I. INTRODUCTION It has been proposed recently that the large N limit of maximally supersymmetric SUèNè Yang-Mills theory may be described by supergrav

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1 Two-Form Fields and the Gauge Theory Description of Black Holes SLAC-PUB-7767 hep-thè March 1998 Arvind Rajaraman æy Stanford Linear Accelerator Center, Stanford, CA Abstract We calculate the absorption cross section on a black threebrane of two-form perturbations polarized along the brane. The equations are coupled and we decouple them for s-wave perturbations. The Hawking rate is suppressed at low energies, and this is shown to be reæected in the gauge theory by a coupling to a higher dimension operator. Submitted to Physical Review D æ address: arvindra@dormouse.stanford.edu y Supported in part by the Department of Energy under contract no. DE-AC03-76SF00515.

2 I. INTRODUCTION It has been proposed recently that the large N limit of maximally supersymmetric SUèNè Yang-Mills theory may be described by supergravity on an AdS background ë1ë. This proposal was motivated by the agreement between computations of Hawking radiation from black threebranes and the corresponding expectation from gauge theory ë2í5,7,9ë. Based on this conjecture, Gubser, Klebanov and Polyakov ë7ë and Witten ë8ë have given a concrete proposal for how to relate correlation functions in the gauge theory to supergravity computations. In their approach, one calculates the supergravity action in the AdS space subject to certain boundary conditions on the æelds. The boundary conditions are treated as the sources for operators on the boundary. One can then read oæ the correlation function of these operators from the supergravity action. For practical calculations, one would like to know the solutions at linearized level èat leastè for all the bulk æelds. This is the ærst step to being able to compute multipoint correlation functions in the gauge theory. In this note, we shall consider perturbations of two-form potentials which are polarized parallel to the brane. These æelds have coupled equations of motion èas was pointed out in ë4ë.è We are able to decouple these equations in the case of s-wave perturbations, and extract the absorption cross section for quanta of these æelds incident on the black hole. We ænd that the absorption cross-section for these æelds is suppressed at small frequencies relative to minimally coupled scalars. This is somewhat surprising because these scalars are not æxed in the sense of ë10,11ë and therefore are not expected to have suppressed absorption rates. We ænd that the suppression of the absorption rate is reæected in the gauge theory in that these scalars are coupled to higher dimensional operators in the gauge theory, which naturally leads to lower absorption rates. One can formulate a conformally invariant coupling èalong the lines of ë8ëè to describe this interaction. The results are in agreement with the semiclassical calculation. Related issues have been discussed in ë16í34ë. II. THE SEMICLASSICAL ANALYSIS A. The black hole The black hole background is deæned by the metric ds 2 = H,1=2 è,dt 2 + dx a dx a è+h 1=2 èdx 2 i è H =1+ R4 r 4 where a = 1;2;3 labels the coordinates parallel to the brane, i = 4æææ9 labels the coordinates perpendicular to the brane. The four-form æeld strength is è! R 4 F 0123r = H,2 è2è 1 r 5 è1è

3 We are considering waves of two-form potentials. The relevant æeld equations at the linearized level are ë14ë r ç H ççç = r ç F ççç =, ç 2 F çççç ç F 3ç çç ç ç 2 F çççç ç H 3ç çç ç è3è where we have denoted the NSNS two-form æeld strength as H ççç, and the RR two form æeld strength as F ççç. We shall denote the corresponding potentials as B çç and A çç respectively. The above equations show that the perturbations of the two two-form potentials are mixed. In particular, a perturbation of A 12 = æ mixes with perturbations of H 03r. In the case of s-waves, i.e. when there is no angular dependence, these equations can be decoupled, and an equation for æ can be obtained. By symmetry, similar equations can be obtained for A 13 ;A 23 ;B 12 ;B 13 and B 23 : B. Decoupling the equations of motion We start with the equation èwe shall always assume the diagonal form of the metricè 1 p gg rr 0è p ç 2 gg 00 g 33 g rr H 33 0r3 è= F r3ççç F 3ç ççç which gives èassuming everything goes as 0 =,i!è =4F r3012 F 012 è4è èi!èg 00 H 0r3 = ç 4 3ç F r3012 g 11 g 22 g 00 èi!èæ è5è that is, H 0r3 = ç 4 3ç F r3012 g 11 g 22 æ è6è We now turn to the equation 1 p gg 11 g rè p gg rr g 11 g r æè,! 2 g 00 æ=12f 120r3 H 0r3 è7è Using è6è, we can simplify this r èhr r æè +! 2 H 1=2 æ=16 R8 r 5 H æ è8è 2

