Chiral boundary conditions for singletons and W-branes

Journal of High Energy Physics, Jul 2017

We revisit the holographic dictionary for a free massless scalar in AdS3, focusing on the ‘singleton’ solutions for which the boundary profile is an arbitrary chiral function. We look for consistent boundary conditions which include this class of solutions. On one hand, we give a no-go argument that they cannot be interpreted within any boundary condition which preserves full conformal invariance. On the other hand, we show that such solutions fit naturally in a generalization of the Compère-Song-Strominger boundary conditions, which preserve a chiral Virasoro and current algebra. These observations have implications for the black hole deconstruction proposal, which proposes singleton solutions as candidate black hole microstate geometries. Our results suggest that the chiral boundary condition, which also contains the extremal BTZ black hole, is the natural setting for holographically interpreting the black hole deconstruction proposal.

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Chiral boundary conditions for singletons and W-branes

Received: May Chiral boundary conditions for singletons and W-branes Joris Raeymaekers 0 1 4 Dieter Van den Bleeken 0 1 2 3 0 3001 Leuven , Belgium 1 34342 Bebek , Istanbul , Turkey 2 Physics Department, Bogazici University 3 Institute for Theoretical Physics, KU Leuven 4 Slovance 2 , 182 21 Prague 8 , Czech Republic We revisit the holographic dictionary for a free massless scalar in AdS3, focusing on the `singleton' solutions for which the boundary pro le is an arbitrary chiral function. We look for consistent boundary conditions which include this class of solutions. On one hand, we give a no-go argument that they cannot be interpreted within any boundary condition which preserves full conformal invariance. On the other hand, we show that such solutions t naturally in a generalization of the Compere-Song-Strominger boundary conditions, which preserve a chiral Virasoro and current algebra. These observations have implications for the black hole deconstruction proposal, which proposes singleton solutions as candidate black hole microstate geometries. Our results suggest that the chiral boundary condition, which also contains the extremal BTZ black hole, is the natural setting for holographically interpreting the black hole deconstruction proposal. AdS-CFT Correspondence; Black Holes in String Theory - 3 4 5 6 2.3 2.4 3.1 3.2 4.1 4.2 4.3 5.1 5.2 1 Introduction 2 Revisiting the free massless scalar on AdS3 2.1 2.2 Classi cation of solutions Representation theory 2.2.1 2.2.2 2.2.3 Dirichlet modes with integer weights Singleton modes Zero-mode representations Variational principle Example: Dirichlet boundary conditions A no-go argument Comments on the extreme IR limit Conformal boundary conditions for singleton modes? Chiral boundary conditions Chiral boundary conditions Example: multiple vortices Asymptotic symmetry algebra 4.4 Including gravity Examples with point-like sources Helical particles W-branes Discussion: a re ned deconstruction proposal A Explicit massless scalar modes B Details on holographic renormalization C First order solution 1 3 In this work we revisit the boundary conditions and holography for a massless complex scalar eld in AdS3. Concretely, we will be interested in the holographic interpretation of solutions of the wave equation t = 0 which are given by an arbitrary holomorphic function of a complex variable v: Here, we work in global coordinates in terms of which the AdS3 metric is '. From (1.1) we can of course generate other classes of solutions depending on a free function by using discrete symmetries such as parity, x charge conjugation, t $ t. We will focus on the set (1.1), since they can be shown to preserve supersymmetry in parity-breaking theories where supersymmetry resides in the `left-moving' sector. Expanding the solutions (1.1) near the AdS3 boundary, one sees that they don't vanish there, but asymptote to an arbitrary right-moving function g(e ix ). This behaviour is reminiscent of what happens in the `alternate' quantization, where the leading part of the scalar is allowed to uctuate. The case at hand is subtle however, since the alternate quantization is only generically de ned for massive scalars in the range [1] 1 < m2l2 < 0; where it corresponds to assigning an operator of dimension = 1 holographic dual. The massless scalar of interest lies at the upper boundary of the window (1.3) and requires special care, since the would-be dual operator dimension = 0 then saturates the unitarity bound. One might expect that in this case the eld propagates a `short' multiplet. Indeed, as we will review below, the solutions of interest (1.1) carry a three-dimensional singleton [2] representation, which is a small part of a larger representation which is reducible but not decomposable. Our goal will therefore be to verify a) whether a version of the `alternate' boundary condition can still be consistently1 imposed so as to allow the solutions of the type (1.1) and b) if so, if it preserves conformal symmetry. While the answer to a) will be a rmative, we will present a simple argument that the answer to b) is negative: there exists an obstruction to consistent boundary conditions which allow the solutions (1.1) while preserving the global SL(2; R) SL(2; R) symmetry of AdS3. Interestingly, the boundary condition we propose in our answer to a) does preserve a purely right-moving Virasoro symmetry, combined with a right-moving U( 1 ) current algebra: they are a natural generalization of the boundary conditions proposed by Compere, Song and Strominger [19] in the context of pure gravity. While this generalization of [19] to include matter is of interest in its own right, our main motivation for studying this question came from black hole physics, namely from a puzzle in the holographic interpretation of the black hole deconstruction proposal of Denef et al. [26]. The idea of this work was to deconstruct stringy black holes in terms bound states of zero-entropy D-brane centers, with a large moduli space to account for the black hole entropy. A concrete realization for the IIA D4-D0 black hole of [27] was proposed to involve certain D2-brane con gurations enveloping a D6-anti-D6 pair. The D2's experience a magnetic eld in the internal space and their large lowest Landau level degeneracy was 1With consistent, we mean such that it leads to a consistent variational principle, i.e. the action functional is di erentiable. { 2 { (1.2) $ x+, or (1.3) p 1 + m2l2 as the shown to account for the entropy of this particular black hole. Upon taking an M-theory decoupling limit, these D2-brane con gurations become particle-like objects in AdS3, which source a complex scalar eld. They can be seen as simple examples of so-called W-branes, the M-theory lift of open strings connecting D6-brane centers, which were conjectured to capture the entropy of a class of D-brane systems2 [28, 29]. As was argued in [7{9], the scalar pro le in the black hole deconstruction solutions is precisely of the form (1.1). Therefore, our results can be interpreted as a no-go argument for the holographic interpretation of the black hole deconstruction solutions within a standard CFT possessing two Virasoro copies (such as the MSW CFT of [27]). On the other hand, these solutions do belong to a dual chiral theory with a right-moving Virasoro and current of the black hole deconstruction proposal. This paper is structured as follows. In section 2, we revisit the classi cation of solutions to the massless wave equation and their representation content. We also discuss boundary conditions and derive a criterion to check whether a given boundary condition follows from a consistent variational principle. In section 3 we present our no-go argument for interpreting the singleton modes of the scalar within conformally invariant boundary conditions, and in section 4 we propose chiral boundary conditions which do include those solutions. In section 5 we consider examples: after the warm-up example of multi-centered solutions in pure gravity we turn to the W-brane solutions which also source a complex scalar. In the Discussion we focus on the implications of our results for the holographic interpretation of the black hole deconstruction proposal. 