aa r X i v : . [ h e p - t h ] M a y WIS/06/08-MAR-DPP
Holographic Duals of Long Open Strings
Ofer Aharony a and David Kutasov ba Department of Particle PhysicsThe Weizmann Institute of Science, Rehovot 76100, Israel
[email protected] b EFI and Department of PhysicsUniversity of Chicago, 5640 S. Ellis Ave., Chicago, IL 60637, USA [email protected]
We study the holographic map between long open strings, which stretch between D -branes separated in the bulk space-time, and operators in the dual boundarytheory. We focus on a generalization of the Sakai-Sugimoto holographic modelof QCD, where the simplest chiral condensate involves an operator of this type.Its expectation value is dominated by a semi-classical string worldsheet, as forWilson loops. We also discuss the deformation of the model by this operator,and in particular its effect on the meson spectrum. This deformation can bethought of as a generalization of a quark mass term to strong coupling. Itleads to the first top-down holographic model of QCD with a non-Abelian chiralsymmetry which is both spontaneously and explicitly broken, as in QCD. Otherexamples we study include half-supersymmetric open Wilson lines, and systemsof D -branes ending on N S . Introduction and summary
Holographic dualities (generalizing the AdS/CFT correspondence [1]) have provento be very useful, both for studying quantum gravity in backgrounds with appropriateboundaries, and for studying the dual theories living on these boundaries. However, thedictionary between the boundary theories and the corresponding quantum gravity duals isnot yet complete.There are two types of objects which we know how to translate between the bulk andboundary theories (at least in the limit in which the bulk geometry is weakly curved, andthus well described by supergravity). Local fields in the bulk map to local operators inthe dual boundary theory; sources for these fields map to sources for the correspondingoperators [2,3]. Extended p -dimensional branes in the bulk can end on closed ( p − N gauge theory, a Euclidean closedfundamental string worldsheet in the bulk, which ends on a closed loop C on the boundary,maps to (a locally supersymmetric version of) a Wilson loop in the dual field theory [4-6].The latter can be thought of as associated with external (infinitely massive) W -bosons inthe gauge theory.In this note we add another entry to this dictionary. When the bulk backgroundincludes branes extending to the boundary, it is possible for other branes to end on thesebranes, and give additional observables in the theory. We will focus on the case where thebulk contains D -branes, and the branes ending on them are fundamental strings, but thediscussion can be generalized to other systems.String theory in the bulk contains in this case operators corresponding to open stringsstretched between D -branes near the boundary. There are two qualitatively differentclasses of such operators. One corresponds to strings which can shrink to zero size (“shortstrings”). These are very similar to the closed string operators mentioned above; theirduals in the boundary theory are local operators, which contain the degrees of freedomassociated with the D -branes. The second class corresponds to “long strings,” that arestretched between D -branes which are separated by a finite amount near the boundary.Such operators depend on the choice of an open contour ˜ C , which ends on the two D -branes on the boundary. We propose that their duals in the boundary theory are certain“line operators.”In the case of large N gauge theories, D -branes ending on the boundary are associatedwith fields in the fundamental representation of SU ( N ), and the line operators are openWilson lines starting from a field in the fundamental representation associated with the first D -brane, and ending on a field in the anti-fundamental representation associated with thesecond one. We propose that an insertion of such an open Wilson line in the field theory1orresponds in the bulk to an insertion of an open string ending on the correspondingcontour on the boundary, as in the Wilson loop case. As there, some correlation functionsof these operators are dominated by semi-classical string worldsheets with the appropriateboundary conditions, and can thus be computed in the supergravity limit.A case in which long open string operators play an important role is the Sakai-Sugimoto holographic model of QCD [7], and its generalizations studied in [8,9]. Thismodel shares with QCD the phenomena of confinement and non-Abelian chiral symmetrybreaking. As we discuss below, open Wilson lines play an important role in understandingthe latter. Previous attempts to study them in this model appeared in [10,11], but ourmethods are different.The Sakai-Sugimoto model describes a 4 + 1 dimensional SU ( N c ) maximally super-symmetric Yang-Mills (SYM) theory, with ’t Hooft coupling λ , compactified on a circleof radius R ( x ≡ x + 2 πR ) with anti-periodic boundary conditions for the fermions, andcoupled to N f left and right-handed fermions in the fundamental representation of SU ( N c )localized at x = − L/ x = L/
2, respectively.The three parameters with dimensions of length, λ , R , and L , can be thought of asproviding an overall scale and two dimensionless couplings on which the dynamics depends[8,9]. In the region of parameter space λ ≪ L ∼ R , the 4 + 1 dimensional gauge theoryis weakly coupled at the scale L ∼ R , and the model is equivalent at long distances (muchlarger than L , R , which can be viewed from this perspective as a UV cutoff), to massless3 + 1 dimensional QCD.For large λ , the 4+1 dimensional gauge theory is strongly coupled and needs to be UVcompleted. In string theory this is achieved by realizing the gauge theory as a low energytheory on a stack of N c D x circle, intersecting N f D D , , at x = ± L/ N c ), one can replace the D U ( N f ) L × U ( N f ) R global chiral symmetry associated with the D D U ( N f ), due to the fact that the eightbranesconnect in the bulk. To study this breaking in more detail, one would like to identify anoperator in the field theory that transforms non-trivially under U ( N f ) L × U ( N f ) R , andhas a non-zero vacuum expectation value (VEV) that preserves the diagonal U ( N f ), i.e. an order parameter for the symmetry breaking.Since the left and right-handed fermions are separated in x , there are no local gauge-invariant operators in the D U ( N f ) groups.The simplest operators with the desired flavor quantum numbers are open Wilson lines of2he type discussed above, such as (for a specific choice of the contour ˜ C ) OW ji ( x µ ) = ψ † jL ( x µ , x = − L P exp "Z L/ − L/ ( iA + Φ) dx ψ Ri ( x µ , x = L , (1 . P denotes path-ordering.In the weak coupling regime, the gauge field A and scalar Φ are weakly coupled at thescale L , and the Wilson line in (1.1) can be neglected. Thus, the operator OW ji reducesin this case to the local operator ψ † jL ψ Ri , which is the familiar order parameter of chiralsymmetry breaking in field theory. In the QCD regime, its VEV is expected to be of orderΛ QCD , where the QCD scale Λ
QCD also sets the scale of masses of mesons and glueballsin the theory.At strong coupling, the Wilson line can not be neglected, since the 4 + 1 dimensionalgauge theory degrees of freedom are strongly interacting at the scale L . We will computethe expectation value of OW ji (1.1) below and find that it is exponentially large. Forexample, in the original model of [7] (in which L = πR ), it scales like exp( λ / πR ). Weinterpret this exponentially large value as associated with the Wilson line contribution to(1.1), rather than with the fermions, since such exponentially large values do not appear inthe effective action of the Nambu-Goldstone bosons (the “pions”) and of the other mesons.Moreover, we will see that the expectation value depends strongly on the choice of contour˜ C connecting the two intersections.Another interesting question in the Sakai-Sugimoto model is how to give a mass tothe quarks. As explained above, local operators which couple the left and right-handedfermions are not gauge-invariant in this model. The best we can do is to add to theLagrangian the non-local operator (1.1). This breaks the chiral symmetry explicitly, andin the region in parameter space in which the model reduces to QCD, becomes equivalentto the quark mass deformation.On the other hand, at strong coupling where we can use supergravity, this deformationis highly non-local and irrelevant ( i.e. it grows in the UV). At low energies it leads to achange in