HHolographic renormalization in no-boundaryquantum cosmology
Lihui Liu
Instituut voor Theoretische Fysica, KU Leuven, Celestijnenlaan 200DB-3001 Leuven, Belgium [email protected]
Abstract
Recent results of Hartle-Hawking wave functions on asymptotic dS boundaries show non-normalizability, while thebulk origin is not clear. This paper attempts to addresse this problem by studying (Kerr-)dS cosmology in Einsteingravity deformed by a minimally coupled scalar field. Various saddle-point contributions exp[ i (saddle-point action)]to the Hartle-Hawking wave functions are computed with mini-superspace formalism. The saddle-point actions arefirst obtained in the spacetime bulks by direct computation, then on the asymptotic dS boundary by holographicrenormalization. It is found that the imaginary part of the saddle-point actions, as functions of scalar field de-formation, are generally bounded in the bulk, but can diverge to −∞ on the boundary. This can probably be asource of the scalar field-related non-normalizability of the boundary Hartle-Hawking wave functions. For Kerr-dS cosmology, some saddle-point actions have imaginary parts diverging to −∞ both in the bulk and on the boundary,when the boundary T is stretched to infinitely long. This can be an origin of non-normalizability of the boundaryHartle-Hawking wave functions related to temperature divergence. Finally the holographic renormalization is ex-tended to scalar-deformed dS d +1 ( d ≥
3) cosmologies for the imaginary part of the saddle-point actions. The resultis tested on a 5d example, and saddle points leading to divergent contribution to the boundary Hartle-Hawkingwave function are shown to exist. a r X i v : . [ h e p - t h ] D ec ontents T S S d × S d
61B Asymptotic expansions near future boundary of asymptotic dS 63 Introduction
Over the past dozen years with the observation of early cosmology attaining higher and higher sophistication, therehas been a continuous trend of theoretical effort based on gauge/gravity duality to explore the quantum gravitymechanism behind the observational results and their implications. Among all the theoretical frameworks proposedfor this purpose, the dS/CFT correspondence [1, 2, 3] is perhaps the most straightforward one, since its bulk sideis suitable for directly describing an inflationary universe. Compared to the famous AdS/CFT correspondence, oneother widely used holographic framework in cosmology, the groundwork of dS/CFT is far less settled. While forthe former, general consent has been comfortably granted to the formulation of [4, 5, 6, 7], a similar consensualpoint for the latter has not yet been reached by the holography community [2, 3, 8, 9, 10, 11, 12, 13].One branch among all the lines of thinkings which has been fruitful, advocates a simple transplantation of theAdS/CFT principle into the dS/CFT realm. It proposes the identification of the Hartle-Hawking wave function[14, 15] of the bulk side, with the partition function of the dual field theory on the boundary [3], where the lattercan be non-unitary [1, 16]. A concrete microscopic realization of this prescription was set forth recently in [17, 18],which relates the Vasiliev higher spin gravity theories [19] in the dS bulk to a critical grassmannian scalar fieldtheory of Sp ( N ) symmetry on the boundary, which is the dS/CFT cousin of the Vasiliev/ O ( N ) correspondencebased on AdS holography [20, 21].Soon after, this concrete prescription was used to obtain the Hartle-Hawking wave function of the Vasilievuniverse via computing the functional determinants of the boundary Sp ( N ) theory [22, 23]. With some properregularization procedure, the wave functions are obtained on dS boundaries of various topologies, and one intriguingfeature uncovered is the omnipresent non-normalizability of the Hartle-Hawking wave functions. In particular, theycan diverge either as a function of scalar deformation (scalar divergence) or as a function of the geometry of thedS boundary (temperature divergence).Another recent result that can be placed in the same line is [24], where the one-loop Hartle-Hawking wavefunction of the empty Kerr-dS universe is obtained on its T -boundary, and is argued to be full-loop exact.The wave function depends on the modular parameter τ = τ + iτ of the boundary torus. It turns out to beexponentially divergent with τ tending to ∞ (temperature divergence), where the divergence arises at the looplevel.One tempting interpretation of these divergences is the instability of dS spacetimes, already suggested by studiesfrom different approaches (see discussion and references in [24]). However this speculation is much based on thequantum mechanics knowledge about a quantum system prepared in a false vacuum, which states that its decayprocess is described by an exponentially divergent wave function (see for example [25]). Indeed without explicitcomputation it is hard to see to what extent this quantum mechanical common sense applies to the wave functionof the universe. Quantitative study is thus needed for unveiling the actual bulk mechanism responsible for thedivergence of the boundary wave function.The recent development in another field of research can pave the way for moving towards this goal. It is theseries of work on the quantum probability measure of classical universes [26, 27, 28, 29], part of the decades-longquest for a probabilist explanation of why inflation happens [30]. Powerful tools have been developed for computingHartle-Hawking wave functions at tree-level in the spacetime bulk. Therefore it appears a promising direction toapply these tools to produce boundary results from the bulk, in order to gain insight into the results from directdS boundary computations.Indeed some efforts in this regard have already been made on the analytical level. In [22], having obtainedthe Hartle-Hawking wave function on the S × S boundary of the Vasiliev universe, divergent with the shrinkingof S (temperature divergence), a bulk computation at tree level in Einstein gravity is performed, with the dSboundary of the same topology S × S . The result shows that some bulk saddle-point contribution diverges in3 very similar way that the Vasiliev universe wave function does. This computation is also performed in [16] ina different context. In [31] a more elaborated analysis of this Einstein gravity model shows that this divergenceis related to some unphysical classical bulk configuration. These works in the bulk shed light on the origin of theboundary wave function divergences.The work to be presented in this paper is in the continuation of the work mentioned above, with a substantialuse of the numerical methods in [29] (the numerical scheme can be traced back to [32, 33, 34]). Yet it tackles theproblem somewhat from another angle, in that the attention will be focused on 3d models. The advantage of thisis that one can enrich the physical content of the model to quite some extent, while still keep exact computationspossible. In the following, I present a summary of the key logic of the work. What to compute (saddle-point actions) and why (their imaginary parts matter)
Concretely, the models to be considered are 3d closed universes. Supposed to be at weak coupling, the bulkphysics is governed by Einstein gravity with a positive cosmological constant. A minimally coupled scalar field isintroduced, and moreover, the universes should have an asymptotic dS region. Two topologies for the asymptoticboundary will be considered: T and S . For each case, the essential part of the work can be encapsulated in thefollowing two successive steps: • First, the Hartle-Hawking wave function is computed in the spacetime bulk directly with the defining pathintegral [14]. However, due to the difficulty in defining and evaluating such path integrals in general, inthis paper the work is focused on the saddle-point contributions, assuming that loop contributions are highlysuppressed. The problem now reduces to computing the saddle-point actions, to be denoted by S . Theassociated tree-level contribution to the wave function is thus e i S . Generically, S is complex in the contextof no-boundary quantum cosmology, so that the saddle point contribution acquires a non-trivial factor ofamplitude in addition to the factor of a pure phase: e i S = e i Re S × e − Im S [15, 27, 41]. • Second, the bulk result is sent onto the asymptotic dS boundary. This step is supposed to yield the version ofwave function to be directly identified with the partition function of the dual boundary field theory, in casethe boundary dual exists. If we focus on the tree level, the problem is a simple application of the standardholographic renormalization scheme to the dS bulk action S , which eliminates the infrared (IR) divergenceand yields a finite renormalized action ˜ S . The associated tree-level contribution to the boundary Hartle-Hawking wave function is therefore e i ˜ S = e i Re ˜ S × e − Im ˜ S . For the 3d models, ˜ S can be worked out explicitlyand the results can be numerically computed. In this step an important thing to examine for the purposeof the paper, is the behavior of Im S and Im ˜ S . Their divergence to −∞ is considered as possible source ofnon-normalizability of the Hartle-Hawking wave function. About the saddle points: how to find them and what do they look like
Since it is clarified in the first step above that the Hartle-Hawking wave functions will be computed at tree-level,there is now one more step to be prepended to the whole agenda: the search for contributing saddle points. Thelatter are compact classical spacetimes with on-shell fields living in them and matching the pre-ordered values onthe spacetime boundary. The search for saddle points will be done numerically using the numerical scheme in [27].Simply stated, one first assigns the wanted values of the fields on the boundary, and then integrate the classical See discussions in [12, 36, 37]. The examples of exact computation on models based on empty (A)dS spaces [24, 38, 39] should notbe considered as counter examples. First it is a one-loop computation whose result is claimed to be full-loop exact using a symmetryclaim based on [40]. Second although the computation deals directly with degrees of freedom in the bulk, it is done over the full (A)dSspace and the result is still a wave (partition) function on the asymptotic (A)dS boundary. S and ˜ S , among some other physical quantities. Some important features of this part ofthe work are: • The search will be carried out in mini-superspace formalism in order for it to be possible. That is, we requirethe spacetime have a time coordinate which realizes homogeneous isotropic slicing. This will make us loosesaddle points but will not cause error to the result of S and ˜ S . • Then it turns out that the mini-superspace coordinate time is necessarily complex so that the entire historyof a saddle point is a complex curve. • Given a boundary condition, infinitely many saddles exist with time contours containing different amount ofEuclidean history and/or ending up on different layers in the scalar field’s Riemann surface.The rest of the paper will be organized as follows. Sec.2 will present the frame work and technicalities forstudying the scalar-deformed Kerr-dS , including the action principle for the saddle points, the computation ofsaddle-point actions in the bulk, and the holographic renormalization, all formulated in mini-superspace formalism.In Sec.3 the search of saddle points and the computation of their actions will be done analytically, with scalardeformation set perturbative. It is shown how different mini-superspace time contours lead to different saddlepoints fitting the same boundary conditions.Sec.4 is the prolongation of the work of Sec.3 into the domain of finite scalar deformation. The saddle-pointactions S (bulk) and ˜ S (boundary) are traced both against scalar deformation and against boundary geometrydeformation, and divergence to −∞ of Im S and Im ˜ S are observed. Meanwhile the Riemann surfaces of the scalarfield are studied, which shows empirically some intriguing connection between the scalar divergence of Im ˜ S andthe pattern that singularities move around in the Riemann surfaces.In Sec.5, all the study of the scalar-deformed Kerr-dS in the previous sections are carried out on scalar-deformeddS , including the perturbative analytical study, the exact holographic renormalization and the non-perturbativenumerical computations. Possible orgin of scalar divergence in Im ˜ S is also found.Sec.6 gives a very brief look into two immediate extensions of the work, which can be interesting possible futuredirections. It is first shown that the work can be extended to models with potentials other than quadratic. Thesecond part explores the possibility of extension to higher dimensions, where a concrete model, scalar-deformeddS is studied and similar features as the 3d models are uncovered.The summary and perspectives are presented in Sec.7. T universes deformed by a minimally coupled scalar field, which have an asymptotic boundaryof topology T . Universes with asymptotic spacelike boundaries of topology S d ( d = 2 , , . . . ) will also be consideredlater in the paper. However adapting the formalisms developed in this section to the S d -cases will be straightforwardenough.Sec.2.1 presents the spacetime geometry near the T -asymptotic boundary, and then introduces the Hartle-Hawking wave function for the very model in question. Sec.2.2 is focused on the tree-level parts of the Hartle-Hawking wave function, and establishes numerically operable formalisms for their computation. In Sec.2.3, thesaddle-point actions are computed with the boundary of the saddle points sent to the asymptotic dS boundary.5olographic renormalization is then performed to remove the infrared (IR) divergences to obtain the saddle-pointactions on the asymptotic dS boundary. Action and spacetime in general
With bulk physics set at weak coupling, its action should be that of Einstein gravity with a positive cosmologicalconstant and a minimally coupled scalar field: S = 116 πG ˆ M d x √− g (cid:2) R − (cid:96) − + ( ∂ Φ) − V (Φ) (cid:3) + S b , (1)where M denotes the spacetime manifold, g µν ( µ, ν = 0 , ,
2) is the 3-metric with R its Ricci scalar, (cid:96) is the dSradius, and V (Φ) is the scalar potential. We will also need to include some appropriate boundary term S b definedon ∂ M in accordance with the boundary conditions.The universes to be considered are supposed to asymptote to a (locally) dS space with asymptotic boundary oftopology T . It will be assumed that some Fefferman-Graham type expansion (see for example [42]) is available inthe asymptotic dS region with the presence of the scalar field: (cid:96) − g µν dx µ dx ν ∼ − dλ + q ij ( λ, x k ) dx i dx j , where q ij ( λ, x k ) ∼ e λ q (0) ij ( x k ) + (sub-leading orders) , (2)where λ is some coordinate time and the asymptotic dS boundary, to be denoted by I + , is located at λ = ∞ ; i , j and k are spatial indices taking the values 1 or 2; q (0) specifies the geometry of a 2-torus. The subleading termsare not shown explicitly in Eq.(2). Due to the scalar field back reaction their fall-off is slowed down with respect tothose in the standard Fefferman-Graham series. Such behavior has already been described in the AdS/CFT context[32, 43, 44, 45]. The sub-leading terms are presented in appendix B and will later be very useful for holographicrenormalizaion. Throughout the paper the dS radius (cid:96) will be used as the length unit, and factorized as the globalcoefficient of the metric, so that x µ , λ , q ij is dimensionless. The Hartle-Hawking wave function in the bulk field representation
We can now move on to the quantum aspect to introduce the Hartle-Hawking wave function. As stated in theintroduction, we will first need to compute directly the Hartle-Hawking wave in the spacetime bulk at tree level.Therefore the function will be defined on some spacelike hypersurface in the bulk of the universe, and is a function(al)of the field configuration and the geometry of this hypersurface. To be relevant to the holographic computationlater, this hypersurface is chosen in the asymptotic dS region of the universe and is just one of the equal- λ surfaceas given in Eq.(2). Let this hypersurface correspond to λ = Λ < ∞ and be denoted by Σ ∗ , and let the induced2-metric of this slice be denoted by h ij and the scalar field value confined on it by ϕ . An illustration of this setupis shown in the left part of Fig.1. It is still necessary to have some more specifications of the hypersurface Σ ∗ toavoid too abstract talking about the Hartle-Hawking wave function.It will be convenient to use the language of complex geometry to parameterize h ij . Therefore Σ ∗ will becharacterized in terms of its area, given by the K¨ahler modulus A , and its shape, specified by the complex structure6igure 1: Left:
The asymptotic dS region of the of 3d universe considered. The spacetime manifold is denoted by M , which supportsa 3-metric g µν and a scalar field Φ. The asymptotic boundary I + has topology T , and the asymptotic region is sliced into a stack of T against some time parameter λ . The Hartle-Hawking wave function will be computed with respect to the slice λ = Λ, denoted byΣ ∗ , whose induced 2-metric and scalar field are denoted by h ij and ϕ , so that the wave function is Ψ[ h ij , ϕ ]. Right:
The spacetime N to be summed up in the defining path integral of Ψ[ h ij , ϕ ]. They are compact and have Σ ∗ as boundary: ∂ N = Σ ∗ , and have fieldcontents ( g µν , Φ) regular everywhere in N , fitting the values ( h ij , ϕ ) on the boundary. N do not necessarily have a slicing into a stackof T . modulus τ = τ + iτ . By a reparameterization within the slice Σ ∗ , its 2-metric can be rendered to the form h ij dx i dx j = (cid:96) A τ − (cid:16) dξ + 2 τ dξ dξ + | τ | dξ (cid:17) , (3)where ξ and ξ are both periodic with periodicity 2 π . Due to the factorization of (cid:96) , A is dimensionless. Theadvantage of using the complex structure modulus τ is to easily formulate the T-duality of the 2-torus: a symmetryof the geometry of the torus when its complex structure modulus undergoes the following transform τ −→ τ (cid:48) = eτ + fcτ + d , where c, d, e, f ∈ Z , and ed − f c = 1 , (4)These transforms form the group SL (2 , Z ). The path integral to be introduced immediately below, Eq.(5), shouldpick up the contribution from all inequivalent 2-tori lying on the SL (2 , Z ) orbit.With the above preliminaries, we can introduce the Hartle-Hawking wave function [14] on a concrete basis. Itwill be denoted by Ψ, and computed against Σ ∗ , and is thus a function(al) of h ij and ϕ . Its formal definition isΨ[ h ij , ϕ ] = Ψ[ A , τ, ϕ ] = ˆ D [ g, Φ] exp (cid:16) i S [ g, Φ] (cid:17) . (5)Here S [ g, Φ] is given by Eq.(1) with a proper boundary term to be specified very shortly. The integral D [ g, Φ] runsover all compact 3-spacetimes bounded by Σ ∗ , with the fields g µν and Φ fitting the boundary values ( h ij , ϕ ). Suchspacetimes are different from the actual physical universe M in Eq.(1), and they will be denoted by N to mark thedifference. Therefore we have ∂ N = Σ ∗ , and if we further let F be the imbedding map of Σ ∗ into N , then F ∗ g = h ,and F ∗ Φ = ϕ . These conditions for the path integral are just the no-boundary proposal [14]. An illustration ofsuch no-boundary spacetimes is given in the right half of Fig.1.Now the action in Eq.(5) can be fully specified. Given that we are fixing h ij and ϕ on the boundary Σ ∗ , the In doing so the resulting h ij as in Eq.(3) will generically no longer be that appearing in the Feffmann-Gramm form because thereparameterization can produce a nonzero shift vector. However this does not prevent us from formally defining the Hartle-Hawkingwave function(al) in terms of h ij . S = 116 πG ˆ N d x √− g (cid:2) R − (cid:96) − + ( ∂ Φ) − V (Φ) (cid:3) + 18 πG ˆ Σ ∗ d x √ h K, (6)where K = h ij K ij with K ij the extrinsic curvature of Σ ∗ . Let us move on to the computational level. The goal is to compute the path integral (5) at tree level, and thatto achieve this we need to first find out the saddle points. This subsection will introduce the formalism for thesearch of saddle points. Recall that the saddle points are on-shell compact spacetimes satisfying the no-boundaryproposal. Therefore the formalism to be presented is essentially about setting up a calculable action principle.An important restriction has to be introduced to make the actual computation feasible: the mini-superspacetruncation of degrees of freedom. This means we only deal with the saddle points whose spacetime has a globallydefined coordinate time which slices the spacetime into homogeneous isotropic hypersurfaces, all having the sametopology T as the boundary (see Fig.2).In the contents that follow, it will be explained how to formulate the variation principle of the saddle points inthe mini-superspace, including the implementation of the mini-superspace formalism, the corresponding boundaryconditions, the action and the equations of motion. Finally a preliminary simplification of the saddle-point actionwill be given for the use of next subsection after a holographic setup is introduced. Mini-superspace reduction
