Exact Half-BPS Flux Solutions in M-theory II: Global solutions asymptotic to AdS_7 x S^4
aa r X i v : . [ h e p - t h ] N ov UCLA/08/TEP/29CPHT-RR079.100823 October 2008
Exact Half-BPS Flux Solutions in M-theory II:Global solutions asymptotic to
AdS × S † Eric D’Hoker a , John Estes b , Michael Gutperle a and Darya Krym a a Department of Physics and AstronomyUniversity of California, Los Angeles, CA 90095, USA b Centre de Physique Theorique, Ecole Polytechnique,F91128 Palaiseau, France
Abstract
General local half-BPS solutions in M-theory, which have SO (2 , × SO (4) × SO (4)symmetry and are asymptotic to AdS × S , were constructed in exact form by theauthors in [arXiv:0806.0605]. In the present paper, suitable regularity conditions areimposed on these local solutions, and corresponding globally well-defined solutionsare explicitly constructed. The physical properties of these solutions are analyzed, andinterpreted in terms of the gravity duals to extended 1+1-dimensional half-BPS defectsin the 6-dimensional CFT with maximal supersymmetry. † This work was supported in part by NSF grants PHY-04-56200 and PHY-07-57702.
E-mail addresses : [email protected]; [email protected]; [email protected];[email protected].
Introduction
One realization of the AdS/CFT correspondence [1, 2, 3] in M-theory is the duality of the
AdS × S vacuum and the 6-dimensional CFT which is obtained by a decoupling limit ofthe M5 brane world-volume theory [4, 5, 6]. The nonabelian world-volume theory of multipleM5-branes is presently unknown and the 6-dimensional CFT has been formulated in the lightcone gauge [7]. One interesting class of deformations in this theory is given by the insertionof local half-BPS chiral operators, where half-BPS means the operators preserve sixteen ofthe thirty-two supersymmetries. The gravitational duals of these operators are the half-BPSsolutions of Lin, Lunin, and Maldacena [8].In our recent paper [9] (see also [10, 11, 12, 13] for earlier work), new exact solutionsof 11-dimensional supergravity were constructed which preserve sixteen of the thirty-twosupersymmetries. In addition, the solutions preserve a SO (2 , × SO (4) × SO (4) bosonicsymmetry. Correspondingly, the 11-dimensional metric is constructed as a warped productof AdS × S × S over a 2-dimensional base space Σ. These solutions can be interpretedas the gravity duals of extended supersymmetric defects in the CFT. The solutions are localin the sense that for a bosonic background, the vanishing of gravitino variation as well asthe bosonic equations of motion and Bianchi identities are satisfied point wise, except atpossible singularities.In general, the local solutions of [9] contain a large variety of solutions many of whichcontain singularities. An important problem is to pick out solutions which are asymptotic toeither AdS × S or AdS × S and everywhere regular so that the supergravity approximationis valid. In particular this requires one to examine the global structure of the solutions.Similar analysis have been carried out in Type IIB supergravity in [8, 15, 18]. An amazingresult in all cases is that the regularity conditions in addition to the general local solutionadmit a superposition principle for the half-BPS objects in the theory. We expect such aprinciple to emerge from the analysis here. There are two distinct classes of solutions whichwere found in [9]:Case I contains solutions asymptotic to AdS × S . The superalgebra of symmetriesis OSp (4 ∗ | × OSp (4 ∗ | OSp (4 ∗ |
2) is a particular real form of
OSp (4 | AdS × S is dual to a 3-dimensional CFT which is obtained by a decoupling limit of theworld-volume theory of M2 branes. The local BPS solution is dual to 1 + 1-dimensionalconformal defects in the 3-dimensional CFT, analogous to the half-BPS defect solutionsobtained in type IIB string theory [14, 15] (see also [16] for earlier work).Case II contains solutions asymptotic to AdS × S . The bosonic isometries together with2he supersymmetries form a superalgebra which is given by OSp (4 | , R ) × OSp (4 | , R ). Thesupergroup OSp (4 | , R ) is a different real form of OSp (4 |
2) than the one appearing in caseI. In the classification of [17] this solution is case VII of table 12. M-theory on