4 C. Solving the equations We will solve this in various regions. Far from the horizon èr ç Rè, we can set H = 1, and R =0. The equation is then 1 r rèr r æè +! 2 æ=0 Using the standard substitution æ = r,5=2,, we get with the usual solution ë3ë 2 r, 15 ç 4r + 2!2,=0 æ=c 1 r,2 J 2 è!rè+c 2 r,2 N 2 è!rè è9è In the intermediate region èr ç R ç!,1 è, we set! =0. The equation is then with the r èhr r æè=16 R8 r 5 H æ æ=c 3 H+c 4 H,1 è10è Finally, near the horizon èr ç Rè, we approximate The equation then becomes with the solution r@ r èr@ r æè + H = R4 r 4 è! 2 R 4 r 2, 16! è! è!!r 2!R 2 æ=j 4 + in 4 r r æ=0 è11è where we have chosen the solution for an ingoing wave. To match the intermediate solution, we need c 3 =!4 R 4 16,è5è c 4 =,96i çè!rè 4 è12è Matching the intermediate to the outer solution, we get c 1 = ç 8! 2 ç èc 3 + c 4 è c 2 =, è!! 2 R 4 ç èc 3, c 4 è 4 è13è 3

5 The solution for large r tends to c 1 cosè!rè+c 2 sinè!rè. The absorption cross section is then A =1,k c 2+ic 1 c 2, ic 1 k 2 which is easily evaluated to be A = è ç ! è!rè 12 The cross-section is then obtained by the formula è! ç = 32ç2 ç 4! A =! 7 R 12 è14è It may seem odd that we get a cross-section that goes to zero more quickly than! 3 èthe behaviour exhibited by minimal scalarsè, which is reminiscent of the behaviour of æxed scalars and intermediate scalars ë11í13ë. This may be unexpected since the scalar we are considering is not expected to be æxed èin the sense of ë10ëè, since it can take on any value in the black hole background. It nevertheless has a Hawking rate that is suppressed, essentially because in the near horizon region it has an eæective mass term similar to the eæective mass term of æxed scalars. The presence of this eæective mass term is conærmed by the analysis of ë15ë, who have worked out the wave equations of all the supergravity æelds in an AdS background. In the next section, we shall see that the suppression of the Hawking rate is reæected in the gauge theory by the fact that the scalar we are considering couples to a higher dimension operator on the threebrane worldvolume. III. THE GAUGE THEORY ANALYSIS We now turn to the extraction of gauge theory correlators from the absorption amplitudes. This is a straightforward extension of the analysis of ë7,8ë.we shall attempt to clarify the relation between the two procedures. We will focus on the near horizon equation z@ z èz@ z æè + ç! 2 z 2, 16 ç æ=0 è15è where we have deæned z = R2 r We analytically continue to spacelike momenta, in which region the solutions are K 4 è!zè and I 4 è!zè. We keep the solution K 4 è!zè, which is the one which decays exponentially at the horizon z!1. The idea of ë7,8ë is that one solves the above equation for a given choice of boundary conditions for small z, which is taken to be the value of the æeld on the boundary. We then 4

6 treat the boundary æeld as the source for an operator O which lives only on the boundary. The Green's functions of O are then generated by the functional obtained by substituting the full solution of è15è into the supergravity action. In this case, the full solution is K 4 è!zè, which for small z diverges as z,4. Accordingly, we need to specify a cutoæ. The problem is then that the boundary value is highly sensitive to the choice of cutoæ. It is natural, therefore, to associate the boundary æeld not to æ directly, but rather to the boundary value of æ 0 = z 4 æ. This is stable in the sense that if we move the cutoæ from z = z 0 to èsayè z =2z 0,the value of æ 0 does not change drastically. This is the same setup as in section 2.5 of ë8ë. We can then proceed as before, by coupling the æeld æ 0 to a boundary operator O, and extracting the correlation functions of O from the supergravity action. Let us see explicitly how this works for the case of a massive scalar. We shall diæer slightly from the method of ë7ë, in that we shall set ç =1,and explicitly follow all factors of R. The equation of motion is which has the solution In ë7ë, the boundary condition chosen was We shall modify this condition in our case to for small!z, which æxes the solution to be ç ç 1 z z zç, z 2! 2 ç, m 2 ç =0 3 ç = z 2 K ç èwzè ç 2 = m 2 +4 ç ç 1 at z = R ç ç Rç,2 z ç,2 ç = R ç,2! ç z 2 K ç èwzè We now substitute this solution into the supergravity action. As shown in ë7,8ë, this can be reduced to a surface term at the boundary z = R, çç ç ç 1 Iëçë ç R 8 ç@ z 3 z ç R To extract the absorption cross-section, we need the nonanalytic part of Ièçèas in ë5ë, which is provided by the logarithm in the expansion of K ç èzè, è 1 K ç èzè ç 2 n,1,ènèz,ç è1 + æææè+è,è n+1 2 n,èn +1è! ç ç 1 ln 2 z èz ç + æææè where æææ represent higher orders in z. The leading nonanalytic term in è16è then scales as 5 è16è

