrability of both the string and the gaugetheory side has played a key role. The semiclassical string states in the bulk side hasbeen used to look for suitable gauge theory operators in the dual side, in establishingthe correspondence. This important observation makes one to believe that perhaps asimilar structure of integrability can be employed in the recently proposed AdS /CFT tounderstand it better. Indeed in [2, 3, 4] there has been attempts along this and it seemsquite interesting to study the semiclassical rotating strings in particular sectors of thetheory (see for example [5]-[16]). For example, the giant magnon solution and spike stringsolutions has been studied and they would certainly correspond to the trace operators inthe three dimensional CFT .Further in the gauge theory, Wilson loop operators are non-local gauge invariant op-erators 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 for more related work see [17]-[37] – 1 –38], and are somewhat less interesting than in four dimensions, where they can be usedas 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, wherethe definition of these operators involves the scalar fields in a non-trivial way. In fact, onedefines 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 inspacetime. Since the beginning of the proposed AdS/CFT correspondence [39], it is knownthat Wilson loops in N = 4 SYM theory can be calculated in dual description using macro-scopic strings [40, 41]. This prescription is based on a picture of the fundamental stringending on the boundary of AdS 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 establishedpart of the AdS/CFT dictionary. In a recent interesting paper [42] a class of new openstring solutions in AdS 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 mo-tion corresponding to fundamental strings, they describe Wilson loops in the fundamentalrepresentations.Motivated by the recent development of AdS /CFT duality, we would like to findout these Wilson line solutions in the AdS bulk, which will correspond to non-local gaugeinvariant operators in the CFT side. However, we will look at various D-branes in theAdS side and argue for the Wilson loop solutions. Some time back it was argued in veryinteresting paper [43], for type IIB theory, that Wilson loops have a gravitational dualdescription in terms of D5-branes or alternatively in terms of D3-branes in AdS × S background . More precisely, in [57] it was argued that a Wilson loop with matter inthe rank l symmetric representation is better described as a D3-brane embedded in AdS with l units of electric flux. Further, it was argued in [58, 59, 66] that a Wilson loop withmatter in the rank l antisymmetric representation is better described by a D5-brane whoseworld-volume is a minimal surface in the AdS part of the geometry times an S inside the S 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 theorywhich correspond to Wilson lines in the dual 3 d CFT. In fact various particle like braneshas been found out in [1] wrapping various cycles in CP . There are D0-branes, D2-branewrapped CP ⊂ CP , D4-branes wrapped on CP ⊂ CP and D6-brane wrapped on CP and so on. We will however, find out solutions for the D-branes which correspond to Wilsonlines in the 3 d boundary theory. We study the D2-brane completely in the Euclidean AdS in analogy with the recent paper [65]. Moreover, by adding a surface term, in accordancewith [57] we find that this solution has similar form as D3-brane configuration in AdS and it corresponds to line operator in N = 4 SYM theory. We also argue that the inducedgeometry on the world-volume of D2-brane is AdS × S . Moreover we also show that theaction evaluated on the classical configuration vanishes and hence we can expect that thevacuum expectation value of dual line operator in 3 d theory is equal to one. Then ourresults can be interpreted as a support for an existence of line operators in 3 d theory. It For closely related works, see [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 59, 55, 56, 58]. – 2 –ould be really interesting to find their explicit form.As the next step we propose another D2-brane configurations with topology AdS × S ,corresponding to the straight and circular Wilson lines. These after adding the boundaryterms to cancel the divergence of the action while evaluated on a straight surface haveshown 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 basicfacts about ABJM theory. Then in section (3) we review description of Wilson loops usingfundamental string in dual geometry. Then in section (4) we study the D2-brane insidethe Euclidean AdS . We show that this configuration corresponds to the dual surfaceoperator of dimension one. Then we study the D2-brane configuration which has AdS × S topology. We find the D2-branes which correspond to straight and circular Wilson linesolutions. We study the boundary terms in both the cases. We conclude in section (5).
