Brane backreactions and the Fischler-Susskind mechanism in conformal field theory
aa r X i v : . [ h e p - t h ] S e p Brane backreactions and the Fischler-Susskindmechanism in conformal field theory
Christoph A. Keller ∗ Institut f¨ur Theoretische Physik, ETH Z¨urichCH-8093 Z¨urich, Switzerland
Abstract
The backreaction of D-branes on closed string moduli is studied in perturbed conformalfield theory. To this end we analyse the divergences in the modular integral of the annu-lus diagram. By the Fischler-Susskind mechanism, these divergences lead to additionalterms in the bulk renormalisation group equations. We derive explicit expressions forthese backreaction terms, and follow the resulting renormalisation group flow in severalexamples, finding agreement with geometric expectations. ∗ E-mail: [email protected]
Introduction
A lot of recent work in string theory has dealt with the question of moduli and modulistabilisation in realistic compactifications. In such setups two kinds of moduli appear.Closed string moduli correspond to deformations of the bulk theory, i.e. in geometriclanguage to deformations of the compactification manifold. Open string moduli on theother hand correspond to deformations of the branes of the configuration.String compactification can also be considered in the framework of two dimensional confor-mal field theory. The compactification is then no longer given by a Calabi-Yau manifold,but by a worldsheet CFT of the correct central charge. The branes of such a configura-tion are described by conformal boundary conditions. Closed string moduli are given byexactly marginal bulk operators, open string moduli by exactly marginal boundary opera-tors, and the theory is deformed by inserting such integrated operators in the correlators.Arguably the CFT point of view is more fundamental, as it includes all α ′ corrections. Onthe other hand only for very few geometric configurations the corresponding worldsheetCFT is known explicitly.To determine the moduli space of the theory, one needs to find all exactly marginaloperators. A marginal operator is exactly marginal if it remains marginal in the perturbedtheory, or, to put it another way, if it does not run under the renormalisation group flow.Criteria for this have been worked out for bulk [1] and boundary operators [2].More recently, [3] considered the interplay between bulk and boundary operators. Inparticular, renormalisation group flow equations were derived which describe the effectsof bulk perturbations on the boundary. These equations describe how the open stringmoduli space changes as bulk perturbations are turned on (see also [4] for a discussionof this question). More generally, they show that the boundary conditions flow to afixed point which is compatible with the new, perturbed bulk theory. The bulk theory,however, remains fixed and is not affected by the boundary conditions — the brane doesnot backreact on the bulk.The aim of this paper is to extend the RG equations of [3] to include the backreaction ofbranes on the bulk theory. The idea for the underlying mechanism goes back to [5, 6, 7,8, 9]: in string theory, to calculate amplitudes one considers not only the disk diagram,but also diagrams of higher genus. The total amplitude is obtained by summing over alltopologies and integrating over the moduli of the conformal structure of the diagrams.This integration can lead to new divergences at the boundary of the moduli space M , i.e. when the surface degenerates. More precisely, the spectrum of the theory may containtadpoles, i.e. massless modes, which give logarithmic divergences when integrated over M . According to [7, 8], these can be absorbed by a suitable shift of the coupling constantsin lower genus diagrams, thus contributing to the RG flow of the bulk couplings. Since thenature of the tadpoles depends on the boundary condition that is imposed, this describesthe backreaction of the brane on the bulk.We show that this prescription works for the annulus diagram, i.e. that the tadpoledivergences can be compensated by local counterterms on the disk diagram, leading toadditional terms in the bulk RG equations of [3]. The brane backreaction can thus beincorporated quite naturally in the language of renormalisation group flows.The RG equations so obtained can be used to study various examples. In many cases,we know already from geometric considerations how the brane should deform the bulk2heory, so that we can compare our results. For instance, we expect that a D1-branewrapping a circle should shrink its radius. This is confirmed by the RG analysis. Inother, more complicated examples we also find agreement between the RG analysis andgeometric expectations or supergravity calculations.This paper is organised as