7 R 8 R 2ç,4 ç 1 z 3 ç è! 2ç z 4 èè!,n z,ç,1 èlnèwzèè!zè ç ç R 2ç+4! 2ç lnèwzè Upon Fourier transforming to position space, we ænd that the two point function of the boundary operator scales as hoèxèoè0èi ç@ 2ç 1 x 4 indicating that the dimension of the operator is æ=2+ç=2+ p 4+m 2 è17è in exact agreement with ë8ë. The reason that the coupling is still conformal is that the boundary value ç 0 has now acquired a dimension. In this case, this can be seen in that if we shift the position of the boundary from z = R to z = çr, the relation between ç and ç 0 changes from to ç 0 = r ç,2 ç ç 0 = ç ç,2 r ç,2 ç Since ç itself was a canonical scalar æeld, ç 0 is not èas otherwise the relation would not change under this rescaling.è In fact, it is a conformal density of dimension 2, ç ë8ë. The coupling ç 0 O therefore has dimension 4, and is a conformally invariant term. This is in spite of the fact that we haveintroduced a higher dimension operator which will result in a suppressed Hawking rate. It is simple to repeat this analysis for massive p-forms. The equation of motion is with the solution ç ç 1 z z zç èpè + z 2! 2 ç èpè, m 2 ç èpè =0 3,2p ç èpè = R ç+p,2! ç z 2,p K ç èwzè ç 2 = m 2 +è2,pè 2 è18è where we have normalized the solution to go as ç èpè ç Rç+p,2 z ç+p,2 for small!z. We can reduce the action to a surface term as before! è ç z ç èpè Iëç èpè ë è "è z 2p,3 R 2p,8 R è19è Approximating the behaviour of ç èpè at small z as before, we ænd the leading nonanalytic term 6

8 I ç z2p,3 R 2p,8 èr2ç+2p,4! 2ç z 4,2p èè!,ç z,ç,1 èè! ç z ç lnèwzèè ç R 2ç,4! 2ç lnèwrè Hence the dimension of the operator on the boundary is æ=2+ç=2+ q m 2 +è2,pè 2 è20è which is the correct result ë8,25ë, as the m 2 refers not to the eigenvalue of the Laplacian, but to the eigenvalue of the Maxwell operator è ~m 2 in ë25ë.è There is however a puzzle in the comparison of the gauge theory to the semiclassical calculation. The problem is that in è19è, if we explicitly substitute the solution è18è, the leading nonanalytic term cancels! This is because I z ç 2 and in ç 2, the leading coeæcient of lnè!zè is z 0. Hence, upon diæerentiation, we ænd that the nonanalytic term lnè!è disappears. Another way ofsaying this is that if we treat p as a continuous variable, the coeæcient of the action is proportional to èp,2è and hence vanishes for two-forms. It is clear that the true answer in the gauge theory cannot be zero, since the absorption cross section is nonzero. Also, the procedure of ë8ë does not seem to give zero for this case. This may be a problem of our normalization. If one treats p as continuous, it is possible that the normalization of ç èpè should be taken to involve inverse powers of èp, 2è which will cancel the apparent zero in the above expression. Other possibilities may exist. We will treat this as an overall coeæcient in the correlation function that we cannot determine, since we cannot normalize the operators unambiguously. In particular, in the case we are considering, we have p =2,and m 2 = 16, which yields æ=6.hence we have a coupling to a dimension 6 operator. The exact form of this operator has been discussed in more detail in ë34ë. We also ænd that I ç R 12! 8 lnèwrè and since the cross section is related to the discontinuity of the above function near! =0, we ænd èfrom ë5ëè ç ç i! R12! 8 èlnè,s + iæè, lnès, iæèè ç! 7 R 12 which agrees with è14è. We therefore get results in agreement with the semiclassical calculation. The exact coeæcient is, however, undetermined. We emphasize that this is because the exact normalization of the operators has not been æxed. It may be necessary to calculate a three-point correlation function in order to resolve the ambiguity. In conclusion, we have extracted the absorption rate for a two-form æeld incident on a black threebrane. We have shown that the Hawking rate is proportional to! 7, a fact which follows from a coupling to a dimension 6 operator on the brane world volume. We havethus found that a non-æxed scalar can also have a suppressed cross-section. 7

9 IV. ACKNOWLEDGEMENTS We would like to thank I. Klebanov and J. Rahmfeld for discussions. This work was supported in part by the Department of Energy under contract no. DE- AC03-76SF

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