2 Revisiting the free massless scalar on AdS3 In this section we revisit some of the physics of a free, massless, complex scalar on a xed AdS3 background. Our main motivation is the application to the black hole deconstruction proposal in the following sections, but as this analysis might be of independent interest we have tried to make this section self-contained. We will review in some detail the classi cation of bulk solutions and their transformation under the action of the SL(2; R) SL(2; R) global symmetry of AdS3, slightly expanding on the results of [3] (see also [2, 4]). We then go on to discuss in general the compatibility of a consistent variational principle with nite on-shell action and a choice of boundary conditions for this free scalar theory, reviewing the example of the standard Dirichlet boundary conditions. In the next two sections we will focus on boundary conditions which allow for the solutions (1.1) and their representation-theoretic content. 2.1 Classi cation of solutions We consider a free, massless complex scalar on a xed AdS3 background with action:3 S = 1 (2.1) 2Some puzzles with the W-brane idea were pointed out in [30]. 3In the remainder of the paper we will work in units where GN = 161 . { 3 { The equation of motion obtained by varying t is simply It is convenient to introduce Fe erman-Graham coordinates (y; x+; x ), with y = 4e 2 , in terms of which the global AdS3 metric takes the form ds23 = l 2 dy2 4y2 1 y dx+dx + (dx2+ + dx2 ) + dx+dx y 2 16 that form the algebra L 1 = ie ix+ L 1 = ie ix y2 16 16 8y 8y y2 These isometries can be conveniently used to classify the solutions to the wave equation (2.2) as the Laplacian is a Casimir of the algebra above: 2 l = 2 (L2 + L2) L2 = 1 2 (L 1L1 + L1L 1) L 2 0 In particular it follows that solutions to this equation will form representations of the symmetry algebra with vanishing values for both L2 and L2. Furthermore as L0 and L0 commute with the Laplacian, we can label di erent solutions by their eigenvalues h and h under these operators, leading to solutions of the form th;h(x+; x ; y) = e i(hx++hx )fh;h(y) Using this separation of variables (2.2) can be solved exactly in terms of hypergeometric functions. We give some details in appendix A, but it is simpler and also su cient for our purposes to do a near-boundary analysis. Note that we will consider general solutions to (2.2), without imposing any restrictions such as boundary- or regularity conditions for the moment. The near-boundary behaviour of a general solution to (2.2) is4 [18] t = t0(x+; x ) + y t2(x+; x ) + y log y t~2(x+; x ) + O(y2 log y): The equation of motion near the boundary leaves t0 and t2 arbitrary, while t~2 is related to t0 as 4The last, `logarithmic', term is special to the massless case and would be absent for a scalar with a mass in the window 1 < m2l2 < 0. { 4 { (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) Combining this information with the separation of variables (2.8) one nds that for given h; h there are two types of solutions: the class th;h for which the leading t0 part in (2.9) vanishes, and the class th;h for which it doesn't. Their near-boundary expansion is + th;h = e i(hx++hx )y + O(y2 log y) th;h = e i(hx++hx ) 1 hhy 1 2jhj t0;h; th;0; = e ihx = e ihx+ Note that we introduced an arbitrary parameter , which can be set to any preferred value by adding a certain multiple of t0+;h or th+;0 respectively. In addition to the solutions discussed so far, there are also the zero-mode solutions which are not eigenstates of both L0 and L0, namely t = x+ and t = x : Like the t solutions, these also don't vanish near the boundary. 2.2 Representation theory As discussed above, the set of solutions to the wave equation is guaranteed to carry a representation of the AdS symmetry group. It is well-known that this representation is neither unitary nor irreducible. In fact, the full set of scalar solutions carries a reducible but nondecomposable representation as we will now discuss in detail. Before we begin, let us recall, and introduce some notation for, the unitary irreducible representations of SL(2; R), see [3, 5] for more details. We will mainly restrict attention to the primary (lowest weight) and anti-primary (highest weight) representations. We will use a simpli ed notation where both are denoted by (h), with the understanding that for h positive, we mean the primary representation built on a state with L0-eigenvalue h and annihilated by L1, while for h negative we mean the anti-primary representation built on a state with L0-eigenvalue h and annihilated by L 1 . The trivial representation will be denoted5 by (0). The corresponding representations of the product group SL(2; R) SL(2; R) will be denoted as (h; h). The three-dimensional singleton representations of [2] are those which are trivial with respect to one of the factors, i.e. of the type (h; 0) or (0; h). These representations are annihilated by L 1 resp. L 1 and therefore short compared to the generic (h; h) representation.6 5In the notation of [3], (h) with h > 0 corresponds to D+(h; h), (h) with h < 0 to D (h; h) and (0) to D(0). 6We should alert the reader that a di erent de nition of a 3D singleton was given in [11], namely as a representation carried by the Fock space of a single harmonic oscillator. In 4D both de nitions of the 3=4, which are not short in the sense described here. acting on a corresponding solution repeatedly with any of the generators (2.6). In case h0 and h0 are non-integer the t+ and t fall in separate representations that contain all h and h such that h h0; h h0 2 Z. However when one (or both) of h0; h0 is integer things are somewhat di erent. Note that this includes the solutions of interest (1.1), so we will focus on this case in what follows. In particular if one starts with th0;h0 one will at some point end up at a state of the form t0;h; or th;0; , which then via the last line of the rules above feeds into states of the form t+, meaning that we get a huge representation containing both t+ and t solutions. Note however that this argument passed through a one way street: one cannot produce t states from t+ states by acting with the generators. Therefore the t+ states form a subrepresentation on their own. 2.2.1 Dirichlet modes with integer weights Let us describe in more detail the representation carried by the t+ modes when both h and h are integer. These can all be obtained from acting with the generators on the solution Acting with L 1 gives t1+;0, but acting with L1 on t1+;0 gives zero instead of bringing us back to t0+;0. Therefore acting with the generators on t1;0 we obtain an invariant subspace, whose + complement is not invariant. The representation is therefore reducible but nondecomposable. Similar remarks apply when acting on t0;0 with the generators L1; L 1 . The full structure of the representation is illustrated in gure 1, and can be schematically denoted + in terms of irreducible representations as Returning to the massless scalar solutions, the particular choice of basis (2.11) is useful as it brings the representation of the algebra (2.6) into a rather simple and disentangled form: ( 1; 1) { 6 { ! ! (1; 1) (1; 0) : " # vh = t0;h; 1 t0;1; 1 = v; { 7 { SL(2; R) action on the th+;h solutions for integer h and h. Left/right arrows show non-trivial L actions and down/up arrows indicate non-trivial L actions. The states in dark red form four unitary subrepresentations built on (anti-) primary states. The total representation is indecomposable as there are irreversible lowering/raising actions starting from states with either h or h vanishing, as can be seen from the presence of one-way arrows. We should note that the states in blue and green are often not considered as they are singular in the center of AdS3. 