2. Basic Facts About ABJM Theory
The purpose of this section is to outline the basic facts about the ABJM theory. It is a3 d superconformal Chern-Simons-matter theory with explicit N = 6 supersymmetry andit can be interpreted as a theory describing N coincident M2-branes at the singularity ofthe orbifold C /Z k . This theory has following basic properties: • Gauge and global symmetries:gauge symmetry : U ( N ) × U ( N ) , global symmetry : SU (4) . (2.1)We also denote trace over U ( N ) and U ( N ) as Tr and Tr respectively. • The on-shell fields are gauge fields, together with complexified Hermitian scalars andMajorana spinors ( A = 1 , , , A µ : Adj( U ( N )) , A µ : Adj( U ( N )) , µ, ν = 0 , , ,Y A = ( X + iX , X + iX , X − iX , X − iX ) : ( N , N ; ) ,Y † A = ( X − iX , X − iX , X + iX , X + iX ) : ( N , N , ) , Ψ A = ( ψ + iχ , − ψ − iχ , ψ − iχ , − ψ + iχ ) : ( N , N ; ) , Ψ † A = ( ψ − iχ , − ψ + iχ , ψ + iχ , − ψ − iχ ) : N , N ; ) . (2.2) • The action of ABJM theory takes the form ( κ = k π ) S = κ Z d x (cid:20) ǫ µνλ Tr (cid:18) A µ ∂ ν A λ + i A µ A ν A λ (cid:19) − ǫ µνλ Tr (cid:18) A µ ∂ ν A λ + i A µ A ν A λ (cid:19) ++ 12 Tr (cid:16) − ( D µ Y ) † A D µ Y A + i Ψ † A γ µ D µ Ψ A (cid:17) + 12 Tr (cid:16) − D µ Y A ( D µ Y ) † A + i Ψ A γ µ D µ Ψ † A (cid:17) −− V F − V B ] , (2.3)– 3 –here the covariant derivatives are defined as D µ Y A = ∂ µ Y A + iA µ Y A − iY A A µ , ( D µ Y ) † A = ∂ µ Y † A + iA µ Y † A − iY † A A µ (2.4)and similarly for fermions Ψ A , Ψ † A . Finally, the potential terms are V F = i Tr h Y † A Y A Ψ † B Ψ B − Y † A Y B Ψ † A Ψ B + ǫ ABCD Y † A Ψ B Y † C Ψ D i −− i Tr h Y A Y † A Ψ B Ψ † B − Y A Y † B Ψ A Ψ † B + ǫ ABCD Y A Ψ † B Y C Ψ † D i (2.5)and V B = −
13 Tr h Y † A Y B Y † B Y C Y † C Y A + Y † A Y A Y † B Y B Y † C Y C ++ 4 Y † A Y B Y † C Y A Y † B Y C − Y † A Y A Y † B Y C Y † C Y A i (2.6)
3. Fundamental string as Wilson line
In this section we review the description of Wilson loop using fundamental string in thebulk of
AdS . We are interested in the configuration when string sigma model is embeddedin AdS only. Since AdS can be trivially embedded in AdS it is clear that many resultsthat were derived for Wilson loops in N = 4 SYM are valid also for Wilson loops in N = 6CS. We begin with Nambu-Gotto action S F = τ F Z dτ dσ p det a µν , (3.1)where a µν = ∂ µ X m ∂ ν X n g mn , µ, ν = τ, σ . (3.2)and where τ F = π in our units. In order to describe infinite Wilson line we consider anansatz X = τ , Y = σ , X ≡ X = X ( σ ) (3.3)while remaining fields are constant. For this ansatz we have a ττ = ˜ R σ , a σσ = g yy ∂ σ Y ∂ σ Y + g xx ∂ σ X∂ σ X = ˜ R σ (1 + X ′ ) (3.4)and hence the action (3.1) takes the form S F = τ F ˜ R Z dτ dσ σ p X ′ . (3.5)Further, the equation of motion for X takes the form ∂ σ [ X ′ σ √ X ′ ] = 0 (3.6)– 4 –hat has clearly solution as X ′ = 0 ⇒ X = const . Then this ansatz above describes infinitelong Wilson line. For this line the action takes the form S F = τ F ˜ R Z T/ − T/ dτ Z dσ σ p X ′ = τ F ˜ R T Z ∞ ǫ dσ σ = τ F T ˜ R ǫ , (3.7)where in order to derive finite result we imposed condition that Wilson line