To implement the mini-superspace formalism, we can start with the 3-metric of the no-boundary spacetime N ,requiring that the saddle points have the metric which takes the form of 2 + 1 decomposition into a stack ofhomogeneous isotropic 2-tori: g µν dx µ dx ν = − (cid:96) dχ + q ij ( χ ) dx i dx j = − (cid:96) dχ + (cid:96) A ( χ ) T ( χ ) − (cid:12)(cid:12) dξ + T ( χ ) dξ (cid:12)(cid:12) . (7)Here the spatial coordinates ξ and ξ are just those in Eq.(3), which have periodicity 2 π separately. A global timecoordinate χ is introduced such that the lapse function is 1, and later on whenever the term “coordinate time”or simply “time” is used for some saddle point, it always refers to this mini-superspace coordinate time. For each χ -slice the 2-metric is q ij ( χ ) dx i dx j = (cid:96) A ( χ ) T ( χ ) − (cid:12)(cid:12) dξ + T ( χ ) dξ (cid:12)(cid:12) , a flat metric of a torus of area (K¨ahlermodulus) A ( χ ) and complex structure T ( χ ) = T ( χ ) + i T ( χ ). Yet a further restriction turns out to be necessaryfor the quantitative computation to be possible: we need T to be constant in χ , or can be transformed to constantin time through Eq.(4). The 3-metric thus becomes (cid:96) − g µν dx µ dx ν = − dχ + A ( χ ) T ( χ ) − (cid:104) dξ + 2 T dξ dξ + (cid:0) T + T ( χ ) (cid:1) dξ (cid:105) = − dχ + A ( χ ) T ( χ ) dξ + A ( χ ) T ( χ ) − d (cid:0) ξ + T ξ ) . (8)A simpler form can be achieved if we let a ( χ ) = A ( χ ) T ( χ ) , b ( χ ) = A ( χ ) T ( χ ); (9) ζ = ξ , ζ = ξ + T ξ , (10)8ith which Eq.(8) further reduces to (cid:96) − g µν dx µ dx ν = − dχ + a ( χ ) dζ + b ( χ ) dζ . (11)The periodicities of the angular coordinates are( ζ , ζ ) ∼ ( ζ , ζ ) + 2 π (1 , T ) ∼ ( ζ , ζ ) + 2 π (0 , . (12) Coordinate time range: between the boundary and the south pole
Then let us take a closer look at the coordinate time, supposing it elapses from a starting point χ o to an endpoint χ ∗ . With the lapse function being set to 1, it follows that we cannot universally assign the same endpoints χ o and χ ∗ to all cases when solving for classical solutions, but should have them determined case by case usingthe boundary conditions. Later we will see that for each classical solution, χ ∗ − χ o is different and is genericallycomplex. One end of the time interval should correspond to the spacetime boundary ∂ N = Σ ∗ and we can let itbe χ ∗ . Thus at χ ∗ , the 2-metri q ij ( χ ) in Eq.(7) should evaluate to h ij , and the scalar field Φ should evaluate to ϕ .That is A ∗ = a ∗ b ∗ = A , ( T ) ∗ = b ∗ a ∗ = τ , Φ ∗ = ϕ, (13)where the subscript “ ∗ ” means evaluated at χ ∗ : a ∗ = a ( χ ∗ ) etc. There is still the spectator parameter ( T ) ∗ ≡ T ( χ ) = τ which can be arbitrarily chosen without having concrete effect on any other results. On the other hand,there should be no boundary present at the starting point χ o , and we can implement this condition by requiring a ( χ o + (cid:15) ) ∼ i(cid:15), b ( χ o ) = b o for b o , (cid:15) ∈ C and | (cid:15) | ∼
0; (14)Φ( χ o ) = Φ o ∈ C , (15)where b o and Φ o are finite complex numbers. Demanding Eq.(15) is because the scalar field should be regular allover N . Moreover, when Eq.(14) is satisfied, the leading behavior of the 3-metric Eq.(11) near χ o is (cid:96) − g µν dx µ dx ν (cid:12)(cid:12)(cid:12) χ = χ o + (cid:15) ∼ − (cid:0) d(cid:15) + (cid:15) dζ (cid:1) + b o dζ . (16)This states that the a -circle smoothly shrinks to zero size at χ o , therefore ensuring the absence of spacetimeboundary. In full generality, the smooth capping of any linear combination of the a -circle and the b -circle at χ o can remove the spacetime boundary. However since these cases can be obtained by doing an SL (2 , Z ) transformEq.(4) on the case of shrinking a -circle, we do not lose generality by only considering Eq.(14). By the way Eq.(14)shows that the fields and the time coordinate are necessarily complexified as a result of no-boundary proposal.The starting moment of minisuperspace cosmic time χ o is named “ south pole ” in the literature, c.f. for example[12, 27], and this paper will also use this term. In Fig.2 a sketch of such no-boundary spacetime configuration ispresented, whose difference from the right half of Fig.1 is that now N allows a homogeneous isotropic slicing into2-tori. Had we left the lapse function N general, then we should fix the time range χ ∗ − χ o , and let the lapse N be determined by boundaryconditions, and N is generally complex. ction principle for mini-superspace saddle points Now we are at the point of putting the metric Eq.(11) and the scalar field Φ on-shell in order to let ( N , g µν , Φ)become the saddle point. For this purpose let us derive the action principle. Inserting the metric ansatz (11) intothe total action (6), assuming the scalar field be homogeneous on the equal- χ surfaces, after partial integral toeliminate second order derivatives, we have S = 1 κ ˆ χ ∗ χ o dχ (cid:20) − ˙ a ˙ b − ab + ab (cid:16) ˙Φ − (cid:96) V (Φ) (cid:17)(cid:21) − κ (cid:16) ˙ ab + a ˙ b (cid:17) χ o , (17)where we introduced the coupling constant κ defined as π(cid:96) G = 1 κ . (18)From now on, we set κ = 1 for simplicity. In Eq.(17) the second term on the righthand side is a “boundary” term(or rather “south pole term”) at χ o produced by partial integral. There is also a same boundary term at χ ∗ , butit is canceled by the Gibbons-Hawking term. The equations of motion obtained from varying the action (17) are˙ aa ˙ bb − −
12 ˙Φ − (cid:96) V (Φ) = 0 , (19)¨ aa − − (cid:96) V (Φ) = 0 , (20)¨ bb − − (cid:96) V (Φ) = 0 , (21)¨Φ + ˙ aa ˙Φ + ˙ bb ˙Φ + (cid:96) V (cid:48) (Φ) = 0 , (22)where the dots stand for the derivative with respect to χ : ˙ a = dadχ etc. In appendix A the action and the equationsof motion are derived for a more general case where the equal-time slices of the mini-superspace model have thetopology S d × S d . This will be useful in Sec.5, and also here Eqs (17)—(22) can be obtained by setting d = d = 1in Eqs (154)—(158).In looking for the saddle points, in principle we first assign the wanted boundary values of fields at Σ ∗ : ( h ij , ϕ ) =( A , τ, ϕ ). We then integrate the equations (19)—(22) for { a ( χ ) , b ( χ ) , Φ( χ ) } starting from the south pole χ = χ o tothe boundary χ = χ ∗ . At the south pole χ o we need to input the starting-up data ( b o , Φ o ), as introduced in Eqs(14) and (15), such that the solutions fit the boundary condition at the final moment χ ∗ as Eq.(13). There arethree complex conditions at Σ ∗ given by Eq.(13), exactly what is needed to determine the three complex parameter:( b o , Φ o ) and χ ∗ − χ o . Saddle-point actions preliminary
The mini-superspace formulation used so far makes the evaluation of saddle-point actions very simple. After partialintegral on a and b in the action (17) to recover their second order derivatives: ˙ a ˙ b = ( a ˙ b + ˙ ab )˙ − (¨ ab + a ¨ b ), itbecomes S = 12 ˆ χ ∗ χ o dχ (cid:104) ¨ ab + a ¨ b − ab + ab (cid:16) ˙Φ − (cid:96) V (Φ) (cid:17)(cid:105) − (cid:20)(cid:16) ˙ ab + a ˙ b (cid:17) χ ∗ + (cid:16) ˙ ab + a ˙ b (cid:17) χ o (cid:21) . (23)10 outh Pole ... (complex) Figure 2:
No-boundary spacetime N allowing a mini-superspaceformulation as Eq.(11). The coordinate time χ elapses from χ o to χ ∗ , with each equal- χ surface a homogeneous isotropic 2-torus.Some sample slices χ = χ , . . . , χ are shown. The induced 2-metrics q ij ( χ k ) are as given in Eq.(7). The χ = χ ∗ slice Σ ∗ musthave real metric components h ij and real scalar field ϕ , while forother slices the fields q ij ( χ k ) and Φ( χ k ) ( k = 1 , . . . ,
4) can becomplex. At χ o , one of the circles of the torus caps smoothly,so that there is no spacetime boundary there. The locus χ o isreferred to as “south pole”. Note its difference from the right halfof Fig.1. Here N has a special time coordinate globally givingrise to a homogeneous isotropic T -slicing, while in Fig.1 the onlyrequirement is that ∂ N be Riemannian and of topology T . Kerr-dS contoursEBTZ contoursEBTZ horizons
IIIIII IV
Figure 3:
The metric of empty spacetime saddle points Eq.(50)represented on the complex time plane. When Eq.(50) is confinedon the dashed blue or red lines it takes the form of Kerr-dS orEBTZ. The south pole χ o = iπ is the horizon of an EBTZ con-tour. Saddle points are obtained by confining Eq.(50) to a contourstarting from χ o and terminates at some point χ ∗ on a Kerr-dScontour, such as the contours I through IV. The saddle points I andII terminate in the bulk of the Kerr-dS spacetime with Re χ ∗ < ∞ ,while the saddles III and IV terminate on the future boundary I + of the Kerr-dS space with Re χ ∗ = ∞ . This paper only covers thelatter type of saddle points. Saddle points like I and III (Im χ ∗ = 0)are most commonly considered in the literature for Hartle-Hawkingwave functions, here referred to as “fundamental”. The bulk part is precisely the sum of the equations of motion (20) and (21), vanishing on-shell. Therefore thesaddle-point action becomes simply a sum of two boundary terms: S [ a, b, Φ] on-shell = − (cid:16) ˙ a ∗ b ∗ + a ∗ ˙ b ∗ + ˙ a o b o + a o ˙ b o (cid:17) := S ( A , τ, ϕ ) , (24)where a ∗ = a ( χ ∗ ), a o = a ( χ o ), etc is understood. Also since a , b and Φ are the solutions to Eqs (19)–(22) andsatisfy the boundary condition Eqs (13), the resulting on-shell action is a function of A , τ and ϕ , and thus can bedenote by S ( A , τ, ϕ ). Then since a o = 0 and ˙ a o = i as in Eq.(14), Eq.(24) is further reduced to S ( A , τ, ϕ ) = − (cid:16) ˙ a ∗ b ∗ + a ∗ ˙ b ∗ + i b o (cid:17) . (25)Note that the results Eqs (24) and (25) requires no knowledge of the detail of the scalar potential.On the other hand, we can still apply the commonly used approach in the literature to compute the on-shellaction, where one inserts the constraint (19) into the action and obtains S [ a, b, Φ] on-shell = − ˆ χ ∗ χ o dχ ab (cid:2) (cid:96) V (cid:0) Φ (cid:1)(cid:3) − (cid:16) ˙ ab + a ˙ b (cid:17) χ o = S ( A , τ, ϕ ) . (26)However the numerical error when using this integral expression can go easily out of control in certain cases basicallybecause it counts on the cancellation between two large terms, ab and abV (Φ), to produce a finite result. In thenumerical results presented later, only Eq.(24) will be actually used.11 .3 Saddle-point actions and holographic renormalization Last subsection has offered a preview of saddle-point action evaluation, showing that it can be drastically sim-plified, reduced to merely some boundary terms with no integral involved. This subsection will pursue furtherthe computation of the saddle-point actions to work out their formal results in a holographic setting, which arefurthermore numerically computable.In order to introduce the holographic context, the defining surface Σ ∗ of the Hartle-Hawking wave function willbe sent to I + . Then it will turn out that the infrared (IR) divergences will emerge in the bulk saddle-point actions S under the limit Σ ∗ → I + . Then holographic renormalization will be performed which removes the IR divergenceswith local counter terms, and leads to a finite saddle-point action on the boundary ˜ S .In the following paragraphs, first the asymptotic behaviors of the fields of the saddle points will be analyzed,so as to display the IR divergences in S , and second the counter terms will be introduced to have them removed.A numerically implementable result is then obtained and the numerical calculation will be presented in Sec.4. Infrared divergences of the on-shell action
In Sec.2.2 it is mentioned that the time coordinate χ is generally complex (we will see concrete examples only inthe next section). Therefore we need to assume that the future boundary of the asymptotic de Sitter space I + corresponds to the coordinate time χ = iδ ∗ + ∞ with δ ∗ a finite real number. We will come to the point of how todetermine δ ∗ in later sections. With I + thus set, the limit of Σ ∗ → I + can be formulated asΣ ∗ → I + ⇔ χ ∗ = iδ ∗ + Λ (Λ → ∞ ) , (27)and this formulation together with the notations δ ∗ and Λ will be used to all models to be covered. The real partΛ can be identified with that used in Fig.1, and the imaginary part δ ∗ will be called the “ Euclidean shift ” of theboundary Σ ∗ .The expression Eq.(25) makes the analysis of IR divergences extremely simple. We will only need the asymptoticexpressions of the involved fields evaluated at χ ∗ = iδ ∗ + Λ for large Λ. They are presented in the appendix B. Weneed to be notified that they are obtained for any minimally coupled scalar field with potential of the form V (Φ) = 12 (cid:96) − m Φ + O (Φ ) , √ < m < , (28)where m is the mass parameter of the scalar field. This paper only focuses on the mass range d − < m < d ,where d is the spatial dimension which is 2 for the current model. The upper and the lower bounds are not included,although these are very interesting cases. I would like to leave them to future work and only give some punctualcomments when there is something interesting to mention. With the formulation Eq.(27) for the near dS boundarylimit, the behaviors of the fields on Σ ∗ when Σ ∗ → I + are given by Eqs (175)–(178) with u = e − Λ . The expressionswe will need areΦ ∗ = ϕ ∼ ˆ α e − ∆ − Λ + ˆ β e − ∆ + Λ + . . . , with ∆ ± = 1 ± (cid:112) − m , (29) A = a ∗ b ∗ = C a C b e (cid:18) − ˆ α e − − Λ + ˆ α e − − Λ − m ˆ α ˆ βe − (cid:19) + . . . , (30)12 (cid:0) a ∗ ˙ b ∗ + ˙ a ∗ b ∗ (cid:1) = C a C b e (cid:20) − −
1) ˆ α e − − Λ + (1 − − ) ˆ α e − − Λ (cid:21) + . . . , (31) τ = b ∗ a ∗ = C b C a + . . . , (32)12here 0 < C a , C b < ∞ , and ˆ α , ˆ β are constants which can be complex. Due to the range of scalar mass √ < m < < ∆ − − ∆ + <
1, ensuring that Eq.(29) has indeed included the leading terms. Also there is one short-coming regarding the expansion Eq.(29), in that the coefficients ˆ α and ˆ β are not SL (2 , Z )-invariant, because Λ isnot. We can improve this by rewriting this asymptotic expansion against A : ϕ ∼ α A − ∆ − + β A − ∆+2 , where (33) α = ( C a C b ) ∆ − ˆ α, β = ( C a C b ) ∆+2 ˆ β. (34)Now since both ϕ and A are modular invariant, so are α and β .One important property that Eq.(31) implies is the reality of the IR divergences in the on-shell action Eq.(25).Here the IR divergences in S are those arising when we take the limit Λ → ∞ keeping the other parameters fixed.The fixed parameters include τ , α and β , or equivalently, b o and Φ o . Therefore from Eq.(25) we see that all IRdivergent terms are included in Eq.(31) and are manifestly real. Therefore Im S ( A , τ, ϕ ) is finite under the limitΛ → ∞ , and it tends to a function of τ and α . Therefore if we letRe S (cid:0) A , τ, ϕ (cid:1) = S R (cid:0) A , τ, ϕ (cid:1) , Im S (cid:0) A , τ, ϕ (cid:1) = I (cid:0) A , τ, ϕ (cid:1) , (35)then we can express the reality of IR divergence withlim A→∞ I (cid:16) A , τ, α A − ∆ − + β A − ∆+2 (cid:17) := I (cid:0) τ, α (cid:1) . (36)The disappearing of A -dependence is obvious, while the ϕ -dependence being replaced by α -dependence is becauseof the expansion (33) as well as the fact that when Φ is on shell, β is a function of τ and α . In the followingup sections when it comes to the imaginary part of the on-shell action in the bulk, it will always be referring to I (cid:0) τ, α (cid:1) . I for finite A will not be studied since it is not relevant to the purpose of the paper. An example of thisstudy can be found in the appendix of [27].Now we can write down the expression of the on-shell action which display the IR divergences: S (cid:0) A , τ, ϕ (cid:1) = − C a C b e (cid:20) − −
1) ˆ α e − − Λ + (1 − − ) ˆ α e − − Λ (cid:21) − i b o + . . . , (37)or, taking the real and imaginary parts: S R ( A , τ, ϕ ) = − C a C b e (cid:20) − −
1) ˆ α e − − Λ + (1 − − ) ˆ α e − − Λ (cid:21) + 12 Im b o + . . . (38) I ( τ, α ) = −
12 Re b o , (39)where the dots in Eqs (37) and (38) are exponentially suppressed terms. Cancellation of infrared divergence
With Eq.(37) we are able to carry out the holographic renormalization procedure. The steps to follow is standard:adding the local counter term needed to cancel the leading IR divergence; updating the coefficients of the rest ofthe IR divergences; adding counter terms which cancel the now leading IR divergence, and so on until all divergenceis removed.Regarding Eq.(37), we need first add to it A to cancel the volume divergence C a C b e , where A should becarefully expanded into power series of e Λ using Eq.(30). After this step, we will find that the next-to-leading order13s − ∆ − ˆ α e (2 − − )Λ . Therefore we continue the procedure adding ∆ − A ϕ , which will cancel − ∆ − ˆ α e (2 − − )Λ and meanwhile induce an infinite sum of terms each of order e (2 − n ∆ − )Λ with n = 2 , , , . . . Due to the setting ofmass range in this paper, these terms are all exponential suppressed and hence the holographic renormalization isdone. The counter term action is thus the well familiar result of holographic renormalizaion with the presence ofa scalar field [55]: S ct = A + ∆ − A ϕ . (40)Adding this to Eq.(37), we have the renormalized saddle-point action, to be denoted by ˜ S here (the same notationwill be used for other models to be studied later):˜ S := S + S ct = − i b o − C a C b e (cid:20) − −
1) ˆ α e − − Λ (cid:21) + C a C b e (cid:18) ∆ − ˆ α e − − Λ + ∆ − ˆ α ˆ β (cid:19) + C a C b e (cid:18) − ˆ α e − − Λ − m ˆ α ˆ β e − (cid:19) + . . . = − i b o + (cid:0) ∆ − − m (cid:1) C a C b ˆ α ˆ β + · · · = − i b o + (cid:0) ∆ − − m (cid:1) α β + . . . (41)Here the dots in each line represent the exponentially suppressed terms. In the first line after the second equalityis the bulk action S , and the second line are terms from A and ∆ − A ϕ . In the last line, Eq.(34) was used. Thefinal result for the renormalized on-shell action, when A → ∞ is fully implemented, is ˜ S ( τ, α ) = − i b o ( τ, α ) + (cid:0) ∆ − − m (cid:1) α β ( τ, α ) . (42)It will be useful to introduce the notations in parallel with Eq.(35):Re ˜ S (cid:0) τ, α (cid:1) = ˜ S R (cid:0) τ, α (cid:1) , Im ˜ S (cid:0) τ, α (cid:1) = ˜ I (cid:0) τ, α (cid:1) , (43)and thus Eq.(42) is split into ˜ S R (cid:0) τ, α (cid:1) = 12 Im b o ( τ, α ) + (cid:0) ∆ − − m (cid:1) α Re β ( τ, α ) , (44)˜ I (cid:0) τ, α (cid:1) = −
