Instantons and the Hartle-Hawking-Maldacena proposal for dS/CFT

Abstract

Abstract

We test the Maldacena proposal for the Hartle-Hawking late time quantum state in an asymptotically de Sitter universe. In particular, we calculate the on-shell action for scalar instantons on the southern hemisphere of the four-sphere and compare the result with the renormalized on-shell action for scalar instantons in EAdS^sub 4^. The two results agree provided the corresponding instanton moduli as well as the curvature radii are analytically continued. The instanton solutions in de Sitter are novel and satisfy mixed boundary conditions. We also point out that instantons on S ^sup 4^ calculate the regularized volume of EAdS^sub 4^, while instantons on EAdS^sub 4^ calculate the volume of S ^sup 4^, where the boundary condition of the instanton in one space is identified with the radius of curvature of the other. We briefly discuss the implications of the above geometric property of instantons for higher-spin holography.

Published for SISSA by Springer Received: July 15, 2014 Revised: October 5, 2014 Accepted: October 24, 2014 Published: November 24, 2014

Instantons and the Hartle-Hawking-Maldacena proposal for dS/CFT

Sebastian de Haroa and Anastasios C. Petkoub

aInstitute for Theoretical Physics and Amsterdam University College,

University of Amsterdam Science Park 113, 1098 XG Amsterdam, The Netherlands

bInstitute of Theoretical Physics, Aristotle University of Thessaloniki,

54124 71003 Thessaloniki, Greece

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We test the Maldacena proposal for the Hartle-Hawking late time quantum state in an asymptotically de Sitter universe. In particular, we calculate the on-shell action for scalar instantons on the southern hemisphere of the four-sphere and compare the result with the renormalized on-shell action for scalar instantons in EAdS4. The two results agree provided the corresponding instanton moduli as well as the curvature radii are analytically continued. The instanton solutions in de Sitter are novel and satisfy mixed boundary conditions. We also point out that instantons on S4 calculate the regularized volume of EAdS4, while instantons on EAdS4 calculate the volume of S4, where the boundary condition of the instanton in one space is identied with the radius of curvature of the other. We briey discuss the implications of the above geometric property of instantons for higher-spin holography.

Keywords: AdS-CFT Correspondence, Models of Quantum Gravity

ArXiv ePrint: 1406.6148

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP11(2014)126

Web End =10.1007/JHEP11(2014)126

JHEP11(2014)126

Contents

1 Introduction 1

2 Instanton solutions in Euclidean AdS4 and dS4 32.1 The HH state and holographic partition functions 32.2 Action and solutions 42.3 Boundary conditions 52.4 On-shell action and the HH wave function 6

3 Geometric interpretation 8

4 Discussion and conclusions 11

A Explicit bulk solutions and holography 12

1 Introduction

The task of extending the holographic principle to an explicitly time-dependent, cosmological, setting proves to be as non-trivial as it is interesting. A particularly simple approach is Maldacenas proposal [1] (see also [2]) for the evaluation of the Harte-Hawking (HH) late time quantum state in an asymptotically de Sitter universe.1 The HH state is specied [7] by a Euclidean wave function of the schematic form:

HH[[vector]] =

Z[notdef]@M [notdef] D

eS[] , (1.1)

where @M is a 3-dimensional spacelike hypersurface near the future innity of an asymptotically de Sitter space-time with radius [lscript]dS. We have collectivelly denoted the bulk elds as . Then, Maldacenas proposal entails that the HH state can be obtained if one calculates the corresponding renormalized on-shell action on Euclidean AdS4 (EAdS4) with radius [lscript]AdS, and performs the analytic continuation [lscript]AdS ! i[lscript]dS. Since the EAdS4 on-shell action

is reasonably well dened it gives the partition function of a Euclidean 3-dimensional CFT Maldacenas proposal gives a way to make sense of and calculate the HH wave function from AdS/CFT. When analytically continued to Lorentzian signature, (1.1) gives the Bunch-Davies wave function in de Sitter space.

AdS/CFT works best when there is an explicit string theory realisation of the bulk physics. At present there is no satisfactory string theory description for gravity with a positive cosmological constant, but an alternative and more adventurous set up for a concrete realisation of dS/CFT was proposed in [8]. This is based on Vassilievs higher-spin

1Important earlier attempts for dS/CFT include [36].