is extended intime interval ( − T / , T / Y and X , then include the ansatz given above and finally evaluate it on thesurface Y = σ = ǫδS F = − Z T/ − T/ dτ δ L δ∂ σ Y Y == − τ F T g yy ∂ σ Y Y p g ( ∂ τ X ) p g yy ( ∂ σ Y ) + g xx ( ∂ σ X ) = − τ F T ˜ R ǫ (3.8)and we obtain familiar result that S F + δS F = 0 (3.9)that shows that the expectation value of corresponding Wilson line is < W > = 1 so thatthis Wilson line preserves some fraction of supersymmetry. We discuss the number ofunbroken supersymmetries below.It is important property of superconformal theories that straight Wilson loop can beconformally transformed to space-like circular Wilson loop that has following string theorydescription. To begin with let us consider the metric in the form ds = ˜ R y ( dy + dr + r dφ + d ( x ) ) . (3.10)Then we consider an ansatz Y = σ , X = const, τ = φ , R = R ( σ ) . (3.11)Then a σσ = ˜ R y (( ∂ σ Y ) + ( ∂ σ R ) ) , a ττ = ˜ R R Y (3.12)and hence the action (3.1) takes the form S F = τ F ˜ R Z dτ dσ RY p ( ∂ σ Y ) + ( ∂ σ R ) . (3.13)¿From this action we easily determine the equation of motion for R Y p ∂ σ R ) − ∂ σ [ RY ∂ σ R p ∂ σ R ) ] = 0 . (3.14)– 5 –hen it can be easily shown that (3.14) can be solved with the ansatz R = K − σ , (3.15)for constant K . Let us now evaluate the action for the solution (3.15) S F = τ F ˜ R Z π dτ Z Kǫ dσ Rσ p ∂ σ R ) = − πτ F ˜ R K [ 1 K − ǫ ] . (3.16)Further, we add to the action the boundary term in the form δS F = − Z π dτ δ L δ σ Y Y = − πτ F ˜ R RY ∂ σ Y Y p ( ∂ σ Y ) + ( ∂ σ R ) = − πτ F Kǫ (3.17)Then we obtain that the whole action is finite and equal to S F + δS F = − πτ F ˜ R = − π √ λ . (3.18)using the fact that τ F = π and ˜ R = π √ λ .Let us now briefly review the explanation why the expectation values of the straightand circular Wilson lines are different [67], at least in case of CF T /AdS correspondence.As was argued there the origin in the difference is in the application of the conformaltransformation that maps line to the circle. In fact, in order to map the line to thecircle we have to add the point at infinity to the line. Then there is a small differencein the calculation of the perturbative theory when we have to add a total derivative tothe propagator. Then it was shown in [67] that the perturbative calculation on SYM sideagrees with the string theory description of this Wilson line.Let us now briefly discuss the space-time symmetries that are preserved by these clas-sical string solutions. It can be shown that each string configuration wraps an appropriate AdS submanifold in AdS so that it preserves SL (2 , R ) × SO (2) symmetry of isometry SO (2 ,
3) of
AdS . On the other hand the string is localized at CP so that it breaks theoriginal isometry U (4) of CP to U (1) × U (3). Further, it can be shown that both thesestring configurations preserve 12 supercharges from 24 supercharges of original background.With analogy with AdS /CF T correspondence we suggest that this string configura-tion is dual straight and circular Wilson line in dual theory. However we leave the analysisof the properties of these objects for further works.