follows. In section 2 we first rederive the bulk-boundary RGequations of [3] using a different regularisation scheme which is more suitable for furtheranalysis. We then derive the backreaction term to first order in the string couplingconstant g s by analysing divergences of the annulus diagram. In section 3 we apply theextended RG equations to the free boson and WZW models. Section 4 discusses bosonicstring theory in flat space and its relation to supergravity solutions. Finally, section 5contains our conclusions. Let us first derive the renormalisation group equations on the disk [10, 3]. Consider thepartition function h e − S i , where S is the perturbed action, S = S ∗ − ∆ S = S ∗ − X i λ i ℓ h φi − Z φ i ( z ) d z − X j µ j ℓ h ψj − Z ψ j ( x ) dx . (2.1)We have introduced the length scale ℓ to keep the coupling constants dimensionless.Expanding h e − S ∗ +∆ S i in powers of λ i and µ j gives terms of the form λ l · · · µ m · · · l ! · · · m ! · · · Y i ℓ ( h φi − l i Y j ℓ ( h ψj − m j × Z h φ ( z ) φ ( z ) · · · φ ( z ) · · · ψ ( x ) · · · i Y d z ik Y dx jk . (2.2)Here the bulk fields φ i are integrated over the entire disk, and the boundary fields ψ j overits boundary. The disk has the conformal symmetry group SU (1 , dµ must transform with conformal weight ( − , −
1) under such transformations,so that integrals of marginal (1 ,
1) fields R dµφ (1 , are invariant. Clearly, d z satisfiesthis property. Since we can use SU (1 ,
1) to map any point to 0, it follows that up to aconstant factor this is the only possible measure.Because of the symmetry group, the integrals in (2.2) are infinite. To render them finite,we use SU (1 ,
1) to fix the position of one bulk and one boundary insertion. Alternatively,we can (formally) divide by the volume of SU (1 , a around all operators. 3nstead we will use a scheme which resembles dimensional regularisation. To evaluatediverging integrals, we change the conformal dimension of the fields involved to such valuesthat the integral converges, and evaluate the original integral by analytic continuation.One motivation for using this scheme comes from the spacetime interpretation of thedivergences that will show up in the modular integrals: they can be interpreted as infrareddivergences due to massless modes, so that a natural regularisation is to introduce asmall mass term. In the worldsheet theory, this corresponds to a shift of the conformaldimension of the field. From a more technical point of view, it is favourable to keepconformal covariance of all expressions, which is destroyed if we cut out small disks.Let us shift the conformal weight of boundary fields as h ψ h ψ − ǫ , and that of bulkfields as h φ h φ − ǫ . † As an example for how the scheme works, consider two marginalbulk fields φ i , φ j that come close to each other to produce another marginal field φ k , λ i ℓ − ǫ λ j ℓ − ǫ φ i ( z ) φ j (0) ∼ λ i λ j ℓ − ǫ φ k (0) C ijk | z | h i + h j − h k = λ i λ j ℓ − ǫ φ k (0) C ijk | z | − ǫ . (2.3)For simplicity, we have fixed the position of φ j to 0. We perform the d z integral up tosome IR cutoff L to obtain λ i λ j ℓ − ǫ φ k (0) 2 πC ijk ℓ − ǫ ǫ L ǫ . (2.4)We have pulled out a factor ℓ − ǫ which will be absorbed in the shift of λ k (see (2.6)). Thesecond factor ℓ − ǫ gives ℓ − ǫ ǫ L ǫ = 12 ǫ − log ℓ + log L + O ( ǫ ) . (2.5)In the limit ǫ → ∞ , only the second term gives a dependence on ℓ which contributes to theRG flow. We see that the regularisation scheme has introduced an implicit dependenceof the integral on ℓ . As h e − S i must be independent of ℓ , we must compensate a shift inlog ℓ by shifting λ i and µ j . A combinatorial analysis shows that the shift needed is λ k ℓ − ǫ λ k ℓ − ǫ + λ i λ j ℓ − ǫ πC ijk · log ℓ . (2.6)In a similar way, we treat the other types of divergences. The resulting renormalisationgroup equations are [3]˙ λ k = (2 − h φ k ) λ k + πC ijk λ i λ j + O ( λ ) , (2.7)˙ µ k = (1 − h ψ k ) µ k + 12 B ik λ i + D ijk µ i µ j + O ( µλ, µ , λ ) , (2.8)where the dot indicates a derivative with respect to the flow parameter t = log ℓ . Toobtain higher order terms in µ and λ , one would have to analyse the situation when threeor more fields come close to each other. In the following, we shall never consider suchterms. † Note that for bulk fields in a theory with boundary h = h L + h R . φφ φφ q → χ (0) φφ φφ Figure 1: Divergences of the annulus diagram