1

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(HS) theory2 which provides the only known consistent classical description of interacting higher-spin gauge elds in a de Sitter background. Its suggested holographic dual is the Euclidean Sp(N) vector model with anti-commuting scalars and it is a free CFT3 when the higher-spin symmetry is unbroken. Since the quantum properties of free theories are known, the HS version of dS/CFT o ers some hope of understanding quantum gravity (plus HSs) on dS4. Moreover, an analytic continuation is also at work here, as N [mapsto]! N,

where N [lscript]2dS/GN, maps the Sp(N) anti-commuting vector model to usual (commuting)

O(N) vector model which is believed to be the holographic dual of HS theory on AdS4.

Notice that Newtons constant GN is held xed.

The Hartle-Hawking-Maldacena (HHM) proposal has been tested in a number of cases. The most explicit check in [10] involves pure gravity only: the on-shell action on Euclidean de Sitter (the four-sphere) is shown to analytically continue to the nite part of the volume of EAdS4, after dropping counterterms which are argued to give only imaginary phases which do not contribute to the measure [notdef] HH[notdef]2 [1, 10]. It has also been tested in the presence of scalars and gauge elds by showing that their generating functionals for two-point functions in EAdS4 and dS4 are related by suitable analytic continuations [2, 11, 12]. However, the HHM proposal and the relevant analytic continuation has not been tested for cosmological theories with matter elds and generic potentials. This would entail a discussion of exact non-trivial solutions which come with a moduli space, and then one wonders how the latter transforms under the analytic continuation in the HHM proposal. Also, it is important to know how boundary deformations (such as multiple trace deformations) of the CFT dual to EAdS4 carry over to de Sitter space, and whether some complexication of the deformation parameter is involved. Finally, one may ask whether there exists an analytic continuation at the level of elds rather than particular solutions. For recent important progress, see [13], where it was shown that holography correctly reproduces both the spectra and the non-gaussianities for general inationary space-times,i.e. for any potential that supports inationary FRW solutions.

In this short note we study exact solutions in a scalar theory conformally coupled

to gravity and check that the HHM proposal works, namely the HH state is given by an analytic continuation of the holographic EAdS4 partition function. These solutions are instantons: they are zero energy, exact solutions of the Euclidean equations of motion with nite action. Such solutions for scalar elds exist in EAdS4 [13, 19] and, as we show here, also in 4-dimensional de Sitter space. Further investigation of these solutions is relegated to a companion paper [20]. We test the HHM proposal for these solutions and nd exact agreement, provided the EAdS4 radius as well as the moduli of the solution and their boundary conditions are analytically continued. We also note there exists a simple geometric description of our results. In particular, the renormalized on-shell action of scalar instantons on EAdS4 evaluates the volume of the four-sphere, while the corresponding on-shell action of instantons on S4 evaluates the regularized volume of EAdS4. In both cases, the instanton moduli serve as regulators of the corresponding volume forms. The above geometric description allows us to interpret the on-shell action of scalar instantons as the

2For a recent review of HS theories see [9].

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free-energy of a theory on S3 and this in turn may have implications for the holography of HS theories.

2 Instanton solutions in Euclidean AdS4 and dS4

2.1 The HH state and holographic partition functions

A central object for the holographic principle is the properly renormalized bulk on-shell action. This can be evaluated as a functional of the boundary conditions for a suitably regular solution of the bulk equations of motion. In this case, depending on the choice of boundary conditions it yields either a generating functional for quantum correlation functions or an e ective action for a putative CFT living on the boundary.

However, given a particular solution of the bulk equations of motion, namely one where the bulk elds assume xed boundary values, the renormalized bulk on-shell action evaluates the free energy of the boundary theory and therefore the partition function as:

Z = eF eSon-shell . (2.1)

For example, the renormalized Einstein-Hilbert action using the Poincar patch of EAdS4 gives zero, while the corresponding quantity evaluated using a global parametrization of the bulk metric with conformal boundary metric S3 yields the non-zero free energy of a CFT on S3.

In the presence of bulk matter elds, things are less clear as exact solutions of the corresponding nonlinear equations of motion are uncommon. Notable exceptions are instantons in EAdS4 [13, 19]. In particular, as we we review in appendix A, for conformally coupled scalars in EAdS4 with boundary behaviour (z, [vector]x) ! z (0)([vector]x) + z2 (1)([vector]x), where (0)([vector]x)

takes a xed form, the bulk on-shell action Son-shell[(0)] = ln

0 gives the logarithm of the partition function0 of the dual boundary CFT, namely the CFT having in its spectrum the operator O1 of dimension = 1 and [angbracketleft]O1[angbracketright] (0). Moreover, having this result one

can calculate by a Legendre transform the partition function of the usual boundary CFT, namely the one having an operator O2 of dimension = 2 and [angbracketleft]O2[angbracketright] (1). In [19] the

partition function0 was interpreted as giving the probability for the nucleation of the instanton vacuum in the boundary theory.