2.2.2 Singleton modes Next we would like to describe the representation which contains the solutions of interest (1.1), which in Fe erman-Graham coodinates read If t is a periodic function, it can be formally expanded in integer powers of v, as well as the zero-mode combination x+ + x . The powers of v are easy to identify in the classi cation above, namely We will restrict our attention to the case of positive powers, h 0, which are regular in the `center' of AdS3 which is at y = 4 in our coordinates. We would like to study the smallest representation of SL(2; R) SL(2; R) that contains the states (2.18). It consists of all the states which can be obtained by acting with the generators on (2.17) (2.18) (2.19) SL(2; R) representation which contains the solutions (2.18). The solutions on the y-axis are t modes, while the solutions in red are t+ modes. and we will denote the resulting vector space by K. This carries an in nite dimensional representation, whose structure we will discuss in more detail in a moment, but it is still much smaller than the representation one would obtain by acting on an arbitrary starting point th;h. This because L+ vanishes on t0;h; 1 while it doesn't vanish on th;h when h 6= 0, in summary K is a short representation. + The representation carried by the vector space K is once again reducible but indecomposable: acting with L 1 connects to the t+ modes, i.e. L 1t0;h; 1 = ht1+;h, while L1t1;h = 0. This structure is illustrated in gure 2 and can be schematically summarized as The vector space obtained by acting with symmetry generators on t1+;1 forms an invariant subspace N shown in red, whose complement in K is not invariant. We remark that by acting by a parity transformation x representation containing the solutions ! x on the above representation we get the smallest uh = th;0; 1; u = 4 4 + y y e ix+;h 2 N The invariant subspace in this representation is again N , which is parity invariant. We note that the representation on K, is non-unitary, since it follows from the algebra that the all states in N should have zero-norm in a unitary representation [2]. The standard way around this to consider instead the quotient space K=N , on which the induced representation is unitary. In physics terms, one only considers the t part of a solution as physical, while any t+ component of a solution is considered a gauge artifact and hence unphysical. The result of the quotienting procedure is to keep only the singleton representation (0; 1) spanned by the modes (2.18), which we will henceforth refer to as the singleton modes. The standard physical realization of this quotienting is to introduce a suitable gauge symmetry. Remarkably, this can be accomplished by replacing the equation { 8 { of motion (2.2) by the fourth-order equation7 2t = 0 as discussed8 in [2]. Here, we will not consider such a modi cation of the bulk Lagrangian but rather investigate which consistent boundary conditions can be imposed on the theory so as to include the singleton modes (2.18). We end with a remark which will be important in what follows: even though the singleton modes vh are part of a nondecomposeable representation from the point of view of the full symmetry group SL(2; R) SL(2; R), they carry a unitary irreducible representation of the subgroup U( 1 ) SL(2; R), where the U( 1 ) is generated by L0. This representation can be denoted as ( 1 )0, where the subscript refers to the U( 1 ) charge. We will see below that the singleton modes vh naturally t within boundary conditions which break SL(2; R) down to U( 1 ). Similarly, the parity-related set of modes (2.21) also carries a unitary irreducible representation, but of a di erent subgroup, namely SL(2; R) U( 1 ). 2.2.3 Zero-mode representations So far we didn't include the zero-mode solutions (2.12) in our discussion of representation theory. Acting with the symmetry generators on them, they connect to both the t+ modes and the singleton modes discussed above. For example, acting with generators on t = x we obtain L0x L x = 0; = i + 2 t 1;0; L0x L x = it0;0 = it0; 1;0: (2.22) (2.23) 2.3 Variational principle So far we have discussed the full solution space of the massless scalar, without imposing any boundary conditions. If we impose boundary conditions which are SL(2; R) SL(2; R) invariant, they will select certain subrepresentations within the full solution space. In the next sections we will investigate which subspaces of solutions are selected by imposing consistent boundary conditions, focusing especially on the boundary conditions which allow the singleton modes (2.18), and discuss their representation content. In investigating possible boundary conditions we require the usual physical conditions motivated by holography: the boundary conditions should lead to a consistent variational principle, meaning that the variation of the action is proportional to the equations of motion without additional boundary terms, and that this action is on-shell (os) nite. Our starting point is the bulk action (2.1) and we will now investigate which possible boundary terms can be added to realize these physical requirements. As is well known, the bulk action (2.1) is divergent when evaluated on generic solutions due to the in nite volume of AdS3. One regularizes this infrared divergence by putting the boundary at y = , for some small , we'll denote this clipped AdS3 space by M . One can 7Note that here the leading term is fourth order, so it is not simply a quadratic theory plus higher 8Since this equation is parity-invariant, the resulting theory also contains the (1; 0) singleton represenderivative corrections, which is the typical situation. tation carried by the solutions (2.21). { 9 { then compute that the divergent part of the on-shell action is The divergence can be cancelled by adding a boundary term of the form nite Sbnd = where the boundary Lagrangian density Lbnd 0 is a function of the boundary elds t0; t2 and their derivatives that remains nite in the ! 0 limit. The action Stot = Sbulk + Sbnd is thus by de nition the most general boundary extension of (2.1) that is on-shell nite. We can then investigate the second requirement of consistency of the variational principle. One computes that Lbnd 0) + O( log ) (2.26) By de nition of a consistent variational principle the above should vanish. This then gives us the following relation between the choice of boundary Lagrangian and choice of boundary conditions on the elds: In what follows we will discuss di erent solutions to the above equation and their representation content. boundary: 2.4 Example: Dirichlet boundary conditions We start by reviewing the standard Dirichlet boundary condition, where t is xed on the os l 2 l 2 Re Z t0 = c0;0 : (2.24) (2.25) (2.27) (2.28) (2.29) (2.31) This provides a solution to (2.27) for the trivial choice Lbnd 0 = 0. Assuming that t0 is single-valued, we can expand it in Fourier coe cients as t0 = X ch;he i(hx++hx ) + c~0(x+ + x ) h;h which via our classi cation above implies that the most general solution satisfying the boundary condition (2.28) is t = X ch;hth;h + c~0(x+ + x ) + X ah;hth+;h; h;h h;h ah;h arbitrary . (2.30) If we want to preserve the full SL(2; R) SL(2; R) symmetry, we must restrict t0 to be invariant under the action of the symmetries in (2.13){(2.14); this is only the case for constant t0, i.e. With this boundary condition a basis for the set of allowed solutions is given by the t + modes. These form the indecomposable representation which was discussed in paragraph 2.2.1, see gure 1 and (2.16). If in addition one imposes regularity in the interior of AdS one ends up with four unitary9 representations built on top of the solutions t+1; 1 and t+1; 1. One can also show that the boundary condition (2.31) is invariant under the extension of the global symmetry algebra to the asymptotic Virasoro symmetry Vir Vir. There is also a standard and simple holographic interpretation of the boundary condition (2.28). From the CFT point of view we added an operator l Z 2 SCFT = dx+dx t0O (2.32) and when t0 satis es the condition (2.31) the theory is conformally invariant, so that O must be a weight (1; 1) primary operator. Therefore, changing the xed constant value of t0 corresponds to marginally deforming the dual CFT. Using the standard holographic dictionary we can compute the VEV of the dual operator via the variation of the classical action of the bulk theory: hOi = t0 Stot = lt2 : Note that if we don't impose conformal invariance and allow t0 to be a non-constant function, the dual interpretation is that of a state in a theory deformed by a term (2.32) which breaks conformal invariance. In this sense the singleton modes (2.18) can be interpreted within Dirichlet boundary conditions, as in the analysis of [9], but each mode represents a state in a di erent theory. 