AdS × S is dual to a 6-dimensional CFT which is obtained by a decoupling limit of the world-volumetheory of M5 branes. The local BPS solution is dual to 1 + 1-dimensional conformal defectsin the 6-dimensional CFT, analogous to the half-BPS Wilson loop solutions obtained in typeIIB string theory [18] (see also [12, 19] for earlier work).The local solutions presented in [9] are very similar for the case I and II as the underlyingintegrable system is the same. However, the analysis of the regularity and the global structureis quite different. In this paper we will focus on the solution of case II. The analysis of theregularity and global structure for case I will be analyzed in a separate paper [20].The structure of the paper is as follows. In section 2 the features of the local solution forcase II which are important for the present paper will be reviewed. In section 3 the boundaryconditions on the solution implied by regularity are analyzed. A general solution whichsatisfies suitable boundary condition is constructed and it is shown that the solution is regulareverywhere. In section 4 the global structure of the solutions as well as its interpretation interms of the dual 6-dimensional conformal field theory is discussed. In appendix A detailedproof of the regularity of our solution is presented. In this section we review the local half-BPS solution of [9]. Derivations and more calculationaldetails can be found in that paper. The 11-dimensional metric is a fibration of
AdS × S × S over a 2-dimensional Riemann surface Σ, ds = f ds AdS + f ds S + f ds S + ds (2.1)The four form field strength is given by F = g a ω AdS ∧ e a + g a ω S ∧ e a + g a ω S ∧ e a (2.2)Here ω AdS and ω S , are the volume forms on AdS and S , respectively. In addition e a , a = 1 , w, ¯ w on the Riemann surface Σ such that the 2-dimensional metric in (2.1) is given by ds = 4 ρ | dw | (2.3)The metric factors f , f , f , ρ , as well as the flux fields g a .g a , and g a only depend on Σ.3he Ansatz respects SO (2 , × SO (4) × SO (4) symmetry which can be interpreted asthe symmetries of a 1+1-dimensional conformal defect in the 6-dimensional M5 brane CFT.The condition that 16 supersymmetries are unbroken is equivalent, for a purely bosonicbackground, to the statement that the gravitino supersymmetry variation δ ǫ Ψ M vanishes for16 linearly independent supersymmetry variation parameters.In [9] the BPS conditions were solved, and it was shown that the half-BPS solution iscompletely determined by the choice of a 2-dimensional Riemann surface Σ, a real harmonicfunctions h ( w, ¯ w ) on Σ and a complex function G ( w, ¯ w ) which is a solution of the followinglinear equation, ∂ w G = 12 ( G + ¯ G ) ∂ w ln h (2.4)In order to express the local half-BPS solution in terms of G and h it is useful to define thefollowing quantity W = −| G | − ( G − ¯ G ) (2.5)The metric factors in (2.1) are then given by f = 4 h (1 − | G | ) W (cid:16) | G − ¯ G | + 2 | G | (cid:17) f = 4 h (1 − | G | ) W (cid:16) | G − ¯ G | − | G | (cid:17) f = h W − | G | ) (2.6)The metric factor in (2.3) is given by ρ = ( ∂ w h∂ ¯ w h ) h (1 − | G | ) W (2.7)The fluxes g i are defined by conserved currents as follows( f ) g w = − ∂ w b = 2( j + w + j − w )( f ) g w = − ∂ w b = − j + w − j − w )( f ) g w = − ∂ w b = 18 j w (2.8)where the conserved currents can be expressed in a compact way by defining J w = hG + ¯ G (cid:18) ¯ G ( G − G + 4 G ¯ G ) ∂ w G + G ( G + ¯ G ) ∂ w ¯ G (cid:19) (2.9)4nd are given by j + w = 2 i J w (cid:18) ( G − ¯ G ) − G ¯ G (cid:19) W − j − w = 2 GJ w (cid:18) − G ¯ G + 3 ¯ G − G + 4 G ¯ G (cid:19) W − j w = 3 ∂ w h W G (1 − G ¯ G ) − J w (1 + G ) G (1 − G ¯ G ) (2.10)It was shown in [9] that the equations of motion of as well as the Bianchi identities aresatisfied for a harmonic h and a G which solves (2.4). G=i h=0y x0 π/2
G=0 h=0G=0 h=0
Figure 1: Σ and boundary conditions for
AdS × S solutionThe simplest solution is the maximally symmetric AdS × S itself. The Riemann surfaceis the half strip Σ = { ( x, y ) , x ≥ , ≤ y ≤ π/ } . Denoting the holomorphic coordinate as w = x + iy , the functions h and G are given by h = − i ( cosh(2 w ) − cosh(2 ¯ w )) G = − i sinh( w − ¯ w )sinh(2 ¯ w ) (2.11)Plugging this into (2.6) the metric factors become f = 2ch( x ) f = 2sh( x ) f = sin(2 y ) ρ = 1 (2.12)Note that the Riemann surface Σ has three boundary components. The boundary is char-acterized by the vanishing of the harmonic function h = 0. Furthermore, taking y = 0 or y = π/