the masses of the mesons, and in particular to a non-zero mass for the Nambu-Goldstone bosons associated with the symmetry breaking. We will study this deformationto leading order in the deformation parameter, and comment briefly on higher order effects.In addition to this main example, we present two other examples of long open stringoperators. One involves a system of k N S N f Dp and ¯ Dp -branes a distance L apart ending on them. For a critical value of the distance, the branes and anti-branes canconnect, and form a single curved D -brane, the hairpin brane of [12-14]. In the process, So far there are no top-down holographic examples of quark masses in theories with a non-Abelian chiral symmetry. U ( N f ) L × U ( N f ) R symmetry acting on the D -branes breaks to the diagonal subgroup,as before. The long open string stretched between the branes and anti-branes near theboundary can again be viewed as an order parameter for the breaking. It has a non-zeroVEV that can be computed in the same way as for the generalized Sakai-Sugimoto model,and one can again study the deformation that breaks the symmetry explicitly.The main advantage of this example compared to the previous one is its tractability.The near-horizon limit of the N S N = 2 superconformalsymmetry on the worldsheet. One can write down explicitly the open string vertex operatorcorresponding to the long string, and compare the results of our semi-classical analysis tothose obtained from the effective action of the stretched open strings, and to the exactsolution of the worldsheet CFT.A second example which we present briefly is of a long open string operator in typeIIB string theory on AdS × S with D -brane defects, which preserves half of the super-symmetry, and is analogous to the circular closed Wilson loop in the d = 4 N = 4 SYMtheory. It is easy to construct many other examples of supersymmetric open string opera-tors, and it would be interesting to study them in more detail, generalizing the studies ofsupersymmetric closed string operators. It would also be interesting to understand if thereis any relation (along the lines of [15-17]) between open Wilson lines of the type studiedhere and scattering amplitudes of quarks and gluons.The organization of this paper is as follows. We begin in section 2 with a generaldiscussion of open Wilson line operators, and their holographic description. In section 3we discuss some holographic computations of their correlation functions in the D − D − ¯ D D -branes ending on N S
2. Holographic open Wilson lines
As mentioned in the introduction, in this paper we will discuss certain non-localobservables in the context of the AdS/CFT correspondence and its generalizations [1-3].A class of such observables that has been widely studied is Wilson loops in large N gaugetheories with only adjoint fields. Locally supersymmetric Wilson loops in the fundamental4epresentation dressed with scalar fields Φ i , W [ C ] = tr (cid:26) P exp (cid:20)I C ds (cid:0) iA µ ( x ν ( s )) ˙ x µ ( s ) + n i ( s )Φ i ( x ν ( s )) | ˙ x | ( s ) (cid:1)(cid:21)(cid:27) , (2 . C (parameterized by x ν ( s ) in the non-compact space-time, and by the unit vector ~n ( s ) in thecompact space). Thus, an insertion of the operator W [ C ] on the boundary corresponds inthe bulk path integral to summing over configurations which include a string worldsheet ending on the loop C on the boundary. Wilson loops with generic (or no) couplingsto scalar fields are more subtle; in particular, their correlation functions have perimeterdivergences (unlike (2.1)) that need to be regularized. Nevertheless, the operators (2.1)already give a large amount of information about the theory. For instance, they can beused as a diagnostic for confinement.When the boundary theory is a large N gauge theory with a finite number of fieldsin the fundamental representation, the corresponding bulk description involves adding D -branes to the gravity background created by the adjoint fields. The gauge symmetry onthe D -branes corresponds to a global flavor symmetry in the dual field theory (which mayor may not be a symmetry of the vacuum). From the (bosonic or fermionic) fundamentaland anti-fundamental fields ψ i ( x ), ¯ ψ j ( x ), one can form local gauge invariant operatorssuch as ¯ ψ j ( x ) ψ i ( x ). Such operators typically map under holography to local fields in thebulk, arising from short open strings stretching from the i ’th to the j ’th D -brane [22].The situation is different when the D -branes are localized in some of the dimensionsin which the gauge theory lives, and thus give rise to defects. Examples include the D − D − ¯ D D − D N = 4 SYM [25,26].In these cases there are no local gauge-invariant operators that involve fundamentalfields from different brane intersections (separated in space-time). The best one can do isto consider generalizations of (1.1), OW ji [ ˜ C ] = ¯ ψ j ( x j ) P exp (cid:20)Z ˜ C ds (cid:0) iA µ ( x ν ( s )) ˙ x µ ( s ) + n k ( s )Φ k ( x ν ( s )) | ˙ x | ( s ) (cid:1)(cid:21) ψ i ( x i ) , (2 . C is a contour between the point x j in the intersection at which the field ¯ ψ j lives,and the point x i in the intersection at which the field ψ i lives. This contour is topologically For Wilson loops in higher dimensional representations of the gauge group, the dominantconfigurations do not look like strings but rather like other branes carrying the same charges[18-20]. This follows from ’t Hooft’s [21] mapping of Feynman diagrams to string worldsheets, inwhich loops of fields in the fundamental representation correspond to holes in the worldsheet. C the operator (2.2) looks just like (2.1), when ~n isa unit vector this operator is locally supersymmetric and its correlation functions do notexhibit divergences proportional to the length of the path ˜ C . The holographic dual of(2.2) must involve a string worldsheet ending on the open contour ˜ C on the boundary.Thus, we propose that an insertion of the operator (2.2) into the path integral of theboundary gauge theory corresponds in the bulk to summing over configurations whichinclude an open string worldsheet which approaches the contour ˜ C at the boundary of thebulk space-time, and near the boundary looks like a strip whose ends lie on the i ’th and j ’th D -branes.As we will see, in some cases the computation of correlation functions of these opera-tors is dominated by a saddle point corresponding to a semi-classical string worldsheet, justlike for many holographic closed Wilson loop computations. In particular, the one-pointfunction h OW ji i is given to leading order in the semi-classical expansion by exp( − A/ πα ′ ),where A is the minimal area of the worldsheet of such a string. If a finite area string world-sheet does not exist, the one-point function of the OWL vanishes.A few comments about the preceding discussion are in order:(1) Just like for other holographic operators, in order to obtain finite correlation functionsone needs to introduce a UV cutoff, and renormalize the OWL operators describedabove. In particular, the string worldsheet that enters the calculation of the one-pointfunction must only have finite area for finite UV cutoff.(2) When performing the bulk path integral in the presence of the open string worldsheet,one has to include all the couplings of the string to the background fields, such as theNS-NS B µν field, and the gauge fields that live on the D -branes.