A Below the BF bound in global AdS

Beyond the unitarity bound in AdS / CFT

Abstract

In this work we expand on the holographic description of CFTs on de Sitter (dS) and anti-de Sitter (AdS) spacetimes and examine how violations of the unitarity bound in the boundary theory are recovered in the bulk physics. To this end we consider a Klein-Gordon field on AdS conformally compactified such that the boundary is (A)dS, and choose masses and boundary conditions such that the corresponding boundary operator violates the CFT unitarity bound. The setup in which the boundary is AdS exhibits a particularly interesting structure, since in this case the boundary itself has a boundary. The bulk theory turns out to crucially depend on the choice of boundary conditions on the boundary of the AdS slices. Our main result is that violations to the unitarity bound in CFTs on dS and AdS are reflected in the bulk through the presence of ghost excitations. In addition, analyzing the setup with AdS on the boundary allows us to draw conclusions on multi-layered AdS/CFT-type dualities.

1 Introduction

The AdS/CFT correspondence relates the dynamics of the fields in a -dimensional gravitational theory with asymptotically anti-de Sitter (AdS) boundary conditions to that of operators in a non-gravitational -dimensional conformal field theory (CFT) on the boundary [1, 2, 3]. The prime examples involve gravity on global and Poincaré AdS, which are dual to CFTs on the cylinder and the plane, respectively. There are, however, also cases of interest beyond these dualities with flat-space CFTs, since the study of quantum field theory in curved spacetimes is of general interest and it is natural to approach it holographically. Moreover, CFTs on manifolds with boundary (BCFT) [4, 5] have received attention recently, e.g. in the context of brane configurations with branes ending on branes [6, 7, 8]. In this work we will consider certain aspects of the holographic study of CFTs defined on the maximally symmetric de Sitter (dS) and AdS spacetimes. This is not only a natural first step from flat to generic curved spacetimes, but also provides a link to BCFT, since the global AdS on which the CFT is defined is conformally related to half of the Einstein static universe. Furthermore, the case with AdS on the boundary offers an interesting possibility for multi-layered AdS/CFT dualities.

The holographic description of a CFT on a specific background involves gravity on an asymptotically-AdS space with that prescribed boundary structure. The geometries for a dual description of CFTs on dS and AdS have been discussed recently in [9] and [10], respectively, and earlier related works can be found in [11, 12, 13, 14, 15, 16]. It is sufficient in these cases to choose different coordinates on global AdS such that it is sliced by (A)dS hypersurfaces, and perform the conformal compactification adapted to these coordinates. For the AdS slicing this results in two copies of AdS on the boundary and a single AdS boundary is obtained by taking a quotient of AdS. The bulk theory then depends on boundary conditions on the hypersurface which is fixed under the action, and the resulting geometry resembles the general construction for BCFT duals outlined in [17, 18].

Facilitated by the matching of bulk isometries and boundary conformal symmetries, the AdS/CFT correspondence provides a concrete map between the bulk and boundary Hilbert spaces. For a free scalar field with mass on AdS with unit curvature radius there are in principle two dual operators with conformal dimensions , up to corrections. This is related to the fact that solutions to the second-order Klein-Gordon equation are characterized by two asymptotic scalings near the conformal boundary. Imposing boundary conditions such that the slower/faster fall-off is fixed, which we shall refer to as Dirichlet and Neumann boundary conditions below, yields a bulk field dual to an operator of dimension / [19, 20]. Note that the conformal dimensions are real so long as the Breitenlohner-Freedman (BF) stability bound [21, 22] is respected. For Dirichlet and Neumann boundary conditions yield well-defined theories [21, 22], and in fact even more general boundary conditions can be imposed [23, 24]. On the other hand, as noted in [19, 20], Neumann boundary conditions for lead to , in conflict with unitarity bounds in the CFT [25, 26, 27]. Consequently, the freedom in the choice of boundary conditions was expected to break down for . This expectation was recently confirmed for global and Poincaré AdS in [28]. A crucial point is that normalizability of the Neumann modes requires a modification of the symplectic structure [29], sacrificing manifest positivity of the associated inner product. Interestingly, the pathologies in the bulk theory show up in different ways for the two cases. While in global AdS the Neumann theories contain ghosts for , such that unitarity in the bulk is explicitly violated, in Poincaré AdS there is no manifest violation of bulk unitarity. Instead, the 2-point function for the Neumann theories is found to be ill-defined even at large separations.