2, we find G = 0, while taking x = 0, we find G = + i . So G takes the values 0 and + i on the boundary of Σ. The boundary of AdS × S on the other hand is located at x = ∞ .5 Regularity
In this section, we analyze the regularity conditions and the global structure of the solution.We look for solutions which are everywhere regular and which are asymptotic to
AdS × S .This leads to the following three assumptions for the geometry.1. The boundary of the 11-dimensional geometry is asymptotic to AdS × S .2. The metric factors are finite everywhere, except at points where the geometry becomesasymptotically AdS × S , in which case the AdS metric factor and one of the spheremetric factors diverge.3. The metric factors are everywhere non-vanishing, except on the boundary of Σ, inwhich case at least one sphere metric factors vanishes. In addition, both sphere metricfactors may vanish only at isolated points.The second requirement guarantees that all singularities in the geometry are of the sametype as AdS × S . The third requirement guarantees that the boundary of Σ correspondsto an interior line in the 11-dimensional geometry.It follows from (2.6) that a particular combination of metric factors is very simple( f f f ) = h (3.1)The metric factor f is positive definite and cannot vanish [9]. Hence the condition h = 0(which defines a 1-dimensional subspace in Σ) occurs if and only if at least one of the metricfactors for the spheres f or f vanishes. It follows from assumption 3, that h = 0 definesthe boundary of Σ.Note that the equation for G (2.4) is covariant under conformal reparamaterizations.This freedom allows one to choose local conformal coordinates u = r + i s, r = h ( w, ¯ w ) , s = ˜ h ( w, ¯ w ) (3.2)Here, ˜ h is the harmonic function dual to h so that u is holomorphic, i.e. ∂ ¯ u ( h + i ˜ h ) = 0. Thedomain of the new conformal coordinate u is the right half plane and the boundary of Σ isat r = 0, i.e. the vertical axis.In the coordinates r, s , it is useful to decompose G into its real and imaginary parts, G ( r, s ) = G r ( r, s ) + iG s ( r, s ) (3.3) We use a slightly different notation: the coordinate s in this paper was called x in [9]. G r , G s real functions. The real and imaginary parts of equation (2.4) are respectively, ∂ r G r + ∂ s G s = G r r (3.4) ∂ r G s − ∂ s G r = 0 (3.5)Equation (3.5) can be solved in terms of a single real potential G r = ∂ r ( r Ψ) G s = ∂ s ( r Ψ) (3.6)Equation (3.4) becomes a second order partial differential equation on Ψ, (cid:16) ∂ s + ∂ r + 1 r ∂ r − r (cid:17) Ψ( r, s ) = 0 (3.7)The general local solution of (3.7) can be obtained by a Fourier transformation with respectto s , which produces an ordinary differential equation which can be solved by [9]Ψ( r, s ) = Z ∞−∞ dk π ψ ( k ) K ( kr ) e − iks (3.8)Here K is the modified Bessel function of the second kind. There is a second linearlyindependent solution of the form (3.8) which involves the modified Bessel function of thefirst kind I ( kr ). However this solution has the wrong behavior for large r and fails to obeythe regularity condition | G | ≤
1. In [9] an explicit expression for Ψ and G was found bydefining C ( v ) C ( v ) = Z ∞ dk π ψ ( k ) e − kv (3.9)and using the following integral representation of K K ( kr ) = Z ∞ t dt √ t − e − tkr (3.10)Using (3.8)- (3.10) Ψ can be expressed in terms of C Ψ( r, s ) = Z ∞ t dt √ t − (cid:18) C ( tr + is ) + C ( tr + is ) ∗ (cid:19) (3.11)and (3.6) can be used to write G as follows G ( r, s ) = r Z ∞ dt √ t − (cid:18) (1 − t ) C ′ ( tr + is ) + (1 + t ) C ′ ( tr + is ) ∗ (cid:19) (3.12)7 .1 Boundary conditions on G In this section we analyze the boundary conditions G has to satisfy at h = 0 or in the r, s coordinates at r = 0 in order for the solution to be regular near the boundary. It will beuseful to have the following expressions for the metric factors obtainable from (2.6)( f f ) = − r (1 − | G | ) Wf = − rW − | G | ) (3.13)The behavior near the boundary (the regularity in the bulk of Σ will be discussed in thenext section) is exhibited by expanding G r , G s , at fixed s , in a power series in r , G r = γ r + γ r + γ r + O ( r ) G s = γ + γ r + γ r + O ( r ) (3.14)where the γ i are all functions of s . Note that the reality of the solution implies that G is bounded | G | ≤ r can appear in the series expansion(3.14). Equation (3.4) and (3.5) impose the vanishing of even/odd powers of r in G r /G s respectively. Furthermore these equations impose differential equations in s between thedifferent γ i ( s ) but these relations will not be needed in the following. The following powerseries expansions will be useful in the following analysis, W = 4 γ (1 − γ ) + 8( γ γ − γ γ − γ γ ) r + O ( r )1 − | G | = (1 − γ ) − ( γ + 2 γ γ ) r + O ( r ) (3.15)When γ = ±