(3) When the i ’th and/or j ’th D -branes give rise to more than one fundamental fieldin the gauge theory, the distinction between the corresponding bulk operators in thesemi-classical calculation described above arises from quantization of zero modes onthe worldsheet of the string.The example that motivated this investigation is the Sakai-Sugimoto model of holographicQCD. In this model, the large N gauge theory lives on D ψ L and ψ R , whichare localized at 3 + 1 dimensional defects – the intersections of the D N f D D D D L in thedirection x along the D U ( N f ) L × U ( N f ) R global symmetrycorresponding to the gauge symmetry on the D D λ ≫ L, R , the vacuum of this model correspondsto a brane configuration in which the D D ( N f ) L × U ( N f ) R symmetry is dynamically broken to the diagonal U ( N f ). Most of thework on the model involved light open 8 − U ( N f ) L × U ( N f ) R chiral symmetry currents ψ † L ( x ) σ µ ψ L ( x ), ψ † R ( x )¯ σ µ ψ R ( x ).These “short string” operators are useful for analyzing the low lying spectrum ofmesons, but in order to study chiral symmetry breaking it is better to consider operatorssuch as (1.1), which transform as ( N f , ¯N f ) under the chiral symmetry. In the next sectionwe will use holography to show that the expectation value of these operators is non-zeroat strong coupling; thus, they are natural order parameters for chiral symmetry breaking.In QCD one can break the chiral symmetry explicitly by adding a mass term for thequarks. The closest analog of this at strong coupling is to add (1.1) to the Lagrangian.We will describe some results about this deformation in section 4 below.OWL operators of the form (2.2) can in principle be also defined for theories in whichthe fundamental fields are not localized at defects, but they seem to be less useful in suchcases. Consider, for example, the D − D N = 4SYM a massless hypermultiplet in the fundamental representation of the gauge group. Inthe dual description this corresponds [22,27] to adding a D AdS × S totype IIB string theory on AdS × S , where the S is a maximal three-sphere inside S .The one-point function of an open Wilson line operator (2.2) involves in this case aconfiguration with a string ending on the contour ˜ C connecting the points x i and x j inIR , . The worldsheet of such a string can always reduce its area by contracting towards theboundary. Therefore, the corresponding one-point function depends on the UV regulator,and does not appear to be well-behaved. This is reasonable from the general perspectivedescribed in the introduction. The open string in question is really a short D − D , . A more natural basis for describing such strings is in terms of excited perturbative D − D D -branes. For a given pair of branes there is a preferred contour ˜ C that has minimal length,and it is natural to study the OWL operator associated with it. For the D − D − ¯ D , and/or vary non-trivially in the interior, as in (2.2), but these are lessnatural. They can be alternatively described by adding string oscillators to the operatorcorresponding to the minimal contour (1.1). This is also true at weak coupling, due to divergences associated with the screening by thefundamental representation fields. . Open Wilson lines in the D − D − ¯ D system To demonstrate the general discussion of the previous section, we will consider herethe following intersecting brane system in type IIA string theory. We start with N c D , labeled by ( x , x , x , x , x ), and add to them N f D x = − L/
2, as well as N f ¯ D x = + L/ ψ L , ψ R , which are localized at the 4 − − ¯8 intersections, respectively, and interact viaexchange of modes living on the D The strength of the interaction is determinedby the ’t Hooft coupling λ = (2 π ) g s N c l s . When the interaction at the scale L is strong( λ ≫ L ), one can replace the D ds = (cid:18) uR D (cid:19) / " − ( dx ) + X i =1 ( dx i ) + (cid:18) R D u (cid:19) / (cid:2) du + u d Ω (cid:3) , (3 . R D ≡ πg s N c l s . The RR four-form and dilaton are F (4) = 2 πN c Vol( S ) ǫ , e Φ = g s (cid:18) uR D (cid:19) / . (3 . D D N f connected eightbranes. They areextended in the IR , labeled by ( x , x , x , x ), wrap the four-sphere labeled by Ω , andform a curve u ( x ) in the ( u, x ) plane, which is a solution of the first order differentialequation u q (cid:0) R D u (cid:1) u ′ = u . (3 . U -shaped brane, with the distance between the two arms ap-proaching L at large u . The minimal value of u to which the D u , isdetermined by L , L = 14 R / D u − / B ( 916 ,
12 ) . (3 . D u → ∞ , but in the ’t Hooft large N c limit there is a The Sakai-Sugimoto model [7] is obtained by compactifying x on a circle, with twistedboundary conditions for the fermions on the D u in which it is small, and we can restrict attention to thatregion by placing a UV cutoff on u , u ≤ u max .Since the D D U ( N f ) L × U ( N f ) R symmetry acting on them is spontaneously broken to its diagonal subgroup. The fermions ψ L , ψ R , which correspond in the brane picture to strings stretching from the bottom ofthe curved D u = 0, obtain a dynamically generated “constituent mass” m = u / πα ′ ∼ λ /L .One can use the effective action for the D N f massless Nambu-Goldstone mesons correspondingto the breaking of the chiral symmetry. The rest of the spectrum is massive; the masses ofthe lowest lying mesons are of order 1 /L . They are much lighter than the fermions, andcan be thought of as tightly bound states of two fermions.As explained above, the simplest operator which can serve as an order parameterfor chiral symmetry breaking in this theory is the OWL operator (1.1). We next turnto a calculation of its one-point function at strong coupling, the chiral condensate, andcomment on more general operators of the form (2.2). In order to compute the expectation value of the OWL operator (1.1) at strong cou-pling, we need to perform the gravitational path integral in the closed string background(3.1), (3.2), in the presence of the curved D u = u max stretches along a straight line in the x direction between the two arms of the curved D , × S , and fills the region in the ( u, x ) plane betweenthe boundary at u = u max and the D α ′ the action of such a string is proportional to its area, S str = 12 πα ′ Z dx Z u max u ( x ) du √ g uu g = 12 πα ′ Z dx (cid:2) u max − u ( x ) (cid:3) . (3 . S str = u max L πα ′ − R / D u / πα ′ B ( 716 ,
12 ) = u max L πα ′ − C λ L , (3 . C is given by C = B ( , ) B ( , ) / π ≃ . u max → ∞ . The term proportional to9 x LD8 u max u Figure 1:
The semi-classical worldsheet which gives the chiral condensate in the D − D − ¯ D the UV cutoff u max on the right-hand side is independent of the coupling λ , and can beabsorbed in the definition of the operator (1.1). Thus, we conclude that at strong coupling the expectation value of the operator (1.1)is given by h OW ji i ≃ δ ij exp ( − S str ) ≃ δ ij exp( C λ /L ) . (3 . U ( N f ) L × U ( N f ) R → U ( N f ) diag , in agreement with our earlier discussion.It grows exponentially with the coupling λ /L in the region λ ≫ L in which our calcula-tion is reliable. At first sight this might seem surprising, since in the weakly coupled fieldtheory regime the chiral condensate is closely related to the dynamically generated fermionmass, whereas here this is not the case – the fermion mass scales like λ /L , those of themesons scale like 1 /L , while the condensate (3.7) is exponentially large. The differencebetween the two regimes is that for strong coupling most of the contribution to (3.7) ap-pears to be due to the Wilson line in (1.1) rather than to the fermion bilinear part of theoperator, while for weak coupling this Wilson line gives a negligible contribution. As explained in [6], this term is naturally canceled by a Legendre transform which is part ofthe definition of locally supersymmetric Wilson line operators. The chiral condensate we find is also widely separated from the pion decay constant f π ,which will be discussed in the next section. C . Consider, for instance, the one point function of an open Wilson line(2.2) connecting two points in IR , , x µ and x µ , (space-like) separated by a distance muchlarger than L . A class of contours connecting these points that is useful for our purposesinvolves moving first in x (at a fixed value of x µ , x µ = x µ ) from − L/ x , thenvarying x µ from x µ to x µ at fixed x , and finally moving again in x to L/
2. Such contourshave cusps, but these can be smoothed out (and in any case the divergences they lead toare well understood and can be subtracted out).Finding the precise shape of the string worldsheet which minimizes the action withthese boundary condition is rather complicated. However, when the two points x and x are widely separated, we expect the main contribution to this expectation value to comefrom the part of the worldsheet at x = x . In the special case x = 0, this part of theworldsheet is easy to analyze. Its contribution to the regularized action is given by S str = − u | ~x − ~x | πα ′ ∝ − λ L | ~x − ~x | . (3 . x approaches (say) L/