12 Re b o ( τ, α ) + (cid:0) ∆ − − m (cid:1) α Im β ( τ, α ) . (45)This is an exact result of holographic renormalization for finite scalar deformation, with the limitation that thedeformation must be homogeneous and isotropic. If the computation is correct, then we should have the relationof “one-point function” generation [6] ∂ ˜ S ∂α = (∆ − − ∆ + ) β. (46)This is not at all straightforward seen from Eq.(42) but it will be verified, either perturbatively in Sec.3 or non-perturbatively by numerics in Sec.4. When the scalar mass saturates the lower bound m = √ , it seems that the ˆ α e (2 − − )Λ -term is finite and we do not know whetherto introduce further counter term to remove it or not. However straightforward computation shows that when ˆ α e (2 − − )Λ -divergenceare cancelled, the ˆ α e (2 − − )Λ -term is automatically removed. Thus the counter terms presented below Eq.(40) actually works for m = √ . This is the result in α -representation, which is related to that of the β -representation by a Legendre transform. The β -representationresult is actually ˜ S ( τ, β ) = − i b o ( τ, β ) + (cid:0) ∆ + − m (cid:1) α ( τ, β ) β. olographic renormalization as generalized Legendre transform So far the result of holographic renormalization ˜ S is automatically considered as the quantity to be directly identifiedwith the boundary field theory generating function (in case boundary dual available), which seems natural enough.There is however an issue arising, if we remember that the initial motivation is to address the problem raised in[22]. The latter have a seemingly different opinion, which states that the bulk-computed quantity to be identifiedwith boundary QFT partition function, is the Hartle-Hawking wave function, but the one having been Fourier-transformed into the representation of dS boundary data (as if τ and α for our case).Therefore, suppose our bulk wave function Ψ( A , τ, ϕ ), when transformed to the boundary data representation,becomes Ψ I + ( τ, α ) with its tree-level approximation Ψ I + ∼ exp( i S I + ), then we need to have ˜ S = S I + , in order thatthe conclusions about divergence drawn from ˜ S are on the same comparing ground with those drawn in [22]. Indeedalthough a rigorous proof is not yet found for the very model being studied, the properties of ˜ S strongly suggeststhat the equality ˜ S = S I + should hold.Basically we see that on one hand S I + is supposed to be the on-shell value of an action well-defined for thevariation principle with ( τ, α ) fixed on the boundary I + , while on the other hand ˜ S fits the main criteria to besuch a quantity. First, it is finite and second it satisfies the relation (46) (to be verified in the next two sections),obligatory for the on-shell action in the α -representation, knowing that (∆ − − ∆ + ) β is the conjugate momentumof α . A very relevant work in this regard has been presented in [47] which reveals that the holographic renormalizationapplies to a wide category of theories possessing an asymptotic boundary, regardless of whether a boundary dualtheory exists or not; from the bulk point of view, it is a procedure of figuring out the right action which have thevariation principle well defined in terms of the asymptotic data characterizing the classical solutions. Thereforeaccording to [47] the holographic renormalization just performed to arrive at Eq.(42) well corresponds to theprocedure for figuring out the right action for the variation principle in terms of τ and α . A theoretical frameworkhas been set forward in [47], and probably it can be applied here to achieve a solid demonstration of ˜ S = S I + . In the previous section the framework has been worked out for quantitatively computing the physical properties ofthe no-boundary saddle points. However before having them implemented numerically, it is important to obtainan analytic preview by treating the scalar field perturbatively. The result will serve as the starting points for thenumerical search of saddle points subjected to finite scalar deformations. This section is devoted to this task andthe numerical computation will be carried out in the next section. A key point is that since the perturbationtheory is for approximating a theory that takes scalar backreaction into full account, the spacetime backgroundshould not be set strictly rigid. Instead, its deformation should be consistently truncated to the order preset in theperturbative framework, and here it is truncated to the quadratic order. The following contents will be presentedin this section.Sec.3.1 works out the saddle points of empty spacetime. Emphasis will be on the necessary complexification ofthe coordinate time, and also on the existence of infinitely many saddle points that fit some boundary condition For the range of mass chosen in this paper, the ( α, β ) space inherits a natural symplectic structure from the phase space of the bulkscalar field. Let Π Φ be the conjugate momentum of Φ and thus Π ϕ = Π Φ ( χ ∗ ). Using the expression of the action (17) and Eq.(33),we have asymptotically Π ϕ = a ∗ b ∗ ˙Φ ∗ ∼ − ∆ − α A ∆ + / − ∆ + β A ∆ − / , from which we can derive explicitly the transform between( ϕ, Π ϕ ) and (cid:0) α, (∆ − − ∆ + ) β (cid:1) , and we will find that it is a canonical transform since the transformation matrix has unit determinant.We can further on to derive the type one generation function of this transform and will find that it is, up to negative powers of A ,just − ∆ − A ϕ , one of the counter terms in the holographic renormalization (c.f. for example [48] for the relation between generatingfunctions of canonical transforms and the kernels of Fourier transforms). All this is still true when m saturates the upper bound m = 1,and the scalar asymptotic behavior becomes ϕ ∼ α A − ∆ − / ln A + β A − ∆ + / . In the absence of the scalar field, the wave function is reduced to a function of the geometry of Σ ∗ : Ψ( A , τ ), sois its tree-level approximation is e i S ( A ,τ ) with S ( A , τ ) the on-shell value of Eq.(6) evaluated on the saddle pointwith Φ = 0 and boundary Σ ∗ characterized by ( A , τ ). Infinite number of saddle points and the “fundamental” saddle point
In the mini-superspace setup, as summarized in the second paragraph below Eq.(22), the saddle points are obtainedby integrating Eqs (19)—(21) with Φ = 0 from the south pole χ o respecting the conditions (14) and (15), untilsome final moment χ ∗ where the field values match the assigned boundary data ( A , τ ) as Eq.(13). The south poledata b o , as well as the final moment χ ∗ are determined through this matching. Since it is the difference χ ∗ − χ o that really matters, we can fix the south pole. Here it is fixed at χ o = iπ , and the solution to Eqs (19)—(21) istrivial: a ( χ ) = cosh χ, b ( χ ) = − ib o sinh χ, (47)where the absence of spacetime boundary at χ o is seen from a ( χ ) = cosh χ , in that a ( χ o + (cid:15) ) ∼ i(cid:15) ( (cid:15) ∼
0) is satisfied.Now we can enforce the boundary condition at Σ ∗ to determine b o and χ ∗ , which are, inferred from Eq.(13): A = a ∗ b ∗ = − ib o cosh χ ∗ sinh χ ∗ , τ = b ∗ a ∗ = − ib o tanh χ ∗ . (48)In the context of Σ ∗ → I + , with χ ∗ parameterized by Eq.(27), we have A ∼ e , and the only possibility forEq.(48) to hold is δ ∗ = kπ, where k ∈ Z ; and b o = iτ coth Λ ∼ iτ , (49)where δ ∗ = Im χ is the Euclidean shift as in Eq.(27). Therefore there are infinitely many saddle points of emptyspacetime, labeled by integer k in the Euclidean shift δ ∗ , corresponding to different amount of Euclidean time thatthe saddle point experiences in completing its complex history in terms of χ . However the metric of all these saddle16oints are the same, given by Eq.(11) with a ( χ ) and b ( χ ) just obtained above: (cid:96) − ds = − dχ + cosh χ dζ + τ sinh χ dζ , (50)where the coordinates ζ , are periodically identified according to Eq.(12).To summarize, a complete specification of a saddle point of empty spacetime consists of the metric Eq.(50),and a χ -contour which starts from the south pole and ends at χ ∗ = ikπ + ∞ with some specific integer k , and thiscontour can be arbitrarily deformed between the fixed two ends since with no singularity present on the χ -plane,such different contours, starting from the same south pole data b o , will result in the same boundary data at Σ ∗ and the same on-shell action. A schematic illustration of these saddle points is given in Fig.3.Among all the saddle points labeled by k , we can single out one type of them that are commonly consideredin the literature as the contributing saddle point of the Hartle-Hawking wave function (for example in the originalpaper [12, 14, 27]). These are saddle points experiencing a minimum amount of positive Euclidean time, such asthe saddle points I and III in Fig.3 which have δ ∗ = 0. Such saddle points are to be referred to as “fundamentalsaddle points” in this paper. Later we will see, when scalar field is switched on, that they yield very differentphysical outcomes from the non-fundamental saddle points. Kerr-dS universe and EBTZ black hole
An important property to mention about the saddle points is that its spacetime can be either asymptotically dSor asymptotically AdS according to the choice of complex time contour.We first notice that when χ runs on the horizontal lines χ = ikπ + λ where k ∈ Z is fixed and λ ∈ R varies, themetric Eq.(50) becomes (cid:96) − ds = − dλ + cosh λ dζ + τ sinh λ dζ , (51)and this is just the Kerr-dS metric. To see more clearly we introduce the time variable t defined ascosh λ = t + τ | τ | or sinh λ = t − τ | τ | (52)as well as the spatial coordinates ( θ, φ ): ζ = | τ | + τ τ θ + τ φ, ζ = τ τ θ + φ, (53)whose periodicities are, referring to Eq.(12):(∆ φ, ∆ θ ) = 2 π (1 ,
0) or 2 π (cid:18) − τ | τ | , τ | τ | (cid:19) (54)In terms of these variables, the 3-metric (50) reads (cid:96) − ds = − t dt ( t − τ )( t + τ ) + ( t − τ )( t + τ ) t dθ + t (cid:16) dφ − τ τ t dθ (cid:17) , (55)which is nothing but the Kerr-dS metric in its standard form.Another type of contours lead to asymptotic AdS behavior, which are the horizontal lines given by χ = i (cid:0) k + (cid:1) π + λ where k ∈ Z is fixed and λ ∈ R varies. Along these contours the metric becomes Euclidean17ut with an overall negative sign, which can be absorbed into the dS length by defining (cid:96) = i(cid:96) (cid:48) : (cid:96) (cid:48)− ds = dλ + sinh λ dζ + τ cosh λ dζ , (56)This is the EBTZ black hole metric, whose event horizon is at λ = 0 where the ζ -circle shrinks smoothly to zerosize. In fact the south pole χ o = iπ lies just on the k = 0 contour and is the event horizon of that very EBTZblack hole. To cast Eq.(56) into the standard form, we introduce the radial coordinate r such thatsinh λ = − r + τ | τ | or cosh λ = − r + τ | τ | , (57)as well as the same angular coordinates as in Eq.(53). This leads to (cid:96) (cid:48)− ds = r dt ( r + τ )( r − τ ) + ( r + τ )( r − τ ) r dθ + r (cid:16) dφ + τ τ r dθ (cid:17) , (58)which describes a Euclidean BTZ black hole whose horizon is at r = τ . With Eq.(54) it is clear that τ is ratherthe modular parameter of its dual thermal AdS space [50], while it is ˜ τ = − τ − that is the modular parameter ofEBTZ black hole. Therefore we have the black hole temperature ˜ τ − and angular momentum ˜ τ .The possibility of analytic continuation between Kerr-dS metric and EBTZ metric is just an example of a moregeneral feature that saddle points having an asymptotic dS region can be continued into one with an asymptoticEAdS region. This feature is used in [3] to obtain correlation functions on the dS boundary. Also it is used in [24]to obtain the locally dS wave function from the locally AdS partition function. In a more general context it hasbeen used in [12, 13, 49] to work out a holographic probability measure of classical dS cosmology, where the dualCFT lives on the EAdS boundary. In this paper, this feature can relate the cosmological problem to that of thethermodynamics of BTZ black holes with scalar hair. Saddle-point action
Plugging the solutions for a ( χ ) and b ( χ ) into Eq.(17) putting Φ = 0, we obtain the saddle-point action S ( A , τ ) = ˆ χ ∗ χ o dχ (cid:16) − ˙ a ˙ b − ab (cid:17) − (cid:16) ˙ ab + a ˙ b (cid:17) χ o = ˆ ikπ +Λ iπ dχ (cid:16) − τ sinh χ cosh χ − τ cosh χ sinh χ (cid:17) − τ sinh iπ − τ sinh Λ = − (cid:112) τ sinh Λ cosh Λ ( τ sinh Λ cosh Λ − τ tanh Λ)= − (cid:112) A ( A − τ ) = −A + τ O ( A − ) . (59)The result consists of a volume divergent term and a finite term and other terms suppressed by negative powers of A . Since it is real, the wave function at tree level Ψ ∼ e i S ( A ,τ ) is a pure phase. Also, note that all the differentsaddle points, specified by χ ∗ = ikπ + Λ with k ∈ Z , give rise to the same on-shell action.Adding the counter terms Eq.(40) to Eq.(59), in fact only the first term A due to the absence of scalar field,we obtain the holographically renormalized saddle-point action:˜ S ( τ ) = S ( A , τ ) + A = − (cid:112) A ( A − τ ) + A = τ O ( A − ) . (60)The corresponding saddle-point contribution to the boundary Hartle-Hawking wave function is e i ˜ S = e iτ , and isjust a pure phase. This is unlike the 4d saddle points of empty spacetime of asymptotic boundary S × S considered18n [22, 16], whose actions have non-trivial imaginary parts. Note that once we recover the coupling constant κ as inEq.(18), we see that Eq.(60) is just saddle-point action of the exact wave function of empty Kerr-dS obtained [24],which can also be obtained in different ways along the EBTZ contour, for example in [50, 51, 52], in the contextof black hole thermodynamics. Indeed it is an interesting aspect to explore whether the result (42) can be usedaddress problems of black hole with scalar hair. Now let us turn on a small scalar perturbation to the no-boundary saddle points. As a result the scalar field shouldalso be an entry in the boundary conditions to impose on Σ ∗ . Since Σ ∗ → I + , it is convenient to assign the boundaryconditions in terms of ( A , τ, α ). With A → ∞ , implying ϕ = α A − ∆ − / + β A − ∆ + / ≈ α A − ∆ − / , we can let α be real to have ϕ real. Note that without holographic renormalization, using ( A , τ, α ) is just a reparameterizationof the boundary conditions, while the wave function is still in the ( A , τ, ϕ )-representation. Then we can solve thescalar equation Eq.(22) using its leading order in the expansion in terms of Φ:¨Φ + 2 coth(2 χ ) ˙Φ + m Φ = 0 . (61)Since the south pole is χ o = iπ/
2, for convenience we define z = χ )2 = cosh χ , so that the south pole is at z = 0. The equation (61) now becomes a hypergeometric equation z (1 − z )Φ (cid:48)(cid:48) zz + (1 − z )Φ (cid:48) z + m . (62)The solution of regular behavior at the south pole z = 0 isΦ( χ ) = C F (cid:18) ∆ − , ∆ + , , z (cid:19) = C F (cid:0) u, v, w, cosh χ (cid:1) . (63)where C = Φ( χ o ) = Φ o is just the south pole data of the scalar field, and is to be determined by the boundarydata α . In the second step above we have set u = ∆ − − √ − m , v = ∆ + √ − m , w = 1 . (64)An important property of the analytical structure of the expression (63), is that it has branch points at χ = ikπ ( k ∈ Z ), where the b -circle shrinks to zero size. When χ ∼ ikπ for any integer k , the hypergeometric function inEq.(63) has the asymptotic behavior F (cid:0) u, v, w, cosh χ (cid:1) = − ψ ( u ) + ψ ( v ) + 2 γ + iπ + ln(cosh χ − u )Γ( v ) + O (cosh χ − , (65)where we define the phase to be − π < arg(cosh χ − ≤ π . With the choice of south pole at χ o = i π the expression(63) is valid only in the strip 0 ≤ Im χ < π . When the scalar field is continued beyond this strip, the value of Φdepends on the complex χ -contour that we choose connecting the south pole χ o and the evaluation point of Φ.For the moment let the time be restricted to the Re χ > χ < ranch cutsEBTZ contoursKerr-dS contoursEBTZ horizonsBranch points(south pole) Figure 4:
Riemann surface of the perturbative homogenousscalar field Φ( χ ) as a function of complexified χ . The expressionis given by (66). The Kerr-dS contours and the EBTZ contoursare still as in Fig.3. The south pole is still χ o = iπ . Thereare logarithmic branch points on each Kerr-dS contour at ikπ ( k ∈ Z ) and the branch cuts are set to open towards Re χ = −∞ .The figure shows only the Riemann surface layer where the southpole is situated (fundamental layer). On other layers, accessibleby crossing the brach cuts, χ = i (cid:0) k + (cid:1) π ( k ∈ Z ) can alsobecome branch points. Fundamental saddle
IV IIIIII
Figure 5:
Examples of saddle points with perturbativescalar field deformation, labeled by I through IV. The χ -contours can either avoid all branch cuts and stay in the samelayer of the Riemann surface as I and II; or circle around somebranch point as III and IV and finish up in another layer. Thesaddle point I has the feature of the saddle points most com-monly considered in the Hartle-Hawking wave function lit-erature: it has minimum amount of positive Euclidean timewith the contour ending up in the same layer as the southpole. It will be qualified as the fundamental saddle point. towards the Re χ < F ( χ ) = F (cid:0) u, v, w, cosh χ (cid:1) , ≤ Im χ < π, Continuation such that all branchcuts open towards Re χ < , otherwise . (66)The scalar field on the whole complex χ -plane is thus expressed asΦ( χ ) = Φ o ˆ F ( χ ) . (67)In Fig.4 a graph of the Riemann surfaces of the homogeneous perturbative scalar field is presented.When matching Φ to its boundary condition at I + , we will need the asymptotic behavior of ˆ F ( χ ) for Re χ → + ∞ .We can derive this asymptotic behavior from the following identity of hypergeometric function [53] F ( u, v, w, x ) = Γ( w )Γ( v − u )Γ( v )Γ( w − u ) F (cid:18) u, w − v, u − v + 1 , − x (cid:19) + Γ( w )Γ( u − v )Γ( u )Γ( w − v ) F (cid:18) v, w − u, v − u + 1 , − x (cid:19) , (68)together with the expansion F ( u, v, w, x ) = 1 + uvw x + O ( x ), where the phase of the argument in Eq.(68) isunderstood as − π < arg(1 − x ) ≤ π . Therefore we have for Eq.(66),ˆ F ( χ ) ∼ Γ( w )Γ( v − u )Γ( v )Γ( w − u ) (cid:18) e − iπ e χ (cid:19) − u + Γ( w )Γ( u − v )Γ( u )Γ( w − v ) (cid:18) e − iπ e χ (cid:19) − v , Re χ → + ∞ , (69)20here we should be careful with the phase of 1 − cosh χ to let the above expression match Eq.(68) when − π ≤ Im χ < π . It can be easily figured out that when the scalar field is perturbative, the χ -contours characterizingthe saddle points are still as in the case of empty space, since their perturbative deformations merely result inbeyond-leading-term contributions to the quantities to be computed. Therefore the χ -contours start off from thesouth pole χ o = iπ and ending up at the asymptotic boundary of a Kerr-dS contour χ ∗ = ikπ + Λ with k ∈ Z andΛ → ∞ . Therefore we have ϕ = Φ ∗ = Φ o ˆ F ( χ ∗ ) ∼ Φ o Γ( w )Γ( v − u )Γ( v )Γ( w − u ) (cid:20) e i (2 k − π e (cid:21) − u + Φ o Γ( w )Γ( u − v )Γ( u )Γ( w − v ) (cid:20) e i (2 k − π e (cid:21) − v . (70)Fitting this expression into the standard form Eq.(33), provided that A = a ∗ b ∗ ∼ τ e according to Eqs (48) and(49), we obtain the coefficients: α = Φ o τ u e iu (1 − k ) π Γ( w )Γ( v − u )Γ( v )Γ( w − u ) , β = Φ o τ v e iv (1 − k ) π Γ( w )Γ( u − v )Γ( u )Γ( w − v ) . (71)The first relation serves to determine the south pole data Φ o with the boundary value of α assigned at (near) I + :Φ o = α τ − u e iu (2 k − π Γ( v )Γ( w − u )Γ( w )Γ( v − u ) . (72)Now the saddle point is fully specified, recalling that the other defining parameters: b o and δ ∗ have already beenobtained in Eq.(49). Once again like in the previous subsection, infinite number of saddle points are obtained, herelabeled by the integer k .In Fig.5 some example saddle points are shown, represented by complex χ -contours. Those relevant to thecurrent and the next subsections are saddle points I ( k = 0) and II ( k = − χ -contours always stay inthe same Riemann layer. Other saddle points have loops in their χ -contours around the branch cuts, and thiswill be considered in Sec.3.4. Again we can single out the fundamental saddle point, which shares the features ofthe saddle points very commonly used in the literature. It is represented by the contour I and it lives through ashortest positive Euclidean history. However in the presence of the branch cuts in the χ -plane, the fundamentalsaddle points should also have the time contours that do not circle around any branch point.Plugging Eq.(72) back into the second relation in Eq.(71), we obtain β , which is complex in general. Howeveras explained in the beginning of this subsection, since the contribution to ϕ from β is negligible when A → ∞ , ϕ can be practically considered as real. For later use, let us put βα = (cid:104) τ e i (1 − k ) π (cid:105) v − u Γ( v )Γ( w − u )Γ( u − v )Γ( u )Γ( w − v )Γ( v − u ) = τ √ − m ρ ( m, k ) , (73)where we denoted the τ -independent factor by ρ : ρ ( m, k ) = exp (cid:104) i π (1 − k ) (cid:112) − m (cid:105) Γ (cid:0) −√ − m (cid:1) Γ (cid:16) √ − m (cid:17) Γ (cid:0) √ − m (cid:1) Γ (cid:16) −√ − m (cid:17) . (74) Note that when the scalar mass saturates the lower bound m = √ , we have 2 v − u = 1. Then the sub-leading order of the firstline of Eq.(68) is of the same order as the leading order of the second line, and therefore the second term in Eqs (69) and (70) shouldreceive an extra contribution. However this paper will not cover this case. .3 Saddle-point actions of homogeneous scalar peurtubations In the naive bulk field representation