4. D-branes in Euclidean AdS × CP We start by writing down the metric for
AdS × CP , which in a particular parametrizationreads ds = ˜ R ( ds AdS + 4 ds CP ) ds AdS = − cosh ρ dt + dρ + sinh ρ (cid:0) dθ + sin θdφ (cid:1) – 6 – s CP = dξ + cos ξ sin ξ (cid:18) dψ + 12 cos θ dφ −
12 cos θ dφ (cid:19) + 14 cos ξ (cid:0) dθ + sin θ dφ (cid:1) + 14 sin ξ (cid:0) dθ + sin θ dφ (cid:1) , (4.1)where 0 ≤ ξ ≤ π , ≤ φ i ≤ π , ≤ θ i ≤ π , , i = 1 , , (4.2)and where ˜ R = R k , e = R k . (4.3)The ’t Hooft coupling constant is λ ≡ N/k where k is the level of the 3-dimensional N=6ABJM model. The relation between the parameters of the string background and of thefield theory are (for α ′ = 1) ˜ R = π r Nk = π √ λ . (4.4)At the same time the two-form field strength is given by F (2) = k ( − cos ξ sin ξdξ ∧ (2 dψ + cos θ dφ − cos θ dφ ) −− k (cid:0) cos ξ sin θ dθ ∧ dφ + sin ξ sin θ dθ ∧ dφ (cid:1) (4.5)and the four-form field F (4) = 3 R d Ω AdS , (4.6)where d Ω AdS is unit volume form of the AdS . There exist freedom in determination ofthe three-form C (3) and we choose following one C (3) = R y dx ∧ dx ∧ dx . (4.7)The dynamics of Dp-brane in general background is governed by following Dirac-Born-Infeld type of action including the Wess-Zumino term: S = S DBI + S W Z ,S DBI = − τ p Z d p +1 ζe − Φ √− det A , A αβ = ∂ α x M ∂ β x N G MN + (2 πα ′ ) F αβ , F αβ = ∂ α A β − ∂ β A α − (2 πα ′ ) − B MN ∂ α x M ∂ β x N ,S W Z = τ p Z e (2 πα ′ ) F ∧ C , (4.8)where τ p is D p -brane tension, ξ α , α = 0 , , . . . , p are the ( p + 1) world-volume coordinatesand where A α is gauge field living on the world-volume of D p -brane. Note also that C inthe last line in (4.8) means collection of Ramond-Ramond fields.– 7 – .1 D2-brane Our goal is to find D-brane description of Wilson lines in dual 3d theory. In order to dothis we will consider Euclidean version of
AdS and also write it in the following form ds AdS = ˜ R y [ dy + ( dx µ ) ] = ˜ R y [( dx ) + dy + dr + r dα ] , µ = 0 , , , (4.9)where the boundary of AdS is at y = 0. Let us consider following D2-brane configuration x = ξ , r = ξ , α = ξ , y = y ( r ) ,ξ = const , θ i = const , φ = const , φ = φ ( α ) , (4.10)where ξ, θ , θ , φ , φ are the coordinates of CP . This configuration should correspond toa topological two dimensional operator in the dual CFT on the boundary of the AdS .Further, for the ansatz (4.10) the matrix A takes the form A = ˜ R y , A rr = ˜ R y (1 + y ′ ) , A αα = ˜ R y (cid:16) y (cos ξ sin ξ cos θ + cos ξ sin θ ) ˙ φ (cid:17) , (4.11)where y ′ = dydr , ˙ φ = dφ dα . Then it is easy to see that the D2-brane action takes the form S D = τ Z d ζe − Φ √ det A − τ Z C (3) == τ R Z dx drdα y r (1 + y ′ ) (cid:16) r + y (cos ξ sin ξ cos θ + cos ξ sin θ ) ˙ φ (cid:17) − ry ! . (4.12)In order to simplify the analysis we note that the equation of motion for θ for non-zero ξ has two solution θ = 0 and θ = π and we choose θ = 0. Then the equation of motionfor ξ has solutions for ξ = 0 , π or ξ = π and in order to find non-trivial configuration wechoose ξ = π . Then the action (4.12) simplifies considerably S D = τ R Z dx drdα y s (1 + y ′ ) (cid:18) r + y φ (cid:19) − ry ! . (4.13)Then the equation of motion