3 Conformal boundary conditions for singleton modes? Rather than interpreting each singleton mode (2.18) in a di erent theory, we would like to nd boundary conditions which allow all the singleton modes, so that they represent di erent states in the same theory. In the mass range (1.3), this would be similar to the Neumann boundary condition imposed in the alternate quantization, but because the massless scalar is on the boundary of this range a separate analysis needs to be made. We will argue in this section that there are no consistent SL(2; R) SL(2; R) invariant boundary conditions which allow the full class of solutions (1.1), unless we take a scaling limit of the scalar theory which keeps only the extreme infrared modes, as was advocated in [12]. On the other hand, we will argue in the next section that there exist consistent boundary conditions preserving the subgroup U( 1 ) SL(2; R) and which do allow the solutions (1.1). 3.1 A no-go argument In section 2.2.2 we saw that the singleton modes are part of a space K which carries a nonunitary and nondecomposable representation as illustrated in gure 2 and (2.20). 9Meaning that for each of these representations we can nd an inner product with respect to which it is unitary, they will not all be unitary with respect to the same inner product however. Note that there are no unitary representations with L2 = L2 = 0 and non-integer weights [3]. Putting worries about unitarity aside for the moment, we would like to address the following question: do there exist conformally invariant boundary conditions, which are consistent in the sense discussed in section 2.3, and which allow the states of K? The problem, as we will now see, lies not in the existence of conformal boundary conditions but in their consistency with a variational principle. From the near-boundary expansion (2.11) we see that all the solutions in K satisfy while the more general th;h solutions with h; h 6= 0 do not. This is therefore a natural conformally invariant boundary condition to impose in order to allow for the solutions in K. Because (3.1) is parity and charge-conjugation invariant it includes also the representations related to K by these discrete symmetries. It furthermore includes all the Dirichlet t+ modes illustated in gure 1: both the red modes (which lie in N and its images under discrete symmetries) and the modes on the horizontal and vertical axes. In order for the boundary condition (3.1) to be consistent with a variational principle for an action that is on-shell nite, we have to nd a boundary Lagrangian which satis es (2.27). In this case this reduces to the requirement10 Lbnd 0 =os t2 t0 + t2 t0; but one realizes rather directly that there exists no Lbnd 0 that can satisfy the above constraint. Indeed, we can formally identify the variation with an exterior derivative on the space of boundary elds. The fact that (3.2) has no solutions is then the simple observation that the l.h.s. is exact and hence closed, while the r.h.s. is not closed. In other words applying to (3.2) we nd t2 ^ t0 + t2 ^ t0 =os 0 But from our classi cation of the on-shell solutions we see that even in the presence of the boundary condition (3.1), t2 remains an arbitrary complex function on-shell. Of course (3.1) implies that @ @+ t0 = 0 on-shell, but this still has an in nite number of independent solutions. In other words all of t2, t2, t1, t1 remain linearly independent even on-shell and so (3.2) is false, implying there exists no boundary completion of the bulk action (2.1) that is on-shell nite and has a variational principle consistent with the boundary condition (3.1). Of course, while (3.1) seems to be a natural boundary condition which selects the states in K, we have not proven that there doesn't exist a di erent boundary condition which is consistent with a variational principle. 3.2 Comments on the extreme IR limit In making our no-go argument, we have focused on modes of the eld whose energy doesn't scale with the IR cuto as we take to zero. The no-go conclusion can be avoided if we 10Note that this requirement is non-trivial as t2 6= 0 at least for some solutions if one wants conformal invariance. Indeed, imagine imposing t2 = 0, this would select the states th;0;0 and t0;h;0 only. Rewriting th;0;0 = 12 (th;0;1 + th;0; 1) it follows from (2.14) that acting with L would generate a t + state which has non-zero t2. (3.1) (3.2) (3.3) instead take a type of extreme IR limit of the scalar theory, in which only boundary modes survive, and is essentially the limit considered in [12]. This limit is obtained by considering the bulk action (2.1) without adding any boundary terms, i.e. we take Sbnd = 0 instead of (2.25). Because of the divergent boundary term, the solutions for which the on-shell action is nite are the modes of the rescaled eld which stay nite as ! 0. From (2.26) the variation of the action is HJEP07(21)49 Stot =os lRe Z dx+dx w0 + O( log ) j log j This vanishes in the limit ! 0 if we impose the boundary condition analogous to (3.1) w = pj log j t which therefore allows for the (rescaled) solutions in K. It was shown in [12] that the natural norm of states in N goes to zero in the limit, thereby keeping only the singleton representations. Note that in this limit, the rescaled eld lives purely on the boundary and doesn't backreact on the bulk metric. 4 Chiral boundary conditions In the previous section we argued that, unless one takes the IR scaling limit discussed above, it is not possible to impose consistent boundary conditions which preserve the full SL(2; R) SL(2; R) symmetry as well as include the singleton modes of the form t = v with h > 0. This suggests that including these modes in the two-derivative bulk theory11 will entail breaking at least part of the symmetry. We also observed that the singleton h modes do carry an irreducible representation ( 1 )0 of the subgroup U( 1 ) SL(2; R). In the rst subsection below we will show that there exist consistent boundary conditions, which arise from a boundary term which breaks SL(2; R) to U( 1 ), which allow for the singleton modes. We will then go on to show that these boundary conditions are a generalization of the `chiral' boundary conditions for pure gravity studied in [19] and extend to an asymptotic symmetry algebra which includes both a right-moving U( 1 ) current algebra and a right-moving Virasoro algebra. After, this, we extend the discussion to include gravitational backreaction and show that our scalar boundary conditions can be consistently combined with the boundary conditions on the metric discussed in [19] to describe fully backreacted singleton modes. 