For computing the on-shell action perturbatively we cannot use the expression (25) or (26), because they areobtained using the full equations (19)—(21), while the perturbative solutions of saddle points do not satisfy themexactly. We need instead to follow the steps in the appendix of [27], dealing with gravity part S g and the scalarpart S Φ separately: S = S g + S Φ where (75) S g = ˆ χ ∗ χ o dχ (cid:16) − ˙ a ˙ b − ab (cid:17) − (cid:16) ˙ ab + a ˙ b (cid:17) χ o , (76) S Φ = 12 ˆ χ ∗ χ o dχ ab (cid:16) ˙Φ − (cid:96) m Φ (cid:17) = 12 (cid:16) ab Φ ˙Φ (cid:17) χ ∗ . (77)In the last line the expression of S Φ , the scalar field has been integrated by parts and the equation of motion (22)as well as the south pole condition a ( χ o ) = 0 is used.The scalar part is easier to work out. We insert into Eq.(77) the asymptotic expansion Eq.(33), and we have S Φ = 12 A (cid:2) − ∆ − α A − ∆ − − (∆ − + ∆ + ) αβ A − ∆ − − ∆ + − ∆ + β A − ∆ + (cid:3) = − ∆ − α A √ − m − ∆ − + ∆ + αβ − ∆ + β A −√ − m = − α (cid:20) ∆ − A √ − m + (∆ − + ∆ + ) τ √ − m ρ (cid:21) + . . . , (78)where ρ is as in Eq.(74). In the last line, the first term in the bracket is IR divergent, the second term complexand finite, and the dots stand for terms suppressed by negative powers of A .Next step we need to work out S g , the contribution from gravity, to the second order in α . Let some perturbativecorrection be induced to the metric components due to scalar deformation: a ( χ ) → a ( χ ) + δ Φ a ( χ ) , b ( χ ) → b ( χ ) + δ Φ b ( χ ) . (79)where a ( χ ) and b ( χ ) are the unperturbed results Eq.(47) with b o given by Eq.(49). Since the south pole conditions(14) should always hold, we must have δ Φ a o = δ Φ ˙ a o = 0 , (80)where the subscript o means evaluated at χ o . Varying the equations (19)–(22) with δ Φ we find that δ Φ a and δ Φ b must start from the order α in order to have the south pole conditions Eq.(80) respected, and we can also derivethe condition δ Φ ˙ b o = 0 . (81)Now plugging the perturbed quantities Eq.(79) into the gravity sector action Eq.(76), using the explicit form ofunperturbed a ( χ ) and b ( χ ), and also using the south pole conditions Eqs (14), (80) and (81), we get S g ( A , τ, α ) = S ( A , τ ) − ˙ a ∗ δ Φ b ∗ − ˙ b ∗ δ Φ a ∗ , (82)22here the subscript ∗ means evaluated at the final moment χ ∗ . Also, S ( A , τ ) is just the Φ = 0 result Eq.(59),where A is taken to be the product of unperturbed a ∗ and b ∗ . Now we need to find out δ Φ a ∗ and δ Φ b ∗ . For thispurpose we turn to the result obtained in appendix B, where our perturbative results here correspond to putting C a = , C b = τ and also u = e − Λ in Eqs (175)–(178). From these expressions we can simply read off the variationsof a ∗ and b ∗ up to the second order in α : δ Φ a ∗ = − u (cid:32) ˆ α u − − ˆ a u + ˆ β u + + . . . (cid:33) , (83) δ Φ b ∗ = − τ u (cid:32) ˆ α u − − ˆ b u + ˆ β u + + . . . (cid:33) , (84)where u = e − Λ = (cid:113) τ A , ˆ α = α (cid:0) τ (cid:1) − ∆ − / , ˆ β = β (cid:0) τ (cid:1) − ∆ + / andˆ a + ˆ b + m ˆ α ˆ β = 0 . (85)We also note that the variations of a and b in Eqs (83) and (84) start from quadratic order in α , consistent withthe discussion under Eq.(81). Now we can insert Eqs (83) and (84) into Eq.(82) to obtain S g , which yields S g ( A , τ, α ) = − (cid:112) A ( A − τ ) + α (cid:18) A √ − m + 2 m τ √ − m + τ √ − m ρ A −√ − m (cid:19) (86)The total saddle-point action Eq.(75) is therefore the sum of Eqs (78) and (86): S ( A , τ, ϕ ) = S g ( A , τ, ϕ ) + S Φ ( A , τ, ϕ )= − (cid:112) A ( A − τ ) + α (cid:20) (1 − ∆ − ) A √ − m + (cid:0) m − ∆ − − ∆ + (cid:1) τ √ − m ρ (cid:21) + . . . = (cid:18) − A + 1 − ∆ − A √ − m α (cid:19) + (cid:18) τ m − ∆ − − ∆ + τ √ − m ρ α (cid:19) + . . . . (87)where in the last line the first parenthesis contains all the IR divergent terms, the second parenthesis contains thefinite terms, and the dots are all higher order terms in α and those suppressed by negative powers of A . The totalaction is obviously a complex number since ρ is complex, but the IR divergences are manifestly purely real, just asdiscussed around Eq.(36). Therefore the imaginary part under the limit A → ∞ is I ( τ, α ) = 12 Im (cid:104)(cid:0) m − ∆ − − ∆ + (cid:1) τ √ − m ρ α (cid:105) = − (cid:0) − m (cid:1) τ √ − m Im ρ α , (88)where terms suppressed by negative powers of A are discarded. Saddle-point actions holographically renormalized
To do holographic renormalization on Eq.(87), we simply apply the counter terms Eq.(40) perturbatively, where A should be understood as ( a ∗ + δ Φ a ∗ )( b ∗ + δ Φ b ∗ ) = a ∗ b ∗ + a ∗ δ Φ b ∗ + b ∗ δ Φ a ∗ = A + δ Φ A , and where δ Φ A = a ∗ δ Φ b ∗ + b ∗ δ Φ a ∗ = − α (cid:18) A √ − m + 2 m τ √ − m (cid:19) + . . . (89) Had we regarded the spacetime background as strictly rigid, then the result would be I = − τ √ − m Im ρ α , which however, doesnot approximate the model in case scalar deformation is finite. S ( τ, α ) = S ( A , τ, ϕ )+ (cid:16) A + δ Φ A (cid:17) + ∆ − A (cid:32) A − ∆ − α + 2 A − ∆++∆ − αβ (cid:33) + . . . = (cid:18) − A + 1 − ∆ − A √ − m α (cid:19) + (cid:18) τ m − ∆ − − ∆ + τ √ − m ρ α (cid:19) + A + α (cid:34)(cid:0) ∆ − − (cid:1) A √ − m + 2 (cid:0) ∆ − − m (cid:1) τ √ − m ρ (cid:35) + . . . = τ (cid:0) ∆ − − ∆ + (cid:1) τ √ − m ρ α + . . . , (90)where the dots are terms suppressed by negative powers of A or by terms of higher orders than α . The resulting˜ S is well defined and is supposed to approximate Eq.(42). This will be numerically verified in the next section.Now an immediate test is taking the derivative with respect to α : ∂ ˜ S ∂α = (∆ − − ∆ + ) τ √ − m ρ α = (∆ − − ∆ + ) β. (91)We obtain precisely the “one-point function generation” Eq.(46) on the perturbative level. Splitting the real andthe imaginary parts, we have˜ S R ( τ, α ) = 12 (cid:0) ∆ − − ∆ + (cid:1) τ √ − m Re ρ α ; ˜ I ( τ, α ) = 12 (cid:0) ∆ − − ∆ + (cid:1) τ √ − m Im ρ α . (92)Comparing the second expression and Eq.(88), we find the relation I ( τ, α ) = (cid:112) − m ˜ I ( τ, α ) (scalar perturbation) . (93)Assuming that the higher orders in the loop expansion are highly suppressed with respect to the tree level, thisrelation seems to imply that the holographic renormalization preserves the relative importance of different saddles.In particular, a dominating saddle for the Hartle-Hawking wave function in the bulk field representation staydominating for the wave function in the boundary data representation. However we will see that Eq.(93) is utterlyabandoned when scalar deformation goes non-perturbative. Observations concerning the sign of Im ρ Summarizing the results of the saddle-point contributions to the Hartle-Hawking wave functions: e i S = e i [ A + O ( α )] × exp (cid:104)(cid:0) − m (cid:1) τ √ − m Im ρ α + . . . (cid:105) , (bulk); (94) e i ˜ S = e i [ τ + O ( α )] × exp (cid:104)(cid:112) − m τ √ − m Im ρ α + . . . (cid:105) , (boundary) . (95)It seems that the sign of Im ρ plays an important role in deciding whether these saddle point contributions areexponentially divergent or suppressed with the scalar deformation characterized by α . However this is obviouslynot the case since these results are valid only for α ∼
0. Whatever the sign of Im ρ , we will always need to tracethe saddle-point contributions to finite α to decide whether they are divergent.However the sign of Im ρ is relevant to the “high temperature”, or large τ behavior of these results. Note thatEqs (94) and (95) are perturbative only in terms of α never in τ . Therefore they are viable for whatever positivevalue of τ . Especially in case the saddle point has Im ρ >
0, its tree-level contributions to the wave functiondiverges exponentially as τ → ∞ , both in the bulk or on the boundary. Thus very likely Im ρ > τ , in case theis divergence is not cancelled by thecontribution from other saddles. It will be important to see if this divergence still persists when α increases to thenon-perturbative domain, and this will be investigated in the next section.In the left half of Fig.(6) Im ρ is plotted for k = 0 until k = −
5, and it shows that for k = 0, Im ρ < m between √ and 1, while for other k , Im ρ can be positive for certain ranges of m . This subsection looks into the cases where the saddle points have complex time contour containing loops aroundsome branch points of the scalar field figured out in Sec.3.2. Quantitative results will only be presented for thecases where only one branch point is involved. When several branch points are involved the discussion will bequalitative, since a detailed analysis will be tedious and there is not yet perspective seen that quantitive result canbring any conceptually important contribution to the purpose of this paper.
Loops around one branch point
As is stated earlier in the paragraph under Eq.(65), the branch points are χ [ k ] = ikπ with k ∈ Z . Here let us considerthe case where the χ -contour circles around the n -th branch point p times, where p > p < χ -contour of the saddle point III circles the singularity χ [ − clockwisely twice ( n = − p = 2); the χ -contour of the saddle IV circles χ [1] clockwisely once ( n = 1, p = 1) andthen circles χ [2] counterclockwisely once ( n = 2, p = − χ [ n ] until point χ a and then make p loops around it, stopping atpoint χ b (c.f. the saddle III in Fig.5). For simplicity of reasoning, define − π < arg(cosh χ a − ≤ π and − π < arg(cosh χ b − ≤ π . Then according to Eq.(65) at χ a the scalar field behaves asΦ( χ a ) = Φ o ˆ F ( χ a ) ∼ − Φ o × ψ ( u ) + ψ ( v ) + 2 γ + iπ + ln(cosh χ a − u )Γ( v ) . (96)After completing the p loops around χ [ n ] , the asymptotic expansion becomesΦ( χ b ) ∼ − Φ o × ψ ( u ) + ψ ( v ) + 2 γ + iπ + ln(cosh χ b − u )Γ( v ) − Φ o × p πi Γ( u )Γ( v ) . (97)That is, the expansion acquires a constant − Φ o × p πi Γ( u )Γ( v ) . It is 4 πi in the numerator because cosh χ − ∼ (cid:0) χ − χ [ n ] (cid:1) when χ ∼ χ [ n ] , so that when the phase of χ − χ [ n ] increases by 2 π , that of cosh χ − π .Thus at this point the expression for Φ( χ b ) is not Φ o ˆ F ( χ b ), but should be Φ o ˆ F ( χ b ) plus an extra term, and theleading term of this extra term when expanded, is − Φ o × p πi Γ( u )Γ( v ) . Since this extra term should be a solution ofEq.(62), the only choice is (cf [53] 9.153-7) − p πi Γ( u )Γ( v ) Φ o ˆ F (cid:18) χ b − χ [ n ] + iπ (cid:19) , (98)and hence Φ( χ b ) = Φ o ˆ F ( χ b ) − p πi Γ( u )Γ( v ) Φ o ˆ F (cid:18) χ b − χ [ n ] + iπ (cid:19) . (99)25o summarize, we have the following expression for the scalar fieldΦ( χ ) = Φ o ˆ F ( χ ) , χ earlier than χ a ;Φ o ˆ F ( χ ) − p πi Φ o Γ( u )Γ( v ) ˆ F (cid:18) χ b − χ [ n ] + iπ (cid:19) , χ later than χ b . (100)The asymptotic behavior near I + isΦ( ikπ + Λ) ∼ ˆ α e − u Λ + ˆ β e − v Λ = α A − ∆ − / + β A − ∆ + / , where (101) α = Φ o τ u e iu (1 − k ) π Γ( w )Γ( v − u )Γ( v )Γ( w − u ) (cid:20) − p πi Γ( u )Γ( v ) e i (2 n − u π (cid:21) , (102) β = Φ o τ v e iv (1 − k ) π Γ( w )Γ( u − v )Γ( u )Γ( w − v ) (cid:20) − p πi Γ( u )Γ( v ) e i (2 n − v π (cid:21) . (103)Thus if we specify α as the boundary condition, the south pole value of the scalar field Φ o is determined by theassigned value of α through Eq.(102):Φ o = α τ − u e iu (2 k − π Γ( v )Γ( w − u )Γ( w )Γ( v − u ) (cid:20) Γ( u )Γ( v )Γ( u )Γ( v ) − p πi e i (2 n − u π (cid:21) . (104)The proportionality relation between the coefficients still formally reads β = α τ √ − m ρ but with ρ = ρ ( m, k, p, n ) = e i ( u − v )(2 k − π Γ( v )Γ( u − v )Γ( w − u )Γ( u )Γ( v − u )Γ( w − v ) Γ( u )Γ( v ) − p πi e (2 n − v πi Γ( u )Γ( v ) − p πi e (2 n − u πi . (105)The on-shell actions of these saddle points are still formally given by Eqs (87) and (90), but with ρ given by theabove expression. The numerical calculation in the next section will cover the cases where the contour circles χ [0] ( n = 0) from p = − p = 2 times. In the frame in the right part of Fig.6 Im ρ is plotted for these cases. Whenthe χ -contour circles χ [0] = 0 clockwisely or counterclockwisely, Im ρ is positive or negative for the whole range ofscalar mass of √ < m <
1. Here the remarks concerning the sign of Im ρ by the end of the previous subsectionare still valid. Loops around several branch points
When the χ -contour circles around more than one branch point, the analysis proceeds in the same way as thatis used to obtain Eq.(100), where in figuring out the extra term that Φ acquire we need to expand the wholeexpression of Φ( χ ) including the initial Φ o ˆ F ( χ ) and the extra terms acquired after circling around the previousbranch points. One important subtlety is that the new terms produced by circling around branch point can inducenew branch points in the χ -plane. Circling around these newly induced branch points will also bring Φ( χ ) to otherRiemann surface layers. For example the new term Eq.(98) induces logarithmic branch points at χ = i (cid:0) k + (cid:1) π ( k ∈ Z ), which are not present on the Riemann surface layer in Fig.5. Stated more accurately, the reality is that theRiemann surface layer shown in Fig.5, is the one where the south pole lies. On this sheet, branch cuts are presentat χ [ k ] = ikπ ( k ∈ Z ), while on other layers which can be reached by passing through some branch cut shown inFig.5, there can be branch points also at χ = i (cid:0) k + (cid:1) π ( k ∈ Z ). Whatever the different ways the χ -contour windsaround the brach points, the bulk and boundary saddle-point actions always take the form of Eqs (87) and (90),but with ρ to be derived case by case. 26 - I m r H m , k , , L - m k =0 - - - - - p =0 (cid:45) (cid:45) (cid:50)(cid:51) (cid:45) (cid:45) (cid:45) m I m Ρ (cid:72) m , , p , (cid:76) Figure 6:
Plots of the coefficient Im ρ of the quadratic term in the on-shell actions ( I ∝ ˜ I ∝ Im ρ α for α ∼
0) as shown in Eqs (88)and (92). The expression of ρ is given by the general formula Eq.(105). Left: the stack of three plots shows Im ρ for saddle pointswhose contours do not cross any branch cut, such as I ( k = 0) and II ( k = −
1) in Fig.5. In these cases p = 0 and n is irrelevant,so that ρ is simply given by Eq.(74). For the fundamental saddle point k = 0, Im ρ < √ and 1 thusperurbatively the saddle’s individual contribution to the Hartle-Hawking wave function is suppressed when τ → ∞ . When k = −
1, asshown in the upper frame, Im ρ can become positive for certain ranges of mass; the same thing is true when k becomes more negative. Right:
The value of Im ρ for saddle points of contour containing minimum positive Euclidean history ( k = 0) and circling the brachpoint χ [0] = 0 ( n = 0) from counterclockwisely twice ( p = −
2) to clockwisely three times ( p = 3). The contours have the look of thegreen contour in the left half of Fig.7, but the detail of the loop should be readapted according to the value of p . When we treat the scalar field perturbatively, solving the scalar equations against the spacetime background givenby Eq.(50), it is by all means feasible to consider inhomogeneous perturbations. To do this we need to solve thewhole Klein-Gordan equation, and below is a sketch of the calculation, while in fact the calculation is alreadywell known in the literature (cf for example [54]). For the simplicity of solving the equation, let r = sinh χ and˜ ζ = τ ζ , so that the metric Eq.(50) takes the form (cid:96) − g µν dx µ dx ν = − dr r + (1 + r ) dζ + r d ˜ ζ . (106)The periodicities of the spatial coordinates are ζ + i ˜ ζ ∼ ζ + i ˜ ζ + 2 π ∼ ζ + i ˜ ζ + 2 πτ . The Klein-Gorden equationfor our minimally coupled scalar field Φ is ( (cid:3) − m )Φ = 0, whose explicit form against the metric (106) is (cid:34) − r ∂ r (1 + r ) r∂ r + ∂ r + ˜ ∂ r − m (cid:35) Φ = 0 , (107)where ˜ ∂ = ∂∂ ˜ ζ and ∂ = ∂∂ζ . Expanding the scalar field against Fourier modes on the 2-torus:Φ( r, ˜ ζ , ζ ) = (cid:88) p,q Φ pq ( r ) exp (cid:20) i q ζ + i (cid:18) pτ − τ qτ (cid:19) ˜ ζ (cid:21) , (108)which we insert into the equation (107) to obtain the equations for Φ pq ( r ):(1 + r )Φ (cid:48)(cid:48) pq + (cid:18) r + 3 r (cid:19) Φ (cid:48) pq + (cid:20) q r + ( p − q τ ) τ r + m (cid:21) Φ pq = 0 . (109)27ere the primes indicate the derivative with respect to r . The equation can be further converted into a moretangible form if we let Φ pq ( r ) = (1 + r ) q r iξ y pq ( r ) , where ξ = p − q τ τ , (110)and introduce the variable z = r + 1 = cosh χ since we are interested in the solution regular at r = −
1. Theequation (109) thus gives rise to a hypergeometric differential equation of y pq : z (1 − z )( y pq ) zz + [1 + q − ( q + iξ + 2) z ] ( y pq ) z − (cid:2) m + ( q + iξ )( q + iξ + 2) (cid:3) y pq = 0 , (111)of which the solution regular at the south pole r = − pq ( r ) = (1 + r ) | q | r iξ F (cid:0) u, v, w, r (cid:1) , (112) u = 12 (cid:16) (cid:112) − (cid:96) m + | q | + iξ (cid:17) ,v = 12 (cid:16) − (cid:112) − (cid:96) m + | q | + iξ (cid:17) ,w = 1 + | q | , where the zero mode Φ is just the homogeneous result in Sec.3.2, Eq.(67). The whole solution for the scalar field,in terms of the variable χ , is thusΦ (cid:16) χ, ˜ ζ , ζ (cid:17) = (cid:88) p,q A pq cosh | q | χ sinh iξ χ F (cid:0) u, v, w, cosh χ (cid:1) exp (cid:20) i q ζ + i (cid:18) pτ − τ qτ (cid:19) ˜ ζ (cid:21) , (113)where the coefficients A pq are to be determined by the boundary condition for Φ on Σ ∗ .Without going through every detail I would like to state qualitatively the main properties and results which arein line with the purpose of the work of the paper: • All modes inhomogeneous in the ζ direction vanish at the south pole: Φ pq ( χ o ) = 0 for q (cid:54) = 0. Therefore thesouth pole value of the scalar field is Φ( χ o , ˜ ζ , ζ ) = (cid:80) p A p exp (cid:16) ipτ ˜ ζ (cid:17) = (cid:80) p A p exp( ip ζ ). This is perfectlyconsistent with the setup that ζ -circle should cap smoothly at the south pole, implying that ζ dimension isabsent there. • Branch points are present on the Riemann surface of each mode Φ pq at χ [ k ] = ikπ , which can be studied byexpanding Eq.(112) around them. • The perturbative computation of saddle-point actions in Sec.3.3, as well as the holographic renormalizationfor homogeneous scalar perturbation, can be easily extended to the inhomogeneous case (holographic renor-malization is perturbative). Now instead of having only one pair of ( α, β ), we will have an infinite number ofthem, each from the asymptotic expansion of Φ pq given by Eq.(112). Thus they can be denoted by ( α pq , β pq ),where ( α , β ) is just ( α, β ) introduced in Eq.(33). The imaginary part of the saddle-point actions are I = (cid:80) ( p,q ) B pq α pq , and ˜ I = (cid:80) ( p,q ) ˜ B pq α pq where B pq = √ − m ˜ B pq . Further, for q (cid:54) = 0, B pq and ˜ B pq depend explicitly on τ . • With the presence of explicit τ -dependence just mentioned, we can show, taking into account the coordinatechange and boundary condition change, that the resulting on-shell action is invariant under the modulartransform with c = d = 0 and e = 1 in Eq.(4), so that when summing up all the inequivalent configurationson the SL (2 , Z ) orbit, we only sum over different ( c, d ), same as for homogeneous scalar perturbation.28 u c li d ea n r un w a y abcde a b cd e f Figure 7:
Complex time contours used for the saddle points studied in Sec.4, along which Eqs (20)–(22) are numerically integrated.