for y takes the form − y s (1 + y ′ ) (cid:18) r + y φ (cid:19) − ddr y ′ q r + y ˙ φ y p y ′ ++ 14 y p y ′ ˙ φ q r + y ˙ φ + 3 ry = 0 (4.14)– 8 –hile the equation of motion for φ is equal to ddα p y ′ ˙ φ y q r + y ˙ φ = 0 (4.15)and it can be easily shown that the equations (4.14) and (4.15) can be solved with theansatz y = κr , φ = 2 α . (4.16)Similar half-BPS configuration was previously studied in [64] in the case of AdS × S background. The induced metric on the world-volume of D2-brane is equal to ds = g MN ∂ α x M ∂ β X N = ˜ R κ (1 + κ )( dξ ) + ˜ R κ ( ξ ) [( dξ ) + (1 + κ )( dξ ) ] (4.17)that clearly shows that this D2-brane configuration has a topology AdS × S . Let us nowevaluate the action on this ansatz S = τ R κ πT Z ∞ ǫ drr = R κ πT ǫ , (4.18)where T is regularized interval in x 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 finiteextend we should take the boundary terms into account. We will discuss the boundarycontributions in more details bellow. Here we only stress that, following [57], that weshould add to the action the boundary contribution in the form δS = − y δ L δy ′ (cid:12)(cid:12)(cid:12)(cid:12) y = κǫ = − τ R πT y y ′ q r + y ˙ φ y p y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = ǫκ = − τ R πT κǫ . (4.19)Consequently we obtain the action after adding the boundary contribution S = S + δS = 0 . (4.20)We see that the action vanishes and hence the vacuum expectation value of the dualline operator corresponding to the D2-brane explained above, is equal to one. This isa strong indication that the dual line operator is stable and preserves some fractions ofsupersymmetry. It would be really interesting to study these operators from the point ofview of 3 d QFT, following [60, 61]. – 9 – .2 D2-brane with topology AdS × S In this section we present another example of D2-brane solution that has a topology AdS × S . Once again we start with the metric on AdS as ds AdS = ˜ R Y [ dY + ( dX µ ) ] , µ = 0 , , . (4.21)This D2-brane can be described by the following parametrization: X = ξ , Y = ξ ≡ σ , X ≡ X ( σ ) , ξ = φ ,θ = θ = θ = ψ = φ = const , F = − F = F . (4.22)The DBI part of the D2-brane action is written as S DBI = τ e − Φ Z d ζ √ det A . (4.23)Further, there is a WZ term in the following form S W Z = − iτ Z F ∧ C (1) = − iτ k Z d ζ (2 πα ′ ) F cos ξ cos θ , (4.24)where the non-zero one form R-R field is C (1) φ = k ξ cos θ . (4.25)The fact that the WZ term in Euclidean signature contains factor i implies that we haveto consider imaginary electric flux so that we can write A = ˜ R σ , A = ˜ R σ (1 + X ′ ) , A = − A = i (2 πα ′ ) F , A = ˜ R (cos ξ sin ξ cos θ + cos ξ sin θ ) , (4.26)where X ′ = ∂X/∂σ . Hence we obtain S DBI = τ Z d ζ ˜ R p A s σ (1 + X ′ ) − (2 πα ′ ) ˜ R F . (4.27)Let us now solve the equations of motion coming from the above DBI action includingthe WZ term. First of all the variation of the action with respect to θ implies τ e − Φ ˜ R cos ξ sin θ cos θ p sin ξ cos ξ cos θ + cos ξ sin θ s σ (1 + X ′ ) − (2 πα ′ ) ˜ R F −− τ k πα ′ ) F cos ξ sin θ = 0 . (4.28)When cos ξ = 0, the equation above is solved with θ = 0 , π and we choose for ourconvenience θ = 0. Further the equation of motion