In this article we take a further step towards a holographic understanding of (A)dS CFTs. We consider a scalar field with on AdS and choose coordinates and compactification such that the boundary is (A)dS. Imposing Neumann boundary conditions in this mass range is dual to a CFT on (A)dS with an operator of scaling dimension . We will investigate the precise way in which this violation of the CFT unitarity bound is reproduced by the dual bulk theory. In the setup with the boundary CFT defined on AdS, the bulk theory depends not only on the boundary conditions on the AdS conformal boundary, which we refer to as Neumann/Dirichlet, but also on the orbifold boundary conditions and on the boundary conditions on the boundary of the AdS slices, referred to as Neumann/Dirichlet in the following. Furthermore, due to the fact that the AdS boundary itself has a conformal boundary, the structure of divergences is more involved than for global or Poincaré AdS. Thus, in order to properly deal with this configuration we have to adapt the well-established procedure of holographic renormalization [30, 31, 32]. The choice of Neumann/Dirichlet turns out to be quite crucial. For Dirichlet the adaption of regularization and renormalization is straightforward, and we find the complete sets of Dirichlet and Neumann modes normalizable with respect to the renormalized inner product. On the other hand, our construction of the theory with Neumann boundary condition leads to a drastically reduced spectrum of normalizable modes, making the AdS theory equivalent to an AdS theory in a trivial way. This will allow us to draw some conclusions on the possibility of multi-layered holographic dualities, which were speculated to arise for boundaries with negative cosmological constant in [29]. The setup with dS on the boundary, on the other hand, is obtained from global AdS by a coordinate transformation which merely results in a rescaling of the boundary metric, such that this setting is more closely related to global AdS. However, the dS slicing covers only a patch of AdS bounded by a horizon, analogous to the Lorentzian Poincaré AdS. We will investigate whether there is a similarly tricky manifestation of the pathologies as found for Poincaré AdS in [28].

The paper is organized as follows. In section 2 we introduce the setups for a holographic description of CFTs on (A)dS and give the relevant properties of the Klein-Gordon field in these settings. Unitarity of the bulk theories for AdS and dS on the boundary is studied in sections 3 and 4, respectively, and we conclude in section 5. In an appendix we discuss a scalar field with tachyonic mass below the BF bound on global AdS.

2 (A)dS slicings of AdS

In this section we introduce the foliations of AdS that will be relevant for the subsequent analysis and discuss some generic features of the Klein-Gordon field in these coordinates. We consider AdS with curvature radius in global coordinates such that the line element takes the form

(2.1)

In the following we discuss coordinate transformations resulting in a metric of the form

(2.2)

with and the conformal boundary of AdS at . The slicing by dS hypersurfaces with Hubble constant is obtained by the coordinate transformation with and

(2.3)

The resulting metric is of the form (2.2) with

(2.4)

Note that (2.3) implies , which restricts the range of to . The coordinates therefore cover a patch as shown in figure 1. The patch is bounded by a causal horizon at , which is an infinite-redshift surface as vanishes there. The conformal boundary of the patch at is part of the AdS conformal boundary, and from (2.4) we see that the boundary metric at is that of global dS, as desired.

The foliation of AdS by AdS hypersurfaces with curvature radius is obtained from the transformation with , and

(2.5)

The resulting metric again is of the form (2.2) but with

(2.6)

As we have to choose the domain for the sine in the 2 equation in (2.5) to be either or we need two patches to cover the full AdS. The patches are ‘joined’ at , the equator of . This is realized in [10] by letting run on and choosing the appropriate domains for on the two half lines. To obtain a holographic description of a CFT on a single copy of AdS we consider the quotient of global AdS identifying the two patches, as discussed in [10]. This quotient is covered by the coordinates discussed above for any of the two choices for the domain of . In turn, this implies that the fields under consideration should have definite parity, which imposes boundary conditions at , as will be discussed in section 3.1. Furthermore, note that the resulting single copy of AdS at the conformal boundary of AdS has itself a conformal boundary, which, in the coordinate system (2.6), corresponds to the locus .

[

[

Figure 1: The slicing of global AdS by (A)dS hypersurfaces. 1 shows the Poincaré disk representation of AdS sliced by AdS in the coordinates used in (2.2), (2.6). Horizontal/vertical curves have constant /. The boundary consists of two copies of AdS joined at their boundaries at . 1 shows the dS slicing (2.2), (2.4) of AdS as cylinder with radial coordinate and the part suppressed. Horizontal/vertical curves have constant /.