Left: the form of the contours when Φ is perturbatively small. The χ -plane is just the Riemann surface in Fig.4. Two types of contourswill be considered colored in blue and green. The blue contours stay always in the same layer of the Riemann surface. They start from χ o along the Lorentzian direction χ o → a , embark on a Euclidean evolution along a → e and can take any Lorentzian exit at Im χ = kπ ( k ∈ Z ). The green contour circles around the branch point χ = 0 and finishes up taking the k = 0 exit on another Riemann sheet. Theexample shown here circles clockwisely twice the branch point, that is p = 2 in the language of Sec.3.4. Right: the generic patternof deformation of the Riemann surfaces and the contours when Φ is continuously tuned up. The Riemann surfaces have their singularpoints moving away from ikπ ( k ∈ Z ); the contours need to be deformed accordingly. The blue contour may have its Euclidean partadjusted to the left or right to avoid singularities, for example a → d or c → f . The green contour should have its loops continuouslyre-adapted in order not to let the branch point inside escape out nor outside in. For both contours the Euclidean shift Im χ ∗ = δ ∗ willdivert from kπ and should be determined by boundary conditions at Σ ∗ . Based on the perturbative result obtained previously, this section will set out into the realm of finite scalar defor-mation. Concretely, we will start from a certain saddle point found in Sec.3, tuning up the scalar field, measuredby Φ o and α , and meanwhile tracing its resulting deformation continuously by numerical means. The focus of thisstudy are • First the quantities worked out perturbatively in Sec.3 ( β , S , ˜ S , b o , Φ o , δ ∗ ), are now traced deep into thenon-perturbative realm of scalar deformation, and their diversions from the perturbative behaviors are to bedisplayed. • Second the various saddles will provide a vast test field for verifying the relation Eq.(46). This is done toall the saddle points covered in this section. This confirms that Eq.(42) is the right result of holographicrenormalization on the non-perturbative level. • Third, the saddle-point actions before ( S ) and after ( ˜ S ) holographic renormalization will be compared. Espe-cially according Eq.(45), holographic renormalization introduces a non-trivial imaginary part to the saddle-point action. And it will turn out that ˜ I can be drastically different from I for some saddle points.The study will be carried out in the following steps:Sec.4.1 will explain the key elements of the numerical scheme employed to trace the saddle points in the non-perturbative regime of scalar deformation. Also some useful properties of the physical quantities will be given,which decide the way that they are presented in the later subsections. The most important aspect is that we only29eed to trace various quantities as a function of α with τ fixed (the shape of T fixed), because the results as suchalready contain all the information of the τ -dependence.Sec.4.2 numerically studies the saddle points at finite scalar deformation in the order in which the perturbativecases were investigated in Sec.3, where the scalar potential are taken to be quadratic potentials. First presented arethe results of the saddle points whose time contours do not circle around any branch points but contain differentamount of Euclidean history. Next presented are the results for those whose time contours contain the minimumamount of positive Euclidean history but circle around one branch point several times. It will especially be shownthat the renormalized imaginary part of saddle-point action ˜ I can have severely different behavior and diverge to ±∞ with α for non-fundamental saddle points.Whereas Sec.4.2 is concentrated on the physical outcomes of the different types of saddle points, Sec.4.3 studiesthe intrinsic characteristics of them, completing the full account of the properties of saddle points. These includethe south pole values Φ o and b o , the Euclidean shift δ ∗ of the boundary Σ ∗ (see Eq.(27)), and the structure of theRiemann surfaces. Some characteristic pattern of Riemann surface deformation are empirically observed in case ˜ I presents scalar divergence. In the numerical study of the saddle points, the general logic to follow stays the same: find out the saddle point thatmeets with the boundary data ( τ, α ) assigned on Σ ∗ , and then compute its various physical quantities. This timeall steps are to be done numerically, that is, the equations (20)–(22) will be integrated numerically to obtain a ( χ ), b ( χ ) and Φ( χ ). The integration starts from the south pole χ o to the final moment χ ∗ = iδ ∗ + Λ (see Eq.(27)), withΛ fixed to a large number to approximate I + and δ ∗ to be determined. By adjusting the south pole data ( b o , Φ o )as well as δ ∗ , we can fit the numerical solutions to the boundary data ( τ, α ) assigned on Σ ∗ . In this procedurethere are five real conditions Im ( a ∗ b ∗ ) = 0, b ∗ a ∗ = τ and ( a ∗ b ∗ ) ∆ − / Φ ∗ = α , exactly what is needed to determinethe five real defining parameters of the saddle points: ( b o , Φ o ) and Im χ ∗ = δ ∗ . We can do this continuouslyover a domain in the ( τ, α )-space and thus establish the continuous mappings b o = b o ( τ, α ), Φ o = Φ o ( τ, α ) and δ ∗ = δ ∗ ( τ, α ), which represent a continuous family of saddle points.However for a fixed set of boundary data ( τ, α ), there can be infinitely many results of ( b o , Φ o , δ ∗ ), which wehave already seen in the previous section where the scalar field is perturbative (Φ o ∝ α ∼ τ, α ), we have infinitely many Euclidean shifts δ ∗ = kπ ( k ∈ Z ), further accompanied byinfinitely many ways for the complex time contour to circle around the singularities of the scalar field, leading to Note that it is not possible when concretely doing numerics, to start integrating the equations (19)—(22) exactly from the southpole χ o , since a ( χ o ) = 0, which produces 0 in the denominator. Instead, the initial conditions will in practice be assigned infinitesimallyclose to the south pole χ o + (cid:15) with (cid:15) ∼
0. Referring to appendix A, the initial conditions to be actually used are a ( χ o + (cid:15) ) = i (cid:15), b ( χ o + (cid:15) ) = b o = | b o | e iγ , Φ( χ o + (cid:15) ) = Φ o = | Φ o | e iθ ;˙ a ( χ o + (cid:15) ) = i, ˙ b ( χ o + (cid:15) ) = (cid:104) V (Φ o ) (cid:105) b o (cid:15) , ˙Φ( χ o + (cid:15) ) = − V (cid:48) (Φ o )2 (cid:15) , where (cid:15) ∈ C , | (cid:15) | (cid:28) b o , Φ o ∈ C , and γ, θ ∈ R , and since Eq.(19) is used in deriving these conditions, when integrating the equations we only need to integrate the second order ones(20)—(22) and the solutions will automatically satisfy the constraint Eq.(19). Here the story of saddle point searching is told with the narrative “given ( α, τ ), we determine ( b o , Φ o , δ ∗ ) to find out the saddlepoint”, rather for pedagogical reason, while the actual steps are more subtile with the numerical code used here. Instead of leaving( b o , Φ o , δ ∗ ) totally free and have them determined by the boundary conditions, we need to pre-determine values for | Φ o | and | b o | , onlyleaving the phases θ = arg Φ o and γ = arg b o , and also the Euclidean shift δ ∗ free to vary. The field values at χ ∗ : ( a ∗ , b ∗ , Φ ∗ ), obtainedfrom integrating the equations of motion, are thus functions of δ ∗ , γ and θ , recalling that Re χ ∗ = Λ is fixed once for all. The nextstep is to numerically solve the equations Im a ∗ Re a ∗ = Im b ∗ Re b ∗ = Im Φ ∗ Re Φ ∗ = 0 which are due to the requirement that all quantities should bereal on Σ ∗ , and this determines the three real parameters δ ∗ , θ, γ that have been left free in the beginning. When this is done, with thepre-determined | Φ o | and | b o | , all the characterizing parameters of the saddle point are obtained. Then from the numerical functions of { a ( χ ) , b ( χ ) , Φ( χ ) } describing the saddle point, we read off the boundary data ( τ, α ) that we should have imposed to obtain this verysaddle point. All these steps are realized with Mathematica for the work in this paper. o . When the boundary data sweeps through a region in the ( τ, α )-space starting outfrom each distinct result of ( b o , Φ o , δ ∗ ), we obtain infinitely many different families of saddle points, each gives riseto distinct saddle-point action as function of ( τ, α ). For instance in the case of scalar perturbation, we have Eqs(94) and (95) with different family of saddle points resulting in different ρ given in Eq.(105).The strategy of this section is to obtain the different families of non-perturbatively deformed saddle points basedon the perturbatively deformed ones obtained in the previous section. Concretely, in the perturbative domain where α ∼
0, the approximative south pole data to be suggested to the numerical code are Φ o according to Eq.(72) and b o ≈ i according to Eq.(48). Once the exact numerical result for small α is established, we can tune up α graduallywith fixed τ , and meanwhile, the continuous deformation of the time contour, as well as the change in south poledata ( b o , Φ o ), are carefully traced. However we should be aware that it is possible that we miss saddle points inthis way. Actually it can happen that there are families of saddle points that do not contain a perturbative regime,that is, when α ∼ β and Φ o stay finite. An example of this will be given in Sec.6.2 for 5d models, while I havenot yet found such family of saddle points in 3d models.Fig.7 shows schematically the generic pattern of Riemann surface deformation and complex time contour defor-mation with the increase of scalar field backreaction. The search for saddle points starts with some perturbativesaddle point whose contour is as shown in the left half of the figure, blue or green, and then when the scalar fieldis tuned up, the Riemann surfaces and the contours are deformed to something else, generically like what is shownin the right half of the figure. Especially δ ∗ is no longer kπ as in the perturbative case, and, since the singularitiesmove with the increase of scalar deformation, the contours need to be carefully re-configured in order to avoidrunning over them.Before setting out to present the numerical results, there are some properties of the model to mention, whichwill more or less directly determine the way that the numerical results are plotted. i ) The infrared divergences in the bulk saddle-point action S are real This discussion is already carried out in Sec.2.3, leading to Eqs (36) and (39), due to which in the followingsubsection I will be plotted against α . We should keep in mind that this does not mean that I is in the α -representation but rather this is a reparameterization of boundary condition. As explained in the beginning ofSec.3.2, when A is very large and fixed, ϕ ≈ α A − ∆ − / .An additional remark is that using the integral expression for the saddle-point action Eq.(26) can also lead tothe same conclusion. If we use the asymptotic expansion Eqs (175)–(177) we find that the integrand evaluated at χ ∗ = iδ ∗ + Λ (Λ → ∞ ) contains only exponentially suppressed imaginary terms. Therefore increasing Re χ ∗ = Λin Eq.(26) does not introduce any contribution to the imaginary part of the result. The same logic also applies tothe models with boundary topology S d ( d = 2 , , . . . ) ii ) Due to the rescaling invariance of the equations of motion, the two variable problem of τ and α can be reduced to a one variable problem of only α . The set of equations (19)–(22) are invariant under the rescaling of the scale factors a ( χ ) or b ( χ ). It turns out thatthis property can suppress the variable τ of our problem. For the quantities of our interest: b o , Φ o , β , I and ˜ S which are initially functions of τ and α , their τ -dependences can be derived from their α -dependence at a fixed τ As explained in footnote 10, in the numerical code that is actually used here, we have no direct control on α and τ but on | Φ o | and | b o | . Tuning up α is realized by increasing | Φ o | . For each | Φ o | , we fix | b o | to 1, and we search for the saddle point. Once the saddlepoint is found for the | Φ o | and | b o | we have chosen, we read off τ , α and β and we compute the on-shell action. After having covereda certain range of | Φ o | , we would obtain various quantities as functions of | Φ o | , especially α , β , τ , I and ˜ I , which we then convertinto functions of α using the α - | Φ o | relation. However for each value of | Φ o | , τ takes the value of what we read off, and is genericallynot a constant. Then we can simply use the relations to be introduced shortly after, Eqs (117) (120) and (121), to obtain the value ofvarious quantities as functions of α for a fixed value of τ , which is set to 1 in this paper. χ o with the south pole values ( b o , Φ o ), to thefinal moment χ ∗ = iδ ∗ + Λ with very large Λ, and that we have found the solutions { a ( χ ) , b ( χ ) , Φ( χ ) } of boundaryvalues ( A , τ , τ , ϕ ). Then we do the same procedure again only with b o rescaled by c , so that the south pole valuesare ( c b o , Φ o ). By the rescaling-invariance of the equations, the solutions should be { a ( χ ) , c b ( χ ) , Φ( χ ) } , whoseboundary values become ( A (cid:48) , τ (cid:48) , τ (cid:48) , ϕ (cid:48) ) = ( c A , τ , c τ , ϕ ). Then we can write down the boundary data of the scalarfield before and after rescaling as: ϕ = α A − ∆ − + β ( τ , τ , α ) A − ∆+2 ; (114) ϕ (cid:48) = α (cid:48) A (cid:48)− ∆ − + β ( τ (cid:48) , τ (cid:48) , α (cid:48) ) A (cid:48)− ∆+2 = α (cid:48) ( c A ) − ∆ − + β ( τ , cτ , α (cid:48) )( c A ) − ∆+2 . (115)Equating the above two since the rescaling has no effect on Φ, we obtain β (cid:0) τ , τ , α (cid:1) = c − ∆+2 β (cid:0) τ , cτ , α c ∆ − (cid:1) , (116)from which we have the following relations which suppress the τ -dependence of β : β (cid:0) τ , τ , α (cid:1) = τ ∆+2 β (cid:0) τ , , α τ − ∆ − (cid:1) ; β (cid:0) τ , , α (cid:1) = τ − ∆+2 β (cid:0) τ , τ , α τ ∆ − (cid:1) . (117)Then let us examine the property of the saddle-point actions inferred from the rescaling invariance. Suppose beforethe rescaling, the solutions { a ( χ ) , b ( χ ) , Φ( χ ) } lead to the bulk saddle-point action S ( A , τ , τ , ϕ ). Then it followsfrom Eq.(17) that after the rescaling b o → c b o , S is rescaled by the same amount, and becomes c S ( A , τ , τ , ϕ ).On the other hand, from the point of view of the boundary data on Σ ∗ , which now becomes ( c A , τ , c τ , ϕ ), theaction after rescaling is also S ( c A , τ , c τ , ϕ ). Thus we establish the equality S ( c A , τ , c τ , ϕ ) = c S ( A , τ , τ , ϕ ) . (118)For the imaginary part I = Im S , since it does not have A dependence when A → ∞ , Eq.(118) implies I (cid:16) τ , c τ , α c ∆ − (cid:17) = c I ( τ , τ , α ) (119)from which by properly choosing c and rescaling α , we obtain the relations I ( τ , τ , α ) = τ I (cid:0) τ , , α τ − ∆ − (cid:1) , I ( τ , , α ) = τ − I (cid:0) τ , τ , α τ ∆ − (cid:1) . (120)We can further on do the same discussion on Eq.(42), where we take into account the rescaling of b o and β whichhave already been explained above, we will find the same relation for ˜ S :˜ S ( τ , τ , α ) = τ ˜ S (cid:0) τ , , α τ − ∆ − (cid:1) , ˜ S ( τ , , α ) = τ − ˜ S (cid:0) τ , τ , α τ ∆ − (cid:1) , (121)and automatically the same relation holds for ˜ S R and ˜ I .The relations Eqs (117) (120) and (121) allow us to obtain the value of the various physical quantities of interestfor arbitrary τ from its value at τ = 1 (at some other α ) and vice versa. Therefore in presenting the numericalresult of those quantities, only the curves against α with τ fixed to 1 will be plotted. iii ) Large τ (“high temperature”) behavior of the on-shell actions τ behaviors of the saddle-point actions are determined by their perturbative forms. This is immediatelyseen when we take the τ → ∞ limit in the first equalities in Eqs (120) and (121) where the third argument α τ − ∆ − on the righthand side becomes small with the limit, so that their perturbative results apply and we obtain Eqs(88) and (90). With this observation, now we can finish up the suspended discussion at the very end of Sec.3.3:at large τ , corresponding to the very stretched T or very “high temperature”, the saddle point contributionsto the Hartle-Hawking wave function takes the perturbative form whatever the value of α . Therefore whether acertain saddle point leads to temperature divergence depends on the sign of Im ρ , regardless of the extent of scalardeformation. In this subsection, the saddle points are studied in the order that perturbative saddle points are studied in Sec.3.Only the quadratic potentials are used, while non-quadratic potentials will be briefly studied in Sec.6.1. The saddlepoints involved are now non-perturbative in scalar deformation; however since they are always obtained by givingfinite scalar deformation to the perturbative saddle points, it makes sense to have them classified according totheir characteristics when they are still perturbative. The results to be presented in the following are the physicalquantity outputs of the saddle points: β , S and ˜ S . The focus points are as announced in the introduction of thissection, briefly: comparing the perturbative/non-perturbative results; testing Eq.(46); comparing I and ˜ I mainlyto check scalar divergence. Fundamental saddle points; results in Fig.8; contour in Fig.7 χ o → a → b → χ ∗ The first case computed are fundamental saddle points of several different masses. The notion of “fundamentalsaddle point” was introduced in the perturbative regime by the end of Sec.3.2, referring to the saddle points withtime contours containing a minimum amount of positive Euclidean history and having no loop around branchpoints. Here in the non-perturbative context, it refers to those continuously deformed out of the perturbativefundamental saddle points. Therefore their contours are χ o → a → b → χ ∗ in Fig.7 when Φ is perturbative. Themasses investigated are m = 0 . , . , .
98 and 0 .
99. The features worth attention (see immediately relevantdetails in the figure and in the caption) are, regarding Fig.8: • With the increase of α , the imaginary parts of β (first lower frame) increase from 0, reach a peak, and thenfall off and asymptote to 0, in contrast to the families of non-fundamental saddle points to be shown in thenext figures, where Im β generally diverges. • The imaginary part of saddle-point actions in the bulk field representation (second column upper frame), I , are bounded as function of scalar deformation ( α ). They differ slightly from ˜ I , their counterpart in theboundary data representation, which is the consequence that Im β asymptotes to 0 with large α . • The “one-point function” relation Eq.(46) holds as shown in the right two frames in the lower row. Moredetail is shown in Fig.9 for the saddle points of scalar mass 0 . • Perturbative results are shown with dotted red lines when they apply. Here they are shown in the inset framesfor the sake of visibility. We see that they are in excellent match with the numerical results when α ∼ Longer Euclidean history for several masses; results in Fig.10; contour in Fig.7 χ o → a → c → χ (cid:48)∗ The second case studied is when the saddle points have just next-to-minimum amount of Euclidean history, cor-responding to the blue contour in Fig.7 which ends up at the k = − m = 0 . , . , (cid:112) / , . , .
96. Below are the noticeable features of these results,in contrast to those of the fundamental saddle points: • For α ∼
0, Im β starts off increasing or decreasing from 0, following Im ρ being positive or negative, with ρ = ρ ( m, k ) given in Eq.(74) where we put k = −
1. For m = 0 . , .
94, Im ρ >
0, and Im β grows, while for m = 0 . , .