Regarding now the HH state for an asymptotically dS4 universe, a simple example arises in the absence of matter elds when it is given by the on-shell value of the EH action with a positive cosmological constant. After the analytic continuation of dS4 to the 4-sphere, the result is nite and proportional to the volume of S4. It is then easily seen [10] that this is analytically continued to the nite part of the holographic partition function on S3. As in the EAdS4 case, non-trivial results for the HH state in the presence of bulk matter are uncommon. In this note we will improve on this situation by providing results for dS4 instantons which, as we will see, are intimately related to the usual EAdS4 ones.

Our main aim is therefore to evaluate both the HH state and the holographic partition functions of systems involving gravity and matter elds and to test whether they are still related by analytic continuation. In doing so, we will obtain some interesting new results.

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2.2 Action and solutions

In the context of AdS/CFT there are few examples where particular exact solutions of the bulk equations of motion of gravity coupled to scalars are known. Exact solutions are needed in order to calculate the exact partition function of the CFT. A non-trivial example was studied in [13, 19]. In this paper we write down similar solutions for de Sitter space and use them to test the HHM proposal.

The Euclidean bulk action we will consider contains the Einstein-Hilbert term, a conformally coupled scalar with a 4 potential, and the Gibbons-Hawking term modied by a coupling to the conformal scalar. It should be noted from the outset that for < 0 and the special value of the quadratic coupling = cr = 2 2/[lscript]2AdS, this action is a consistent

truncation of N = 8 sugra to the diagonal of the Cartan subgroup U(1)4 of the SO(8)

gauge group [19, 21]. We will encounter this special value of later on.

As usual, we regulate the theory introducing a large distance cuto [epsilon1] which we send to zero at the end of the calculation [22]:3

S = 1

2 2

ZM[epsilon1] d4x pg (R + 2 ) + [integraldisplay]M[epsilon1]d4x pg

. (2.3)

For the Euclidean dS case, no counterterms are needed because the wave function (1.1) only includes contributions from congurations that are asymptotically regular.

The instanton solutions are constructed using the Weyl invariance of the matter part of the action, hence it is useful to use global coordinates that are conformal to I [notdef] @M,

where I is a (nite or innite) interval. In the cases at hand @M = S3. In EAdS4 we will use conformal cylinder coordinates:

ds2EAdS = 1 sinh2 [lscript]

1 cosh2 r

[lscript]dS

dr2 + [lscript]2dS d 23

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12 (@[notdef])2 +R12 2 + 4! 4[parenrightbigg]

Z@M[epsilon1] d3x p 2K [parenleftbigg]1

26 2[parenrightbigg]

, (2.2)

where 2 = 8GN and the cosmological constant can be either positive or negative. The di erence between a positive and a negative cosmological constant = 3[lscript]2 is in the location and orientation of the regulated conformal boundary @M[epsilon1] and in the counterterms:

SEAdSct = 1

2

1

2 2

Z@M[epsilon1] d3x p [parenleftbigg]4[lscript] + [lscript] R[ ]

d2 + [lscript]2AdS d 23

[parenrightbigg]

[parenrightbig]

, 2 (0, 1) . (2.4)

For the purposes of the HHM proposal we will work with half4 Euclidean de Sitter, i.e. the southern hemisphere of S4:

ds2S4

=

[parenrightbig]

, r 2 (1, 0] . (2.5)

Lorentzian de Sitter is obtained by Wick rotating back r ! i[lscript]dS .

3See also [23].

4Since our scalar instantons do not back-react, the calculation of the HH wave functional involves gluing the Euclidean half S4 in the past to Lorentzian dS4 in the future. Since we are interested in the norm of the HH wave functional, considering half S4 su ces as the conguration along the imaginary (i.e. Lorentzian) path just gives a purely imaginary contribution to the on-shell action.