The setup for a holographic description of CFTs on AdS discussed above resembles the holographic description for generic BCFT proposed in [17, 18]. For a CFT on a -dimensional manifold with boundary it was proposed there to consider as dual a gravitational theory on a -dimensional asymptotically-AdS manifold with conformal boundary and an additional boundary , such that . The AdS slice at in our setup corresponds to , the -fixed hypersurface at to , and imposing even/odd parity translates to Neumann/Dirichlet boundary conditions on . This similarity of the setups can be understood as a consequence of the relation of CFTs on AdS to BCFTs discussed in the introduction.

2.1 Klein-Gordon field

We consider a free, massive Klein-Gordon field on AdS foliated by (A)dS and discuss the features that apply to both slicings in parallel. Our starting point is the ‘bare’ bulk action for a free scalar field,

(2.7)

which will later be augmented by boundary terms. For a metric of the form (2.2) the resulting Klein-Gordon equation reads

(2.8)

We separate the radial and transverse parts by choosing the ansatz , such that are the modes on the (A)dS slices and are the radial modes. Introducing as separation constant, (2.8) separates into the radial equation

(2.9)

and the (A)dS hypersurface part . The latter is a Klein-Gordon equation for the transverse part with ‘boundary mass’ . Note that (2.9) can be written in Sturm-Liouville form,

(2.10)

Fixing , and reproduces (2.9) with . The inner product defined from the ‘bare’ symplectic current associated to (2.7) is the standard Klein-Gordon product

(2.11)

Choosing for an (A)dS slice, such that , the inner product (2.11) can also be factorized. In fact, with such that is normalized w.r.t. , (2.11) becomes

(2.12)

where is the Klein-Gordon inner product on the (A)dS slice and is the Sturm-Liouville inner product

(2.13)

Using partial integration and (2.9) yields1

(2.14)

The inner product (2.11) is finite and conserved for Dirichlet and Neumann boundary conditions if . However, for larger masses, the holographic renormalization of the bulk action introduces derivative terms on the boundary, which in turn induce a renormalization of the inner product [29]. We shall discuss this issue in detail in section 3.

2.2 Asymptotic solutions

The covariant boundary terms introduced by the holographic renormalization of the bulk theory are crucial for the construction of the renormalized inner product. The construction of these terms involves the asymptotic expansion of the on-shell bulk field, which we shall now discuss. The relevant computations are most conveniently carried out with the metric in Fefferman-Graham form. For the dS slicing (2.2), (2.4) this form is obtained by the coordinate transformation , resulting in the metric

(2.15)

Likewise, for the AdS slicing (2.2), (2.6) the transformation yields

(2.16)

The conformal boundary of AdS is at in both cases. The asymptotic expansion of in these coordinates is obtained by solving the Klein-Gorden equation expanded around the conformal boundary. With we obtain

(2.17)

where have regular power-series expansions around , and in particular

(2.18a)
(2.18b)

Here we have defined  , with the curvature of the hypersurface metric . The curvature convention is such that and .

3 AdS on the boundary

In this section we study the case of AdS on the boundary. After setting up the regularization and renormalization procedure we discuss the Dirichlet theory in the mass range dual to a CFT beyond the unitarity bound and discuss the special properties of the Neumann theories.

3.1 Renormalization and boundary conditions

We consider the AdS slicing of AdS using the coordinates such that the metric is of the Fefferman-Graham form (2.16). The action (2.7) evaluated on-shell is divergent as a power series in a vicinity of the boundary at , and we also expect divergences from . To renormalize the divergences we introduce cut-offs at , and boundary counterterms to render the asymptotic expansions in , finite as the cut-offs are removed by . The form of the cut-offs is the standard prescription adapted to the current slicing, and is illustrated in figure 2.

[

[

Figure 2: The boundaries of the regularized geometries. 2 shows the regularization of the AdS slicing discussed in section 3.1. The intersection of and is . 2 shows the regularization of the dS slicing discussed in section 4.

We use the notation and parametrize the boundary of the regularized as follows: is the hypersurface which is invariant under the orbifold action, denotes the (regularized) AdS part at large , consists of the boundaries of the AdS slices and is the boundary of , see figure 2.

