arXiv:hep-th/0612159v1 15 Dec 2006NSF-KITP-06-124hep-th/0612159Anyonic strings and membranes in AdS spaceand dual Aharonov-Bohm effectsSean A. HartnollKITP, University of California Santa BarbaraCA 93106,

[email protected] is observed that strings in AdS5× S5and membranes in AdS7× S4exhibit longrange phase interactions. Two well separated membranes dragged around one anotherin AdS acquire phases of 2π/N. The same phases are acquired by a well separated Fand D string dragged around one another. The phases are shown to correspond to boththe standard and a novel type of Aharonov-Bohm effect in the dual field theory.1IntroductionIt is by now well established that point particles in 2+1 dimensions may have fractional,or anyonic, statistics [1, 2, 3, 4]. These are particles that do not obey either Bose-Einsteinor Fermi-Dirac statistics. Under exchange of two identical anyons, the wavefunction of thesystem does not get multiplied by ±1, but may rather change by an arbitrary, fixed, phase.A physical example of anyons are the quasiparticle or quasihole excitations of a fractionalquantum hall fluid [5].Anyons may be physically understood as particles with a long range phase interaction.In 2+1 dimensions a particle may be charged under an abelian Chern-Simons gauge poten-tial. The Chern-Simons interaction attaches a magnetic flux to all electric point charges.Therefore when two such charges are dragged around one another, at arbitrarily large dis-tances, they acquire a phase through the Aharonov-Bohm effect.The possibility of anyonic phases in two spatial dimensions is directly associated withthe topology of the two particle configuration space. In particular, H1(R2\ {0}) = Z, sothe path of a particle that is dragged around another is topologically nontrivial. In higherdimensions, these paths are homologically trivial and hence anyonic phases are not possiblefor point particles. However, higher dimensional objects such as strings and membranesdo have nontrivial configuration spaces in higher dimensions. In particular, H2(R4\ R) =H3(R6\ R2) = Z, which implies that strings in 4+1 dimensions and membranes in 6+1dimensions can describe topologically nontrivial cycles as they are dragged around oneanother. Some previous work on the configuration space of strings in 3+1 dimensions maybe found in [6, 7]. The 3+1 dimensional analogue of the physics to be considered here wouldinvolve a string linking a point source, c.f. [8].The next observation is that two of the most basic backgrounds of string and M theory,the AdS5×S5solution of type IIB string theory and the AdS7×S4solution of M theory, re-alise the potential for anyonic behaviour of strings and membranes. The strings/membranesneed to be well separated relative to the size of the sphere in the background, so thatthe physics is effectively five/seven dimensional, respectively.The two ingredients thatmake nontrivial phases possible are firstly the nonlinear Chern-Simons like terms in thesupergravity actions and secondly the presence of a background flux in the solution. Theresulting mechanism is essentially identical to that for point charges in 2+1 dimensionalChern-Simons theory.These AdS backgrounds are of special interest because they are supergravity duals ofconformal field theories [9].After explicitly exhibiting the advertised long range phase1interactions in these backgrounds below, we will show that, in the case when one of thestrings or membranes reaches the boundary of AdS, then the bulk anyonic behaviour has adual interpretation as a nonabelian Aharonov-Bohm effect. If neither of the objets reach theboundary, we show how the phase may be dually understood as a novel type of Aharonov-Bohm effect involving intersecting flux tubes.2Anyonic membranes in AdS7× S4Start with the case of a single M2 brane coupled to eleven dimensional supergravity. Thephysics we are studying only depends upon the action for the three form C field. LettingG = dC, the required action isSC field= −1 4κ211Z ? G ∧ ⋆G +1 3C ∧ G ∧ G? + TM2ZM2C .(1)Using 2κ211= (2π)8l911and T−1M2= (2π)2l311one obtains the equations of motiond ⋆ G +1 2G ∧ G − (2πl11)6δ(x) = 0.(2)Here l11is the eleven dimensional Planck length and δ(x) is the Poincar´ e dual eight formto the worldvolume of the membrane. Recall that the Poincar´ e dual is defined by ZM2C =Z C ∧ δ(x),(3)for any C. The second integral here is over the whole spacetime.We wish to consider the effect of a single membrane in the AdS7×S4background. Thisbackground has fluxG(0)= 3πNl311volS4,(4)where volS4is the volume form on a unit radius four sphere. The effect of the membraneis to shift this background flux by a small amount. Let us set G → G(0)+ G and work tolinearised order in G. The equation of motion becomesd ⋆ G + N(2πl11)33 8π2volS4∧ G − (2πl11)6δ(x) = 0.