4

The relevant solutions of the scalar sector [20] are obtained by solving the Klein-Gordon equation:

2

3

6 3 = 0 (2.6)

together with the requirement that the stress-energy tensor vanishes. The latter requirement turns out to give [24]:

r[notdef]r 14 g[notdef]

1 = 0 . (2.7)

The solutions on EAdS4 and S4 are then given by:

",bIEAdS4(, 3) =

" sinh [lscript]

b0 cosh [lscript] + b5 sinh [lscript] + bi i

",aIS4 (r, 3) =

" cosh r[lscript]

a0 sinh r[lscript] + a5 cosh r[lscript] + ai i

, (2.8)

where i = 1, . . . , 4 and i is a unit vector normal to the three sphere. " can take the values " = [notdef]1. The requirement that these are solutions to the equation of motion (2.6) renders

the moduli space non-trivial,

on EAdS4 : b20 + b25 + b2i =

AdS

12 [lscript]2AdS , i = (1, . . . , 4) (2.9)

on S4 : a20 a25 + a2i =

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12 [lscript]2dS . (2.10)

From (2.9) and (2.10) we see that the moduli spaces of instantons on EAdS4 and dS4 are themselves EAdS4 or dS4 depending on the sign and values of the quartic couplings AdS and dS; more specically, the curvatures depend on the particular combinations

AdS

12 [lscript]2AdS b25 and

S4

12 [lscript]2dS a20. In the next subsection we will see the reason for this: b5

and a0 are not moduli of the solution; they parametrize boundary conditions instead. From the boundary point of view, b5 is the marginal coupling of a triple-trace deformation of the CFT. Therefore the moduli space of the solutions is EAdS4 if AdS12 [lscript]2AdS < b25, which is

the condition required for regularity of the bulk solution. At the critical value AdS

12 = b25

the e ective potential of the dual eld theory becomes unbounded from below, which was interpreted as an instability of the dual theory against marginal deformations, decaying via quantum tunneling [19].

2.3 Boundary conditions

Not all of the parameters aI, bI (I = 0, . . . , 5) are moduli as we now show. Two of them are boundary conditions relating the leading and subleading modes of the scalar:

S4

([epsilon1] [lscript], ) =

"

a5 + ai i +

S4

[epsilon1] " a0(a5 + ai i)2 + O([epsilon1]2) = a5,ai(0) + [epsilon1] a5,ai(1) + O([epsilon1]2) (2.11)

EAdS([epsilon1] [lscript], ) = [epsilon1] "

b0 + bi i

[epsilon1]2 " b5

(b0 + bi i)2 + O([epsilon1]3) = [epsilon1] b0,bi(0) + [epsilon1]2 b0,bi(1) + O([epsilon1]3) .

5

The leading and subleading terms in the expansion of the eld are related by:

a,ai(0)( )

"

a + ai i

a,ai(1)( ) = [notdef] " [parenleftBig]

2, EAdS/S4 , (2.12)

where = b5 for EAdS4 and = a0 for S4. Thus, b5, a0 parametrize marginal triple

trace deformations that change the boundary conditions from Dirichlet to mixed [19] (see also [24]).

2.4 On-shell action and the HH wave function

We stress that, given the action (2.2), the solutions (2.8) are exact: they have zero stress-energy tensor hence the EAdS/dS background stays unmodied. Thus we can compute the on-shell e ective action including its nite part.5 For simplicity we set the spherical modes ai = bi = 0 and get:

Son-shellEAdS = 42[lscript]2AdS

2

a,ai(0)( )

AdS 2[lscript]4AdS

12

2b0 + b5

3b30(b0 + b5)2

+ 2[lscript]2AdS b5 b30

b20 + b0b5 + b25b0 + b5 + O([epsilon1]), (2.13)

where in the last line we used (2.9). In the special case of a Neumann boundary condition b5 = 0, which requires negative coupling AdS < 0, we reproduce the result in [13],

= 42[lscript]2AdS

2 +

22[lscript]2AdS

3b30

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+ O([epsilon1])

82 AdS . (2.14)

Notice that this is a positive quantity. The case AdS > 0 was described in [19] where it was found that the instanton solution signals an instability of EAdS against deformations by mixed boundary conditions that can be described via a Coleman-de Luccia scenario. The decay rate calculation was presented in [19, 25].

The result of the corresponding calculation (1.1) for the de Sitter case is:

log HH = Son-shellS4

=

42[lscript]2S4

2

Sb5=bi=0

on-shell, EAdS = 42[lscript]2AdS

2

dS 2[lscript]4S4

12

2a5 a0

3a35(a5 a0)2

2[lscript]2S4

a0 a35

=

42[lscript]2S4

2 +

22[lscript]2S4

3a35

a25 a5a0 + a20 a5 a0

, (2.15)

and again we used the condition for the moduli space (2.10). If we take a0 = 0, we get a25 = S412 [lscript]2dS which can only be the if case S

4 < 0. In that case, the on-shell value of the

action is:

82 S4 , (2.16)

5As mentioned before, the EAdS4 solutions can be embedded in M-theory. We are not considering 1/N corrections here, but to the given order in the bulk coupling we can trust the result for the nite part of the on-shell action including its -dependence, and this is only possible because the solution is exact.