11As outlined in paragraph 2.2.2, a way to obtain a theory of the singleton modes is to modify the bulk Lagrangian to a fourth-order one. In this theory, which we will not consider here, both the (1; 0) modes t = uh and the (0; 1) modes t = vh are physical while the (1; 1) modes in (2.20) are pure gauge. (3.4) (3.5) (3.6) Without further ado, we propose the following chiral boundary conditions: B1 B2 i 2 It is straightforward to check that they allow general solutions of the type t = g(v). In terms of the mode solutions discussed in section 2, our boundary conditions include: the solutions t0;h; 1 = vh which for h > 0 are the singleton modes. a linear combination of the zero-mode x and the mode t0+;0, namely Note that this is an allowed eld con guration only if the real part of t is compact. It then directly follows from the action of the generators (2.13) and (2.23) that the L 1symmetry is broken by these boundary conditions while the other generators L0; L1; Lm are compatible with them, however with L1 acting trivially on all states. To obtain a consistent variational principle one can add to the regularized bulk action the boundary term The consistency and niteness follows from observing that this boundary Lagrangian satis es (2.27) when the boundary conditions (4.1) are imposed. Indeed, from (2.26), we nd for the variation of the total action dx+dx t2 + t0 + O( log ) (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) Stot =os lRe Z which indeed vanishes when (4.1) holds. 4.2 Example: multiple vortices 1 2 i 2 t t + 1; As we will explore in more detail in section 5, the boundary conditions (4.1) allow for a class of interesting topological defects in the bulk. Indeed, suppose that t has a periodicity we can then consider topological line defects located at constant v, around which t has a nontrivial monodromy. In terms of the single-valued eld M = e2 it, these are vortices for the global U( 1 ) charge. More concretely, such solutions look like t = i 2 i v 1 vi viv where vi are the locations of the defects and qi are the corresponding integer winding numbers. For later convenience, we have introduced image charges such that Imt = 0 on the boundary jvj = 1. Generically these solutions are linear combinations of the constant mode and the singleton modes vh; h > 0, with coe cients obtained by expanding (4.6) around v = 0. In the particular case that a defect is present in the center v = 0 of AdS3, the expansion includes the mode (4.2) as well. We now address the question whether our boundary conditions (4.1) preserve an extended asymptotic symmetry algebra. It will turn out that they are a natural extension of the chiral boundary conditions proposed in [19] for pure AdS3 gravity to include matter. These boundary conditions preserve the subgroup U( 1 ) SL(2; R) of the global AdS symmetry, which gets extended to a combined right-moving Kac-Moody and right-moving Virasoro algebra. As a rst order check, we will now verify that our boundary conditions (4.1) for the scalar in a xed AdS background are preserved by the asymptotic symmetries of [19]. In the next section we will extend the analysis to include dynamical gravity. The asymptotic symmetries of [19] are generated by two sets of asymptotic Killing vectors, depending on two arbitrary rightmoving functions U (x ); V (x ): ~ U B B~=0 = i 2 ~ V B B~=0 1 2 = (4.14) 1 2 + The U vectors generate a right-moving u( 1 ) current algebra, while the V are the standard right-moving Virasoro generators. The near-boundary components of the scalar transform 1 2 Using these expressions, one easily checks that the boundary conditions (4.1) are invariant under the asymptotic symmetries generated by (4.7), (4.8), i.e. U BijB1;2=0 = 0; V BijB1;2=0 = 0: Before we go on to discuss the inclusion of dynamical gravity, we end with a remark. The attentive reader may have noted from (4.4) that, from the point of view of the variational principle, we could have just as well imposed the weaker boundary condition i 2 ~ B t2 + which would also have allowed the singleton modes. The reason we adopted the stronger conditions (4.1) is that (4.13) is not invariant under the symmetries generated by (4.7), (4.8), since (4.7) (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) We will now show that the boundary conditions on the scalar (4.1) can be extended to the case where backreaction of the scalar on the metric is included, in a consistent manner and preserving the asymptotic symmetries generated by (4.7), (4.8). The boundary conditions on the metric will be those proposed for the pure gravity case in [19]. The fact that these remain consistent upon inclusion of the massless scalar is nontrivial, since the scalar backreaction turns on a logarithmic mode in the metric which is absent in the case of pure gravity. We will here only sketch the derivation, referring to appendix B for more details and a generalization to a family of axion-dilaton like kinetic terms and the inclusion of a HJEP07(21)49 U( 1 ) Chern-Simons eld. Both these generalizations will be relevant in the next section. The starting point is the Einstein-Hilbert action for gravity minimally coupled to a massless complex scalar: S = Z M d x 3 p G R + 2 2 l 1 2 2 Z K : (4.15) We have included the standard Gibbons-Hawking boundary term to compensate for the fact that the Einstein-Hilbert term contains second derivatives of the metric. We use Fe erman-Graham coordinates in terms of which the metric reads The near-boundary expansion of the elds is then [18] t = t0 + y t2 + y log y t~2 + O(y2 log y) gij = g0 ij + y g2 ij + y log y g~2 ij + O(y2 log y): As was the case for the scalar eld, the logarithmic term in the expansion of the metric is special for the massless scalar and would be absent for a scalar with mass in the window (1.3). Expanding the equations of motion following from (4.15) near the boundary, one nds that the boundary values t0 and g0 ij are arbitrary, while the logarithmic coe cients t~2; g~2 ij as well as the trace and the divergence of g2 ij are xed in terms of them: t~2 = g2 = 1 4 1 2 0t0; R0 + 1 rj0(g2 ij 1 Here and in what follows, indices are raised and contracted with the g0 metric. Once again we must add boundary terms to the action in order for it to be nite as the IR cuto is taken to zero. This leads to the following generalization of (2.25): Z Sbnd = l d x 2 p g 2 + R 2 Stot =os l Z d x 2 p g0h2Re (t2 + t~2) t0 (4.16) (4.17) (4.18) (4.19) (4.20) log Lbnd 0 + O( log ) (4.21) where Lbnd 0 is a nite part which may depend on both the metric and the scalar eld. For the variation of the action one nds (g2 ij + g~2 ij g2g0 ij ) g ij 0 i Lbnd 0 + O( log ) (4.22) = 1 2 ; t2 = g2 ++ = g0 ++ = 0: k i t0 Here, P (x ) is an arbitrary right-moving function, while is a xed number; di erent values of correspond to di erent theories. We also de ned k = 4 ` = c 6 Note the similarity between the rst two lines (4.23) and (4.24): as for the scalar, one of the leading components of the metric, g0 , is allowed to be a uctuating right-moving function, while one of the subleading components, g2 ++, is xed. Furthermore, the equations following from (4.15) imply As in [19], we will parametrize g2 in terms of another arbitrary right-moving function L(x ) as The boundary conditions (4.23){(4.25) are consistent: indeed, taking the boundary Lagrangian in (4.22) to be 1 k g2 = L(x ) + (P 0)2 : we nd that the variation (4.22) vanishes when the boundary conditions (4.23){(4.25) hold. One can check that, as promised, the boundary conditions (4.23){(4.25) are invariant under the asymptotic symmetries generated by (4.7), (4.8), with the free right-moving functions transforming as We now present our proposed boundary conditions for the scalar and the metric, which allow for the backreacted singleton solutions. These combine the scalar boundary condition (4.1) with the boundary conditions of [19] on the metric: V L = V L0 + 2V 0L V t0 = V t00 k 2 V 000 k 4 V 0jt00j2 U P 0 = U 0 As we already noted in paragraph 4.2, the chiral boundary conditions (4.1) on the complex scalar allow for multi-vortex solutions which would not t in the standard conformally invariant boundary conditions. In this section we explore the holographic meaning of what is essentially the backreacted version of the solutions of paragraph 4.2, with the di erence that we will now replace the free complex scalar with an interacting axion-dilaton eld. Such solutions play an important role in the black hole deconstruction proposal [26] where they describe wrapped M2-branes, which were proposed as microstate geometries which make up the entropy of the M-theory black hole of [27]. They can also be seen as simple examples of so-called W-branes, the M-theory lift of open strings connecting D6-brane centers, which were conjectured to capture the entropy of a class of D-brane systems [28, 29]. As we shall argue below, in a rst approximation these solutions reduce to the geometry produced by multiple conical defects moving on helical geodesics in AdS3, which were considered in [24]. As a warm-up example, we will therefore reinterpret these solutions in the chiral boundary conditions [19] for pure gravity, i.e. (4.23){(4.24). Interestingly, we will see that these boundary conditions allow a wider class of multi-particle con gurations where the total mass exceeds the black hole threshold, already suggesting their relevance for black hole physics. 