2, the regularized action turns out to be proportionalto − λ | ~x − ~x | / ( x − L/ .We see that the expectation values of these operators, proportional to exp( − S str ),grow exponentially with the distance between the endpoints of the contour ˜ C in IR , ,and the coefficient of the distance in the exponent depends on the precise contour wechoose. We conclude that this exponential growth is a property of the contour rather thanof the fermion bilinear at its ends. This also explains why the expectation value underconsideration does not decay exponentially with the distance in IR , , | ~x − ~x | , as onemight have expected due to the fact that the fermions are massive.So far we have discussed the computation of the chiral condensate for the extremal D x liveson a circle of radius R , with anti-periodic boundary conditions for the fermions. The near-horizon D x circle varies between its asymptotic value R at large u andzero at u = u Λ = λ α ′ / πR [28].One can again analyze the shape of the D L and R andcalculate the expectation value of OWL operators such as (1.1), by evaluating the area ofthe corresponding Euclidean string worldsheet. There are some small differences in theprecise form of the solutions for the D x direction and the other in the negative11 direction. For generic L , R , the worldsheets that determine the expectation values ofthe two operators have different areas, so one of the operators has a larger VEV. In thespecial anti-podal case L = πR considered in [7] the one-point functions of both operatorsare given by h OW ji i ≃ δ ij exp( λ / πR ) . (3 . L ≪ λ one finds a result that smoothly interpolates between (3.9) for L = πR and (3.7) for L ≪ R , where the modification of the background at the location of the D x is negligible.The discussion above was restricted to the strong coupling regime λ ≫ L . In theopposite limit, λ ≪ L ∼ R , at energies much smaller than 1 /R one expects the modelto reduce to QCD with massless quarks. In this limit the gauge field A and the scalarfields are expected to decouple [28], so (1.1) should go over to the usual chiral condensateof QCD, which scales as Λ QCD ≃ R exp( − π R/λ ). If there is no phase transition asone varies λ /R , we expect a smooth interpolation between this result and (3.9).For L ≪ R , and in particular in the limit R → ∞ with fixed L , the situation is notcompletely clear. At strong coupling ( λ ≫ L ), one finds in this limit a theory whichbreaks chiral symmetry but does not confine, which can be thought of as a particular UVcompletion of the Nambu-Jona-Lasinio model [8]. Field theoretic intuition suggests thatat weak coupling ( λ ≪ L ) chiral symmetry is not broken, and thus the theory undergoesa phase transition at some critical value of the coupling λ /L , but this has not beenconclusively established yet.In other closely related brane systems, discussed in [23,24], which give rise to 1 + 1dimensional intersections, such as the D − D − ¯ D R ). The strong coupling computation is very similarto that described above, and gives h OW i ∼ exp( ˜ C λ /L ) with some calculable constant˜ C that depends on L/R and approaches a finite value as
L/R → R , one obtains in this case a generalized Gross-Neveumodel which can be analyzed using field theoretic methods and gives h OW i ∼ exp( − L/λ ).For finite R one gets a generalization of the ’t Hooft model of two dimensional QCD thatincludes four-Fermi interactions, and is solvable at large N c , like its two extreme limits –the ’t Hooft and Gross-Neveu models. It would be interesting to compute the chiralcondensate in this model as a function of L/R , and compare it to the strong couplingcalculation described above. We expect a smooth interpolation between the strong andweak coupling limits as one varies the parameters λ /L , λ /R that govern chiral symmetrybreaking and confinement, respectively. 12 .2. Correlation functions of open Wilson lines The computation of the expectation value of a product of several OWLs (2.2) is alsostraightforward in principle, but in practice it is more difficult to find the appropriate semi-classical string worldsheets (if they exist). As in correlation functions of closed Wilsonloops, in some cases a correlation function of a product of OWLs is dominated by asingle semi-classical worldsheet; in other cases it is dominated by several semi-classicalworldsheets connected by propagators in the bulk (at leading order in 1 /N c they must beconnected by propagators of open string fields); in yet other cases there may be no semi-classical contribution at all. In the supergravity limit, there can be sharp phase transitionsbetween the first two possibilities, as in closed Wilson loop correlators [30]. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) x1x0 x1 x0 Figure 2:
The two semi-classical configurations that dominate the computationof F . On the left we have the two-string configuration, with one string (as infigure 1) ending on a D x and the other at x , connected by an openstring propagator inside the D D x and x . A case where the dominant worldsheets are easy to describe is the correlation function F ≡ h OW ii ( x µ )( OW ii ) † ( x µ ) i (no sum over i implied) in the D − D − ¯ D OW ii and( OW ii ) † (the worldsheet corresponding to OW † is the same as the one for OW , but with anopposite orientation), connected by a propagator of an open string field on the D F ≃ h OW ii ih OW ii i † | ~x − ~x | ≃ exp(2 C λ /L ) | ~x − ~x | . (3 . x µ and x µ , byextending into the bulk, as in the right part of figure 2.13f the D D x , this configuration wouldbe precisely the one that appears in the computation of the energy of a quark and an anti-quark separated by a distance | ~x − ~x | , with x playing the role of time (the worldsheetwould simply stretch in this direction and end on the D x = ± L/ D − λ / | ~x − ~x | ), so in thiscase we would obtain F ≃ exp( λ L/ | ~x − ~x | ) . (3 . D x , and (3.11)should be modified by taking their shape into account. When the extent of the string inthe radial direction becomes comparable to u , this modification is significant. However,for short distances, (3.11) is still reliable. As the distance increases, the area of the worldsheet in the right part of figure 2increases, and at some point it becomes larger than that in the left part. At that point,the correlation function in question makes a phase transition from (3.11) to (3.10). Thistransition is expected to occur at | ~x − ~x | ≃ L .An example of a correlator for which there is no obvious smooth worldsheet configu-ration at short distances is h OW ii ( x µ ) OW ii ( x µ ) i . The string ending at x µ would have tochange its orientation in the bulk before coming back to end at x µ . Thus, in this case itseems likely that the two-string configuration on the left of figure 2 always dominates andgives the behavior (3.10).Another interesting correlator is h det( OW ji ( x µ )) i , which is a singlet of the non-Abelian SU ( N f ) L × SU ( N f ) R but carries axial U (1) charge. In the case of finite R , due to theaxial anomaly, this should be non-zero even in phases where the chiral symmetry is notspontaneously broken and the D D h det( OW ji ) i involves in this case N f strings ending on the boundary, on the D D D D D x circle (which are instantons from thepoint of view of the 4 + 1 dimensional gauge theory). Such Euclidean D N f fundamental strings ending on them between the D D [32]. These strings can extend to the boundary and thus contribute to h det( OW ji ) i . Being D -instanton effects, such contributions are exponentially suppressed in the ’t Hooft large N c limit, but they are the leading contribution to h det( OW ji ) i in phases where the non-Abelian chiral symmetry is unbroken. Note that F diverges as ~x → ~x (when the cutoff is sent to infinity). Recall that the D D . Deforming by open Wilson lines In the previous section we computed the expectation value of the OWL operator (1.1)in the generalized Sakai-Sugimoto model. In this section we will study a deformation ofthe model that corresponds to adding this operator to the Lagrangian, δS = κ Z d x N f X j =1 OW jj ( x ) + c.c. . (4 . U ( N f ) L × U ( N f ) R chiral symmetry to the diagonal U ( N f ), in addition to the spontaneous breaking present at κ = 0. The deformation(4.1) can be thought of as a generalization to strong coupling of a “current mass” for thefermions, δS = κ R d x P j ψ † jL ψ Rj + c.c. , that plays a role in QCD. The generalization tonon-equal “masses” κ j for different quark flavors is straightforward.We will study the deformed theory semi-classically at strong coupling, in the hope thatthe strong coupling results are smoothly related to large N c QCD with massive quarks.We will work to first order in the mass parameter κ ; this involves a single insertion of theperturbation, for which we can use our results from the previous section. In QCD this isa good approximation for the u and d quarks, whose current mass is much smaller thanthe QCD scale. It would be interesting to go beyond first order in κ . For this, one needsto evaluate n > OW ii ( x ), which are complicated, asdiscussed in the previous section.To first order in κ , the deformation (4.1) can be described by adding to the