Son-shell

S4 =

42[lscript]2S4

2

6

which after analytic continuation [lscript]S4 [mapsto]! i[lscript]AdS, S

4 [mapsto]! AdS agrees with (2.14), as it should.

Notice that this is negative for S4 < 2 2/[lscript]2S4 and vanishes for S

4 = 2 2/[lscript]2S4, which

coincides with the critical value arising in the consistent truncation of N = 8 sugra, after

the analytic continuation of the dS4 radius.

Comparing (2.13) and (2.15) we see that the rst, pure gravity, term matches under the analytic continuation [lscript]AdS ! i [lscript]dS. This is Maldacenas result in [10]. The result is

non-trivial in part because the EAdS4 needs to be regularized and renormalized, whereas the S4 calculation of the HH wave function is completely nite. In order to match the

matter contributions, however, in the second term we need to analytically continue the couplings as well. This analytic continuation from EAdS4 to S4 is an invertible map :

([lscript]AdS) = i [lscript]dS

(b0) = i a5

(b5) = i a0

(bi) = iai . (2.17)

It follows that the coupling constant does not change, i.e. ( AdS) = dS. Notice that the

fact that the moduli have to be analytically continued is a consequence of the prescription to analytically continue the EAdS4 radius of curvature. Then the two expressions exactly match.

In contrast to [10, 12], we did not need to write absolute value bars around the HH wave function (2.15) because this is the full semi-classical result (1.1) which is real in the Euclidean. This result can be directly compared after analytic continuation, as we have seen, to the Euclidean AdS/CFT partition function because the latter is nite we took into account the correct counterterms, thereby rendering a result that can be precisely matched without the need to take the real part.

The complexication of the moduli can be understood from the fact that they are dimensionful quantities, to be measured in units of the radius. Dening dimensionless moduli yI = aI/[lscript]dS, zI = bI/[lscript]AdS, I = 0, . . . , 5, we nd the the moduli spaces can be

represented as:

z20 + z25 + z2i = AdS12 (i = 1, . . . , 4)

y20 y25 + y2i =

dS

12 . (2.18)

These are all real quantities on both sides. The moduli space is O(1,5) invariant:

IJyI yJ = dS

12 , I, J = 0, . . . , 5 , (2.19)

with IJ the O(1,5) Minkowski metric. The analytic continuation is then simply an SO(1,5) map of the moduli space onto itself:

zI = [epsilon1]IJyJ ; [epsilon1] =

7

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0 1

1 0

[parenrightBigg]

, = [notdef]1. (2.20)

14[notdef]4

We note here that in order to enforce the boundary conditions (2.12) from the bulk equations of motion, one needs to add a further term to the action:

22[lscript]2AdSb5

3b30

3a35 . In the presence of these terms, the on-shell actions are obtained simply by adding (2.21) and the corresponding dS4 result to (2.13)-(2.15).

3 Geometric interpretation

The 3-sphere partition function ZS3 for a three-dimensional CFT measures its number of degrees of freedom. Moreover, it has been argued that for unitary CFTs the corresponding free energy is given (in suitably chosen units) by

F = log [notdef]ZS

g[notdef] = 2 h[notdef] (3.2)

Z@M1 d3x p

h 2K[h] (3.3)

= ZM2 d4x pg [parenleftbigg]12 (@[notdef])2 +R[g]12 2 + 4! 4[parenrightbigg]+ 16 [integraldisplay]@M2d3x p g K[g] 2 =: I(g,)

6To obtain the r.h.s. of (3.3) we used the fact that the trace of the extrinsic curvature for the conformally related metrics (3.2) are related as

K[h] = 1

K[g] +

8

Sbdy def =

2[lscript]2dS a0

b5[lscript]2AdS 3

[integraldisplay]

. (2.21)

This of course agrees, after the analytic continuation (2.17), with the term one gets in the dS4 case, 2

d 3 3(0)( ) =

3 [notdef] , (3.1)

which is positive and satises an F -theorem, namely it decreases along RG ows from the UV to the IR [26, 27]. Holographically, the partition function is usually calculated using the bulk gravitational action on EAdS4 with all other matter elds set to zero. The result is the rst term in (2.14) and it is proportional to the dimensionless ratio [lscript]2AdS/ 2. One

may then wonder what the physical interpretation is of the second term in (2.14), which corresponds to the contribution of the bulk scalar elds. Notice that for AdS < 0 this term

is also positive. A similar question may be asked for the result (2.16), namely whether this can also be interpreted as a partition function of a CFT3 on a 3-sphere. Since (2.16) is not always positive, such a CFT3 need not be unitary.