5.1 Helical particles As explained above, it will be useful to rst consider the warm-up example of pure gravity with point-particle sources. We will focus on multi-particle solutions which represent the backreaction of particles moving on helical geodesics in global AdS, which were studied in [24]. These t in standard Brown-Henneaux boundary conditions when the total mass is below the black hole threshold, but interestingly we will see that they t in the more general boundary conditions (4.24){(4.25) even when the total mass exceeds the black hole threshold. The point particle source terms are in other words, v coincides with the holomorphic coordinate we introduced before in (1.1). where Wi are the timelike worldlines of the particle sources and mi are the point-particle masses. Our ansatz for the metric is where the function and the one-form A are de ned on the base space parametrized by v; v. For example, for global AdS3 these are given by AdS = ln(1 jvj2); The relation with standard global coordinates (1.2) is Ssource = X mi i Z Wi dsi v = tanh e ix ; (5.1) (5.2) (5.3) (5.4) More generally, the Einstein equations imply that must satisfy the Liouville equation12 away from the sources. We will take the particles to follow geodesics of constant v = vi, which corresponds to helical curves = constant, x = constant in global AdS. We then need to solve Liouville's equation in the presence of delta-function13 sources coming from (5.1): vi; v vi): The coordinate v can be taken to run over the unit disk, jvj 1, and to ensure asymptotically AdS behaviour the eld should satisfy the Zamalodchikov-Zamolodchikov boundary conditions of [25] HJEP07(21)49 The solution for the one-form A can be expressed in terms of as where (v) = 1 8 X mi ln i v 1 vi viv The large gauge transformation involving the multivalued function (v) ensures that A is free of Dirac string singularities, as required by the equations of motion. We now show how solutions satisfying the above constraints t in the boundary conditions (4.24){(4.25). We de ne the antiholomorphic Liouville stress tensor T (v) = 2 Using a doubling trick argument, one can show that T must be of the form T (v) = N X i=1 (v i vi)2 + (v i 1=vi)2 + v c i vi + v ~ c i 1=vi where the i are related to the particle masses through i = m8i (1 m8i ), and the ci; c~i are called accessory parameters. Their determination requires solving a certain monodromy problem, and amounts to the knowledge of a particular large c Virasoro vacuum conformal block [24]. One can show that the near-boundary behaviour of is determined by T = ln(1 jvj2) 6 1 e2i arg vT (ei arg v)(1 jvj2)2 + O (1 jvj2)4 : Using this and (5.2), (5.7), one nds that a coordinate transformation which brings the metric into the asymptotic form (4.18) is v = 1 y 2 + y 2 8 e ix + O(y3); T = (x+ + x ) + O(y2): 1 2 12We will provide more details on the derivation of the equations of motion in the next section. 13Our delta function is normalized as R d2v 2(v; v) = 2 R d(Rev)d(Imv) 2(v; v) = 1. In particular 2(v; v). (5.5) (5.6) (5.7) (5.8) (5.9) (5.10) (5.11) (5.12) The metric can be seen to t in the boundary conditions (4.24){(4.25) with = k 4 where the rightmoving functions P 0(x ) and L(x ) are given by P 0 = (eix ); L = + ke2ix T (eix ): Note that, as explained in [24], a further large coordinate transformation can bring these metrics into Brown-Henneaux boundary conditions, where g0 = dx+dx , provided that the sum of the point-particle masses is below the upper bound Pi mi 4 . This bound is essentially the black hole threshold: for a single particle in the origin of AdS3, the black hole threshold is m = 4 . In the current boundary conditions we don't encounter such an upper bound on the point-particle mass, but the causal structure of the boundary metric g0 naively becomes problematic when the total point-particle mass exceeds 4 . It's not clear however how serious a problem this is in the present context: for standard BrownHenneaux boundary conditions, the conformal class of the boundary metric, and therefore its causal structure, has physical meaning, but it is not clear what replaces this in the case of chiral boundary conditions. It will be useful below to work out the explicit solution to rst order in an expansion for small masses mi. As shown in appendix C, the result is The Liouville stress tensor is, to this order, ifLm; Lng = (m n)Lm+n + ifLm; Png = nPm+n ifPm; Png = 2 m m; n: (5.13) (5.14) (5.16) (5.17) (5.19) (5.20) (5.21) T = 1 8 X i mi(1 (v vi)2(1 jvij2)2 viv)2 + O(mi2): and P 0 and L are given by P 0 = 1 4 i L = k 4 + k 8 i The U( 1 ) current algebra and Virasoro charges are de ned as in [19] Pm = They were shown to satisfy the Dirac bracket algebra d'eimx + 2 P 0 ; d'eimx L (P 0)2 : (5.18) Lm = In particular, the zero modes are L0 = k 4 + k 8 X mi + O(mi2): i The next example we will consider is motivated by black hole physics, more speci cally the black hole deconstruction proposal for the construction of microstate geometries in supergravity with M2-brane sources [26]. After dimensional reduction to 3 dimensions, these involve AdS gravity coupled to an axion-dilaton eld [7{9] rather than the free complex scalar considered in the previous sections. The bulk action is now The axion-dilaton equation of motion following from this action is + i Im = 0: (5.22) (5.23) (5.24) (5.25) (5.26) (5.27) (5.28) Evaluated on our solutions, one nds Pm = Lm = 4 k i jvij2) jvij2)2 X n supf0;mg X n supf0;mg vi vi n n m ! n(n m)vin 1vin m 1 + O(mi2) ! = g(v): ary of AdS3, the nary value: A necessary condition to trust the e ective 3D description (5.25) is that, near the boundpro le describes a small uctuation around a large, constant, imagi= iV1 + t; t j j V ; 1 where V 1 is the volume of the internal Calabi-Yau space in 11D Planck units. From (5.25) and (5.26) we see that t behaves like a free scalar near the boundary, so that our earlier free scalar analysis is a good approximation near the boundary. Also, one sees that the stringy SL(2; Z) symmetry of the eld reduces to the periodicity t t+1 of the small uctuation. Furthermore, independent of whether we are near the boundary where (5.27) holds, the equation (5.26) shares, in a global AdS background, a set of solutions with the free scalar eld, namely those for which the two terms in (5.26) vanish separately. This is precisely the case for the singleton solutions When the bosonic system is embedded in a theory with `left-moving' supersymmetry (i.e. where the left-moving SL(2,R) factor gets enhanced to a supergroup), as is the case in the black hole deconstruction context, these solutions preserve the left-moving Once again, one can analyze which consistent boundary conditions for axion-dilatongravity allow for the solutions (5.28). This proceeds largely parallel to the free scalar-plusgravity system and we refer to appendix B for details. The asymptotic behaviour of the elds is unmodi ed and the resulting chiral boundary conditions are again (4.23){(4.25), with