space-timeaction the term δS = κ Vol( S ) Z d x Z d Ω X i e − S ( i )str + c.c. , (4 . S ( i )str is the action of the string ending on the i ’th D and the integral over the four-sphere implements an average over the scalar fieldthat enters the definition of the operator (1.1), restoring the SO (5) symmetry of the model.The deformation (4.2) is non-local , since the action S ( i )str depends on the positionof the D D Of course, this only makes sense in the phase in which the D D h OW i vanishes, and we do not have a semi-classical description of the deformation. Similar non-local mass terms were also recently considered in [33]. The deformation (4.1), (4.2), is of order N c (or 1 /g s ), like any other open stringdeformation, so it is expected to influence open string fields (like the position of the D g s . One thing that is relatively easy to compute is the mass ofthe Nambu-Goldstone bosons (the “pions”) due to the deformation (4.1) at leading orderin κ .For κ = 0 we have a U ( N f ) L × U ( N f ) R global symmetry spontaneously broken to U ( N f ). In the effective field theory on the D U ≡ P exp( i Z L/ − L/ dx ˜ A x ) , (4 . A is the gauge field on the D U transforms as a bifundamentalof U ( N f ) L × U ( N f ) R . It is precisely the matrix appearing in the low-energy chiralLagrangian, which is usually written in terms of pion fields as U ( x ) = exp( iπ ( x ) /f π ).Its low-energy effective Lagrangian is given by L eff = ( f π / ∂ µ U ∂ µ U † ) , (4 . This non-locality could be avoided if instead of deforming by ( OW + c.c. ) we would deform by(ln( OW )+ c.c. ), since this would just shift the action by a multiple of the action S str of the stretchedstring, which is an integral of a local function of the D U (1) symmetry),and it is not obvious that h ln( OW ) i = ln( h OW i ) semi-classically, so we will not consider it furtherhere. Parameterizing the position of the D u, x ) plane by a variable z which goesfrom minus infinity at one boundary of the branes to plus infinity at the other boundary, we canwrite U = P exp( i R ∞−∞ dz ˜ A z ). Naively one might think that the holonomy matrix U could serve as an order parameter forthe chiral symmetry breaking in the full string theory as well. However, while the holonomy isgauge-invariant in the D B field. In order to obtain a gauge-invariant object wemust multiply U by exp( i R B ) where the integral is over a surface bounded by the D D OW as a completion of U to the fullstring theory; in the N f = 1 case U is the phase of OW . f π ≃ λ N c /L . (4 . N f pion fields in U are massless Nambu-Goldstone bosons. The deformation (4.1) explicitly breaks the chiral symmetry to the diagonal subgroup,and is expected to give a mass to all the Nambu-Goldstone bosons. Indeed, the perturba-tion exp( − S ( i )str ) in (4.2) includes a coupling to the gauge field on the D P exp[ − i R L/ − L/ dx ˜ A x ]) ii . This coupling did not play a role in our evaluation of h OW i ,since we assumed that we were expanding around a configuration in which the gauge fieldon the D U , (4.4), is deformed at first order in κ by δL eff = |h OW i| κ tr( U ) + c.c. . (4 . κ is real and positive, this is precisely the same as the change in the low-energyeffective action of QCD when we add to the theory a quark mass proportional to κ (theproportionality constant depends on the chiral condensate). It leads to a pion mass equalto (this is sometimes called the Gell-Mann-Oakes-Renner relation [38]) m π = 4 κ |h OW i| f π . (4 . κ and f π have dimensions of mass, while OW has the dimension of a mass cubed.When κ has an imaginary part (or is negative), the minimum of the pion potential isno longer at π ( x ) = 0 and the deformation (4.1) leads to a change in the phase of the chiralcondensate. We will assume that κ is positive from here on (the other cases are classicallyequivalent to this, since they are related by the axial U (1) symmetry).In addition to giving a mass to the pions, the perturbation (4.2) changes the massesof the massive mesons as well. To calculate their mass shifts, one needs to determine theshape of the D L = T D Z dx u s (cid:18) R D u (cid:19) u ′ + 2 κB exp (cid:18) πα ′ Z dx u (cid:19) , (4 . B is the coefficient of the exponent in the computation of h OW i , which is necessaryto give L the appropriate dimension and to make the second term have the same scaling O ( N c ) as the first term. In the second term in (4.8) we used (3.5). The axial U (1) symmetry is anomalous, and the corresponding pion obtains a mass at order1 /N c [36,37,7,32]. This coefficient also depends on u ( x ) through the coupling of the worldsheet to the varyingdilaton in our background. However, this dependence is suppressed by a power of α ′ in the smallcurvature limit we are working in. T D R D h (cid:0) R D u (cid:1) u ′ i / " uu ′′ − u ′ − (cid:18) uR D (cid:19) = κBπα ′ exp (cid:18) πα ′ Z dx u (cid:19) . (4 . x , so the left-hand side is a constant. Denoting thisconstant by A ≡ κ |h OW i| /πα ′ , (4.9) is equivalent to the first order differential equation H = T D u q (cid:0) R D u (cid:1) u ′ − Au = constant , (4 . { x → x + constant } of (4.8). The constant value of H maybe determined by requiring that u goes to the UV cutoff u = u max at x = ± L/
2. It isrelated to the minimal position u of the D u direction by H = T D u − Au .Equation (4.10) enables us to compute the deformation in the position of the D κ |h OW i| . This may then be used to determine the shift of themeson masses, by analyzing the quadratic fluctuations of the deformed action around thisnew solution. It would be interesting to understand how to go to higher orders in κ .Note that, unlike the QCD mass deformation, the deformation (4.2) in the stronglycoupled D − D − ¯ D u = u max , and demandthat the deformation is small at the cutoff scale.It is easy to generalize the computations above to the Sakai-Sugimoto model in whichthe x direction is compactified. One interesting difference is that, in the special case of L = πR , it seems natural to deform by the sum of the OWL operator (1.1) correspondingto the contour connecting the D D x direction, and the oneconnecting them in the negative x direction. In this case the shape of the D D D D u by the string. In this model the distance between the minimal positionof the D u , and the minimal value of the u coordinate, u Λ , may be interpretedas a constituent quark mass (at least in the context of high-spin mesons [39-41]). We findgenerically (except for the special case discussed above) that increasing the bare quarkmass increases also the constituent quark mass, as expected.18 . D -branes in the background of N S -branes In this section we study another example of a holographic description of operatorscorresponding to long strings stretched between two D -branes. This example is of interestfor the study of D -brane dynamics near singularities of the bulk geometry. It also hasthe advantage that the relevant classical string background is under control, and can beanalyzed exactly in α ′ .Consider the following brane configuration in type II string theory. We start with kN S , labeled by ( x , x , x , x , x , x ), and located at the originin the transverse IR . As is well known from the brane construction of gauge theories (see[42] for a review), Dp -branes which have one direction transverse to the fivebranes can endon them. Thus, we add a Dp -brane stretched in the directions ( x , x , x , · · · , x p − ), andsemi-infinite in the x direction ( i.e. it has x ≥ x = 0). x xDp Dp k NSL Figure 3:
The brane configuration : a Dp -brane and an ¯ Dp -brane ending on N S