96, Im ρ <
0, so that Im β decreases. When m = (cid:112) /
9, Im ρ = 0, Im β = 0 for all α covered bynumerics. • As α grows larger, Im β keeps increasing or decreasing in the same way as when α is small. The monotonyis not always the case as we will see in the next category of saddle points studied. However it is generalfor saddle point containing non-minimum amount of Euclidean history that Im β diverge to ±∞ when α increases. This is contrary to the previous case with fundamental saddle points, where Im β asymptotes to 0when α increases. • The imaginary parts of the saddle point-actions in the bulk field representation I are bounded (the secondupper frame) but their boundary counterparts ˜ I tend to diverge to ±∞ with α , in the opposite directions tothe divergences of Im β . The special case is m = (cid:112) /
9, where Im ρ = 0 and we obtain I = ˜ I = 0 over therange of α covered by numerical computation. It is interesting to see whether this property holds for otherforms of potential. Several lengths of Euclidean history; results in Fig.11; contour in Fig.7 the blue line taking the k = 0 till the k = − Lorentzian exit
The third case studied consists of saddle points of the same scalar mass m = 0 .
95 but living through differentamount of Euclidean histories. These saddle points have, when the scalar deformation is perturbative, the bluecontours in Fig.7 starting from χ o and ending up in the branches of k = 0 , − , . . . , −
5. The content of the figuresare arranged in the same way as the previous two cases, and the features relevant to our purpose are: • k = 0 corresponds to the fundamental saddle points, and their results have similar features as those presentedin Fig.8: at large α , Im β tend to 0, ˜ I differ slightly from I where both are bounded. • Saddle points of k (cid:54) = 0 show similar features as those of the previous case in Fig.10: at small α , Im β increaseor degreases following Im ρ > ρ <
0; for large α , Im β show tendency to diverge to infinities, I arebounded while ˜ I diverge to infinity in the opposite direction that Im β diverges. • Different from the previous case however, Im β have turning point, for instantce for k = −
3: it starts offdecreasing but then turns up and shows tendency of diverging to + ∞ . As a result, ˜ I starts off increasingwhen α ∼ −∞ . With contours circling a branch point; results in Fig.12; contour in Fig.7 in green
This category of saddle points of boundary topology T are those with contours circling around a branch point asshown in Fig.7 in green. When the scalar field is perturbative, the χ -contours circle the first branch point belowthe south pole from counterclockwisely twice to clockwisely tree times, i.e., p = − , − , . . . ,
3. The result in Fig.12shows on the non-perturbative level the different physical outcomes of these different saddle points, where thefeatures worth mentioning are • Similar as for the fundamental saddle points, Im β asymptotes to 0 when α is large. In the whole range of α investigated each keeps the same sign, which is identical to that of Im ρ , with ρ given by Eq.(105). See theright frame of Fig.6 where we can obtain the signs of Im ρ .34 (cid:45) (cid:45) (cid:45) Α R e Β (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α I m Β (cid:45) Α (cid:72) (cid:68) (cid:45) (cid:45) m (cid:76) Α I m Β Α Α (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:182) Α I (cid:142) (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Im Β (cid:182) Α S (cid:142) R (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Re Β Α I Α I (cid:142) Α S R (cid:142) Figure 8:
Results of fundamental saddle points of boundary topology T . The scalar potential is quadratic, with scalar masses m = 0 . , . , . , .
99, marked beside the corresponding curves. The χ -contours of these saddle points are χ o → a → b → χ ∗ inFig.7 when the scalar field is perturbative. The curves plotted are obtained with the “Interpolation” command of Mathematica ondiscrete data with default interpolation order 3 (same for all other results). Column 1: the α -Re β and α -Im β relations, where withthe increase of α , | Re β | grows monotonically, while Im β peaks and then falls back and asymptotes to 0. Column 2: the upper frameshows the imaginary part of the Lorentzian bulk on-shell action I against α . The lower frame shows the boundary term induced bythe holographic renormalization, which when added to I leads to the imaginary part of boundary on-shell action ˜ I . Column 3: theupper frame shows the imaginary part of the boundary Lorentzian saddle-point action, and the lower frame shows their derivatives ∂ α ˜ I , together with the imaginary part of the conjugate momentum of α , i.e., the α -(∆ − − ∆ + )Im β curves, where the two sets of curvesmatch precisely each other. This verifies the imaginary part of Eq.(46). Column 4: the ˜ S R counterpart of the third column, wherewe see again that the (∆ − − ∆ + )Re β curves run exactly over the ∂ α ˜ S R curves, thus verifying the real part of Eq.(46). The thin reddotted lines in the inset windows are the perturbative results obtained in Sec.3.2 and Sec.3.3. • The imaginary parts of the saddle-point actions in the bulk field and boundary data representations, I and˜ I , differ slightly from each other. However different from the case of fundamental saddle points, they bothshow mild tendency of diverging to ±∞ . The previous subsection has focused on the physical outcomes of the saddle points concerned. To give a completeaccount of these saddle points, it is important to also show how their intrinsic structures evolve with the scalardeformation. This subsection will show the results of this respect and try to identify some connection between theseresults and the features of the physical quantities observed in the previous subsection. Especially an interestingcorrelation is observed between the way that singularities move in the scalar field’s Riemann surfaces and thedivergence properties of Im β . South pole data ( b o , Φ o ) and Euclidean shifts δ ∗ Let us first examine the behavior of the defining parameters of the mini-superspace saddle points. They arepresented in Fig.13 and Fig.14, which are from exactly the same saddle points whose physical quantities arepresented in Fig.11 and Fig.12 respectively. The aspects worth mentioning are mainly about the comparison totheir perturbative behaviors: 35 (cid:45) Α (cid:45) Α I (cid:61) (cid:45) Re b o I (cid:142) (cid:73) (cid:68) (cid:45) (cid:45) m (cid:77) Α Im Β (cid:182) Α I (cid:182) Α (cid:65)(cid:73) (cid:68) (cid:45) (cid:45) m (cid:77) Α Im Β (cid:69) (cid:182) Α I (cid:142) (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Im Β (cid:182) Α S (cid:142) R (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Re Β(cid:182) Α (cid:65)(cid:73) (cid:68) (cid:45) (cid:45) m (cid:77) Α Re Β (cid:69) Im b o S (cid:142) R (cid:73) (cid:68) (cid:45) (cid:45) m (cid:77) Α Re Β (cid:182) Α Im b o Figure 9:
Details of the matching ∂ α ˜ S = (∆ − − ∆ + ) β for the fundamental saddle points of scalar mass m = 0 .
96. The left and theright columns show respectively the real and imaginary parts of this relation, which correspond to the third and fourth columns inFig.8. The upper row visualizes the detail of the addition in Eqs (44) and (45) for obtaining ˜ S R and ˜ I , and the lower row shows thederivative of the curves being added in the upper row to show how they add up to a ∂ α ˜ S that matches exactly (∆ − − ∆ + ) β . Α R e Β (cid:45) (cid:45) Α I m Β (cid:45) (cid:45) Α (cid:72) (cid:68) (cid:45) (cid:45) m (cid:76) Α I m Β (cid:45) (cid:45) Α (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α (cid:144) (cid:45) (cid:45) (cid:182) Α I (cid:142) (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Im Β (cid:182) Α S (cid:142) R (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Re Β (cid:144) (cid:144) (cid:144) (cid:144) (cid:144) (cid:144) (cid:144) (cid:45) (cid:45) Α I (cid:142) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α S R (cid:142) (cid:45) (cid:45) Α I Figure 10:
Results of saddle points of boundary topology T whose χ -contrours are χ o → a → c → χ (cid:48)∗ ( k = −
1) in Fig.7. Thescalar masses covered are m = 0 . , . , (cid:112) / , . , .
96. The contents of the figures are arranged in the same way as in Fig.8. Notethe difference between the fundamental saddle points in Fig.8. Here Im β diverge with α and resulting in divergent ˜ I , although I isbounded. The thin red dotted lines are perturbative results. (cid:45) (cid:45) Α R e Β (cid:45) (cid:45) Α I m Β (cid:45) (cid:45) Α (cid:45) (cid:45) (cid:45) Α (cid:182) Α S (cid:142) R (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Re Β(cid:182) Α I (cid:142) (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Im Β (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α (cid:72) (cid:68) (cid:45) (cid:45) m (cid:76) Α I m Β k =0 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) k =0 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) k =0 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) k =0 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) k =0 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) k =0 (cid:45) (cid:45) (cid:45) k =0 (cid:45) (cid:45) (cid:45) k =0 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α I (cid:45) Α I (cid:142) (cid:45) (cid:45) Α S R (cid:142) Figure 11:
Results of saddles points with scalar mass m = 0 .
95 whose contours contain different amount of positive Euclidean timelabelled by the values of k attached to the corresponding curve. The contents of the figures are arranged in the same way as in Fig.8.Note that the general pattern of these curves is similar with those of the non-fundamental saddle points shown in Fig.10, but with theincrease of Euclidean time length, more varied details show up. For example the α -Im β curve has turning point when k = − (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α R e Β (cid:45) (cid:45) (cid:45) Α I m Β (cid:45) (cid:45) (cid:45) Α (cid:72) (cid:68) (cid:45) (cid:45) m (cid:76) Α I m Β (cid:45) Α Α (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:182) Α S (cid:142) R (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Re Β(cid:182) Α I (cid:142) (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Im Β p =0 p =0 p =0 p =0 p =0 p =0 p =0 p =0 (cid:45) (cid:45)(cid:49) (cid:49) (cid:49)(cid:45)(cid:49) (cid:45) (cid:49)(cid:45)(cid:49) (cid:45) (cid:49)(cid:45)(cid:49) (cid:45)
23 2 (cid:49)(cid:45)(cid:49) (cid:45) (cid:45) (cid:45) Α I (cid:45) (cid:45) Α I (cid:142) Α S R (cid:142) Figure 12:
Results of saddle points of boundary topology T and with scalar mass fixed at 0 .
95. The time contours circle around abranch point which, when the scalar field is perturbative, is located at χ = 0. The integers p = − , − , . . . , n = k = 0 in Eq.(105). (cid:45) (cid:45) (cid:45) Α R e (cid:70) o (cid:45) (cid:45) (cid:45) (cid:45) Α I m (cid:70) o (cid:45) (cid:45) Α R e b o Α I m b o k =0 (cid:45) k =0 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) k =0 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) k =0 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) k =0 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45)Π(cid:45) Π(cid:45) Π(cid:45) Π(cid:45) Π Α ∆ (cid:42) Figure 13:
Defining parameters ( b o , Φ o , δ ∗ ) of the saddle points studied in Fig.11, traced as a function of scalar boundary data α .The index k = 0 , − , . . . , − Left:
The fourframes are plots of Φ o and b o . The red dotted lines are perturbative results presented in Eq.(72). Right:
Plots of the Euclidean shift δ ∗ . It turns out that δ ∗ can be increasing or decreasing with the increase of α , according to whether Im ρ < ρ > Α R e (cid:70) o (cid:45) (cid:45) (cid:45) Α I m (cid:70) o (cid:45) (cid:45) (cid:45) (cid:45) Α R e b o Α I m b o p =0 (cid:45) (cid:45)(cid:49) (cid:49) (cid:49) p =0 (cid:45)(cid:49) (cid:45) (cid:49) p =0 (cid:45)(cid:49) (cid:45) (cid:49) p =0 (cid:45)(cid:49) p =0 32 (cid:49) (cid:45)(cid:49) (cid:45)(cid:50) (cid:45) (cid:45) (cid:45) (cid:45)(cid:49) (cid:45)(cid:50)
30 5 10 150.00.20.40.6 Α ∆ (cid:42) Figure 14:
Defining parameters ( b o , Φ o , δ ∗ ) of the saddle points studied in Fig.12. The curves are labeled p = − , − , . . . , δ ∗ are not necessarily monotonicwith the growth of α . Deformation of the Riemann surface of Φ with the increase of α , for the fundamental saddle points with quadratic scalarpotential of scalar mass m = 0 .
95. The physical quantities of this case are presented in Fig.11 by the k = 0 curves. The red solid linesare the χ -contours along which the equations of motion are integrated. In the perturbative regime α ∼ χ = 0 and − π , and the integral contour is just the χ o → a → b → χ ∗ in Fig.7. However the branch point at − iπ/ α increases,we observe the singular points clustering towards the south pole, and they are no longer in the line of Re χ = 0 as they do in the firstcolumn. Meanwhile the length of Euclidean history shortens, indicated by the decreasing length of the vertical segment of the contour;this last behavior is also shown in the right half of Fig.13 by the increasing δ ∗ for the k = 0 curve. • Φ o is proportional to α just as Eq.(72) for α ∼ α -Φ o plots in Fig.13 and Fig.14. Indeed,Eq.(72) is plotted with red dotted lines, which are in very good match with the non-perturbative results,plotted with solid lines, when α ∼
0. When α increases, | Φ o | increases the more and more slowly. However ingeneral | Φ o | does not necessarily grow with α , but such example have not yet been found for the 3d modelsstudied here, while in Sec.6.2 such a case will be shown in the model of scalar-deformed dS . • For different values of α , b o needs to be adjusted differently to keep the value of τ to be 1, making b o anon-trivial function of α . Especially we have already seen that I = − Re b o as in Eq.(39). • The Euclidean shifts δ ∗ start off from the values found in perturbative theory when α increases from 0. InFig.13 they start from 0 , − π, . . . , − π for the saddle points of k = 0 , − , . . . , −
5, while in Fig.14 they allstart from 0. Then with the increase of α for very small α , δ ∗ increases or decreases according to whetherIm ρ is negative or positive, but δ ∗ is not necessarily monotonic when α increases all the way to large values. Riemann surfaces of the scalar field
There is another intriguing aspect that seems to have the potential of providing more insight into the property of thesaddle points. That is the analytical structure of the scalar field Φ as a function of the complex time χ . To exposethis feature in the following contents, the deformation as a function of α of the scalar field’s Riemann surfaces willbe shown by density plots. Note that they have already been studied in Sec.3 in the cases where Φ is perturbative,and are shown in Fig.4. A rough illustration of how they get deformed when Φ becomes non-perturbative has beengiven in Fig.7, while now a precise account will be provided.In the following, three cases are given attention to, which can well summarize all the cases encountered so far,all of them are of scalar mass m = 0 .
95. 39 he first two cases are the saddle points whose contours do not circle around singularities of Φ( χ ) but coverdifferent amounts of Euclidean time. They are the fundamental and the next-to-fundamental saddle points, whosephysical quantities are presented in Fig.11 by the k = 0 and k = − k = 0 and k = − • When α ∼ − π , which are just the blue contours χ o → a → b → χ ∗ and χ o → a → c → χ (cid:48)∗ in Fig.7. • As α increases, the Riemann surfaces for the two cases deform differently, manifestly in the way that thesingular points behave. In Fig.16 depicting the k = − χ -axis towards the positive Lorentzian time direction. If we keep on increasing α , it will collide withthe Euclidean part of the contour. Then we will have to retreat the latter to the right to avoid the collision,in order to ensure the continuous tracing of the saddle points. However for the k = 0 saddle points as shownin Fig.15, the singular points also leave off the Im χ -axis but towards the negative Lorentzian time direction,and no “aggressive behavior” to the integral contour is observed. • In regard of the different ways that singular points behave, and the fact that k = − β with the increase of α but not the k = 0 saddles, it is tempting to associate the presence ofa singularity tending to collide into the integral contour, with the divergence of Im β . Among all the modelstudied in this paper, this association holds. We will see more examples in the following studies. The third case shown in Fig.17 involves the saddle points, when Φ is perturbative, having their contours circlingaround once ( p = 1) the branch point χ = 0, as shown by the solid red contour in the first column. As α increases,the deformation of the Riemann surface and the contour are evident, while here I would like to mention that thesingular points do not leave away from the Im χ -axis towards the right, which corresponds well to the non-divergentbehavior of Im β in terms of α shown in Fig.12. This provides another example of the guess in the last paragraph,that the divergence of Im β is always accompanied with a singular point aggressive to the integral contour. S T , fromwhich many results are drawn. It is thus interesting to carry out a study in parallel on another model with differentboundary topology to see the effect of topology on the results. A very obvious candidate is S . This section willbe dedicated to this model.All the conceptual elements and general formalisms developed for the case of T in Sec.2, including the generalmethod of addressing the problem, spacetime ansatz, the definition of the Hartle-Hawking wave function, compu-tation of its saddle-point contribution, can be adapted to the S case here, and they will not be constructed indetail. What will be given below are the immediately useful formulae and equations.Again for the sake of feasibility of quantitative computation, the search of saddle points will be restricted withinthe range that mini-superspace formalism can reach. Therefore we will only be dealing with the saddle points whosespacetime can be sliced globally into homogeneous isotropic S -hypersurfaces against the coordinate time χ . Thelatter ranges from the south pole χ o to some final moment χ ∗ , where χ ∗ − χ o is generically complex. This time at χ o , which will still be set at iπ/
2, it is an S that is supposed to shrink smoothly to zero size. The final moment χ ∗ is mapped to the spacelike boundary Σ ∗ , and it is still parameterized as in Eq.(27), to formulate the limit Σ ∗ → I + .40igure 16: Density plots of Φ showing the deformation of Riemann surface of Φ( χ ) of m = 0 .
95 with the increase of the boundarydata α . The saddle points concerned are the non-fundamental ones whose time contour is given by χ o → a → c → χ (cid:48)∗ in Fig.7 whenthe scalar field is perturbative, as shown in the first column. The physical quantities are plotted in Fig.11 with the k = − α increases, there is one singular point taking off from the Im χ -axis and tend to collide into the Euclidean part of the contour. This isnot observed in Fig.15 and seems to be related with the divergence of Im β seen in the first column in Fig.11. Figure 17:
Deformation of Riemann surfaces of Φ with m = 0 .