t replaced by . We are interested in solutions in the presence of W-brane source terms, which arise from M2-branes wrapped on an internal two-sphere. These are point-like objects in the e ective three-dimensional description which couple both to gravity and the axion-dilaton eld in the following way: HJEP07(21)49 1 is a sign factor, which in our conventions is one for M2-brane sources and minus one for anti-M2-branes. The one-form C is dual to the axion which is the real part of : 1 dC = (Im )2 ? d(Re ) and the second term in (5.29) represents a magnetic charge for the axion eld. The equations of motion are G (Im ) + (5.29) (5.30) (5.32) (5.33) (5.34) (5.35) 1 `2 G Im 1 2 T dd(Re ) = X qi Z 2 i X q i X q i i i Z Z dwi 3(x p dwi dwi 3(x 3(x p p xi(wi)) GIm p xi(wi)) xi(wi)) q x_ i x_ i G x_ x_ i i x_ i ? dxm (5.31) G x_ i x_ i : T (Im )2 G G G : where G is the Einstein tensor and T the matter stress tensor Our ansatz for the metric is once again (5.2),14 with v running over the unit disk, and should satisfy the Zamolodchikov-Zamolodchikov boundary conditions (5.6). We will once again take the W-branes to follow geodesics of constant v = vi, corresponding to helical curves in global AdS. The equations of motion reduce to X i q i 4Im 2(v vi; v vi) 14Our current conventions are related to those of [9] as: t = tthere=2; v = zthere; A = there=2; = there ln(Im ) + ln 2; qi = 2 qi there. X qi 2(v i vi; v vi) A = Im ln(Im ))dv : 1 2 = iV1 i 2 v 1 vi viv : The solution for the equation (5.34) for A is, up to an exact form which can be absorbed in a rede niton of t, given in terms of and as as one can check using (5.33){(5.37). For = 1, i.e. for M2-brane sources, the solution of equations (5.35){(5.37) for satisfying the condition (5.27) is precisely the multi-centered global vortex solution we encountered in paragraph (4.2): HJEP07(21)49 (5.37) (5.38) (5.39) (5.40) For anti-M2-branes ( = 1), one would nd a similar antiholomorphic solution. Now, let's consider the backreacted metric. Substituting (5.39) into (5.33), we see that, as long as mi 1; q i V 1 as is the case in the regime of interest, the rst order deviation from the AdS background is the same as that produced by point particles on helical geodesics with masses mi, discussed in the previous section. The solution for the metric to this order is simply given by (5.15), and the conserved asymptotic charges are given by (5.23), (5.24). 6 Discussion: a re ned deconstruction proposal In this work we studied boundary conditions in two-derivative scalar-gravity theories which allow for the class of solutions (1.1), and the symmetries preserved by them. We introduced chiral boundary conditions (4.23){(4.25) which include those solutions and follow consistently from a variational principle with a nite on-shell action. We also showed that these preserve the asymptotic symmetries of [19], which form a combined left-moving Virasoro and U( 1 ) current algebra. We didn't however work out the contribution of the scalar eld to the asymptotic charges, which for the pure gravity case [19] were computed in the formalism of [20, 21]. This contribution starts at the second order in an expansion in the scalar pro le, which goes beyond the approximation considered in this work. Obtaining well-de ned charges to all orders may involve a generalization of our boundary conditions since the scalar eld sources the logarithmic mode in the metric. The fact that we obtained a nite on-shell action stems us hopeful that this should be possible. Since the analysis in this work was motivated by the holographic interpretation of scalar-gravity solutions which arise in the black hole deconstruction (BHD) proposal, let us summarize what our results imply, in our view, for this proposal. In [26], certain D2brane con gurations were proposed to semiclassically represent microstates of a 4D stringy black hole, based on an appealing counting argument in the probe approximation which relies on their huge lowest Landau level degeneracy. Upon taking an M-theory decoupling limit, the backreacted BHD solutions become essentially the M2-brane solutions of section 5.2, with an additional constraint that the total M2-charge should vanish imposed by tadpole cancellation. In order to satisfy this constraint while preserving supersymmetry one is led to consider solutions of the type (5.39) where some of the qi are negative, so that Pi qi = 0. Such `negative branes', which have negative tension, were investigated in [32]. In addition, in the BHD con gurations an additional U( 1 ) Chern-Simons eld in the 3D theory is switched. This eld doesn't interact with the metric and axion-dilaton elds in the bulk and therefore our solutions for these elds are unmodi ed, but its presence does in uence the boundary theory as is familiar from discussions of holographic spectral ow [36, 37]. In appendix B, we show that adding a U( 1 ) Chern-Simons eld A and imposing suitable boundary conditions on it changes the value of in (4.24) as follows: The BHD solutions have [39] A0+ = 1 and g2 ++ = 1=4, and therefore belong to the theory with k = g2 ++ + 1 2 4 A0+ = 0: (6.1) (6.2) The same stringy black hole of the BHD proposal was of course originally studied in [27], and its microstates were identi ed as states in a dual CFT, the so-called MSW theory. Our work points to some tension in trying to interpret the BHD solutions as the bulk duals of the MSW microstates.15 We argued in 3 that, unless one takes an IR scaling limit of the theory in which the scalar has no bulk dynamics, it is not possible to t the BHD solutions within conformally invariant boundary conditions. Therefore it seems hard to maintain that the BHD solutions are the bulk duals of the black hole microstates in the MSW CFT or in a marginal deformation thereof. On the other hand, we observed that the BHD solutions do naturally t in a generalization of the chiral boundary conditions of [19], which still allow for a right-moving Virasoro algebra together with a U( 1 ) current algebra. On the bulk side, the new boundary conditions arose from adding additional nite boundary terms to the action, and one expects therefore that the dual eld theory also arises from some deformation of the MSW theory, and it would be interesting to make this more precise. Letting components of the boundary metric uctuate typically means coupling the dual theory to gravity [40, 41], but it is at present unclear how the chiral boundary conditions of [19] arise in this manner. Since the chiral theory still contains an in nite number of conserved charges, it is also possible that it could interpreted as a type of `integrable' deformation [38] of the MSW theory. The proposed chiral boundary conditions still allow for the extremal BTZ black hole with M + J = 0, of which the BHD solutions are the proposed microstates: it is easy to see [19] this solution also lives in the = 0 theory and has P 0 = 0; L = M . These considerations suggest that chiral theory, which includes both the BHD solutions and the 15Another objection to such an interpretation was raised in [30]. extremal BTZ, is the proper setting to interpret the BHD proposal holographically. The main open question which remains is whether the BHD solutions, which have extra structure or hair encoded into the other Virasoro and current algebra modes, can realize the same zero mode charges as the extremal BTZ geometry, and if so, if there is a su ciently large moduli space16 of them to account for the entropy. To answer this question it would also be of interest to study the extra asymptotic structure emerging when our chiral boundary conditions are generalized to the additional elds present in the 3D (4; 0) supergravity theory which governs the black hole deconstruction setup. Acknowledgments It is a pleasure to thank I. Bena, N.S. Deger, M. Guica, O. Hul k, M. Porrati, T. Prochazka, D. Turton and B. Vercnocke for discussions and useful suggestions. The research of JR was supported by the Grant Agency of the Czech Republic under