The above D -brane is localized in the IR − p labeled by ( x p , x p +1 , · · · , x ). We can add asecond D -brane, which is parallel to the first one, but is displaced from it by a distance L in IR − p , and has the opposite orientation, i.e. it is a ¯ Dp -brane. We will label the directionalong which the D and ¯ D -brane are separated by x , with x ( D ) = − L and x ( ¯ D ) = + L .The brane configuration is depicted in figure 3.We will be primarily interested in the physics associated with the two brane intersec-tions in figure 3. As reviewed in [42], each of the two intersections separately preserves 8supercharges, and carries no localized massless modes. One way to see this is to com-pactify some of the directions along the fivebranes, and use U-duality to turn each of theintersections in the system in question to k D D , . If the D x , 3 − The system with both branes and anti-branes of course does not preserve any supersymmetry. ( k ) gauge symmetry on the fivebranes, localized at the intersection. To reach the config-uration of interest to us, one needs to separate the two halves of the D x ) along the fivebranes, and to send the lower half to infinity. Thiscorresponds to giving an infinite mass to the hypermultiplet.The endpoint of the Dp -brane on the fivebranes looks like a charged object in thefivebrane theory. For example, for p = 1, the D N S U ( k ) gaugetheory of k N S p = 3, the D -brane is extended intwo of the directions along the fivebranes (12), and looks like a magnetic monopole in theremaining three.While the system with just one intersection is uninteresting in the infrared, whenboth branes and anti-branes are present, as in figure 3, the situation is richer. Since weare interested in the physics near the intersections, we can replace the fivebranes by theirnear-horizon geometry, the CHS geometry [43]: ds = dx µ dx µ + dφ + d Ω , (5 . φ is related to the radial coordinate in the transverse IR as follows: r = g s √ kα ′ exp (cid:18) φ √ kα ′ (cid:19) , (5 . , whose radius is given by √ kα ′ . Moreprecisely, the angular degrees of freedom are described by a supersymmetric SU (2) WZWmodel at level k . g s is the asymptotic string coupling, far from the fivebranes. Thegeometry (5.1) is obtained from the full fivebrane geometry by taking g s → φ heldfixed; in this limit it describes a “little string theory” (LST) (see [44,45] for reviews). Thedilaton behaves in this limit like Φ = − φ √ kα ′ . (5 . Dp -brane ending on the fivebranes corresponds in the geometry (5.1) – (5.3) to a branestretched in ( x , x , · · · , x p − , φ ), and localized on the three-sphere and in IR − p [46]; the¯ Dp -brane is described similarly. As in the full geometry, the D and ¯ D -branes are a distance L apart in IR − p . Note that unlike the previous cases we discussed, here this distance doesnot grow as we move out in the radial direction.The Dp and ¯ Dp -branes attract each other via exchange of closed string modes, but wewill ignore this effect, and work just at leading order in the string coupling. We will view In brane constructions of gauge theories, such systems do give interesting infrared physicswhen embedded in richer brane configurations; this will not play a role in our discussion below. Dp and ¯ Dp -branes at φ → ∞ , L , as a fixed (= non-normalizable)boundary condition. Our focus here will be on the classical dynamics of normalizable openstring modes.It turns out that for L larger than a certain critical value, L crit = π √ kα ′ , (5 . L → L crit , a light mode appears. For L < L crit this modebecomes tachyonic and destabilizes the brane configuration of figure 3. A heuristic way ofunderstanding this instability is the following. The endpoints of the Dp and ¯ Dp -branes onthe N S Dp -branes goes like the inverse string coupling,while the attractive force due to exchange of a particular fivebrane mode is of order one,this is a subleading effect in (the local) g s .A classical instability can only occur if the sum over the exchanges of all modes of theLST diverges. Such a divergence can only be due to the contributions of arbitrarily heavyLST states. The contribution to the attractive force of a given mode of mass m decreasesat large mass like exp( − mL ), while the density of LST states is well known to behave like ρ ( m ) ∼ exp(2 π √ α ′ km ). Thus, superficially it seems that the sum over states diverges for L < L crit (5.4).This factor of two discrepancy is familiar from another, closely related, context –closed string emission from accelerating branes in LST. It was argued in [47] that it isnatural to expect that the density of states that can be emitted by D -branes in LST infact goes like p ρ ( m ). This would certainly be the case in ordinary (critical) string theory,since a D -brane can only emit left-right symmetric closed string states. Assuming that thisis the case in LST as well, we conclude that the exchange of LST modes by the D -branesdiverges precisely for L < L crit .In the regime k ≫ L crit is large in string units, and the above light mode is bestdescribed as a translational mode of the D -brane configuration (which will be described indetail below). For k ∼ , a better description of this mode is as a fundamentalstring stretched between the D and ¯ D -branes. We will consider the geometric regime k ≫
1, but will return to this stretched string below.To exhibit the geometric massless mode for k ≫
1, consider the projection of the D -branes of figure 3 on the two dimensional space labeled by ( φ, x ). This corresponds toa D -string described by a curve x = x ( φ ). The configuration of figure 3 corresponds to x = ± L/
2; the light mode corresponds to deformations to a more general x ( φ ). The DBI Such values of k in (5.3) cannot arise in the near-horizon limit of flat N S D -brane is given by S = − C Z dx exp (cid:18) φ √ kα ′ (cid:19) p φ ′ . (5 . φ = φ ( x ), φ ′ = ∂ x φ , and C is a known constant whose value will not be needed below.The fact that the Lagrangian (5.5) does not depend explicitly on x implies that onecan integrate the Euler-Lagrange equation once. After squaring the resulting equation onegets exp (cid:18) φ √ kα ′ (cid:19) = 1 + φ ′ , (5 . φ . The solution of (5.6) is exp (cid:18) − φ √ kα ′ (cid:19) = cos (cid:18) x √ kα ′ (cid:19) . (5 . k NSDpLcrit x φ Figure 4:
The hairpin D -brane (5.7). It describes a U-shaped connected brane, the hairpin brane of [12] (or, more precisely, itsgeneralization to the fermionic string discussed in [13,14] and other papers). As φ → ∞ ,it approaches a brane and anti-brane a distance L crit (5.4) apart. As φ decreases, the two D -branes bend towards each other; they smoothly connect at φ = 0 (see figure 4).As mentioned above, the position of the bottom of the brane depicted in figure 4 is afree parameter of the solution, as is clear from the form of the action (5.5). Moreover, theenergy of the brane is independent of this parameter. Thus, when the distance betweenthe D and ¯ D -branes at infinity is equal to the critical one (5.4), the mode corresponding22o fluctuations of the bottom of the hairpin brane is massless and has a flat potential (atleading order in the string coupling).When L > L crit , this mode is massive, and the hairpin tends to collapse back to thebrane-antibrane configuration of figure 3. For
L < L crit it is tachyonic and the bottomof the U-shape tends to run to large φ . The resulting time-dependent solutions can bedescribed using techniques similar to those of [48], who studied a closed string analog ofthis problem.The original brane configuration of figure 3 has a U (1) × U (1) symmetry associatedwith the two Dp -branes. This is a local symmetry on the D -branes, but from the point ofview of the LST it is a global one. This symmetry is broken to the diagonal U (1) when thebranes connect. It is interesting to ask whether there is an operator that is charged underthe broken U (1) and has a non-zero expectation value in the configuration of figure 4.Such an operator could serve as an order parameter for the symmetry breaking describedabove, as in our discussion of the D − D − ¯ D Dp and¯ Dp -branes in figure 3. The lowest lying state of such a string is the open string “tachyon”stretched between the two branes. From studies of the hairpin brane, which turns out tobe described by an exactly solvable boundary conformal theory, it is known that such anoperator is indeed turned on in the vacuum. In the bosonic string this was discussed in[12,49], while in the fermionic case of interest to us here in [50].Asymptotically, at large φ , the worldsheet Lagrangian contains a term correspondingto a boundary N = 2 superpotential, which behaves like δS ws = µ Z dtdθ exp " − r kα ′ ( φ + i ˜ x ) + c.c. . (5 . x = x L − x R is the T-dual of x , and the coupling µ is determined by φ IR , the locationof the bottom of the hairpin brane. The dependence can be determined by a scalingargument of the kind familiar from Liouville theory. For the hairpin shape (5.7), one has φ IR = 0, which corresponds to some particular µ = µ ( ∗ ) . If we replace φ → φ − φ IR , suchthat the bottom of the hairpin is at φ = φ IR , we see from (5.8) that µ = µ ( ∗ ) exp r kα ′ φ IR ! . (5 . φ IR → −∞ , the bottom of the hairpin brane descends into the strong couplingregion and one smoothly approaches the parallel brane-antibrane configuration of figure 3. µ (5.9) also goes to zero in this limit. When φ IR becomes too small, we cannot trust the shape of the bottom of the hairpin dueto strong quantum effects, but there is no reason to expect non-smooth behavior there.