95. The complex time contours of the saddle points, when Φ isperturbative, are as the green contour in Fig.7, but circle clockwisely once the branch point χ = 0. In the first column where Φ isperturbative, the south pole should be regular, but we see a branch cut leaving precisely from the south pole. This is due to the fact,referring to the discussion at the end of Sec.3.4, that the Riemann sheet shown in the plots is that of the end point χ ∗ and is not thesame sheet as that of the south pole. On the Riemann sheet of χ ∗ , the perturbative scalar field is given by the second line of Eq.(99)with p = 1 and n = 0, which clearly has branch points at χ = ikπ/ k ∈ Z ), among which there is the south pole. T boundary, since there isno detail in the boundary geometry to take into account. We can refer to the appendix A for the action and theequations of motion, where we put d = d , d = 0 in Eqs (154)–(158). The action for no-boundary saddle points is κS = 12 ˆ χ ∗ χ o dχ a d (cid:34) d ( d − (cid:18) − ˙ a a + 1 a (cid:19) − d ( d −
1) + ˙Φ − V (Φ) (cid:35) − d a d − o ˙ a o , (122)where a ( χ ) is the scale factor or the radius of S , which can be complex but must be real at χ ∗ ; κ = πG(cid:96) d − Ω d , withΩ d the volume of a d -dimensional sphere. We will set κ = 1 for simplicity. Unlike the case of the previous casestudied of T spatial slices, the south pole term − d a d − o ˙ a o vanishes due to the south pole condition a o = 0. Theequations derived from this action are d ( d − (cid:18) ˙ a a + 1 a (cid:19) − d ( d − − ˙Φ − V (Φ) = 0 , (123)2( d −
1) ¨ aa − d − d − (cid:18) ˙ a a + 1 a (cid:19) − d ( d −
1) + ˙Φ − V (Φ) = 0 , (124)¨Φ + d ˙ aa ˙Φ + V (cid:48) (Φ) = 0 . (125)These equations are to be integrated from χ o to χ ∗ in the search for saddle points. At χ o the fields should satisfythe conditions just as Eqs (14) and (15) but b o is not there. On the other hand at χ ∗ , the boundary conditions toimpose is simply the scalar boundary data α , where the expressions of the asymptotic behaviors can be found inappendix B. The notation A will still be used, but now A = a ∗ ∼ C a e measuring the surface area of the sphere S at Σ ∗ . Therefore the asymptotic behavior of the scalar field can still be expressed as Eq.(29) and Eq.(33), butnow α = C ∆ − a ˆ α and β = C ∆ + a ˆ β .The work to be presented in this section will be based on the variation principle presented above. The followingcontent will be covered. Sec.5.1 studies the saddle points with perturbative scalar deformation, of boundarytopology S d keeping d generic. Just as for the case of T in Sec.3, the perturbative results will later serve asthe starting points for numerically tracing the non-perturbative scalar deformation of the saddle points. Theremarkable difference from the T cases on the perturbative level, is the absence of branch cuts in the Riemannsurfaces of all modes of Φ in the expansion against the S d -harmonics. Sec.5.2 studies the on-shell action of thesaddle points of boundary topology S . Its holographic renormalization will be worked out for finite homogeneousscalar deformation. Sec.5.3 carries out the numerical calculation on the saddle points of boundary topology S , inthe same manner as the studies have been presented in Sec.4.2–Sec.4.3. Solution of perturbative scalar fields
Let us directly work out the inhomogeneous result for boundary topology S d ( d = 2 , , . . . ). Again the method isalready known in the literature (cf for example [54]), and what is to be presented below is rather a reformulationusing the notations and expressions adapted to our context.First ignoring the scalar field, setting the south pole at χ o = iπ/
2, we solve from Eq.(123) and obtain the scalefactor a ( χ ) = cosh χ . Therefore the spacetime metric is that of an empty dS d +1 ds (cid:96) = − dχ + cosh χ d Ω d . (126)Just like the empty spacetime saddle points studied in Sec.3.1, we can figure out the dS contours and EAdS contours42 AdS contoursdS contoursIsolated polesSouth pole abc
Figure 18:
Riemann surface of homogeneous perturbative scalar field Φ in the background Eq.(126), as a function of the complexifiedtime χ . The south pole is set at χ o = iπ/ L = 0 component in Eq.(133), with which we locatesingular points at χ = i (2 k − ) π ( k ∈ Z ). They are simple poles of order d − d +1 space whenIm χ = kπ and an Euclidean AdS space when Im χ = (cid:0) k + (cid:1) π ( k ∈ Z ). The defining contours of saddle points start from the southpole χ o and terminate on one of the Lorentzian dS contours: χ ∗ = ikπ + Λ with k ∈ Z and Λ → ∞ . The saddle points studied in Sec.5will have, in the perturbative regime of Φ, the χ -contours as shown by the solid blue and green lines. for this metric, which are respectively Im χ = ikπ and Im χ = i (cid:0) k + (cid:1) π ( k ∈ Z ). An illustration of these contoursis shown in Fig.18 (ignore the red spots for the moment which are the poles to be induced by the scalar field).Therefore we have another example of no-boundary saddle points which can be continued from asymptotic dS intoasymptotic EAdS [12], in addition to the saddle points in Sec.3.1 which allow continuation between EBTZ andKerr-dS spacetimes. The no-boundary saddle points are hereby specified by the metric (126) supplemented witha complex χ -contour. The latter starts off from the south pole χ o and terminates on the asymptotic boundary ofany dS contour: χ ∗ = iδ ∗ + Λ where δ ∗ = kπ ( k ∈ Z ) and Λ → ∞ . Therefore we have an infinite number of saddlepoints labeled by the integer k . Examples of these contours are as shown in Fig.18 by the blue and the green lines(again ignore the red spots).Then let us further on switching on the scalar field perturbation. The Klein-Gorden equation against thisbackground is − d χ ∂∂χ (cid:18) cosh d χ ∂ Φ ∂χ (cid:19) + 1cosh χ ∇ S d Φ − m Φ = 0 , (127)where ∇ S d is the Laplacian on S d . We can expand the scalar field against the d -dimensional spherical harmonics Y jL (Ω) (c.f. [54]): Φ( χ, Ω) = (cid:88)
L,j Φ Lj ( χ ) Y jL (Ω) , with (128) ∇ S d Y jL = − L ( L + d − Y jL , where L = 0 , , , . . . and j = − L, − L + 1 . . . , L , and Ω denote collectively the coordinates of S d . Inserting thisexpansion into the Klein-Gordan equation (127), we obtain the equations for each component Φ Lj ( χ ):¨Φ Lj + d tanh χ ˙Φ Lj + (cid:20) m + L ( L + d − χ (cid:21) Φ Lj = 0 . (129)43o obtain an equation in a more obvious form we can do the following change of variables:Φ Lj = z L (1 − z ) L y Lj , with z = 1 + i sinh χ . (130)Then we obtain from Eq.(129) the equation for y Lj : z (1 − z )( y Lj ) (cid:48)(cid:48) zz + (cid:20) d + 1 + 2 L − ( d + 1 + 2 L ) z (cid:21) ( y Lj ) (cid:48) z − (cid:2) m + L ( d + L ) (cid:3) y Lj = 0 , (131)which is a hypergeometric differential equation. We are interested in the solution regular at the south pole z = 0.Therefore the branch of solution to pick up is y Lj = A Lj F (cid:18) ∆ − + L, ∆ + + L, d + 12 + L, i sinh χ (cid:19) , where ∆ ± = d ± √ d − m , (132)where A Lj are constants to be determined by boundary conditions at Σ ∗ . Thus we have the expression for thescalar field: Φ( χ, Ω) = (cid:88)
L,M A Lj (cid:2) z (1 − z ) (cid:3) L F (cid:18) ∆ − + L, ∆ + + L, d + 12 +
L, z (cid:19) Y jL (Ω) . (133)Note that the field value at the south pole z = 0 is the coefficient of the homogeneous mode A , consistent withthe fact that z = 0 is not a boundary so that the dimensions represented by the angular coordinates Ω do not exist. No branch cuts
The solution, regarded as a function of z , is singular at z = 1, where χ = χ [ n ] := i (cid:0) k − (cid:1) π . Expanding onesingle mode Φ Lj of the scalar field around z = 1, we have the asymptotic behaviorΦ Lj ∼ (cid:104) C (1 − z ) L + C (1 − z ) − d − L (cid:105) × (cid:104) O ( z − (cid:105) , (134)where C and C are some coefficients depending on m , d and L . Therefore we see that on the z -plane z = 1 is abranch point when L or L + d − z = i sinh χ , therefore when χ ∼ χ [ n ] := i (cid:0) k − (cid:1) π thevariable z expands as z ∼ (cid:16) χ − χ [ n ] (cid:17) + O (cid:16) χ − χ [ n ] (cid:17) (135)whose leading term is quadratic. Therefore z = 1 or χ = χ [ n ] are not branch points but isolated poles in the χ -plane of order L + d −
1. Therefore examined perturbatively, the scalar fields are single-valued. An illustrationof the Riemann surface of Φ ( χ ) is given in Fig.18. In the presence of a perturbative scalar field the no-boundarysaddle points are specified by the metric Eq.(126), the scalar field Eq.(133) and a contour on the complex χ -planestarting from the south pole and ending at the asymptotic boundary of a dS contour. Some examples of suchcontours are shown in Fig.18. Since the singular points in the Riemann surface (red spots) are isolated poles, thegreen contour and the k = 0 blue contour represent the same saddle point. However later we will see that they willbecome different saddle points when the scalar field becomes non-perturbative.44 erturbative saddle-point actions We can work out the perturbative result of saddle-point action for inhomogeneous perturbations. However to berelevant with the numerical calculation, let us only focus on the homogeneous case, while the extension to theinhomogeneous case is straightforward. Thus the scalar field is given by the L = 0 component of Eq.(133) and itsasymptotic behavior near the dS boundary is:Φ ∗ = ϕ ∼ Φ o (cid:34) Γ (cid:0) d +12 (cid:1) Γ(∆ + − ∆ − )Γ(∆ + )Γ (cid:0) d +12 − ∆ − (cid:1) (cid:18) − i e χ ∗ (cid:19) − ∆ − + Γ (cid:0) d +12 (cid:1) Γ(∆ − − ∆ + )Γ(∆ − )Γ (cid:0) d +12 − ∆ + (cid:1) (cid:18) − i e χ ∗ (cid:19) − ∆ + (cid:35) , (136)where Φ o = A , and χ ∗ = ikπ + Λ ( k ∈ Z and Λ → ∞ ). Therefore if on the other hand we require the standardasymptotic formula at future boundary:Φ ∗ = ϕ ∼ ˆ α e − ∆ − Λ + ˆ β e − ∆ + Λ ∼ α a − ∆ − ∗ + β a − ∆ + ∗ , (137)where α = C ∆ − a ˆ α , β = C ∆ + a ˆ β and a ∗ = a ( χ ∗ ) ∼ e Λ ( C a = ), we can read off the coefficients α = 2 − ∆ − ˆ α = Φ o ∆ − exp (cid:20) i (cid:18) − k (cid:19) π ∆ − (cid:21) Γ (cid:0) d +12 (cid:1) Γ(∆ + − ∆ − )Γ(∆ + )Γ (cid:0) d +12 − ∆ − (cid:1) , (138) β = 2 − ∆ + ˆ β = Φ o ∆ + exp (cid:20) i (cid:18) − k (cid:19) π ∆ + (cid:21) Γ (cid:0) d +12 (cid:1) Γ(∆ − − ∆ + )Γ(∆ − )Γ (cid:0) d +12 − ∆ + (cid:1) , (139)where − i in Eq.(136) should be understood as e − iπ/ . The ratio between α and β will be useful in the expressionsof perturbative on-shell action, and we denote it again by ρ as previously for the T case: ρ := βα = (cid:18) − ie ikπ (cid:19) ∆ − − ∆ + Γ(∆ + )Γ (cid:0) d +12 − ∆ − (cid:1) Γ(∆ − − ∆ + )Γ(∆ − )Γ (cid:0) d +12 − ∆ + (cid:1) Γ(∆ + − ∆ − )= e iπ ( − k ) √ d − m −√ d − m Γ (cid:16) d + √ d − m (cid:17) Γ (cid:16) √ d − m (cid:17) Γ (cid:0) −√ d − m (cid:1) Γ (cid:16) d −√ d − m (cid:17) Γ (cid:16) −√ d − m (cid:17) Γ (cid:0) √ d − m (cid:1) . (140)The results of the saddle-point actions in the bulk field representation S ( a ∗ , ϕ ) and boundary data representation˜ S ( α ) in terms of power series of α , are respectively: S R ( a ∗ , α ) = [IR divergences] − d − m d Re ρ α + O ( α ) , I ( α ) = I (0) − d − m d Im ρ α + O ( α ) , (141)˜ S R ( α ) = ˜ S R (0) − (cid:112) d − m Re ρ α + O ( α ) , ˜ I ( α ) = ˜ I (0) − (cid:112) d − m Im ρ α + O ( α ) . (142)Here I (0) = ˜ I (0) is the value corresponding to empty spacetime, which depends on spacetime dimension and canbe obtained by integrating Eq.(170) with d = d , Φ = 0 and a ( χ ) = cosh χ . For example, I (0) = ˜ I (0) = (cid:16) k − (cid:17) π , S , − , S , (cid:16) k − (cid:17) π , S , (143)where k ∈ Z is just the integer in the Euclidean shift δ ∗ , appearing also in Eqs (138)–(140). The result of S willbe useful in Sec.6.2. 45 .2 Saddle-point actions and holographic renormalization Starting from the action Eq.(122) we can do as in the case of T , to have it converted into a form more amenablefor the study of asymptotic behavior near the dS boundary. For this reason we need to do a partial integral toturn first order derivatives into second order ones, and use Eq.(124) to eliminate the second-order derivative termin the resulting expression. Denoting the on-shell value of the action of S using S , we have S = − d ( d − a d − ˙ a (cid:12)(cid:12)(cid:12) χ ∗ χ o + 12 ˆ χ ∗ χ o dχ (cid:110) d ( d − d − a d − ˙ a + d ( d − a d − + 12 d ( d − d − a d + 12 (2 − d ) a d (cid:2) ˙Φ − V (Φ) (cid:3)(cid:111) − d a d − o ˙ a o . (144)When we go to 3d spacetime by setting d = 2, the action becomes very simple: S = − a ∗ ˙ a ∗ + ˆ χ ∗ χ o dχ = − a ∗ ˙ a ∗ + χ ∗ − χ o = − a ∗ ˙ a ∗ + i (cid:16) δ ∗ − π (cid:17) + Λ . (145)where the default setting of this paper is used: χ o = iπ/ χ ∗ = Λ + iδ ∗ . The IR divergences are contained inthe boundary term − a ∗ ˙ a ∗ and in the logarithmic term Λ ∼ ln a ∗ . Using the asymptotic behavior Eq.(185), we canfind that the IR divergent terms in − a ∗ ˙ a ∗ are real, and that − a ∗ ˙ a ∗ does not contain finite imaginary terms. Thusthe imaginary part of Eq.(145) is exclusively contained in the second term in its last step: I ( α ) = δ ∗ ( α ) − π , (146)which is just the amount of Euclidean history contained in the saddle point’s whole complex history. The subtractionof the IR divergences in − a ∗ ˙ a ∗ proceeds essentially in the same way as that in the T cosmology studied in Sec.2.3,where the related counter terms are given by Eq.(40) with A = a ∗ . This step yields the result˜ S = − Λ + i (cid:16) δ ∗ − π (cid:17) + (cid:0) ∆ − − m (cid:1) αβ + (other counterterms) (147)Now we still have the logarithmic divergence Λ to take care of. An immediate thinking is to mimic the AdS/CFTrenormalization scheme by directly introducing a term of order − Λ, where a natural choice here is − ln a ∗ . Indeedin AdS/CFT the same operation leads to the bulk computation of Weyl anomaly on the boundary field theory [56].Using the asymptotic behavior Eq.(185) setting u = e − Λ , we haveln a ∗ = ln (cid:2) C a e Λ (1 + . . . ) (cid:3) = Λ + ln C a + . . . (148)When the scalar field is perturbatively small, we have C a = 1 /
2, while for finite Φ, C a is generically a nontrivialfunction of α . Including Eq.(148) among the counter terms of Eq.(147) assuming that no other counter terms willbe necessary, we arrive at a finite expression for the boundary on-shell action˜ S = − ln C a + i (cid:16) δ ∗ − π (cid:17) + (cid:0) ∆ − − m (cid:1) αβ. (149)which is supposed to be a function of α , and indeed on the righthand side, C a , δ ∗ and β are all functions of α .Splitting the real and the imaginary parts, we have˜ S R = − ln C a + (cid:0) ∆ − − m (cid:1) α Re β, ˜ I = δ ∗ − π (cid:0) ∆ − − m (cid:1) α Im β. (150)46 (cid:45) (cid:45) (cid:45) (cid:45) Α R e Β (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α I (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α I (cid:142) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α I m Β Α (cid:72) (cid:68) (cid:45) (cid:45) m (cid:76) Α I m Β Α p =0 (cid:45)(cid:49)(cid:49) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) p =0 (cid:49) (cid:45)(cid:49) (cid:49) p =0 (cid:45)(cid:49) (cid:49) p =0 (cid:45)(cid:49) (cid:49) p =0 (cid:45)(cid:49) (cid:49) p =0 (cid:45)(cid:49)(cid:49) p =0 (cid:45)(cid:49) (cid:49) p =0 (cid:45)(cid:49)(cid:49)(cid:45)(cid:49) p =0 (cid:182) Α I (cid:142) (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Im Β Α S R (cid:142) Α (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Re Β (cid:49)(cid:45)(cid:49) p =0 (cid:182) Α S (cid:142) R Figure 19:
Results from saddle points of boundary topology S with quadratic scalar potential and scalar mass m = 0 .
94. Theperturbative-regime contours are as the green one in Fig.18, where the cases studied are those with the contour circling ± χ = − iπ/
2. The corresponding curves are marked by p = 0 , ± p = 0 case is just thefundamental saddle point. Compared to the previous cases with boundary topology T presented in Fig.12, here the adjustment of thecontour for p (cid:54) = 0 is much more difficult because the movement of singularities is complicated. For this reason, the tracing of p = ± α increases. Therefore the perturbativeresult, showing no branch cuts existing in the Riemann surfaces, do not hold for finite scalar perturbation. Next subsection will show numerically that this is the exact expression for the boundary on-shell action by showingthe generation of “one-point function” Eq.(46).
The numerical computations follow the same method and logic as those in Sec.4.2 and Sec.4.3. That is, theywill carry out the search for saddle points subjected to non-perturbative scalar deformation, starting out fromthe perturbative results obtained in Sec.5.1. When the scalar field is tuned up from perturbative to finite, thedeformations caused to the saddle point are traced continuously. Once the family of saddle points is found, furthercomputation will be done to obtain their physical quantities and intrinsic features as function of scalar deformationmeasured by the boundary data α . In the following content, the results for physical quantities of the saddle points,their south pole data, and the Riemann surfaces will be presented. The saddle points involved all have quadraticscalar potential with mass m = 0 . Physical quantities
The numerical calculation cover two situations in parallel with the situations studied in Sec.4.2 for the case ofboundary topology T : saddle points with time contours circling differently around singularities of Φ, and thosewith time contours containing different amount of Euclidean time. The corresponding contours in the regime ofperturbative scalar deformation are shown in Fig.18 in green and blue respectively. The first situation involves saddle points whose complex time contours, when scalar deformation is perturbative,takes the form of the green contour shown in Fig.18, with the whole complex history containing minimum amountof positive Euclidean time: δ ∗ = 0. The contour circles the singularity χ = − iπ/ − p = − , , (cid:45) (cid:45) (cid:45) Α (cid:45) Α I (cid:61) ∆ (cid:42) (cid:45) Π I (cid:142) (cid:73) (cid:68) (cid:45) (cid:45) m (cid:77) Α Im Β (cid:73) (cid:68) (cid:45) (cid:45) m (cid:77) Α Re Β (cid:45) ln C a S (cid:142) R (cid:182) Α I (cid:182) Α (cid:65)(cid:73) (cid:68) (cid:45) (cid:45) m (cid:77) Α Im Β (cid:69) (cid:182) Α I (cid:142) (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Im Β (cid:182) Α S (cid:142) R (cid:72) (cid:68) (cid:45) (cid:45) (cid:68) (cid:43) )Re Β (cid:45)(cid:182) Α ln C a (cid:182) Α (cid:65)(cid:73) (cid:68) (cid:45) (cid:45) m (cid:77) Α Re Β (cid:69) Figure 20:
Details of the matching ∂ α ˜ S = (∆ − − ∆ + ) β for the fundamental saddle point, whose result has been presented with the p = 0 curves in Fig.19. The left column shows the matching of the imaginary part, and the right column real part, correspondingrespectively to the third and fourth columns in Fig.19. The upper frames show the detail of how terms on the righthand sides ofEq.(150) add up to the boundary on-shell action, and the lower frames shows the addition of the derivatives and the matching betweenthe resulting ∂ α ˜ I with (∆ − − ∆ + ) β . Re Β Im Β (cid:72) (cid:68) (cid:45) (cid:45) m (cid:76) Α Im Β (cid:45) (cid:45) Α - - - - - - - a - - - - a I = d * - p I é ∂ a I ∂ a AI D - - m M a Im b E ∂ a I é H D - - D + )Im b - ln C a S é R I D - - m M a Re b ∂ a S é R H D - - D + )Re b -∂ a ln C a ∂ a AI D - - m M a Re b E - - - - - Figure 21:
Results of the saddle points of boundary topology S , which have contours given by χ o → a → c → χ (cid:48)∗ in the regimeof perturbation scalar deformation. The left-most frame shows the α - β relation (blue) and the imaginary part of the boundary termproduced by holographic renormalization (yellow). The latter is to be added to the imaginary part of the bulk saddle-point action I to obtain its boundary counterpart ˜ I . The middle column shows the plots related to the imaginary part of bulk on-shell action; theright column the real part. The dotted red lines are perturbative results. Since there is only one family of saddle points presented, the“one-point function” generation is shown in detail as Fig.20. a R e F o - - - - - - - a I m F o p =0 -1 1-1 p =0 Figure 22:
The south pole value Φ o of the saddle points in Fig.19,where the saddle points have time contours circling the first singu-lar point under the south pole 0 and ± Re (cid:70) o Im (cid:70) o (cid:45) (cid:45) (cid:45) Α Figure 23:
South pole value Φ o of the saddle points stud-ied in Fig.21. The defining time contour in the perturbativeregime is as χ o → a → c → χ (cid:48)∗ ( k = −
1) in Fig.18. Thedotted red lines are perturbative results.
Figure 24:
The scalar field profile of the fundamental saddle points of S boundary and quadratic scalar potential with scalar mass m = 0 .