1, we have | G | 6 = 1 as r →
0. From (3.13), the product f f goes to zero as r → W ∼ r . It follows from (3.15) that one hasto choose γ = 0 in order to avoid a singularity. If this is the case, f will remain finite, while f → S tends to zero. Comparing with the AdS × S solution, this behavior is associated with the y = 0 , π/ γ = ±
1, we will have f → f will remain finite, as long as the conditions γ = ∓ γ and γ = ∓ γ are satisfied. If this is the case, the volume of the sphere S willtend to zero. Comparing with the AdS × S solution, this behavior is associated with the x = 0 boundary component of (2.11).In summary we have the following boundary conditions on G and the metric factors G (0 , s ) = 0 ⇔ f = 0 f = 0 , Vol( S ) → G (0 , s ) = ± i ⇔ f = 0 f = 0 Vol( S ) → .2 General regular solution We first parameterize the boundary conditions for G at r = 0 which leads to solutionssatisfying regularity conditions 1 to 3, listed at the beginning of section 3. As we shall see, afurther condition needs to be imposed to guarantee the absence of singularities in the bulk,namely the values of G in the second line of (3.16) will be restricted to be either all positiveor all negative. g+1 sr G=0 G=0 G=0G=i Σ G=i η η a b a a b
Figure 2: The surface Σ and boundary conditions for a general regular solution.General boundary conditions for G are given by the choice of g + 2 points on the s axis, −∞ < a < b < a < b < · · · < a g +1 < b g +1 < ∞ (3.17)The AdS × S solution corresponds to g = 0. There are two kinds of intervals which aredistinguished by the boundary condition for G (0 , s ). s ∈ [ −∞ , a ] G ( s,
0) = 0 s ∈ [ a n , b n ] G ( s,
0) = iη n , n = 1 , , · · · gs ∈ [ b n , a n +1 ] G ( s,
0) = 0 , n = 1 , , · · · gs ∈ [ b g +1 , ∞ ] G ( s,
0) = 0 (3.18)where η n = ±
1. The boundary conditions can be implemented as follows: G ( s,
0) = g +1 X n =1 iη n (cid:18) Θ( s − a n ) − Θ( s − b n ) (cid:19) (3.19)9here Θ( s ) is the step function. Since on each segment, G ( s,
0) can take only the values0 , ± i , no two bumps can “overlap”, and this forces the a n and b n to alternate as in (3.17).It remains to calculate ψ ( k ) from the boundary condition in (3.19). To this end, we firsttake the Fourier transform in s with ℓ >
0, of (3.6) with (3.8) plugged in Z ∞−∞ ds e + iℓs ∂ s (cid:16) r Ψ( s, r ) (cid:17) = − iℓrψ ( ℓ ) K ( ℓr ) (3.20)whose r → K , and we find, Z ∞−∞ ds e + iℓs lim r → ∂ s (cid:16) r Ψ( s, r ) (cid:17) = − iψ ( ℓ ) (3.21)Using the boundary condition of (3.19), we thus have ψ ( ℓ ) = X n iη n Z a n b n ds e iℓs = X n η n e iℓa n − e iℓb n ℓ (3.22)The Fourier transform can be done exactly C ( v ) = − π X n η n ln (cid:18) v − ia n v − ib n (cid:19) (3.23)The expression for G ( s, r ) may then be obtained using the integral representation (3.12) G ( s, r ) = Z ∞ dt √ t − (cid:18) r (1 − t ) C ′ ( tr + is ) + r (1 + t ) C ′ ( tr + is ) ∗ (cid:19) (3.24)After some simplifications, we obtain, G ( s, r ) = − π g +1 X n =1 η n Z ∞ dt √ t − " iα n t + iα n + 1 + iα n t − iα n − iβ n t + iβ n − iβ n t − iβ n (3.25)where the quantities α n and β n are defined as α n = ( s − a n ) /r, β n = ( s − b n ) /r, n = 1 , , · · · g + 1 (3.26)are both real. Next we use the integral formula Z ∞ dt √ t − (cid:18) t + z + 1 t − z (cid:19) = π √ − z (3.27)The result is an algebraic expression for G which solves (3.4) and satisfies the boundarycondition (3.19) and is given by G ( s, r ) = − g +1 X n =1 η n r + is − ia n q r + ( s − a n ) − r + is − ib n q r + ( s − b n ) (3.28)It is easy to see that (3.28) is equal to (3.19) in the limit r → .3 Regularity in the bulk In the remainder of this section we give an argument that the general solution (3.28) is regulareverywhere. Note that the general solution was constructed in section 3.2 by demanding thatthe geometry is regular at the boundary of Σ.The general solution (3.28) should also approach the