We briefly discuss the boundary conditions to be imposed on the various parts of the boundary. On the -fixed part at /, definite orbifold parity demands either vanishing function value or vanishing normal derivative . In view of the decomposition discussed in section 2.1, this places restrictions on . Further restrictions are imposed on by Dirichlet/Neumann or more general mixed boundary conditions at /. For non-Dirichlet boundary conditions and , the inner product needs to be properly renormalized, as usual. On the remaining part, which is the boundary of the AdS slices at , Dirichlet/Neumann or mixed boundary conditions can be imposed. We focus on Dirichlet first and discuss non-Dirichlet boundary conditions in section 3.3. Finally, regularity and normalizability at the origin of the AdS slices at places restrictions on the AdS modes . This condition is satisfied by choosing for the modes discussed in [20], which decay as a power-law at the origin. In the following we set .

Focusing on Dirichlet boundary conditions we now construct the counterterms that render the action finite and stationary when both the equations of motion and the boundary conditions hold. Using integration by parts and dropping terms which vanish by orbifold parity and normalizability at the action (2.7) reads

(3.1)

The volume forms are suppressed throughout, they are the standard forms constructed from the (induced) metric on the respective (sub)manifold. Note that both terms are divergent for . The familiar divergence of the first term has to be cancelled by counterterms on . The second term is divergent due to the integral over . Expanding the integrand in and performing the integral order by order we isolate the divergent part, which is to be cancelled by a counterterm on . For we find the counterterms

(3.2a)
where is the Lie derivative along , which is the outward-pointing normalized vector field in normal to . is defined below (2.18) and is the induced metric. For the second term in the first integral in (3.2a) is absent, such that the boundary terms do not contain derivatives. For we find
(3.2b)

Here we have included, with an arbitrary coefficient , a combination of boundary terms which is compatible with all bulk symmetries and finite for . Note that invariance under radial isometries, corresponding to conformal transformations on the boundary, is broken by the terms. We also emphasize that partial integration in the counterterms has to be carried out carefully, since e.g.  itself has a boundary. The counterterms also enter the symplectic structure and the associated inner product. Following [29], we find

(3.3a)
(3.3b)

For calculating CFT correlation functions we need the variations of the action to be finite when evaluated on-shell. The variation of reads

(3.4)

The first boundary term vanishes for solutions with definite parity. The integral is divergent for and the remaining part is

(3.5a)
(3.5b)

for and , respectively. The terms are divergent for and combine with the divergent term in (3.4) to render the variation finite as we remove the cut-off on . For fixed Dirichlet boundary conditions there are no divergences for , such that the limit is finite and independent of the order in which the limits are performed. Thus, we have renormalized the theory such that we have finite variations with respect to the boundary data at , while keeping fixed Dirichlet boundary conditions at . This allows to compute correlators for the dual CFT on AdS with fixed boundary conditions.

In summary, the renormalized action is stationary for solutions of the Klein-Gordon equation with Dirichlet boundary conditions, provided they have definite parity such that the integral in (3.4) vanishes and satisfy either the Dirichlet condition or the Neumann condition

(3.6)

such that the integral in (3.5) vanishes. The remaining finite combination of the and integrals vanishes for Dirichlet boundary conditions. This can be seen as follows, expanding

(3.7)

where is defined in (3.8), and using the fixed Dirichlet boundary condition , bilinears in , scale at least as . does not decrease the order in and the volume forms on , are . Thus, the overall scaling is with a positive power of and as the integrations are performed for fixed the integrands vanish for .

3.2 Dirichlet beyond the unitarity bound

With the renormalization set up in the previous section, we now study the bulk theory in the mass range corresponding to a CFT with an operator violating the unitarity bound. We use the decomposition discussed in section 2.1 and determine the spectrum from the boundary conditions at and parity, which impose restrictions on the radial profiles . This yields a quantization condition on the ’AdS mass’ introduced in section 2.1, which we parametrize by a complex parameter as

(3.8)

Note that modes with respect the AdS BF bound. We start with non-integer and discuss the case separately. For completeness we discuss both Neumann and Dirichlet boundary conditions, but of course expect unitarity violations only for the former.