(5)Place this membrane in the AdS7part of the geometry.Now consider a second membrane in the same background. Let us adiabatically drag thismembrane along a closed spatial curve that is within AdS7and encircles the first membrane,ending up back where it started. This is well defined as the first membrane is occupyingtwo spatial dimensions and is therefore a point in the transverse four spatial dimensions2Figure 1: M2 denotes the first membrane and ∂Σ is the worldvolume traced out by thesecond membrane.in AdS7, that can be encircled by a three volume. Denote by ∂Σ the worldvolume of thissecond membrane. This process is illustrated in figure 1.More precisely, a spatial slice of AdS7has topology R6. Removing the space occupiedby the first membrane gives topology R6\ R2. However H3(R6\ R2) = Z, and it is thisnontrivial three cycle we are dragging the second membrane around.The phase acquired by the partition function of the second membrane upon completionof the closed loop is computed as follows∆ΦM=2π (2πl11)3Z∂ΣC=2π (2πl11)33 8π2Z∂Σ×S4C ∧ volS4[asR → ∞]=2π (2πl11)33 8π2ZΣ×S4G ∧ volS4=−2π N(2πl11)6ZΣ×S4?d ⋆ G − (2πl 11)6δ(x)?=2π N−2π N(2πl11)6Z∂Σ×S4⋆G.(6)In the second step we have required that the minimum spatial distance between the twomembranes, R, be large.At large distances, the C field in AdS7sourced by the firstmembrane is independent of the coordinates on the S4. This fact permits us to wedge Cwith volS4. The higher harmonics of the C field decay exponentially at spatial distancesbeyond the size of the S4, which is the AdS scale, due to the Laplacian term d ⋆ dC in(5). In the appendix we have exhibited this exponential falloffexplicitly by solving a toy3model for (5): a point charge in R1,2×S1with analogous Chern-Simons like couplings. Thisintuition is not confined to flat space. The higher harmonics acquire masses in AdS7, andmassive fields on AdS decay exponentially in geodesic distance, see e.g. [10, 11, 12].In order to evaluate the expression for the phase (6), we should solve the equation ofmotion (5) for G. However, we can immediately note that at long distances from the sourcemembrane, the topological mass term in (5) is important and the field strength dies offexponentially. Hence the flux term in the last line of (6) is negligible. This falloffin Goccurs even for the homogeneous mode on S4(the homogeneous C that survives at longdistance has G = dC = 0). Thus we obtain the phase∆ΦM=2π N[asR → ∞].(7)Rephrasing this result in terms of membrane exchange, in which the second membrane isonly taken ‘half way’ around the first, and adding together the phases acquired by thetwo membranes, we have found that well separated membranes in AdS7× S4obey anyonstatistics with angle 2π/N.At short distances the full eleven dimensional geometry must be considered and there isno effective linking of membranes. Furthermore, the derivative term in (5) will be important.It may seem surprising that the phase depends on the distance between the membranes.However, the same behaviour occurs in Maxwell-Chern-Simons theory in three dimensions[13]. The effect of the Maxwell term is to give the magnetic flux tubes attached to theanyons a finite width. When the charges are close enough together for these to overlap,the fractional statistics is lost. The characteristic feature of anyons in any case is the long(infinite) range phase interaction, independently of short distance physics.To gain intuition for this system, it is useful to solve the equation for a topologicallymassive four form field strength on R1,6. This is done in the appendix. The exponentialdecay of the flux at large distances may be seen very explicitly.The propagator for atopologically massive form in AdS has been computed in [14] and exhibits exponentialdecay with geodesic distance.3Long range phase interaction for strings in AdS5× S5Consider a single D or F string coupled to type IIB supergravity. We are interested in aneffect due to the two form potentials in the theory. The action for these forms, in Einstein4frame, isSB2,C2=−1 4κ210Z ? e−ΦH3∧ ⋆H3+ eΦ˜F3∧ ⋆˜F3+1 2˜F5∧ ⋆˜F5+ C4∧ H3∧ F3?−TF1ZF1B2− µ1ZD1(C2− CB2) ,(8)where H3= dB2, F3= dC2,˜F3= F3− CH3and˜F5= F5− C2∧ H3. The self dualitycondition˜F5= ⋆˜F5is imposed on the equations of motion. We have included contributionsfrom both an F and D string to the action. To start with, we are interested in one or theother.The equations of motion for the two form potentials C2and B2are found to bed ⋆? eΦ˜F3? − F5∧ H3− 2κ210µ1δ(xD1) = 0,(9)andd ⋆? e−ΦH3− CeΦ˜F3? + F5∧ F3− 2κ210TF1δ(xF1) + 2κ2 10µ1Cδ(xD1) = 0.(10)In these expressions, δ(x) denotes the Poincar´ e dual eight form to the string worldvolume.We want to consider the effect of a single D or F string in the AdS5× S5background.This background has five form fluxF(0) 5= 16πNl4s(volS5+ ⋆volS5) ,(11)and furthermore has constant dilaton and axion: eΦ= g and C = θ/2π. Here volS5is thevolume form on an S5of unit radius and lsis the string length. The string source introducesa small amount of three form flux which we describe by linearising the equations of motionabove. Using TF1= µ1= 1/(2πl2s) and 2κ210= (2π)7l8s, and writing the equations in a more symmetric way, givesgd ⋆˜F3−N(2πls)4 π3(1 + ⋆)volS5∧ H3− (2πls)6δ(xD1) = 0,(12)and 1 gd ⋆ H3+N(2πls)4 π3(1 + ⋆)volS5∧˜F3− (2πls)6δ(xF1) = 0.