AdS × S boundary asymptoticallyas r → ∞ . The expression for AdS × S . given in (2.11) can be recovered from the generalregular solution (3.28) by setting g = 0, η = − a = − b = 2. G ( s, r ) = r + is + 2 i | r + is + 2 i | − r + is − i | r + is − i | = − i sh( w − ¯ w )sh(2 ¯ w ) (3.29)where we have used the coordinates r = 2 sin(2 y ) sinh(2 x ) and s = 2 cos(2 y ) sinh(2 x ). Theboundary of AdS is reached by taking x → ∞ . Using the same coordinate change for thegeneral solution it is easy to see that (3.28) approaches AdS × S in the limit x → ∞ .It remains to show that the geometry is regular in the interior for Σ. Since h = r in thechosen coordinate system the regularity of the solution requires that away from the boundary W is required to satisfy the strict inequality W >
0. Note that this condition automaticallyguarantees that we also have 1 − | G | >
0. Furthermore the relation W = − | G | − ( G − ¯ G ) = ( | G − ¯ G | − | G | )( | G − ¯ G | + 2 | G | ) (3.30)shows that if W > | G − ¯ G | − | G | ) = 0 and ( | G − ¯ G | + 2 | G | ) = 0.Examining the explicit formula for the metric factors (2.6) one can see that they are thenfinite and the geometry is thus regular. It is shown in Appendix A if the a n , b n obey theordering (3.17) and the η n are all either +1 or −
1, that W > W = 0 in the bulk then the solutions is singular, we firstnote that for W to vanish, either ( | G − ¯ G | − | G | ) or ( | G − ¯ G | + 2 | G | ) must vanish. Takingthe ratio of f and f in (2.6) we see that one of them must either be vanishing or be infiniteresulting in a singular geometry. g = 1 solution In this section, we examine the g = 1 solution in detail, specifically we choose the parametersfor the general solution (3.28) as follows: g = 1 , a = − , b = − , a = 0 , b = 1 (3.31)11n Figure 3, we show the behavior of the metric factors. The sphere metric factors alter-natingly vanish as r →
0. While as r → ∞ , the metric factors flatten out to those of AdS × S . (cid:1) (cid:0) (cid:2) (cid:3) f (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:9) (cid:10)(cid:11)(cid:12) (cid:13) (cid:14) (cid:15) (cid:16) (cid:17) (cid:18) f (cid:19) (cid:20) (cid:21) (cid:22) (cid:23) (cid:24) (cid:25) f (cid:26) (cid:27) (cid:28) Figure 3: Metric factors for a g = 1 solution.For ρ there are singularities as we approach r → ∞ , but these are coordinate singularitiesand are due to the conformal transformation we made in order to map the half-strip to theupper half plane.An important feature of the solutions with g > g = 1 solution (3.31). Inaddition to the four cycle C which is already present in the g = 0 solution, there are twoadditional nontrivial four cycles C and C .The behavior of the fluxes for the g = 1 solution is very interesting. For comparisonpurposes we first plot the fluxes (2.8) for the AdS × S solution given by (3.28) with g = 0 , a = − , b = 1 (3.32)Note that the fluxes g and g vanish identically and the only nontrivial flux is g . There is12 Σ sr a =0 b =1b =−1a =−2 C C C Figure 4: Nontrivial four cycles for the g = 1 solution -2 -1 0 100.20.40.60.81 sr (cid:29) w b Figure 5: The fluxes for a g = 0 solution.13nly one nontrivial topological cycle forming a four sphere. The integrated flux g is nothingbut the non-vanishing four form flux through the four sphere in AdS × S . -2 -1 0 100.20.40.60.81 sr (cid:30) w b -2 -1 0 100.20.40.60.81 sr (cid:31) w b -2 -1 0 100.20.40.60.81 sr w b Figure 6: The fluxes for a g = 1 solution.In Figure 6, we plot the fluxes for the g = 1 solution (3.31). Due to the complicatedform of the currents (2.8) we have not been able to integrate them to analytically obtain aclosed form of the fluxes. However it is clear from figure 6 that the g = 1 solution has indeednontrivial flux through the cycles C and C for g and g respectively.14 Discussion
In the previous section we found a family of regular half-BPS solutions labelled by an integer g and 2 g + 1 real moduli. In this section we discuss the interpretation of these solutions fromthe point of view of the AdS/CFT correspondence. The AdS × S spacetime is obtainedas the near horizon limit of a large number of M5 branes. The AdS/CFT duality relatesM-theory on this background to the decoupling limit of the M5-brane world-volume theorywhich defines a 6-dimensional CFT with (2 ,
0) supersymmetry [5, 4].A first step towards interpreting the solution is to understand the boundary structure.The only region on Σ where the spacetime becomes asymptotically