To this end, we will point out that the instanton contributions in (2.14) and (2.16) also arise from bulk gravitational actions, and hence can be interpreted as F -functions on the 3-sphere. This follows [28] from the well-known representation of the conformal factor of a metric as a scalar eld with quartic self interaction. Indeed, consider the conformally related metrics

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One can then show6 that

Ih := 1 2 2

ZM1 d4x

ph (R[h] 2 h) +1 2 2

3

2 n[notdef]@[notdef] ,

with n[notdef] the outward normal to the boundary in the metric g[notdef] .

where h and g are the induced metrics of h[notdef] and g[notdef] on the boundary @M1, @M2 and we have dened

:=

[radicalbigg]

6 2 , :=

2 2

3 h . (3.4)

The critical value for mentioned in the previous section arises when we relate the scalar action to a gravitational action. The second line in (3.3) coincides with the matter part of the action (2.2), and from the above it equals minus a gravitational action including the GH term. other words,

The crucial observation now is that the instanton solutions inst on either S4 and

EAdS4, with the moduli set to specic values, correspond exactly to the conformal factor relating the two metrics. Hence, the on-shell action of instantons on EAdS4 corresponds to the volume of (half) S4 and conversely, the on-shell action of instantons on the half S4 corresponds to the volume of EAdS4, where the instanton deformation parameter b (which corresponds to a0 in the previous section) regulates the volume of EAdS4. It is also crucial to point out that in evaluating the on-shell values of the gravitational actions, the boundary GH term does not contribute to the nite part. This is true for AdS4, but it is also true for S4, since in this case the extrinsic curvature vanishes.

Let us see how this arises. On the half S4 with curvature radius [lscript] and metric

ds2 = 4

1 + 2[lscript]2

2 d2 + 2d 23

[parenrightbig]

, (3.5)

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the instanton solutions are given by

inst() = [notdef]

[radicalBigg]

1 +

12 S4

1 b

2

[lscript]2

(3.6)

where S4 > 0. It is important to note the presence of the instanton modulus b which is in principle unrelated to [lscript]. In particular, since the range of the radial coordinate is 2 [0, [lscript]),

if we consider b > [lscript] the solution is everywhere regular.Next we notice that half S4 and EAdS4 are conformally related metrics. In particular

ds2 = 4

1 + 2[lscript]2

2 d2 + 2d 23

1 2b2

2

b2

[parenrightbig]

= 1

, (3.7)

where on the right EAdS4 has a generally di erent radius b which is set equal to the instanton modulus. Hence, the calculation of the on-shell action for instantons on half S4 including an appropriate boundary term as in (3.3) boils down to the calculation of the regularized volume of EAdS4. Notice that it is the presence of the instanton modulus b > [lscript] that gives rise to a particular regularization of the volume of the EAdS4 space with radius [lscript].

Explicitly, we obtain

Ion-shell

(g,) S4

d2 + 2d 23

[parenrightbig]

1 + 2[lscript]2

!24

1 2b2 [parenrightBig]2

[parenrightbig]

= h

2

[integraldisplay]

d4x ph = 82b2

2

4( 2 3) (1 2)3

= 162

S4

4( 2 3) (1 2)3

, = [lscript]

b < 1 . (3.8)

9

This diverges as ! 1, which we shall interpret presently. The nite part of (3.8) for

! 1 coincides with minus the rst term in (2.14), as it should.

Next we consider the instanton solutions on EAdS4 with radius [lscript]. These have the form

inst() = [notdef]

[radicalbigg]

12

1

2

[lscript]2

. (3.9)

with AdS < 0 and b the modulus which we take here again to be b > [lscript]. The on-shell action would give part of the volume of S4 with radius b. We nd

Ion-shell

AdS

1 b

1 + 2b2

4( 2 + 3)

( 2 + 1)3 , (3.10)

where here AdS < 0. For = 1 this is just minus the rst term in (2.16), as it should.