(13)Note that here ˜F3= F3−θ 2πH3.(14)From the structure of the equations of motion (12) and (13), we see that we can obtainnontrivial phases for strings in AdS5in the same way as we did for membranes in AdS7.This will occur if we drag an F string around a D string, or vice versa. However, there is5no phase associated with dragging an F string about another F string or a D string aboutanother D string. For two D strings for instance, we can compute the phase as follows∆ΦD=−2π (2πls)2Z∂Σ? C2−θ 2πB2?=−2π (2πls)21 π3ZΣ×S5˜F3∧ volS5[asR → ∞]=−2π Ng(2πls)6Z∂Σ×S5⋆H3.(15)We have used the same notation as previously for membranes.Now ∂Σ is a two cyclesurrounding the first D string in AdS5. This is well defined as H2(R4\ R) = Z. Unlike theprevious case of membranes, this flux does not decay exponentially at large distances butis instead identically zero. This can be seen by considering the electric and magnetic partsH3= E ∧√−g ttdt + B and˜F3=˜E ∧√−g ttdt +˜B, where t is a static time coordinate forAdS. From the equations of motion we have that1 gd ⋆9E −N(2πls)4 π3(1 + ⋆)volS5∧˜B=0,gd ⋆9√−g tt˜B +N(2πls)4 π3(1 + ⋆)volS5∧√−g ttE=0.(16)There are no source terms in these equations, which are therefore solved by E =˜B = 0. Itfollows from (15) that ∆ΦD= 0.As for the membranes, more complete intuition for the coupled equations of motion(12) and (13) may be gained from solving the equations for similarly coupled topologicallymassive three form field strengths on R1,4. This is done in the appendix.Considering (p,q) strings does not alter the situation. The relative minus sign betweenthe topological mass terms in (12) and (13) means that the effects of the F and D chargescancel and there is no long range phase interaction. Alternatively, this follows from SL(2,Z)duality of the D string results. Therefore, the various strings in AdS5do not have anyonstatistics. However, there is an infinite range phase interaction between F and D strings.In this sense we might call these strings anyonic. The phase shift in the partition functionof a D string dragged around an F string is∆ΦF−D=−2π (2πls)21 π3ZΣ×S5˜F3∧ volS5[asR → ∞]=2π N−2π gN(2πls)6Z∂Σ×S5⋆H3,(17)where ∂Σ is as usual the worldvolume swept out by the D string. The electric flux contri-bution in the previous equation is determined from (16) except that now there is a delta6function source term due to the F string in the background. As for the case of membranes,the topological mass terms imply that the flux decays exponentially at large distances giving∆ΦF−D=2π N[asR → ∞].(18)4Dual field theory implicationsThe various membranes and strings we have discussed are dual to specific states (operators)in the dual conformal field theories. The F and D strings are dual to electric and magneticflux tubes, respectively [15]. The M2 branes are dual to interesting analogous extendedfield configurations in the six dimensional (2,0) theory.A simple case to map to the dual theory is when one of the strings or membranes extendsto the AdS boundary. In this case the endpoint of the string or membrane is an externalpoint or string charge, respectively, in the dual theory. Consider the case of an F string thatextends to the boundary. Dragging this string around a D string in the bulk is mapped ontoan external electric charge that is dragged around a magnetic flux tube on the boundary.Such a charge will acquire a phase due to the (nonabelian) Aharonov-Bohm effect.In fact, we can recover precisely the phase we found from the bulk computation. Thetotal flux along the magnetic flux tube is 2π, which follows from the Dirac quantisationcondition. However, this flux is in the (decoupled) diagonal U(1) of U(N), so we can writeit as 2π/N times the identity N ×N matrix. The external electric charge is in some specificU(1) of SU(N) and therefore is only sensitive to one component of the flux. Thus from theAharonov-Bohm effect we obtain the phase 2π/N, in agreement with the bulk computation.The Aharonov-Bohm setup just described is illustrated on the left hand side of figure 2.Note that the string needs to return to the boundary as it has nowhere to end in the AdSbulk. The figure depicts an F string that extends to the boundary being dragged around aD string in the bulk. Note also that the surface linking the D string is not strictly closed, asit does not close offat the boundary. However, this finite sized ‘hole’ is infinitely far awayfrom the source and no flux escapes through it.The right hand side of figure 2 illustrates the case in which the F string does notextend to infinity. Projecting the motion onto the boundary, this implies that the magneticand electric flux tubes intersect twice, as one is dragged through the other. We will nowunderstand why this process picks up a phase.The first step is to note that an infinitessimally thin electric flux tube dragged