AdS × S is r → ∞ .There is however another boundary component since the AdS factor also has a boundary.This can be seen by rewriting the metric (2.1). ds = 1 z (cid:16) f ( dz + dx − dt ) + z f ds S + z f ds S + z ds (cid:17) (4.1)The boundary of AdS is reached as z → /z . The z factor in front of the metric factors ofthe spheres and Σ implies that the boundary in the limit z → t, x plane. The SO (2 ,
2) isometry of the
AdS factor corresponds to theconformal symmetry of the 1 + 1-dimensional defect, which is contained in the OSp (4 | , R ) × OSp (4 | , R ) supergroup of preserved superconformal symmetries. Note that for all values of g and the moduli there is only one defect.The interpretation of the 1+1-dimensional half-BPS defect from the perspective of thedual CFT is the supersymmetric self dual string solution of the 6-dimensional (2 ,
0) super-symmetric M5-brane world-volume theory [21, 22, 23] which was constructed in [24]. Theselfdual string in the (2 ,
0) theory can also be interpreted as the boundary of an open M2brane which ends on the M5-brane [25, 26] Unfortunately the action for multiple membranesis not well understood and the selfdual string soliton solution has only been derived for theabelian case of a single 5-brane.There is a strong analogy of the selfdual string defect with the BPS-Wilson loop in TypeIIB string theory. While the details of the supergravity solution are somewhat different thegeneral structure of the half-BPS flux solution and its moduli space presented in section 3.2is intriguingly similar to the Type IIB supergravity flux solutions dual to BPS Wilson loopswhich was found in [18].The BPS Wilson loop in
AdS × S also has a probe description. The original proposal[35, 36] identified the Wilson loop in the fundamental representation with a fundamental15tring with AdS world-volume inside AdS . BPS-Wilson loops in higher rank symmetricrepresentation and are identified with a probe D3 brane with electric flux with AdS × S world-volume inside AdS . BPS-Wilson loops in higher rank anti-symmetric representationand are identified with a probe D5 brane with electric flux with AdS × S world-volumeinside AdS × S [27, 28].The 1+1-dimensional BPS defect in the 6-dimensional CFT can be viewed as the insertion“Wilson surface”-operator [29, 30, 31, 32]. In the probe approximation one can use a analogybetween the Wilson loop in N = 4 SYM and the Wilson surface operators: The fundamentalstring is related to M2-brane probe. The D3 brane with electric flux and AdS × S world-volume is related to a probe M5 brane with 3-form flux on its AdS × S world-volume (withthe S embedded in the AdS ). The D5 brane with electric flux and AdS × S world-volumeis related to a probe M5 brane with three form flux on its AdS × S world-volume (withthe S embedded in the S ). These probe branes and their supersymmetry where analyzedin [33, 34, 13]The supergravity solutions we have obtained are the analog of the “bubbling” Wilsonloop solutions [18, 12, 28]. They are fully backreacted and replace the probe branes bygeometry and flux. In particular as the discussion of the g = 1 solution in section 3.4 showedthere are two new nontrivial four cycles C , in the g = 1 solution. The fluxes through thesecycles are the remnants of the probe M5-branes in the backreacted solution.Unfortunately the (2 ,
0) theory for multiple M5-branes is not as well understood as N = 4SYM theory. It is possible that the bubbling solutions can be useful in the understanding ofthe M5-brane theory. It would be interesting to see whether there is an analog of the matrixmodel description of the BPS-Wilson loops (and its relation to the bubbling supergravitysolution) for the Wilson surfaces.The general solution we have obtained has only one asymptotic AdS × S region. Itwould be interesting to investigate whether its possible to have more than one asymptotic AdS region, this would presumably correspond to a harmonic function h with multiplepoles. A similar phenomenon occurs in the case of half-BPS solutions which are asymptoticto AdS × S which we are currently investigating [20]. Acknowledgments
MG gratefully acknowledges the hospitality of the International Center for TheoreticalScience at the Tata Institute, Mumbai and the Department of Physics and Astronomy, JohnsHopkins University during the course of this work.16
Proof of the regularity condition W > In this appendix we shall prove a theorem which is central to establishing the regularity ofthe general solution constructed in section 3.2.