The association of the on-shell instanton actions with minus the EH action in (3.3) points towards a remarkable correspondence. Firstly, it is not hard to see that (3.8) and (3.10) are mapped to each other by the transformation b [mapsto]! i b. This is natural, if

we recall that b is the radius of both the S4 and EAdS4 of the associated EH actions. Next, we should remember that the AdS4 result (2.14) gives (minus the logarithm) of the partition function of a three dimensional CFT on S3 as a sum to two terms. The rst term is the usual holographic F -function that is proportional to dimensionless ratio [lscript]2/ 2. What we have shown here is that the second term too has a geometric origin, since it also comes from a gravitational action which is nevertheless in principle unrelated to the AdS4 action. Namely, if we identify the coupling [prime]2/b2, the second term is the on-shell value

of a gravitational action with Newtons constant [prime] and positive cosmological constant, in other words it gives the volume of an S3 with radius b. Moreover, the ratio of the two radii, = [lscript]/b, which also has a geometric origin, can be viewed as a natural deformation parameter, running from b = 1 ( = 0) to b = 0 ( = 1). It is worth noting then that

such a deformation is monotonic and according to the conjectured F -theorem takes the boundary CFT towards the UV. The case of dS4 is exactly the reverse; here it is the scalar instanton part that has a natural holographic interpretation as a partition function of a CFT on S3, while the gravitational part is geometric. Here too, the parameter deforms monotonically the (non unitary) boundary theory.

We notice here that the instanton (3.6) is of the type (2.8) under the following identication:

b2 = [lscript]2dS a0 a5a0 + a5 , (3.11)

as is easy to verify doing the coordinate transformation. We see that the analytic continuation b [mapsto]!ib indeed corresponds to [lscript]dS [mapsto]! i [lscript]AdS.

The limit ! 1 of the above action, in which the instanton deformation parameter

approaches the curvature radius, corresponds to the limit a25 ! 0+ and a20 !

S4

12 [lscript]2dS+

JHEP11(2014)126

h (EAdS4) = 82b2

2

4( 2 + 3)

( 2 + 1)3 =

162

AdS

which is the critical value at which the moduli space shrinks to zero curvature radius. In this limit, the instanton part of the action (2.15) diverges as 1/a35, which is precisely the

10

divergence of the EAdS4 volume.7 Thus, the instanton is computing the EAdS4 volume, and the boundary deformation parameter a0 regulates this volume. The critical value of the deformation parameter a20 = S412 [lscript]2dS, for which we get the correct divergence, corresponds, via the analytic continuation (2.17), to the critical value at which the dual boundary theory becomes unstable.

4 Discussion and conclusions

In this note we have tested the HHM proposal in the case of scalar instantons. We have calculated the on-shell action for instantons on half-S4, which yields the late-time HH state, and compared it with the on-shell action for instantons on EAdS4. The results match under the HHM prescription of analytically continuing the curvature radii. Additionally, we have found that it is also necessary to analytically continue the boundary condition, which corresponds to a marginal triple trace deformation. This provides new evidence that the HHM proposal works for exact, but non-trivial congurations.

Our goal was to check the HHM proposal but along the way we got a number of results which we believe are relevant for the HS proposals in de Sitter space, in particular for the Sp(N) vector model that has been conjectured to be dual to it. On the EAdS side, the instanton modulus b5 corresponds to the coe cient of a marginal triple trace deformation (2.21) for an operator of dimension 1. Under the analytic continuation to dS space this is again a triple trace deformation, and the free energy has been computed. The free energy as a function of the boundary deformation parameter presents zeroes which usually signal an instability of the theory. The presence of instabilities seems to be connected to the fact that our free energy result (2.16) may be negative, hence it corresponds to a non-unitary CFT3. Similar behaviour has been found in [12]. It would also be interesting to work out the relationship of this result with holographic stochastic quantisation [28, 29].

We also found an interesting geometric realization of the same computation, in which the S4 instantons (for particular values of the moduli) are seen to compute the regularized volume of EAdS4, and the EAdS4 instantons are seen to compute the volume of the four-sphere. The regulator of the EAdS4 volume is a0, with the divergence appearing precisely for the critical value a20 = S412 [lscript]2dS. This might imply that a sector of the Sp(N) model with this marginal deformation is dual to a pure gravitational theory with no scalars, and hence signal a duality between Sp(N) models with di erent values of the deformation parameter. This is reminiscent of similar scenarios as e.g. the dualities in [30, 31].

As stressed in [13, 32], instantons of the type found here seem to have special holo-graphic properties because they parametrize di erent Weyl vacua of the theory. The exact bulk action can be calculated and compared to the boundary e ective potential [19, 24]. It was pointed out in [13] that the boundary values of bulk instantons are also solutions of the equations of motion of a three-dimensional conformally coupled scalar eld theory on the boundary, which was conjectured to be the e ective action for an operator of dimension 1.

7This divergence can be removed by adding boundary terms in a way similar to what was done in section 2, however our aim here is to exhibit the form of this divergence and how it relates to the on-shell value of the instanton.

11

JHEP11(2014)126

In [19, 24] it was found that this action in fact agrees with the e ective action near the critical point, which can be calculated by di erent methods. Given the robust structure of the instanton solutions it is unlikely that this is a mere coincidence. Near the critical point, as we have seen, the instantons in fact just calculate the volume. We point out here that a similar boundary e ective action description applies to the de Sitter instantons as well.