To calculate amplitudes in string theory, we have to take into account higher genus dia-grams as well. For simplicity assume that there is only one type of field φ in our theory. Asbefore, a string amplitude F can be expanded in powers of λ , F = P n λ n F n . Each term F n itself contains contributions from all topologically different diagrams with n insertionsof φ . Moreover, for a given topology we must integrate over all conformal structures,parametrised by modular parameters t i . In full, F n = X k g χ k s Z M k dt i F kn ( t i ) , (2.9)where g s is the string coupling constant and χ k is the Euler characteristic of the diagram F k . Integration over the moduli space M k leads to new divergences due to marginal andrelevant modes in the spectrum of the theory. The divergences have to be regularised,and we must try to compensate for them by introducing counterterms on diagrams oflower genus. These ℓ -dependent terms then give the the backreaction terms in the bulkRG equations. We will now calculate the backreaction terms caused by the annulus diagram A n = F n .The annulus has a single real modular parameter q , its inner radius. The integral over q produces a divergence for q →
0. In this case there is an intuitive way to see how thecounterterm on the disk arises, as shown in figure 1: the divergent part of the annulusdiagram with n integrated insertions corresponds to a disk diagram with an additionalfield χ (0) inserted. A shift λ λ + δλ on the disk diagram D n +1 = F n +1 can thuscompensate the divergence. The corresponding term is of order g s .Although we will only calculate the term of order g s λ , some comments on terms of higherorder in λ are necessary. The analysis on the disk showed that λ terms are produced bytwo fields approaching each other, and that higher order terms appear when n fields comeclose together. In the situation here, higher order corrections arise when additional fieldsmove close to the new field produced on the disk or to the boundary of the annulus. If5or instance a single φ moves close to the centre of the annulus A n , the divergence can becompensated by the disk diagram D n , which produces a contribution of order g s λ . As weare only interested in the lowest order correction, we can thus subtract divergences whicharise from fields moving close to each other or to the boundary.Note that the symmetry group of the annulus is only U (1) — we can fix the position of oneboundary insertion, or alternatively we can divide the amplitude by 2 π . This also meansthat unlike on the disk, the conformal symmetry no longer uniquely fixes the integrationmeasure. Nevertheless, the correct measure is still d z , see e.g. [11].For a given radius q , the integrated n -point amplitude of the annulus is given by A n ( q ) = 1 π n Y i =1 Z q d z i hh B || φ ( z ) . . . φ ( z n ) q L +¯ L − || B ii . (2.10)For simplicity, we have only included one type of marginal field φ . As usual, hh B || isthe boundary state at the outer radius 1. To obtain the boundary state at the innerradius, we transport || B ii to the inner radius q using the propagator π − q L +¯ L − , whosenormalisation is fixed by the construction of the boundary states. By inserting a completeset of states, we expand the boundary state in a sum of fields inserted at the point 0. Theaction of the propagator then gives π − q L +¯ L − || B ii = π − X i q h i +¯ h i − | φ i ih φ i || B ii . (2.11)Here h φ i || B ii is the disk one-point function with φ i sitting at the point 0. Integrating(2.10) over its moduli space using the measure q − dq , we see from (2.11) that divergencesarise for q → h i = ¯ h i ≤
1. In a supersymmetric setup, we expectno relevant, i.e. tachyonic fields. In the bosonic theories we will consider, the only suchfield is usually the vacuum h = ¯ h = 0. The vacuum only changes overall normalisations,so that we will ignore it in what follows. The only divergences are then due to marginalfields h i = ¯ h i = 1 − ǫ . Their contribution is || B ( q ) ii ≃ q − ǫ π X i h φ i || B ii φ i (0) . (2.12)For the moment, let us assume that there are no integrated bulk insertions. The integralof (2.12) over moduli space converges if ǫ <
0, and we will use its analytic continuation, Z q − dq || B ( q ) ii = − π ǫ X i h φ i || B ii φ i (0) . (2.13)The pole in ǫ will then contribute to the RG equations as in (2.4).If the diagram contains integrated bulk insertions, the comparison is a bit more subtle:in the disk diagram, the additional bulk insertions are integrated over the entire disk,whereas on the annulus they are only integrated up to the inner radius q . The divergentcontribution of the tadpole, however, comes from the limit q →