Theorem 1
When all η n are equal to one another, the function G , defined by G = − g +1 X n =1 η n iα n q α n − iβ n q β n (A.1) satisfies W > , for all α n , β n subject to the ordering condition α < β < α < β < · · · < α g < β g < α g +1 < β g +1 (A.2)Numerical analysis suggests that this property holds, and also shows that, when not all η n are equal to one another, the condition W ≥ α n and β n . Weshall prove Theorem 1 for η n = +1 for all n = 1 , · · · , g + 1; the theorem for the opposite case η n = − W > W = − | G | − ( G − ¯ G ) = (cid:16) | G − ¯ G | − | G | (cid:17) (cid:16) | G − ¯ G | + 2 | G | (cid:17) (A.3)The second factor on the right hand side of the last equality is manifestly positive for all G ,and may be dropped in the inequality. Thus, the condition W > | G − ¯ G | − | G | >
0, which is equivalent to the following quadratic inequality X + (cid:18) | Y | − (cid:19) < G = X + iY (A.4)where X, Y are real. In the sequel, it will be convenient to introduce the following notations, p ( α ) ≡ −
12 1 √ α q ( α ) ≡ + 12 α √ α p + q = 14 (A.5)In terms of these functions, we define the following partial sums, for m = 1 , , · · · , g + 1, X m = m X n =1 (cid:16) p ( α n ) − p ( β n ) (cid:17) Y m = m X n =1 (cid:16) q ( β n ) − q ( α n ) (cid:17) (A.6)so that the real and imaginary parts of G , defined in (A.4), are given by X = X g +1 , Y = Y g +1 .The first key ingredient in the proof of Theorem 1 will be the fact that, for α ≥
0, thefunctions p ( α ) and q ( α ) are strictly monotonically increasing as α increases.17 .1 The case ≤ α We begin by proving Theorem 1 for the following special ordering,0 ≤ α < β < α < β < · · · < α g < β g < α g +1 < β g +1 (A.7)Using the fact that p ( α ) and q ( α ) are monotonically increasing with α for α ≥
0, it isimmediate that X g +1 < Y g +1 >
0. Both bounds are sharp, as they can be saturated atthe boundary of the domain (A.2) in the limit where α n − β n →
0. A lower bound for X g +1 and an upper bound for Y g +1 may be obtained by letting β n − α n +1 → α g +2 ≡ + ∞ ).Putting all together, we obtain the following double-sided bounds,0 < − X g +1 < − p ( α )0 < Y g +1 < − q ( α ) (A.8)To prove W >
0, we proceed recursively. Using the definition (A.6), we have, X g +1 = X g + p ( α ) − p ( β ) Y g +1 = Y g + q ( β ) − q ( α ) (A.9)where we use the abbreviations α = α g +1 , and β = β g +1 . Notice that we have X g < Y g >
0. The quantity of interest is W g +1 ≡ X g +1 + (cid:18) Y g +1 − (cid:19) (A.10)Here, we have suppressed the absolute value sign on Y g +1 , as we already know that Y g +1 > W > W g +1 < /
4. Thus, we need to derivean optimal upper bound for W g +1 , and show that this bound is less than 1 / W g +1 as a function of α and β , subject to the conditionthat β g < α < β . To this end, express W g +1 as follows, W g +1 = ( p ( β ) + x g ) + ( q ( β ) − y g ) x g = − X g − p ( α ) y g = − Y g + 12 + q ( α ) (A.11)The bounds established earlier, namely X g < Y g < /
2, guarantee that x g > y g > α ≥
0. We now search for the maximum of W g +1 as a function of β β ∈ [ α, + ∞ ], with α viewed as fixed. To determine it, we investigate thederivative with respect to β , ( W g +1 ) ′ ( β ) = x g β − y g √ β (A.12)This derivative can vanish in the interval β ∈ [ α, + ∞ ] if and only if x g α − y g ≤
0. If this isthe case, the corresponding point is β = y g /x g , which should satisfy β > α .Hence, the extrema of W g +1 as a function of β may be attained either at β = β , or ateither one of the extremities of the interval β ∈ [ α, + ∞ ]. These three values are given by, W g +1 ( β ) = x g + y g + 14 − q x g + y g W g +1 ( α ) = x g + y g + 14 + 2 p ( α ) x g − q ( α ) y g W g +1 ( ∞ ) = x g + y g + 14 − y g (A.13)Since x g , y g >