for ξ implies τ e − Φ (cos ξ − sin ξ ) ˜ R s σ (1 + X ′ ) − (2 πα ′ ) F ˜ R − τ k (2 πα ′ ) F cos ξ sin ξ = 0(4.29)– 10 –ow let us consider the equations of motion for A , A that again imply the existence of aconserved electric flux Π: e − Φ ˜ R ˜ R p A (2 πα ′ ) F q σ (1 + X ′ ) − (2 πα ′ ) F ˜ R − πα ′ k ξ = Π (4.30)Simplifying the above equations one gets: s σ (1 + X ′ ) − (2 πα ′ ) F ˜ R = √ X ′ σ q e ˜ R sin ξ ( Π2 πα ′ cos ξ + k cos ξ ) (4.31)Inserting the above result into the equation of motion for ξ we obtainΠ2 πα ′ = − k . (4.32)Note that this equation is obeyed for any ξ . Notice also that in deriving the above, wehave used the fact that e ˜ R = 4 k . Finally we consider equation of motion for X that takes the following simple form ∂ σ [ X ′ σ p ( . . . ) ] = 0 . (4.33)It is clear that this equation has natural solution X ′ = 0 ⇒ X = const . . Then the metricinduced on the world-volume of D2-brane takes the form ds in = g mn ∂ α X m ∂ β X n dξ α dξ β = ˜ R σ (( dξ ) + dσ ) + ˜ R sin ξ cos ξ ( dξ ) (4.34)that clearly has a form AdS × S .Let us again compute the current J which takes the form J = − τ πα ′ (cid:18) Π + 2 πα ′ k ξ (cid:19) + τ k cos ξ = − τ Π(2 πα ′ ) = τ k . (4.35)Note that it is again proportional to the number of fundamental strings k .Let us now evaluate the action on the solution given above S = τ e − Φ ˜ R Z π dφ Z T/ − T/ dξ Z ∞ ǫ dσ sin ξ cos ξ s σ (1 + X ′ ) − (2 πα ′ F ) ˜ R ++ τ k Z π Z T/ − T/ dt Z ∞ ǫ dσ (2 πα ′ ) F cos ξ – 11 – 2 πT τ e Φ ˜ R sin ξ cos ξ (cid:18) k ξ + k Π4 πα ′ + k ξ (cid:19) cos ξ ×× Z ∞ ǫ dσ √ X ′ σ q e ˜ R sin ξ ( Π2 πα ′ cos ξ + k cos ξ ) = 2 πT τ e Φ ˜ Rk ξ ǫ . (4.36)Where deriving above we have used the following identities e ˜ R = 4 k , e ˜ R sin ξ cos ξ (cid:18) Π2 πα ′ + k ξ (cid:19) = 14 sin ξ cos ξ (4.37)Notice that this action derived above is proportional to the world volume electric flux Π. The D2-brane solution that we found corresponds to D2-brane that extends all the way tothe boundary of AdS and ends there along a one-dimensional curve. Since this D2-braneaction has finite extent we have to discuss the possibility of adding boundary terms to theaction. These boundary terms should not change the equations of motion and hence thesolution will still be the same, but the value of the action when evaluated at this solutionwill in general depend on the boundary terms.As it is well known that when we calculate the Wilson loop using string surfacesthe bulk action is divergent but this divergence can be canceled by boundary term aswe will review in the Appendix. Explicitly, the string that corresponds to Wilson loophas to satisfy three Dirichlet boundary conditions on the three directions parallel to theboundary of AdS , and the seven Neumann ones that combine the radial coordinate ofAdS and coordinates along CP and hence it is the appropriate action assuming that wehave Dirichlet boundary conditions. So we have to add appropriate boundary terms thatchange the boundary conditions.In case of the fundamental string we have the coordinate Y with Neumann bound-ary condition (before imposing the static gauge). Then it is natural to define p y as themomentum conjugate to it p Y = δSδ∂ n Y , (4.38)where n is normal derivative to the boundary. The new action including the term thatchanges the boundary conditions is ˜ S = S − Y Z dτ p Y (4.39)where the integral is over the boundary at a cutoff Y = Y . In fact, the variation of theoriginal action is δS = Z d σ (cid:20) δ L δY − ∂ α δ L δ∂ α Y (cid:21) δY + Z dτ δ L δ∂ σ Y δY = Z dτ δ L δ∂ σ Y δY , (4.40)– 12 –here we used the fact that the field Y obeys the equation of motion in the bulk. Theboundary term above shows (since it is proportional to δY ) that on-shell action is functionalof Y . Then including the boundary term we obtain δ ˜ S = Z dτ [ p Y δY − δp Y Y − p Y δY ] = − Z dτ δp Y Y (4.41)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 stringmentioned above we now return to D2-brane action. In case of D2-brane action there isanother subtlety. The DBI action is a functional of the gauge field, but the Wilson loopobservable should depend on the variable Π. Then in order to find correct form of theaction we have to add to it following boundary term δS = − Z dtdφ (cid:20) δ L δ∂ σ Y Y + δ L δ∂ σ A A (cid:21) == − πT δ L δ∂ σ Y Y + 2 πT τ Z ∞ ǫ dσ Π F σ == − τ e Φ πT ˜ R k ξ ǫ , (4.42)where δ L δ∂ σ A = − 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 whichis the same result as in case of AdS /CF T correspondence. In order to find this solution we consider following form of AdS metric ds AdS = 1 y ( dy + dr + r dφ + d ( x ) ) (4.44)and consider the Wilson line that is siting at x = 0. Our goal is to find Wilson line thatfor y = 0 takes the form of circle with r = R . In order to find such a configuration weconsider an ansatz σ = φ , y = σ ≡ σ , r = r ( σ ) ,F = − F = iF , φ = ξ , θ = θ = ψ = φ = const . (4.45)Then we easily obtain A = ˜ R r y , A = ˜ R y (1 + r ′ ) , A = i (2 πα ′ ) F , A = − i (2 πα ′ ) F , A = ˜ R (cos ξ sin ξ cos θ + cos ξ sin θ ) , (4.46)– 13 –here r ′ = ∂ σ r . Then we obtaindet A = ˜ R A ( r y (1 + r ′ ) − (2 πα ′ ) F ˜ R ) (4.47)so that DBI part of the action takes the form S = τ e − Φ ˜ R Z d σdy p A s r y (1 + r ′ ) − (2 πα ′ ) F ˜ R . (4.48)Further, there is a coupling to C (1) field in the form S W Z = − iτ Z (2 πα ′ ) F ∧ C = − i τ (2 πα ′ ) k Z d ζF cos ξ cos θ . (4.49)As in the case of straight Wilson line we start with the equation of motion for θ and wefound that it is solved for θ = 0, or θ = π and we choose for our convenience θ = 0.Further the equation of motion for ξ implies τ e − Φ (cos ξ − sin ξ ) ˜ R s r y (1 + r ′ ) − (2 πα ′ ) F ˜ R − τ k (2 πα ′ ) F cos ξ sin ξ = 0 . (4.50)Further, the equations of motion for A , A imply a conserved electric flux Π e − Φ ˜ R ˜ R p A (2 πα ′ ) F q r y (1 + r ′ ) − (2 πα ′ ) F ˜ R − πα ′ k ξ = Π (4.51)that allows us to find2 πα ′ F ˜ R = e Φ ˜ R sin ξ ( Π2 πα ′ cos ξ + k ξ ) r √ r ′ y q e ˜ R sin ξ ( Π2 πα ′ cos ξ + k cos ξ ) , s r y (1 + r ′ ) − (2 πα ′ ) F ˜ R = r √ r ′ y q e ˜ R sin ξ ( Π2 πα ′ cos ξ + k cos ξ ) . (4.52)Inserting the above results into the equation of motion for ξ we again obtainΠ2 πα ′ = − k . (4.53)Note that in deriving the above, we have used the fact that e ˜ R = 4 k . Finally we determine the equation of motion for r y p r ′ − ddσ (cid:20) rr ′ y √ r ′ (cid:21) = 0 (4.54)– 