Finally, we wish to point out that our instanton solutions are intimately related to the SO(4) and SO(3, 1) invariant solutions of 4-dimensional HS theory found in [33]. The latter are solutions with all HS gauge elds switched o , except of the metric and the conformally coupled scalar, and they are also related [19] to a consistent truncation of N = 8 gauge

supergravity down to a single scalar of the SO(8) group. In that sense, our results should also provide the partition functions of both the above theories, at the scalar instanton vacua. We leave a more detailed analysis of this intriguing point of view for future work.

Acknowledgments

We thank D. Anninos, J. de Boer, K. Skenderis and P. Sundell for useful discussions, communications and comments on the manuscript. SdH thanks the Institute for Theoretical Physics of the Aristotle University of Thessaloniki for the kind hospitality during the nal stages of this work. The work of SdH was partially supported by EU-COST action The String Theory Universe, STSM-MP1210-16939. The work of A. C. Petkou is partially supported by the research grant ARISTEIA II, 3337 Aspects of three-dimensional CFTs, by the Greek General Secretariat of Research and Technology, and also by the CreteHEPCosmo-228644 grant.

A Explicit bulk solutions and holography

The near boundary behaviour of a conformally coupled scalar eld (z, [vector]x) on a xed EAdS4 background is as

(z, [vector]x) ! z (0)([vector]x) + z2(1)([vector]x) + [notdef] [notdef] [notdef] (A.1)

One can then calculate the renormalized bulk on-shell action Ion-shell[(0)] as a functional

of the boundary conditions (0). This is interpreted as (minus) the generating functional for connected correlation functions of the boundary operator O as:

Ion-shell[(0)] = W [(0)] ,

Dening then the Legendre transform [A] as

W [(0)] = [A] +

we have

JHEP11(2014)126

W [(0)]

(0) = [angbracketleft]O[angbracketright](0) . (A.2)

[integraldisplay] A

(0) , (A.3)

[A]

A

= (0) , A = [angbracketleft]O[angbracketright](0) , (A.4)

12

which shows that [A] is the e ective action of the boundary theory. Knowledge of [A]

allows us to study non-trivial vacua of the boundary theory, which are then given by the set of equations

[A]

A

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]A=A

= 0 , W [(0)] (0)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

(0)=0

= [angbracketleft]O [angbracketright] A [negationslash]= 0 . (A.5)

Now, suppose that we impose mixed boundary conditions by adding a boundary term of the form f((0)) to the bulk action. We then obtain for the variation of the on-shell bulk action:

Ion-shell[(0)] =

[integraldisplay]

(0) ((1) f[prime]((0))) . (A.6)

This means that we are rendering the bulk on-shell action stationary for solutions of the bulk equations of motion satisfying

(1) = f[prime]((0)) . (A.7)

This in turn implies that W [(0)]

(0)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]f[prime][

(0)]=(1)

= 0 . (A.8)

Therefore, we could have interpreted W [(0)] as an e ective action of a dual boundary theory that has a non-trivial vacuum structure. In fact, we can write [19]

W [(0)]

~

[(0)] ,

JHEP11(2014)126

[] =

~

[(0)] +

[integraldisplay]

(0) (A.9)

such that

[]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]=0

= 0 (A.10)

yields the non-trivial vacuum expectation value for the operator[notdef] in the dual bound

ary theory.

The partition function is given by

Z[J] =

[integraldisplay] D

' eS[']+

= [angbracketleft]

[notdef] [angbracketright] [negationslash]

[integraltext]

JO = eW[J] = e [A]+

[integraltext]

JA (A.11)

If the theory possesses non-trivial vacua (A.5), then setting the external source J = 0 we obtain

Z[J = 0] Z0 = e [A ] = eS[' ]+[notdef][notdef][notdef] (A.12)

Hence, the leading part of the e ective action gives the partition function.

From the AdS/CFT point of view, given an exact particular solution of the bulk equations of motion, namely a solution with a given form (0)

(0) in (A.1), then we

would be able to evaluate W [

(0)]. According to (A.9) this would give us the partition function0 of the dual boundary theory as

W [

(0)] = ~

[

(0)] = ln0 (A.13)

13

On the other hand, given W [

(0)] we are able using (A.3) to obtain the value of the e ective action of the boundary theory at its non-trivial vacuum A , as

[A ] = W [

(0)]

[integraldisplay] A

(0) = ln Z0 (A.14)

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Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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