0, it is manifest that W g +1 ( β ) < W g +1 ( ∞ ). Thus, W g +1 ( β ) cannot be theoptimal upper bound for W g +1 ( β ). Comparing the remaining two possible values, we find, W g +1 ( ∞ ) − W g +1 ( α ) = 2 p ( α ) X g + 2 (1 − q ( α )) Y g (A.14)Given that X g < Y g >
0, it follows that the right hand side is positive and so that W g +1 ( ∞ ) is the optimal upper bound. In summary, W g +1 < V g ( α g +1 ) V g ( α g +1 ) ≡ (cid:16) X g + p ( α g +1 ) (cid:17) + (cid:16) Y g − q ( α g +1 ) (cid:17) (A.15)for all values of α g +1 such that β g < α g +1 .Since X g < Y g >
0, it is straightforward to derive an upper bound for the righthand side of (A.15). Indeed, both terms increase as α g +1 decreases. Thus, the optimal boundfor the right hand side is attained when α g +1 assumes its smallest possible value, which is α g +1 = β g . Hence, we have V g ( α g +1 ) < V g ( β g ) α g +1 ∈ [ β g , ∞ ] (A.16)But, using the definitions of X g and Y g in terms of α n and β n , we see that the quantity V g ( β g ) admits a drastic simplification, V g ( β g ) = (cid:16) X g − + p ( α g ) (cid:17) + (cid:16) Y g − − q ( α g ) (cid:17) = V g − ( α g ) (A.17)19ombining all, we get a recursive series of bounds, W g +1 < V g ( α g +1 ) < V g − ( α g ) < V g − ( α g − ) < · · · < V ( α ) (A.18)From their definitions, X = Y = 0, we readily find V ( α ) = 1 /
4, so that W g +1 < /
4. Thisconcludes the demonstration of Theorem 1 for the case 0 ≤ α . A.2 The case β g +1 ≤ Next, we proceed to proving Theorem 1 for the following special ordering, α < β < α < β < · · · < α g < β g < α g +1 < β g +1 ≤ ≤ α , since wecan reduce the present case to the α ≥ α n = − ˜ β g +2 − n n = 1 , · · · , g + 1 β n = − ˜ α g +2 − n (A.20)The ˜ α n and ˜ β n are now all positive and satisfy the following ordering,0 ≤ ˜ α < ˜ β < ˜ α < ˜ β < · · · < ˜ α g < ˜ β g < ˜ α g +1 < ˜ β g +1 (A.21)Denoting the corresponding function by G − , and its real and imaginary parts by X − g +1 and Y − g +1 , we have by definition, X − g +1 = g +1 X n =1 (cid:16) p ( ˜ β n ) − p ( ˜ α n ) (cid:17) = − ˜ X g +1 Y − g +1 = g +1 X n =1 (cid:16) q ( ˜ β n ) − q ( ˜ α n ) (cid:17) = + ˜ Y g +1 (A.22)where ˜ X g +1 and ˜ Y g +1 are given by (A.6) but with α n → ˜ α n and β n → ˜ β n . From the proof ofthe case α ≥
0, it now follows that W > A.3 The general case
Next, we shall prove Theorem 1 for the cases whose ordering is given by α < β < · · · < α N < β N ≤ ≤ α N +1 < β N +1 < · · · < α g +1 < β g +1 (A.23)20or N = 1 , · · · , g . (The proof for the case with the ordering · · · < α N < < β N < · · · follows the same steps, or may be derived by taking the limit α N +1 , β N → X − = N X n =1 ( p ( α n ) − p ( β n )) X + = g +1 X n = N +1 ( p ( α n ) − p ( β n )) Y − = N X n =1 ( q ( β n ) − q ( α n )) Y + = g +1 X n = N +1 ( q ( β n ) − q ( α n )) (A.24)so that the full sums are given by G = X g +1 + iY g +1 X g +1 = X + + X − Y g +1 = Y + + Y − (A.25)To the sums X + , Y + , we apply the results derived for case 0 ≤ α , while to the sums X − , Y − ,we apply the results derived for case β g +1 <
0, namely X + < < Y + <
12 ( X + ) + (cid:18) Y + − (cid:19) < X − > < Y − <
12 ( X − ) + (cid:18) Y − − (cid:19) <
14 (A.26)From the fact that X + and X − have opposite sign, and the fact that Y + − / Y − haveopposite sign, it follows immediately that (cid:16) X + + X − (cid:17) + (cid:18) Y + + Y − − (cid:19) < (cid:16) X ± (cid:17) + (cid:18) Y ± − (cid:19) <
14 (A.27)so that | G − ¯ G | − | G | >
0, and thus W >
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