0. We can thus concentrateon annulus diagrams where q < | ǫ | . Indeed, Z | ǫ | dqq − − ǫ = − ǫ (1 − e − ǫ ln | ǫ | ) = O (ln | ǫ | ) (2.14)6s only a subleading contribution compared to (2.13). We claim then that to lowest orderin λ we can rewrite the annular integral as Z | ǫ | dq Z q d z i h . . . i = Z | ǫ | d z i Z | ǫ | dq h . . . i + O ( ǫ ) . (2.15)This holds because we can estimate the contribution of the fields φ integrated over thesmall disk of radius | ǫ | : since we only calculate the lowest order term in λ , we subtractall singular terms in φ . The remaining expression is then bounded by some constant B ,and we can estimate its contribution as ≤ πǫ B . A similar argument shows that we cancut out the same small disk in the disk diagram without changing the result. This showsthat we can compare annulus diagrams with disk diagrams even if they contain integratedinsertions.So far, the fields φ i introduced by the tadpoles are inserted at the point z = 0. In order tobe able to compensate them with a disk diagram, we need to rewrite them as integratedinsertions. To do this, we use the fact that the disk has a larger symmetry group thanthe annulus. Consider the disk diagram with n integrated fields φ ( z i ) and one additionalfield χ ( z ), each of them marginal. We can use part of the symmetry group SU (1 ,
1) to fixthe position of χ to 0. In particular, for each z choose f z ∈ SU (1 ,
1) such that f z ( z ) = 0.Defining ˆ z i = f z ( z i ), conformal covariance tells us that the z i integral changes as Z d z i φ ( z i ) → Z d ˆ z i (cid:12)(cid:12)(cid:12)(cid:12) ∂z i ∂ ˆ z i (cid:12)(cid:12)(cid:12)(cid:12) − ǫ φ (ˆ z i ) = Z d ˆ z i φ (ˆ z i ) + O ( ǫ ) . (2.16)Up to terms of order ǫ , the resulting integral is thus independent of z , and the additionalfield χ ( z ) is fixed at the position z = 0. Formally, we can write this manipulation as1 | SU (1 , | Z d z Z d z i h χ ( z ) φ ( z ) . . . i = 1 | U (1) | Z d ˆ z i h χ (0) φ (ˆ z ) . . . i + O ( ǫ ) , (2.17)where | G | denotes the volume of the respective symmetry group. On the right handside of (2.17), we divide by | U (1) | because we still have not fixed the entire symmetry:after choosing f z , we can always rotate the disk around its centre. This remaining U (1)symmetry however is exactly the symmetry group of the annulus, so that the right handside of (2.17) is the standard annulus diagram with one fixed insertion.The upshot of this analysis is that the divergent part of A n has the same form as D n +1 ,so that it can be compensated by introducing a counterterm on the disk diagram. Asbefore, we need to split off a factor ℓ − ǫ to be included in λ . The annulus contribution tothe disk diagram is thus − ℓ − ǫ g s π ℓ ǫ ǫ h φ i || B ii Z d z φ i ( z ) = − ℓ − ǫ g s π (cid:18) ǫ + log ℓ + O ( ǫ ) (cid:19) h φ i || B ii Z d z φ i ( z )(2.18)for each marginal field φ i . The usual combinatorial analysis shows that this can becompensated by shifting the coupling constant λ i .Putting everything together we obtain the modified bulk RG equations˙ λ k = (2 − h φ k ) λ k + g s π h φ k || B ii + πC ijk λ i λ j + O ( g s λ, λ , g s ) . (2.19)7 WZW models and the free boson
We now apply equation (2.19) to some examples. First we consider the free boson com-pactified on a circle, subject to Neumann or Dirichlet boundary conditions. Then we turnto Wess-Zumino-Witten models based on compact Lie groups. These models and theirboundary states are very well understood and can be interpreted geometrically. We canthus check RG flow results against geometric expectations.
Let X ( z, ¯ z ) be the free boson compactified on a circle of radius R , X ∼ X + 2 πR . Itsaction is given by S = 12 π Z d z ∂X ¯ ∂X . (3.1)Neumann and Dirichlet boundary conditions are given by identifying on the real axis z = ¯ z ∂X = ¯ ∂X (Neumann) and ∂X = − ¯ ∂X (Dirichlet).As usual, we can switch to the closed string picture by mapping the upper half-plane tothe disk. The boundary condition is then described by the boundary states || N ii and || D ii , respectively.The ground states of the system are parametrised by momentum and winding numbers n, w ∈ Z such that ( p L , p R ) = (cid:16) n R + wR, n R − wR (cid:17) , (3.2)with conformal weight given by ( p L , p R ). At a generic radius R , the only marginaloperator is ∂X ¯ ∂X . Its one-point function is given by h ∂X ¯ ∂X || N ii = 1 and h ∂X ¯ ∂X || D ii = − . (3.3)We will also have to deal with the relevant fields that are present in theory.Let us analyse the Neumann case first. The one-point function vanishes unless p L = − p R , i.e. n = 0, so that only pure winding modes couple. If we take R big enough, (3.2)shows that all these modes become irrelevant. It is thus sufficient to only consider theperturbation by ∂X ¯ ∂X , S = 12 π Z d z ∂X ¯ ∂X − λ Z d z ∂X ¯ ∂X . (3.4)We see that (2.19) yields ˙ λ = g s /π >
0. An increase in λ means that the circle shrinks,as can be seen from (3.4): to maintain the correct normalisation of the action, we haveto introduce rescaled fields X ′ = √ − πλX , which satisfy X ′ ∼ X ′ + 2 πR ′ = X ′ +2 πR √ − πλ .This shows that a Neumann brane that wraps the circle shrinks its radius. Similar rea-soning shows that the D0 brane given by || D ii increases the radius of the circle.When R becomes of the order of the self-dual radius R = 1 / √