23s mentioned above, the large symmetry of the problem ( N = 2 worldsheet supercon-formal symmetry) allows one to solve the boundary conformal field theory correspondingto the hairpin brane exactly, and in particular one can deduce the presence of the boundary N = 2 superpotential (5.8). Thus, it is interesting to study this case in detail, in the hopeof developing techniques which could be useful also in more general circumstances wherethe worldsheet theory is not solvable, such as backgrounds with Ramond-Ramond fieldsturned on.In particular, we would like to understand the origin of (5.8) at large k , where boththe closed string background (5.1) – (5.3), and the shape of the D -brane (5.7), are slowlyvarying, and we can expect semi-classical techniques to be valid. To do that it is usefulto note that the boundary superpotential (5.8) is a normalizable operator at large φ . Asis familiar from holography in general, µ is proportional to the expectation value of thenon-normalizable operator that creates a string stretched between the D and ¯ D -branes atthe boundary. This operator, which is analogous to the OWL operators described in theprevious sections, behaves at large φ like T ≃ exp " r kα ′ − √ kα ′ ! φ − i r kα ′ ˜ x . (5 . µ (5.9).To calculate this expectation value it is useful to note that the tachyon background(5.8) is a non-perturbative effect in the worldsheet theory, whose loop expansion parameteris 1 / √ k (the curvature of the D -brane). Thus, it is natural to expect that it is due toa worldsheet instanton effect, involving an open string ending on the boundary; this alsofollows from our general discussion in the previous sections of the holographic dual of longopen strings. The instanton in question is a map from the worldsheet disk | z | ≤ x, φ ) plane bounded by the hairpin,exp (cid:18) − φ √ kα ′ (cid:19) ≤ cos (cid:18) x √ kα ′ (cid:19) . (5 . φ → ∞ this worldsheet looks like a string stretched between the D -branes, which implies that this configuration contributes to the one point function of thestretched string operator (5.10).The instanton configuration can be constructed as follows. Start with the worldsheetaction S = 1 πα ′ Z d z ( ∂ z φ∂ ¯ z φ + ∂ z x∂ ¯ z x ) . (5 . x, φ )-plane by the coordinate U = exp (cid:18) φ − φ IR + ix √ kα ′ (cid:19) , (5 . U + U ∗ = 2 , (5 . U −
1) = 0.The worldsheet action (5.12) now takes the form S = kπ Z d z | U | ( ∂ z U ∂ ¯ z U ∗ + ∂ ¯ z U ∂ z U ∗ ) . (5 . U − z − z . (5 . A of the Euclidean string worldsheet (5.16): S inst = A πα ′ . (5 . φ → ∞ the hairpin looks like two D-strings a distance L crit (5.4) apart, so there is a divergence from that region. This divergence can be regulated byintroducing an upper bound on φ , φ UV , which can be thought of as a UV cutoff.In any case, we are only interested in the dependence of the area on the position ofthe bottom of the hairpin, φ IR , discussed around (5.9). We can isolate this dependenceby differentiating the area with respect to φ IR . A short calculation leads to ∂A∂φ IR = − L crit (5 . φ UV → ∞ . Therefore, after rescaling the operator (5.10) by a factor whichdepends on the UV cutoff, we conclude that h T i ∼ exp ( − S inst ) ∼ exp (cid:18) L crit φ IR πα ′ (cid:19) ∼ µ , (5 . φ IR to give a non-zero one-point function to the long open stringoperator (5.10). Note that we have only computed the leading exponential contribution to25he one point function. The pre-exponential factor involves contributions from the dilatoncoupling in the worldsheet action, and the determinant of small fluctuations around theinstanton (5.16). These are subleading in the large k limit, and are expected to give riseto a constant contribution to (5.19) (independent of φ IR ).So far we discussed the spontaneous breaking of the U (1) × U (1) symmetry of the braneconfiguration of figure 3 to the diagonal U (1), by the brane configuration of figure 4. Wehave seen that the order parameter for this breaking can be taken to be the stretched stringoperator (5.10), and it indeed has a non-zero expectation value in the hairpin state (5.19).It is natural to ask what happens if we deform the system by adding to the worldsheetLagrangian the non-normalizable operator T (5.10), δS ws = κ Z dtdθT ( φ, x ) + c.c. . (5 . U (1) × U (1) symmetry explicitly. It also breaks the N = 2superconformal symmetry of the hairpin brane; therefore we do not expect the resultingtheory to be exactly solvable. However, one can still ask how the shape of the D -braneand its low-lying spectrum change in the presence of this deformation.To first order in κ and in the semi-classical regime k ≫ S = − C Z dx exp (cid:18) φ √ kα ′ (cid:19) p φ ′ − κBe − S NG . (5 . S NG = A πα ′ , where A is the area of a minimal worldsheetenclosed by the deformed hairpin, and B is the pre-exponential factor in the expectationvalue of T above. It depends on the shape of the deformed hairpin, but for the purpose ofthe calculation below, to leading order in the 1 /k expansion we can neglect this dependence.To calculate the shape of the deformed hairpin to first order in κ we need to solvethe equation of motion of φ ( x ) with the deformed action (5.21). For this we need thedependence of S NG on the shape φ ( x ). It is easy to see that it is given by S NG = − πα ′ Z dxφ ( x ) + · · · , (5 . φ UV , but not on the shape φ ( x ). Varying (5.21) with respect to φ ( x ) and integrating once, we find the first orderequation C e φ √ kα ′ p φ ′ + κ h T i πα ′ φ = D , (5 . D is a function of φ IR (or, equivalently, of the separation between the brane andanti-brane at some UV cutoff φ UV ). This equation generalizes (5.6) to non-zero κ , and itcan be solved by expanding φ as φ = φ + κφ + · · · , and keeping only first order terms in κ . For example, at large φ , the leading deformation of the hairpin from its original formis given by C∂ φ x = (cid:18) D − κ h T i πα ′ φ (cid:19) e − φ √ kα ′ . (5 . φ becomes too large, one has to go beyond the linear approximation in κ described above.It is interesting to compare the deformed shape of the hairpin (5.24) which we foundabove, to the deformed shape implied by the effective action on the Dp -brane coupled tothe “tachyon” field T . In curved space and for curved D -branes (as in the discussion ofthe previous sections) it is not known how to write down such an effective action, but forflat D -branes in flat space we know how to write it down, and this is the situation in theasymptotic region of the hairpin. In this region we know that, if we denote the distancebetween the brane and the anti-brane by L crit − x ( φ ) (where x ( φ ) is small in the UV),the mass of the open string ground state is given by m ( x ( φ )) = − α ′ + (cid:18) L crit − x πα ′ (cid:19) ≃ m − r kα ′ xπ . (5 . D and ¯ D -branes in figure 4 isgiven to quadratic order by S = − C Z dφ exp (cid:18) φ √ kα ′ (cid:19) (cid:2) ( ∂ φ x ) + ( ∂ φ T ) + m ( x ) T (cid:3) . (5 . h T i , and the non-normalizable mode (5.10) is turned on witha coefficient κ , such that at leading order in κ the tachyon field behaves asymptotically as T ≃ βκ h T i exp( − φ √ kα ′ ) , (5 . β is a constant coming from carefully normalizing the normalizable and non-normalizable modes of the “tachyon”.The equation of motion of x ( φ ) with this “tachyon” source, at leading order in κ (andin the UV region where x is