The two independent solutions to the radial equation (2.9) for non-integer are given by

(3.9)

where are the generalized Legendre functions. For the discussion of Dirichlet and Neumann boundary conditions we use the radial variable , see section 2.2. The asymptotic expansions of the radial modes (3.9) around the conformal boundary at are given by , where the ellipsis denotes subleading terms of integer order. Hence, we conclude that modes with radial profile / satisfy Neumann/Dirichlet boundary conditions. Imposing definite parity translates to the conditions for odd and for even parity. For the modes (2.9) we have

(3.10a)
(3.10b)

The expressions on the right hand sides vanish when the appropriate -functions in the denominator have a pole, which is for non-positive integer arguments. The spectrum can therefore be read off from (3.10a) for odd and (3.10b) for even parity, which yields

(3.11)

where and for even and odd parity, respectively. Note that these are real, such that the transverse modes of the bulk field with Dirichlet/Neumann boundary conditions do not violate the AdS BF bound. However, there can be modes with which saturate the BF bound for half-integer and Neumann boundary condition.

For a concrete realization of the transverse modes we use the AdS modes discussed in [20]. Imposing normalizability at the origin and boundary conditions on the conformal boundary of the AdS slices yields a quantization of their frequencies depending on . For the Dirichlet case, all the modes are normalizable with respect to the usual symplectic structure and the frequencies are given by

(3.12)

where denotes the principal angular momentum. Note that the subscripts in (3.12) refer to Dirichlet/Neumann boundary conditions on the conformal boundary of AdS. For the case of Neumann to be discussed in section 3.3, the frequencies are given by (3.12) with .

For Dirichlet boundary conditions the AdS norms are positive [28], so the existence of ghosts depends only on the norms of the radial modes, which we now calculate. With the decomposition the renormalized inner product (3.3a) reads  , where the renormalized SL product is given by

(3.13)

We evaluate (3.13) using (2.14) and the modes / given in (3.9) which satisfy Neumann/Dirichlet boundary conditions for all . The divergence of the bare SL product for Neumann and is cancelled by the boundary term, such that the inner product is finite. Furthermore, the finite contribution from vanishes for all if Neumann/Dirichlet boundary conditions are satisfied. The inner product thus evaluates to

(3.14)

Note that the term in parenthesis on the right hand side vanishes if orbifold boundary conditions are satisfied, and therefore modes with different boundary mass are orthogonal. The expression (3.14) as it stands is not defined for . However, it can be extended continuously to coinciding masses given by (3.11), since in that case the term in parenthesis vanishes as well. Defining , the inner product for AdS fields with Dirichlet/Neumann boundary conditions reads

(3.15)

For the Dirichlet modes with even/odd parity we find

(3.16)

As expected, these are positive for all and . Thus, since is non-negative for Dirichlet boundary conditions, the spectrum is ghost-free. Similarly, for Neumann boundary conditions we find the norms

(3.17)

which are positive for as expected. For we first consider . If , where denotes the integer part, is even, the mode has negative norm since the coefficient of the -function in the denominator is negative while the -function itself is positive. If is odd the mode is of negative norm since the coefficient is positive while the -function is negative. As is non-negative we thus have ghosts in the spectrum for non-half-integer in , such that the non-unitarity of the dual boundary theory is nicely reflected in the bulk. For , the AdS modes have integer and by continuity of (3.17) there are modes with vanishing or negative norm. The and modes are of norm zero and yield the same unless with odd parity2. This degeneracy in the spectrum indicates that the basis of solutions we are using is incomplete, so we expect ‘logarithmic modes’, analogous to those in [33]. Once the log-modes are incorporated, continuity of the spectrum indicates that ghosts must be present [34]. For with odd parity the mode is of negative norm and the others are positive, such that the non-unitarity of the dual theory is reproduced in the bulk.

We close this section noting that the results established explicitly here for extend to higher even without knowledge of the exact counterterms. We consider the expansions of the Dirichlet and Neumann bulk fields near the conformal boundary

(3.18)

where is a non-negative integer. For the only way to get boundary terms which are quadratic in the field and have an integer scaling – finite terms, in particular – is through the combination . However, since and derivative/curvature-terms are subleading with even powers of , while the volume form is , only the boundary term without derivatives can yield a finite part. This implies that there are no extra finite contributions to the norm from the additional boundary terms. For half-integer the combination of two fields with the volume form scales as an odd power of , so such terms can again not yield finite contributions. The results (3.16), (3.17) are therefore also valid for generic .

Saturating the unitarity bound

We now consider the special case , corresponding on the boundary to an operator which saturates the unitarity bound3. The modes (3.9) are not linearly independent for integer , so we instead use the basis of radial profiles

(3.19)

where and , . Since , and , , the modes are independent and /