2, new relevant andmarginal fields appear, and the above analysis breaks down. To analyse this case, wewill use the fact that the free boson at the self-dual radius is equivalent to the SU (2)Wess-Zumino-Witten-model at level 1. We therefore turn our attention to WZW-models.8 .2 Renormalisation group flows in general WZW models Wess-Zumino-Witten models are often described as σ -models on a group manifold ofa Lie group G [12]. A different, more algebraic approach is to define them via theiroperator content and correlation functions. For the moment, we will use this more abstractformulation, before changing to a more geometric picture in the next section.The currents of the WZW model of a Lie group G at level k correspond to elements ofthe Lie algebra g of G and satisfy the operator product expansion J a ( z ) J b ( w ) ∼ kδ ab ( z − w ) + if abc J c ( w )( z − w ) , (3.5)where f abc are the structure constants of g . The marginal fields of the theory are givenby all possible combinations J a ¯ J b of left-moving and right-moving currents. We considerbranes that preserve the affine symmetry up to conjugation by g ∈ G [13, 14, 15]. In theclosed string picture this means that the boundary state || B ii has to satisfy the gluingcondition ( gJ am g − + ¯ J a − m ) || B ii = 0 , (3.6)whereas in the open string picture the left and right moving currents are identified at theboundary as gJ a ( z ) g − = ¯ J a (¯ z ) for z = ¯ z . (3.7)The one-point function is best evaluated in the open string picture and gives [16, 17] h ( J a ¯ J b )( u ) i B = k tr ( J a gJ b g − )( u − ¯ u ) = − k tr ( J a gJ b g − ) | u − ¯ u | , (3.8)so that h J a ¯ J b || B ii = − k tr ( J a gJ b g − ). Note that the currents are normalised such thattr ( J a J b ) = δ ab . The orthonormal marginal fields are thus φ ab ( z ) = k − J a ¯ J b . (3.9)Let us start from the model which is initially unperturbed. To lowest order, (2.19) givesthen ˙ λ ab = − g s π tr ( J a gJ b g − ) . (3.10)Higher order contributions in the bulk come from evaluating connected n -point functions.They are given [16, 17] by the product of traces k tr ( J a . . . J a n ) k tr ( ¯ J b . . . ¯ J b n ), so thatin the normalisation (3.9) they go as k − n . In the limit k → ∞ they only give subleadingcontributions.Let us make a side remark. We can choose an orthogonal basis J a , a = 1 , . . . , r for theleft moving currents, and a corresponding basis ¯ J b := g − J b g, b = 1 , . . . , r for the rightmoving currents. (3.10) then shows that only the fields φ aa are switched on. Note thatthese fields leave the boundary conditions unchanged, as[ t a , g ¯ t a g − ] = [ t a , t a ] = 0 , (3.11)which means that all B ik in (2.8) vanish, so that no boundary fields are switched on [3].The brane changes the bulk without inducing a backreaction on itself.9 .3 Geometric interpretation of SU (2) k To get a geometric picture of the brane backreaction, we switch to a more geometricdescription of WZW models. We will concentrate on G = SU (2). We can write thistheory as a σ -model on the group manifold, using the parametrisation [18] g = e i ( θ +˜ θ ) σ / e iφσ / e − i ( θ − ˜ θ ) σ / , (3.12)or explicitly g = (cid:18) cos φ cos ˜ θ + i sin φ sin θ cos φ sin ˜ θ + i sin φ cos θ − cos φ sin ˜ θ + i sin φ cos θ cos φ cos ˜ θ − i sin φ sin θ (cid:19) . (3.13)At level k the action then becomes S ( φ, θ, ˜ θ ) = k π Z d z (cid:18)
14 ¯ ∂φ∂φ + sin φ ∂θ∂θ + cos φ ∂ ˜ θ∂ ˜ θ + cos φ ∂θ∂ ˜ θ − ¯ ∂ ˜ θ∂θ ) (cid:19) . (3.14)For later use, we also derive explicit expressions for the currents J = − k∂g g − and¯ J = kg − ¯ ∂g , J = − k i √ ∂φ cos(˜ θ + θ ) − ∂θ sin φ sin(˜ θ + θ ) + ∂ ˜ θ sin φ sin(˜ θ + θ ) J = − k i √ ∂θ (1 − cos φ ) + ∂ ˜ θ (1 + cos φ )) J = − k i √ ∂φ sin(˜ θ + θ ) + ∂θ sin φ cos(˜ θ + θ ) − ∂ ˜ θ sin φ cos(˜ θ + θ )and ¯ J = k i √ ∂φ cos(˜ θ − θ ) + ¯ ∂θ sin φ sin(˜ θ − θ ) + ¯ ∂ ˜ θ sin φ sin(˜ θ − θ ))¯ J = k i √ ∂θ ( − φ ) + ¯ ∂ ˜ θ (1 + cos φ ))¯ J = k i √ − ¯ ∂φ sin(˜ θ − θ ) + ¯ ∂θ sin φ cos(˜ θ − θ ) + ¯ ∂ ˜ θ sin φ cos(˜ θ − θ )) . The boundary states are given by || j, g ii . For each gluing condition g there are k + 1possible branes, labelled by j = 0 , , . . . , k . [19] gives a geometric interpretation for thesebranes in terms of conjugacy classes: if g is the identity e , then || j, e ii is the S thatwraps the conjugacy class given by h (cid:18) e πij/k e − πij/k (cid:19) h − . (3.15)In particular, for j = 0 and j = k , the conjugacy class collapses to a point and the branedescribes a D0 brane sitting at the point e and − e , respectively. If the gluing map isgiven by a general g , the position of the brane shifts accordingly.To go to the geometric limit, we fix j and let k → ∞ . Independent of j the brane thusbecomes a D0 brane sitting at the point g . Also, (3.10) shows that the flow induced10epends only on g . We can therefore suppress the index j and parametrise the brane onlyby g = g (Φ , Θ , ˜Θ). Note that we denote its position by capital letters Φ , Θ , ˜Θ, as opposedto small letters for the coordinates of the manifold.In the geometric limit the SU (2) k model corresponds to a non-linear σ -model on S withradius r ∼ √ k . We can read off the target space metric G and the field B from thecoefficients of the action. In the unperturbed case (3.14) this gives G = k/ k sin φ