small), then takes the form ∂ φ (cid:20) exp (cid:18) φ √ kα ′ (cid:19) ∂ φ x (cid:21) = − βκ h T i πα ′ r kα ′ . (5 . /β = C p k/α ′ , this precisely agrees with (5.24) above.27 . Additional issues The main examples we focused on so far were non-supersymmetric, but one can alsoconstruct interesting examples of OWL operators (2.2) in supersymmetric theories, includ-ing examples which preserve some of the supersymmetry. We will describe here just oneexample, leaving a further investigation to future work.Consider the d = 4 N = 4 SU ( N c ) SYM theory coupled to N f three dimensionalmassless hypermultiplets living on the surface x = 0. In the ’t Hooft large N c limitwith ’t Hooft coupling λ and with fixed N f , this is described by type IIB string theoryon AdS × S , with N f D AdS × S subspace [25,26]; if we use thePoincar´e coordinates of AdS (with a boundary at z → ds = p λ α ′ dx µ + dz z , (6 . x = 0 (and wrap some maximal S inside the S ). This theory breaks half of the supersymmetry of the N = 4 SYM theory; it preservesa d = 3 N = 4 superconformal symmetry.Now, consider an OWL starting at a hypermultiplet at x = x = x = x = 0 andstretching to infinity in the x direction. Such an operator is analogous to the “straightWilson line” in the N = 4 SYM theory; it is well-defined if we put appropriate boundaryconditions at infinity. In the holographic dual description, the computation of the one-point function of this operator is dominated by a string sitting at x = x = x = 0 andfilling the z axis and the positive x axis in (6.1) (we assume that the OWL couples toa scalar such that the string lives at a point in the S filled by the D x = x = x = 0, x = a . This trans-formation leaves the field theory described above invariant. However, the contour in theOWL now maps to a semi-circle( x − a + 12 a ) + x = 14 a , x ≥ . (6 . x = a, x = 0) and ( x = a − /a, x = 0).Our derivation of this configuration by a conformal transformation ensures that this OWL28till preserves 8 supercharges, though these are now combinations of standard supersym-metries and superconformal symmetries.The holographic computation of the one-point function of this OWL is straightforward;the dominant solution is just half of the solution for the circular Wilson line [51,6], with astring worldsheet at ( x − a + 12 a ) + x + z = 14 a , x ≥ . (6 . √ λ , so theVEV of the OWL (at leading order in the α ′ expansion) is equal to exp( √ λ / h OW i describedabove should also be computable by a zero-dimensional model of matrices and vectors; itwould be interesting to verify this. Closed supersymmetric Wilson loops are known to have divergences at cusps, whichcan be computed both perturbatively and at strong coupling (with a qualitatively similarbehavior found in both limits [6]). Similarly, in the case that the fields in the fundamentalrepresentation are localized on some subspace, the correlation functions of the open Wilsonline observables (2.2) have a divergence whenever the contour ˜ C ends on that subspace atan angle which is not a straight angle. In this section we describe this divergence both atweak coupling (using perturbation theory) and at strong coupling (using the mapping tostring worldsheets).Let us consider a D -dimensional large N gauge theory, in which some fields in thefundamental representation are localized on a d -dimensional subspace; without loss ofgenerality we can take this subspace to be x d +1 = x d +2 = · · · = x D = 0 . (6 . x = 0, there is now an angle associated with this operator, which is theangle θ between the direction of the Wilson line (near x = 0) and the subspace that thefundamental fields live on. For instance, again without loss of generality, we can assumethat near x = 0 the Wilson line (parameterized by t ) looks like x d +1 = t sin( θ ) , x d = t cos( θ ) , t ≥ , (6 . θ ) A d +1 +cos( θ ) A d .On the other hand, the fields in the fundamental representation couple just to A d andthey do not couple to A d +1 . The one-loop diagram involving the exchange of a gauge fieldbetween the Wilson line and the propagator of the field in the fundamental representationthen has a divergence as t →
0, proportional (near θ = π/
2) to cos ( θ ), going as R dt/t D − .For D = 4 we have a logarithmic divergence (as for a cusp in a closed Wilson line), andfor D = 5 a linear divergence. The only case in which there is no divergence is when theWilson line intersects the surface (6.4) at a straight angle θ = π/ D − D − ¯ D D -brane just sits at x d +1 = · · · = x D = 0 and stretches in the radial direction.We need to find a minimal worldsheet ending on the contour (6.5) at the boundary andtransverse to the D -brane. It is easy to convince oneself that such a worldsheet is the sameas half of the closed string worldsheet ending on the contour x d +1 = t sin( θ ) , x d = | t | cos( θ ) , (6 . t = 0with an angle of 2 θ , so it leads to a divergence which is similar to the cusp divergenceoccurring in closed Wilson loops (whenever θ = π/ D = 4 this is alogarithmic divergence, just as in the previous paragraph, but its precise dependence onthe angle is different from the one found at weak coupling (this is also true for the closedWilson loop cusp divergence) [6].Note that in this computation we assumed that the end of the open Wilson line is atthe same position as the D -brane in the compact directions (otherwise there is no semi-classical worldsheet contributing to the computation of correlation functions of OW ). Ifthe D -brane is partially localized in the compact directions (so that the fundamental fieldscouple to some of the scalar fields of the gauge theory) then this implies that near the end ofthe open Wilson line, the Wilson line couples to different scalar fields than the ones whichthe fundamental fields couple to. Thus, for such Wilson lines there is no contribution fromthe scalar fields at leading order in perturbation theory, and their one-loop computationdiverges as described above.In any case, we showed that both at weak coupling and at strong coupling, whenthe fundamental representation fields are localized on a subspace, one has to choose the30ilson line operators (2.2) such that the direction of the open Wilson line is transverseto that subspace at its beginning and end, in order to avoid cusp-like divergences in thecomputation. Acknowledgements
We would like to thank O. Bergman, N. Drukker, S. Hartnoll, Z. Komargodski, O.Lunin, D. Reichmann, A. Schwimmer, J. Sonnenschein, and S. Yankielowicz for usefuldiscussions. The work of OA was supported in part by the Israel-U.S. Binational ScienceFoundation, by a center of excellence supported by the Israel Science Foundation (grantnumber 1468/06), by a grant (DIP H52) of the German Israel Project Cooperation, bythe European network MRTN-CT-2004-512194, and by Minerva. DK is supported in partby DOE grant DE-FG02-90ER40560, by the National Science Foundation under Grant0529954, and by the Israel-U.S. Binational Science Foundation. DK thanks the WeizmannInstitute for hospitality during part of this work.31 eferences [1] J. M. Maldacena, “The large N limit of superconformal field theories and supergrav-ity,” Adv. Theor. Math. Phys. , 231 (1998) [Int. J. Theor. Phys. , 1113 (1999)][arXiv:hep-th/9711200].[2] S. S. Gubser, I. R. Klebanov and A. M. 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