the grant 17-22899S, and by ESIF and MEYS (Project CoGraDS - CZ.02.1.01/0.0/0.0/15 003/0000437). DVdB was partially supported by the Bogazici University Research Fund under grant number 17B03P1. This collaboration was supported by the bilateral collaboration grant TU BITAK 14/003 & 114F218. JR would like to thank the Galileo Galilei Institute for Theoretical Physics (GGI) for the hospitality and INFN for partial support during the completion of this work, within the program \New Developments in AdS3/CFT2 Holography". A Explicit massless scalar modes After the separation of variables (2.8), the Laplace equation (2.2) on AdS3 is equivalent to the standard hypergeometric equation d 2 z(1 z) dz2 gh;h(z) + [c (a + b + 1)z] d dz gh;h(z) ab gh;h(z) = 0 with c = 0, a = h, b = h and the de nitions fh;h(y) = 4 4 + y y h h gh;h(z) The asymptotic solutions (2.11) are thus extended to the bulk in terms of the solutions of this hypergeometric equation: z = 16y (4 + y)2 16y (4 + y)2 (A.1) (A.2) (A.3) (A.4) (A.5) th;h = th;0; = 16y e i(hx++hx ) (4 + y)2 4 4 + y y h h 1 1 2 2 e ihx e ihx+ 4 4 4 + y 4 + y y h y h + + 1 + 1 + 2 2 2F1(1 + h; 1 h; 2; e ihx e ihx+ 4 4 4 + y h 4 + y h y y 16It will be important to take into account the lowest Landau level degeneracy in the internal space, which results in a large number of M2-particle species in the e ective 3D description. gh;h(z) = gh;h(z) = ah;h = gh;h(z) = ah;h = gh;h(z) = ah;h = th;h = e i(hx++hx ) ah;h = hh(2 1) (1 + h) (1 (1 + h y 4 + y h + h) + hh(2 z(1 1) h h h 2 h h h 2 2 ( 1 )h (1 + h) (1 + h) (1 + h + h) ( 1 )h (1 h + hh(2 h) (1 + hh(2 1 1 i ) i ) z)h h 2F1(1 h; 1 + h; 1 h + h; 1 z) z h 2F1(h; 1 + h; 1 + h + h; z 1) h) zh 2F1( h; 1 h; 1 h h; z 1) when h; h > 0 when h; h > 0 (A.7) gh;h(z) + ah;h th;h z 2F1(1 + h; 1 h; 1 + h h; 1 z) (A.6) Note that the gh;h are analytic series around small 1 z or z 1, not around z = 0 which is the boundary. This way of presenting those functions is more compact, and one can recover the asymptotic expansion around z = 0 via a standard reference on hypergeometric functions, e.g. [42], or a suitable computer algebra system. Those near boundary expansions will have the form of a product of log z with a hypergeometric function in z, plus some additional power series in z. The small z expansion can then be translated to the small y expansion. The leading terms of small y expansion of the precise linear combination appearing in (A.6) coincide with those given in (2.11). B Details on holographic renormalization In this appendix we provide more details on the holographic renormalization procedure for the gravity- scalar system. We will simultaneously treat the cases of a free complex scalar and an interacting axion-dilaton, and will also include a U( 1 ) Chern-Simons eld which is present in the black hole deconstruction setting. The latter does not couple to the other elds in the bulk, but can in uence the asymptotic charges as is known from the holographic realization of spectral ow [36, 37]. We start from the action for a complex scalar minimally coupled to 3D gravity with negative cosmological constant and a U( 1 ) Chern-Simons eld: S = Z M d x 3 p G R + 2 2 l + l 2 A ^ dA 2 Z K: (B.1) The constant lets us interpolate between an axion-dilaton eld for = 1 and a free scalar, which is obtained upon taking the limit ! 0 with t = i xed: (B.2) The equations of motion following from (B.1) are + i = 0; R 2 + 2 G 2 2(Im )2 = 0; dA = 0: We use Fe erman-Graham coordinates in terms of which the metric looks like The near-boundary expansion of the elds is then [18] HJEP07(21)49 (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) (B.9) (B.10) (B.11) (B.12) where the coe cient functions on the r.h.s. are independent of y. Substituting these in the equations of motion (B.3) and working out the leading terms one nds that the logarithmic coe cients g~2; ~2 are completely determined by the boundary values g0; 0: ~2 = 4 Im 0 where indices are raised and covariant derivatives taken with respect to the boundary metric g0. We note that g~2 ij is traceless. For the tensor g2 ij on the other hand, only the trace and divergence are xed in terms of g0; 0: g2 = R0 + 1 2 rj (g2ij g2g0ij ) = 2(Im 0)2 Proceeding as in [18], we regularize the action by cutting o the y integral at y = One nds for the regularized on-shell action Using (B.7) one derives that this contains the following divergent terms as ! 0: Sreg = 2l Z d2x Z dy y2 g + 2 p y g y= Sdiv = l d2xp 2 + g( 2 ) log We propose to add the following additional boundary terms to the action: dy2 gij = g0 ij + y g2 ij + y log y g~2 ij + O(y2 log y) A = A0 + O(y); Z 2 1 2 R 2 2(Im )2 1 2Im log 1 + Sbnd = l d x 2 p g Z (B.13) The role of rst line is to cancel the divergences of (B.12), while the second line serves to get a good variational principle under the boundary conditions to be speci ed below. It breaks boundary covariance and depends on a xed, symmetric, lower index two-tensor wij and a vector vi which will be speci ed below. Note that, unlike in the treatment of e.g. [35], we have not added any boundary terms for the Chern-Simons eld A. The variation of the total action then reads, up to bulk terms proportional to the equations of motion (B.3), Z (S + Sbnd) =l d x 2 p g0 ( g2 ij g~2 ij + g2g0 ij + 1 2 1 2Im 0 2 + 2(Im 0)2 Re 1 + 2p g0 (A0+ A0 1 + A0 A0+) Guided by [19] for the choice of wij and the choice (4.3) for vi in a xed background, we will take vi = i ; wij = + + k i j : We now propose our boundary conditions: 0 = t0(x ); g0 g0 + = 1 2 A0+ = constant; g2 ++ + 2 = 41 A02+ = g0 ++ = 0 i 0 2 0 A0 = A0+P 0(x ): 2 wg0 ij ij (B.14) (B.15) (B.16) (B.17) (B.18) (B.19) (B.20) (B.21) where k 4 l and that the value of is a xed constant which speci es the theory, i.e. = 0. We note changes if we turn on the Chern-Simons eld A. This is analogous to the bulk realization of spectral ow [36, 37] in the case of conformal boundary conditions. Furthermore, the equations following from (B.9) imply g2+ = k P 0 + 41 A02+; 4 2(Im 0)2 i j These boundary conditions (B.16){(B.19) lead to a good variational principle, as one can check that plugging into (B.14) one obtains (S + Sbnd) = 0. They are also invariant under the asymptotic symmetries generated by (4.7), (4.8), with the free functions transforming as U P 0 = U 0 (B.22) V 0 = V 00 C First order solution 2 k 4 U 0 = 0 (B.23) (B.24) In this appendix we construct the backreacted metric of a collection point particles on helical geodesics, to rst order in an expansion in the (assumed small) mass parameters mi. From (5.5) we nd that the rst order correction of the Liouvile eld should satisfy 4 i 2e 2 AdS 1 = vi; v vi); (C.1) HJEP07(21)49 where AdS is the global AdS3 solution (5.3). One way to solve this equation would be to use a Green's function but we will use here a simpler method using conformal mapping. We start by constructing the solution for 1 corresponding to a single mass m in the origin z = 0. As explained in [9], this can be found by using rotational symmetry to reduce the problem to a second order ordinary di erential equation, and to x the integration constants to get the correct delta-function source in (C.1) and the desired near-boundary behaviour 1 ! 0 for jvj ! 1. This leads to To nd the solution corresponding to a point mass away from the origin, say in v = v1, we apply the Mobius transformation which maps the origin to v1: The AdS solution AdS is essentially the conformal factor in front of the metric on the Poincare disk, e 2 AdS dzdz. The Mobius transformation (C.3) is an isometry of this metric and hence leave it form-invariant, which amounts to the following -transformation 1 = m 4 1 + 1 + jjvvjj22 ln jvj : 1 v ! w = v 1 v1 vv1 : e 2 AdS(w) = e 2 AdS(v) as one can easily check from the explicit expression (5.3). 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Joris Raeymaekers, Dieter Van den Bleeken. Chiral boundary conditions for singletons and W-branes, Journal of High Energy Physics, 2017, 1-33, DOI: 10.1007/JHEP07(2017)049