00 0 k cos φ , B = k cos φ − k cos φ . (3.16) Let us now calculate the RG flow and try to interpret it. (3.10) shows that the marginalfields J i ¯ J j are turned on with the respective strength˙ λ ij (Φ , Θ , ˜Θ) = − g s π tr ( J i gJ j g − ) =: − g s π K ij (Φ , Θ , ˜Θ) . (3.17)The coefficients K ij depend on the position of the brane and are given by K ij = 2 cos 2 ˜Θ cos + cos 2Θ sin sin(Θ + ˜Θ) sin Φ sin 2Θ sin − sin 2 ˜Θ cos − sin(Θ − ˜Θ) sin Φ cos Φ cos(Θ − ˜Θ) sin Φsin 2Θ sin + sin 2 ˜Θ cos − cos(Θ + ˜Θ) sin Φ cos 2 ˜Θ cos − cos 2Θ sin . (3.18)This flow has a nice geometric interpretation. The mass of a brane is given by the valueof the dilaton ϕ . Perturbing the metric of S induces a non-constant dilaton and sochanges the mass of the brane. [3] showed that in the case of an induced boundary flow,the brane deformed in such a way as to minimise its mass. We will show that a similarthing happens here: this time, the brane remains at the same place, but it deforms thegeometry in such a way that its mass is minimised.To show this, let us first find the change in geometry that decreases the mass of the braneas much as possible. The most general current-current deformation of the original theoryis S = S − α Z d z X i,j a ij J i ( z ) ¯ J j (¯ z ) , (3.19)where the a ij are real coefficients. This gives a new metric G ′ ( φ, θ, ˜ θ ) = G − αG and anew B -field. The new, nontrivial dilaton ϕ can be calculated by [18, 20] e − ϕ p det G = e − ϕ ( φ,θ, ˜ θ ) q det G ′ ( φ, θ, ˜ θ ) . (3.20)The mass of the brane at g = g (Φ , Θ , ˜Θ) is proportional to g − s ∼ e − ϕ (Φ , Θ , ˜Θ) . We thuswant to maximise the increase of det G − G ′ at the point (Φ , Θ , ˜Θ). Its derivative is givenby ∂ α det(1 − αG − G ) | = − tr G − G . (3.21)A straightforward calculation then showstr G − G (Φ , Θ , ˜Θ) = k X i,j a ij K ij (Φ , Θ , ˜Θ) , (3.22)11here K ij is the same expression as in (3.18). Introducing a Lagrange multiplier term ν P i,j a ij shows that the expression is extremised by a ij = − K ij (Φ , Θ , ˜Θ). Comparing to(3.17) we find perfect agreement.We can try to follow the flow further and describe the geometry of the deformed manifold.By the symmetry of the problem, it is sufficient to consider the brane sitting at θ = 0 , ˜ θ =0 ‡ so that g = (cid:18) e i Φ / e − i Φ / (cid:19) . (3.23)(3.18) then turns on the fields λ ij = − g s π − sin Φ cos Φ . (3.24)They change the metric G by some expression 2 g s π G Φ1 ( φ, θ, ˜ θ ). At the point of the brane, G Φ1 simplifies: G Φ1 (Φ , ,
0) = k / k sin
00 0 2 k cos = 2 kG (Φ , , . (3.25)The effect of the backreaction is simply to rescale the original metric. We can continueto use our original reasoning even away from the point t = 0 to obtain the differentialequation ˙ G Φ (Φ , , ∼ G (Φ , , . (3.26)The geometric analysis only gives the direction of the flow, so that we are free to choosethe actual flow parameter. Writing G Φ µν ( t ) = G µν + 4 g s π ktG Φ1 µν (3.27)we fix t so that it agrees with the conformal field theory flow parameter at t = 0.Note that this analysis agrees with the observation in section 3.2, where we argued thatin the limit k → ∞ , only the zero order term is important, and that thus no new bulkfields are turned on. This translates to the statement that (3.26) remains valid away fromthe starting point.We can now try to understand how the geometry of the S changes as we start to flow,and we can also try to estimate how far we should trust our analysis. Define a new flowparameter t ′ = 4 g s t/π . Take the metric G Φ µν ( t ′ ) and calculate the associated Ricci scalar R ( t ′ ). At the point g it is given by R ( t ′ ) = 6 + 84 kt ′ k (1 + 2 kt ′ ) . (3.28)The curvature thus increases at first, in agreement with the intuition that the brane warpsthe space around it. ‡ We could simply restrict to g = e , but e is a coordinate singularity in the parametrisation chosen. R ( t ′ ) on all of S , it turns out that at kt ′ = the curvature becomessingular at some points. The geometric approximation thus becomes unreliable as soonas kt ′ ∼