14 –ow we will argue that the ansatz r = R − y solves the equation above. Indeed, usingthe fact that r ′ = − yr we one can check that the above equation is identically zero. Let usnow evaluate the action for the D2-brane configuration on this solution S = τ e − Φ Z dφdφ dy ˜ R sin ξ cos ξ s r y (1 + r ′ ) + (2 πα ′ ) F ˜ R ++ τ k ˜ R Z dφdφ dy (2 πα ′ ) F ˜ R cos ξ == 4 π τ D e Φ ˜ R k ξR [ − R + 1 ǫ ] . (4.55)Further, we have the first boundary contribution at σ = ǫ . We again proceed as in previoussection for the boundary contributions. Namely, we evaluate the contribution to the actionfor general y , then insert the ansatz y = σ and finally evaluated the action at y = ǫδS Y = − Z dφ dφ (cid:20) δ L δ∂ σ y y (cid:21) = − π τ e Φ ˜ R k R − ǫ ǫR . (4.56)In the same way the boundary contribution from the gauge fields takes the form δS A = τ Z dφdφ Z Rǫ dσ Π F τσ == 4 π τ ˜ Re Φ k
16 (1 − ξ ) 2 R Z Rǫ dσσ = 4 π τ ˜ Re Φ k R − ξ ) (cid:20) − R + 1 ǫ (cid:21) . (4.57)Collecting all these terms together we obtain that the divergent terms cancel as in the caseof straight Wilson line. On the other hand we find finite contribution to the action in theform S + δS Y + δS A = − π τ e Φ ˜ R k k S F S , (4.58)where we used the convention that τ = π and where S F S is the fundamental stringaction evaluated on circular Wilson line. Borrowing the interpretation of Wilson linesin
AdS /CF T correspondence using D3-branes we can argue that our solution describesWilson line in symmetric k representation where k is level of CS action. It would becertainly very interesting to study the problem whether there exists D-brane description ofWilson loops in arbitrary representations. We hope to return to these problems in future.
5. Conclusions
We have studied in this paper various D-brane configurations corresponding to Wilsonloops and lines in the boundary of AdS . We have taken D2-brane as an example and have– 15 –hown various configurations of this in the Euclidean AdS × CP correspond to straightand circular Wilson line solutions. We analyzed the D2-brane which has a topology ofAdS × S and corresponds to straight Wilson line gives vanishing action after adding theboundary term. We have also studied the circular Wilson line solution and show thatthe action gives a non-zero contribution after adding the boundary term. One can studythe D4-brane which correspond to both straight and circular Wilson line in the boundaryAdS , however we mention that the action will take a similar form like that of D2-braneand 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. Acknowledgements:
We would like to thank an anonymous referee for finding out mis-takes in the first version of the manuscript. KLP would like to thank the hospitality atInstitute of Physics, Bhubaneswar, India where apart of this work was done. This workwas partially supported by the Science Research Center Program of the Korea Science andEngineering Foundation through the Center for Quantum Spacetime (CQUeST) of SogangUniversity with grant number R11 - 2005 - 021. The work of JK was supported by theCzech Ministry of Education under Contract No. MSM 0021622409.
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