1. In particular, one should not trust (3.28) for values kt ′ ∼ , where R ( t ′ )seemingly starts to decrease. The last example we consider is the bosonic string in flat space in the presence of a D p -brane. In this case, one can consider the low-energy supergravity limit of the theory. TheD-brane is then given by a p -brane, a solution of the corresponding supergravity equation.[21] performed a boundary state calculation and found agreement with the supergravityresults. We will reproduce their results using the extended RG equations. The conformal field theory is described by 26 free bosons with ladder operators a µn , ¯ a νn . AD p -brane located at y is described by the boundary state [21] || Dp ; y ii = T p Z d d ⊥ k ⊥ (2 π ) d ⊥ e ik ⊥ y exp " − ∞ X n =1 a µ − n S µν ¯ a ν − n | k k = 0 , k ⊥ i . (4.1)The diagonal matrix S µν is given by S µν = ( η αβ , − δ ij ) , (4.2)where α, β run over the d k = p + 1 dimensions parallel to the brane, and i, j over the d ⊥ = 26 − p − T p = √ π ( d − / (4 π α ′ ) ( d − p − / . (4.3)Again, we will ignore the tachyon and concentrate on marginal fields. The correspondingstates are of the form a µ − ¯ a ν − | k i . (4.4)Here | k i is the ground state of momentum k , normalised as h k | k ′ i = 2 πδ ( k − k ′ ), with(2 π ) d δ ( d ) (0) = V . The conformal weight of (4.4) is (1 + α ′ k / , α ′ k / p -brane centred at y = 0 as A µνk := h k | a µ ¯ a ν || Dp ; 0 ii = − T p δ ( d k ) ( k k ) S µν . (4.5)We see that only states with k k = 0 couple to the brane. It is thus necessary to considerstates with non-vanishing transverse momentum, which means k ⊥ >
0, such that k > k = 0.This analysis indicates that we need to go off-shell to find states that couple to the brane.From the CFT point of view such this means that we need to consider states (4.4) thatare almost marginal. 13 .2 Applying the RG equations We would like to apply (2.19) and find the fixed point to which the theory flows. Althoughwe derived (2.19) only for marginal fields, the argument also works for almost marginalfields with h = 1 + δh . δh then takes the role of ǫ , and the counterterm needed is ∼ ℓ δh ( δh ) − . The contribution to (2.19) is again g s π h φ k || B ii . It is clear however thatseveral steps of the derivation depended on taking ǫ → δh ≪ λ µν = (2 − h ) λ µν + g s π A µνk + O ( g s λ, λ , g s ) , (4.6)so that to lowest order λ µν = 2 g s πα ′ A µνk k ⊥ = g s T p (2 π ) p +1 V p +1 πα ′ S µν k ⊥ . (4.7)To compare to the metric in the supergravity solution, we calculate the expectation valueof the graviton, i.e. its one-point function. Assuming that the fields φ µν ( k ) were orthonor-mal in the original theory, the perturbed one-point function of a µ − ¯ a ν − | k i is h φ µν ( k ) i λ = λ σρ h φ σρ ( k ) φ µν ( k ) i + O ( λ ) = λ µν + O ( λ ) . (4.8)To obtain the expectation value of the graviton, we have to extract the symmetric tracelesspart of (4.7), as has been done in [21]. (4.7) then agrees with their results, up to a constantfactor due to different normalisations.Our analysis is only valid if α ′ k ⊥ ≪
1, since else δh is too big. Moreover g s T p α ′ k ⊥ ≪ We have calculated the backreaction of a brane on the bulk theory. The RG equationsso obtained are a concrete realisation of the Fischler-Susskind mechanism. For the freeboson on a circle and for the SU (2) WZW-model, the resulting flows agree with geometricexpectations. For flat space, we are able to reproduce the long-distance behaviour of thesupergravity solution.An obvious extension of this work is to try to include higher order terms in g s . Techni-cally, this is probably quite challenging. There is however a more fundamental question:the analysis of section 2 shows that annulus tadpoles can be compensated by local coun-terterms, i.e. that their effect can be expressed by standard RG equations. It is not clearthat this will also work for higher order tadpoles, e.g. for the disk with one thin handlethat shrinks to zero thickness.The other natural extension is to generalise the RG equations to theories with worldsheetsupersymmetry. This would allow to consider setups that are phenomenologically more14nteresting. In particular, one could investigate supersymmetric configurations similarto [22]. In such a setup, shifting the closed string moduli away from their original,supersymmetric values will induce a flow in the configuration of branes. These in turnwill backreact on the bulk. It would be interesting to find the end point of this combinedflow and to check if the resulting theory is again supersymmetric. Acknowledgements
I want to thank my advisor Matthias Gaberdiel for suggesting this topic to me, andfor his constant support and advice during the project itself. I would also like to thankCostas Bachas, Ben Craps, Stefan Fredenhagen, Ingo Kirsch, and Stefan Stieberger forhelpful discussions.
References [1] S. Chaudhuri and J.A. Schwartz,
A criterion for integrably marginal operators , Phys.Lett. B (1989) 291.[2] A. Recknagel and V. Schomerus,
Boundary deformation theory and moduli spaces ofD-branes , Nucl. Phys. B (1999) 233 [arXiv:hep-th/9811237] .[3] S. Fredenhagen, M.R. Gaberdiel and C.A. Keller,
Bulk induced boundary perturba-tions , J. Phys. A (2007) F17 [arXiv:hep-th/0609034] .[4] S. Fredenhagen, M.R. Gaberdiel and C.A. Keller, Symmetries of perturbed conformalfield theories , arXiv:0707.2511 [hep-th] .[5] E. Cremmer and J. Scherk, Factorization of the pomeron sector and currents in thedual resonance model , Nucl. Phys. B (1972) 222.[6] L. Clavelli and J.A. Shapiro, Pomeron factorization in general dual models , Nucl.Phys. B (1973) 490.[7] W. Fischler and L. Susskind, Dilaton tadpoles, string condensates and scale invari-ance , Phys. Lett. B (1986) 383.[8] W. Fischler and L. Susskind,
Dilaton tadpoles, string condensates and scale invari-ance. 2 , Phys. Lett. B (1986) 262.[9] C.G. Callan, C. Lovelace, C.R. Nappi and S.A. Yost,
String loop corrections to betafunctions , Nucl. Phys. B (1987) 525.[10] J.L. Cardy,
Conformal invariance and statistical mechanics , in: ‘Fields, strings andcritical phenomena’, proceedings of the Les Houches summer school 1988, NorthHolland (1990).[11] M.B. Green, J.H. Schwarz and E. Witten,
Superstring Theory. Vol. 2: Loop ampli-tudes, anomalies and phenomenology , CUP. (1987).1512] E. Witten,
Nonabelian bosonization in two dimensions , Commun. Math. Phys. (1984) 455.[13] M.R. Gaberdiel, A. Recknagel and G.M.T. Watts, The conformal boundary states forSU(2) at level 1 , Nucl. Phys. B (2002) 344 [arXiv:hep-th/0108102] .[14] C.G. Callan, I.R. Klebanov, A.W.W. Ludwig and J.M. Maldacena,
Exact solu-tion of a boundary conformal field theory , Nucl. Phys. B (1994) 417 [arXiv:hep-th/9402113] .[15] J. Polchinski and L. Thorlacius,
Free fermion representation of a boundary conformalfield theory , Phys. Rev. D (1994) 622 [arXiv:hep-th/9404008] .[16] I.B. Frenkel and Y. Zhu, Vertex operator algebras associated to representations ofaffine and Virasoro algebras , Duke Math. J. (1992) 123.[17] M.R. Gaberdiel and P. Goddard, Axiomatic conformal field theory , Commun. Math.Phys. (2000) 549 [arXiv:hep-th/9810019] .[18] S.F. Hassan and A. Sen,
Marginal deformations of WZNW and coset models fromO(D,D) transformation , Nucl. Phys. B (1993) 143 [arXiv:hep-th/9210121] .[19] A.Y. Alekseev and V. Schomerus,
D-branes in the WZW model , Phys. Rev. D (1999) 061901 [arXiv:hep-th/9812193] .[20] S. F¨orste, D-branes on a deformation of SU(2) , JHEP (2002) 022 [arXiv:hep-th/0112193] .[21] P. Di Vecchia, M. Frau, I. Pesando, S. Sciuto, A. Lerda and R. Russo,
Classical p-branes from boundary state , Nucl. Phys. B (1997) 259 [arXiv:hep-th/9707068] .[22] M. Cvetic, G. Shiu and A.M. Uranga,
Chiral four-dimensional N = 1 supersymmet-ric type IIA orientifolds from intersecting D6-branes , Nucl. Phys. B (2001) 3 [arXiv:hep-th/0107166][arXiv:hep-th/0107166]