94. Their time contour in the perturbative regime is given by χ o → a → b → χ ∗ in Fig.18. In the perturbative regime Φ o ∼ α ∼ χ = − iπ/ ∞ point closely attached to a −∞ point, which is just the characteristic of asimple pole of order 1. This corresponds exactly to the discussion under Eq.(134). As α grows, the two opposite infinity points leaveaway from each other, and very likely there is a branch cut stretching between them. Since the numerical scheme solves the equationsalong horizontal lines, the branch cuts are all horizontal. However it seems that in the third and the fourth column in the lower figures,the branch cuts at the upper and lower edges of the white bands should be joining each other at infinity. Scalar field profile of the saddle points whose contour is χ o → a → c → χ (cid:48)∗ in the perturbative regime (Φ o ∼ α ∼ .
94. The singular point at χ = − iπ/ α , containing closely neighboring + ∞ point and −∞ point. With the increase of α or | Φ o | , the pole splits into two separate branchpoints, one of which moves toward the χ -contour. This seems related to the divergence of Im β and hence ˜ I in Fig.21. Figure 26:
Riemann surfaces for the saddle points studied in Fig.19, where the time contour circles clockwisely once ( p = 1) the firstsingular point under the south pole when the scalar field is perturbative. The singularity being circled around starts off as a pole oforder 1 (first column), while with the increase of α , the ∞ point and the −∞ point are teared away, seemingly having a branch cutlinking them. One of them (the ∞ point) moves towards the positive direction of the Lorentzian time and tends to collide into the χ -contour. This may be related to the tendency of divergence of Im β (the p = 1 curve) in Fig.19. • The absence of branch points in the Riemann surfaces in the perturbative regime, stated below Eq.(135),clearly does not persist in the realm non-perturbative scalar perturbation. In the figure it is shown that theresults for p = 0 , ± α = 0 in the same direction, sharing the same perturbative results (dottedred lines), but soon they divert from one another as α increases. • The “one-point function” generation Eq.(46) is verified as shown in the third and fourth frames in the lowerrow. Fig.20 shows the detail of the verification of Eq.(46) for the fundamental saddle points of p = 0. The second situation involves saddle points with no circling contours around singularity but with more Euclideantime length covered by its whole complex history than the fundamental saddle point. The results are shown inFig.21 where the saddle points studied have complex time contours in the regime of scalar perturbation given bythe k = − χ o → a → c → χ (cid:48)∗ in Fig.18. Again as the non-fundamental saddle points studied in Fig.10and Fig.11, we observe the divergence of Im β with the increase of α . As a result, I ( α ) is bounded as a functionof α while ˜ I diverges to −∞ when α increases. Therefore the k = − South pole data Φ o The south pole data here for the case of S contains only Φ o . Fig.22 and Fig.23 show the α -Φ o relations forthe saddle points covered in Fig.19 and Fig.21 respectively. With the relation I = δ ∗ − π , the Euclidean shiftis not especially plotted since they are already given in Fig.19 and Fig.21. The noticeable properties is that thesmall α behavior match the perturbative result Eq.(138) with ρ given by Eq.(140). Also for saddle points differingin the way the time contours circle around the singularity (Fig.22) different curves share the same perturbativeapproximation but start to divert when α increases. Riemann surfaces
The results of tracing Riemann surface deformations are summarized in Fig.24, Fig.25, and Fig.26, which coverthe case of fundamental saddle points (Fig.19), k = − p = − • For very small α (first columns) we notice the singularity at χ = − iπ/
2, has exactly the look of an isolatedpole of order 1. There is a + ∞ spot closely attached to a −∞ spot. This corresponds perfectly to thediscussion around Eq.(135) in the perturbative context. • Comparing Fig.24 and Fig.26, when α increases we see clearly on the level of Riemann surfaces that the twofamilies of saddle points ( p = 0 and p = −
1) are after all different, although perturbatively they are the same. • In Fig.25 and Fig.26, when α increases we observe a singular point moves towards the positive direction ofLorentzian time with the tendency of collision with the χ -contour. Meanwhile we notice that Im β for thesetwo cases tend to diverge. On the other hand in Fig.24 for the fundamental saddle points, all singularitiesstick well close to the Im χ -axis, and meanwhile the corresponding Im β curve in Fig.19 ( p = 0) asymptotes51o 0 when α increases. This conforms to the speculation in Sec.4.3 that divergent Im β is accompanied by asingular point of Φ that moves towards the positive direction of Lorentzian time with the increase of α , andwhich tend to collide into the integral contour of the equations of motion. This section browses through two topics which are in the immediate continuation of the line of thinking of the workbeing presented so far, in order to give a preview of the possible direction that future work can be oriented to aswell as making up for some important aspect that the previous sections did not cover.Sec.6.1 will show numerically that the holographic renormalization results Eqs (42) and (149) are valid for non-quadratic potentials. Sec.6.2 will try to extend the holographic renormalization in Sec.5.2 for boundary topology S to higher dimensions S d ( d = 3 , , . . . ), where attention will be focused on the renormalization of the imaginarypart of the saddle-point action. An application of the result to the case of S will be shown. The numerical verification of the holographic renormalization via Eq.(46) has so far been carried out for quadraticscalar potentials only, while the derivation only required the potential to take the form (cid:96) V (Φ) = m Φ + O (Φ )as in Eq.(28). This subsection shows with two examples of non-quadratic scalar potentials that the holographicrenormalization works beyond quadratic level. The potentials tested are: the quadratic potential (for reference);the Φ potential, containing a quartic term; and also the cosh potential, containing an infinite series of even orderterms in Φ. They will be denoted by V , V and V respectively: (cid:96) V (Φ) = m ; (cid:96) V (Φ) = m + m
4! Φ ; (cid:96) V (Φ) = m (cosh Φ − . (151)The potentials are chosen as such in order to let all three share the same quadratic term, and let V and V havethe same quartic term.The numerics have been done to both boundary topologies of T and S , but only for the fundamental saddlepoints for simplicity. The results of physical quantities are presented in Fig.27 and Fig.28 respectively. The contentare organized in the same way as results are presented in Sec.4.2. The plots show that all three curves start offfrom α = 0 in much the same way, sharing the same leading order approximation as shown by the dotted red lines.Then as α increases, the higher and higher order terms in the potentials start to enact, and so we see first V curvesstart to divert from V and V curves and then V and V curves start to divert from each other. In Sec.5.1 the perturbative results of the saddle-point actions were derived for boundary of topologies S d ( d =2 , , . . . ) presented in Eqs (141) and (142), while later in Sec.5.2 the holographic renormalization for non-perturbativescalar deformation was worked out only for S in Eqs (149) and (150). It seems that a generalization to the bound-ary topologies S d ( d = 3 , , . . . ) can be very ready worked out, but there is actually the difficulty that analyzing theIR divergence becomes much more complicated than for d = 2. However in the context of no-boundary quantumcosmology where saddle-point actions are generally complex, a partial generalization is still possible: it is possibleto work out Im S ct , and hence we can obtain ˜ I from I .The derivation is to a great extent a repeating of the steps for obtaining Eq.(149) while the important differenceis that when applying the formula Eq.(40) for counter terms, A is understood as the volume of the boundary a d ∗ and the leading counter term A should acquire a coefficient d − A → ( d − a d ∗ [58, 59]. The next-to-leading52 a I a I é - - - a I m b - - a H D - - m L a I m b a a a S é R - - - - a R e b V V V V V V V V V V V V V V V V V V V V V V V V ∂ a I é H D - - D + )Im b ∂ a S é R H D - - D + )Re b Figure 27:
Results of saddle points of boundary topology T , comparing quadratic and non-quadratic scalar potentials. The contentsare organized in the same way as in Fig.8. The potentials V , V and V are given by Eq.(151), and the scalar mass is m = 0 . α = 0 with almost the same behavior. This is because their potentials have the samequadratic term, and the leading order approximation of the curves are determined by the quadratic term. Therfore the perturbativeresults, represented by the dotted red lines, fit all three curves in the frames where they apply. When α becomes larger, the differencesbegin to show up. First it is V -curves that split from the V and V -curves when the effect of quartic term becomes important, andthen V and V -curves begin to divert from each other when the terms beyond quartic terms in V become important. The generalpattern is that larger potential tends to enhance the value of I as well as ˜ I . The result also shows that the relation of “one-pointfunction” generation Eq.(46) is satisfied. - - - - a R e b - - - - a I - - - - a I é a S R é - - - - a I m b a H D - - m L a I m b a a V V V V V V V V V V V V V V V V V V V V V V V V ∂ a I é H D - - D + )Im b H D - - D + )Re b∂ a S é R Figure 28:
Results of saddle points of spatial topology S , comparing quadratic and non-quadratic scalar potentials. Except for theboundary topology, all other settings are the same as in Fig.27. Here we observe the similar feature of diversion of the three curveswhen α increases. Also we have Eq.(46) verified. ∆ − A ϕ = ∆ − a d ∗ ϕ . The further subleasing terms are more difficult to obtain, but in fact to work out˜ I does not need the knowledge about these terms, since they do not contribute to ˜ I . Therefore ˜ I and I differ bythe imaginary part of ( d − a d ∗ + ∆ − a d ∗ ϕ . Using the asymptotic expansions Eqs (182) and (183), we obtain ˜ I ( α ) = I ( α ) + (cid:18) ∆ − − m d (cid:19) α Im β. (152)This formula is valid for scalar potentials of the form V (Φ) = (cid:96) − m Φ + O (Φ ), and scalar mass within the range d − < m < d , where the formalisms in appendix B are valid. A model of boundary topology S In the following I show an application of this result on the model of scalar-deformed dS cosmology, where thescalar field is minimally coupled and has quadratic potential. The scalar mass is m = 1 .
96, which is within therange where Eq.(152) is valid. All the technicalities used in Sec.5 can be very easily transplanted here. The mostimportant difference however, is that this time we have to do the bulk saddle-point action I directly using theintegral formula: Eq.(170) with d = 4 and d = 0. This sometimes makes the numerical error difficult to control.An intriguing aspect of this model is that we can very easily see that there are families of saddle points whichdo not allow the scalar field to be switched off. That is, when α vanishes, β and Φ o are still finite, unlike the saddlepoints studied in the previous sections all of which have α ∝ β ∝ Φ o ∼ α approaches 0. This feature is alsofound in scalar-deformed dS models [35]. Following the same pattern that the results are presented in Sec.4.2 andSec.4.3, I show the results of two families of saddle points of boundary topology S , where one family contains andthe other does not contain a perturbative regime, and they are labeled by A or B respectively.The physical quantities are shown in Fig.29, the south pole data Φ o and the Euclidean shifts δ ∗ are shown inFig.30 and the Riemann surfaces of the scalar field of the two families of saddle points are shown in Fig.31 andFig.32. Here are the features relevant to our purpose: • The saddle points of family A are obtained by augmenting the scalar deformation in the perturbative saddlepoints with contour χ o → a → b → χ ∗ ( k = 0) shown in Fig.18. In 3d when we do so, we obtain fundamentalsaddle points, but here they are not fundamental. From the behavior of the Euclidean shifts δ ∗ in Fig.30, wesee that saddle points of family B actually experience less Euclidean time, and in fact they are the fundamentalsaddle points of the S model. • The behavior of Φ o in Fig.30 shows that for the family A, when α ∼ o . However this is not thecase for family B, showing that family B does not cover a perturbative regime of the scalar deformation where α ∝ Φ o ∼
0. This can also be seen from the α - β relation in Fig.29 where we have α ∝ β ∼ • The relation of “one-point function” generation Eq.(46) is verified for the imaginary part, seen from theforth column of Fig.29. Therefore ˜ I obtained from Eq.(152) is indeed the imaginary part of the boundarysaddle-point action. There are wiggles in the curve ∂ α ˜ I , which is very probably due to the numerical errorintroduced in the computation of I by direct integral Eq.(170) with d = 4 and d = 0. • The behaviors of Im β in Fig.29 have the same feature as the previous models of boundary topologies T and S . When α increases Im β asymptotes to 0 for family B the fundamental saddle points, while for family Aof non-fundamental saddle points, Im β diverges. The result in the β -representation is ˜ I ( β ) = I ( β ) + (cid:16) ∆ + − m d (cid:17) β Im α . - - - - - - a R e b - - - - - - a I m b a a A AA A I é I numerical error ∂ a I é H D - - D + )Im b - - - a R e b a I m b - - - - - - a - - a B B B B I ∂ a I é H D - - D + )Im b I é Figure 29:
Results of two distinct families, labeled by A and B, of saddle points of boundary topology S . They are minimallycoupled to a scalar field of quadratic potential and mass m = 1 .
96. The family A (upper row) contains a perturbative regime where itscomplex time contour is χ o → a → b → χ ∗ ( k = 0) in Fig.18. The family B (lower row) does not contain a perturbative regime. Thefour columns show respectively the α - β curves in the first two columns, the imaginary part of the saddle-point actions in the bulk ( I )and on the boundary (˜ I ) related by Eq.(152) in the third column, and the generation of 1-point function by the boundary saddle-pointaction, i.e., the imaginary part of Eq.(46). In the fourth column there are some wiggles in the ∂ α ˜ I curves, and this is very probablybecause the bulk on-shell action I is computed by direct integration, which is numerically less stable than the indirect way used in T and S cases. The find dotted red lines in the upper row are perturbative results Eqs (138)–(142). • As shown in Fig.29, for both families A and B, I are bounded. For family A, ˜ I diverges to + ∞ , while forfamily B, ˜ I and I differ very little, as expected of the fundamental saddle points. Therefore in the boundarywave function, the contribution from saddles of family A is exponentially suppressed when α increases (incase higher-loop contributions are suppressed). • There are also families of saddle points giving rise to ˜ I diverging to −∞ when α increases. For examplethose that have k = 1 contour (blue contour in Fig.18, with the final Lorentzian segment along Im χ = π )when scalar deformation is perturbative. Such saddle points can lead to exponentially divergent contributionto the Hartle-Hawking wave function. It is not necessary to show this case in detail because all the physicalquantities are just the opposite of those of the family A in Fig.29. • For the deformation of the Riemann surfaces of Φ, Fig.31 shows that for saddle points of family A, with theincrease of α , a singular point moves the more and more to the right and tends to collide into the χ -contour(it is really the case if we further increase α ), while it is not the case in Fig.32. Meanwhile we already noticedin Fig.29 that Im β diverges for family A, while asymptotes to 0 for family B, when α increases. This againsuggests the connection between the divergence of Im β and the presence of a singularity taking off from theIm χ -axis and moving towards the χ -contour. • For both families A and B, we can extend α to negative values. For family A the extension is obvious: I ( α ),˜ I ( α ) and δ ∗ ( α ) are even functions, while β ( α ) and Φ o ( α ) are odd; but for family B these quantities do nothave definite parity as function of α . Indeed Fig.32 (especially the first two columns) shows that when α dropstowards 0, there is a singularity (the dark spot) colliding into the first Lorentzian segment of the χ -contourfrom above, preventing α from decreasing to negative values. It is possible to further adjust the χ -contourto let it avoid this singularity so as to further decrease α . However, I hope to leave this to future work whenthe physical meaning of the result is better figured out.55 (cid:45) (cid:45) (cid:45) Α R e (cid:70) o (cid:45) (cid:45) (cid:45) Α I m (cid:70) o (cid:45) Α ∆ (cid:42) AB A BA B
Figure 30:
Defining parameters of the no-boundary saddle points of spatial topology S with scalar mass m = 1 .
96. The physicalquantities are shown in Fig.29. The family B does not cover a perturbative regime, which is seen from the fact that when α ∼
0, Φ o isfinite. For about 0 < α < | Φ o | decreasing with the increase of α for branch B, which is not seen for other cases studiedso far. The red dotted lines are results of scalar perturbation Eq.(138), applying only to family A. We see in the third frame that thefamily B, which cannot go perturbative, contains less Euclidean history and in fact it is the family of fundamental saddle points. Figure 31:
The scalar profile of the saddle points of boundary topology S in the family A, whose physical quantities are presentedin the upper row in Fig.29. In the perturbative realm as shown in the first, the singular point at χ = − iπ/ −∞ points clustered with three + ∞ points. Then with the growing of α the system becomesnon-perturbative, the ±∞ points are separated away, each becomes a branch point. Meanwhile one of them is attracted the more andmore towards the integral contour of the equations of motion, which seems to be related to the divergent behavior of Im β shown inFig.29. The scalar profile of the saddle points of boundary topology S in family B. The physical qualities of these saddle pointsare presented in the lower row of Fig.29. With the χ -contour configured as in these plots, α cannot decrease below about 0 since asingularity (dark spot) collides into the first Lorentzian segment of the contour from above. Then when α increases, there is a branchpoint (bright spot) to the left of the Euclidean part of the contour that moves towards the right as shown in the second column, andthen falls back to the Im χ axis. This seems to correspond to the presence of the peak of the α -Im β curve at α ∼ α continues to grow, no singular point tends to collide into the χ -contour from the left, and this well corresponds to the behavior of Im β for large α , which asymptotes to 0 as shown in the lower row of Fig.29. What is done
This paper studied several models of no-boundary quantum cosmology in a dS/CFT holographic setting, computingthe Hartle-Hawking wave function on the asymptotic boundary based on bulk computations and holographicrenormalization. The goal is to address the non-normalizability problem of the Hartle-Hawking wave functionsobtained by direct dS boundary computations. The models studied are mainly 3d universes governed by Einsteingravity, with a positive cosmological constant and a minimally coupled scalar field, and they have either T or S as the topology of asymptotic spacelike boundary.An extensive search for the probable contributing saddle points has been carried out in the bulk, where mini-superspace formalism is used for this search to be feasible (at the cost of losing saddles). The admission criteria forthe saddle points is the no-boundary proposal: they should be compact spacetimes accommodating on-shell fieldsthat satisfy the wanted boundary conditions. The search was first carried out for perturbative scalar deformation(Sec.3 and Sec.5.1) and then traced all the way to finite scalar deformation (Sec.4 and Sec.5.3).For some selected saddle points, supposedly representative, their saddle-point actions are computed first directlyin the bulk S = S R + i I and then on the asymptotic boundary ˜ S = ˜ S R + i ˜ I [cf. Eqs (35) and (43)], as functionsof scalar deformation α as in Eqs (33) and (137), and of boundary geometry τ as in Eq.(3) in case the boundarygeometry is T . Therefore the individual tree-level contribution of each saddle is e i S = e i S R e −I to the bulk Hartle-Hawking wave function, and e i ˜ S = e i ˜ S R e − ˜ I to the boundary one. I have resorted to the non-trivial amplitudes e −I and e − ˜ I for indications of the normalizability property of the Hartle-Hawking wave functions.One key step in the above part of the work is the holographic renormalization which computes ˜ S from S .Technically it is just a dS application of the standard AdS/CFT holographic renormalization, where one finds outthe counter term action S ct which cancels the IR divergences in S , so as to obtain ˜ S = S + S ct . This is done in57etail in Sec.2.3, and less in detail in Sec.5.2. The results are applicable to the case of finite homogeneous scalardeformation.Finally, a partial generalization of holographic renormalization to higher dimensional models of boundary topolo-gies S d ( d = 3 , , . . . ) is realized. In particular, the quantity explicitly worked out is the imaginary part of thecounter term action, and therefore we are able to compute ˜ I = I + Im S ct , and hence to infer from e −I and e − ˜ I the normalizability properties of the Hartle-Hawking wave functions. What is found
In the search for saddle points, the coordinate time in the mini-superspace is necessarily complex, so that the wholecoordinate time history of a saddle point is depicted by a generic curve on the complex time plane. This is in factalready well-known in the literature.For a given set of boundary condition, expressed in terms of scalar deformation and boundary geometry ( α, τ ), aninfinite number of saddle points exist which can be described by mini-superspace formalism. These saddle pointsfound are distinguished according to their complex time contours. Different saddle points can have a contourcovering different amount of Euclidean time (blue contours in Fig.7 and Fig.18) and/or winding differently aroundthe singularities (comparing blue and green contours in the same figures) in the Riemann surface of the scalar field.Among all saddle points figured out, we can single out one family which, in the literature where the Hartle-Hawking wave functions are computed in the bulk, are very commonly considered as the contributing saddle points.It is the one whose complex time contour covers a minimum amount of positive Euclidean history and does notcircle around singularities of the scalar field (contour χ o → a → b → χ ∗ in figures 7 and 18). The second propertymeans that the complex time contour always lies in the same layer of the Riemann surface. These saddle pointsare referred to as “fundamental” in this paper. All other saddle points are thus called non-fundamental.When the actions are computed for the fundamental saddle points, it is found that as function of α , I and ˜ I differ very little and are bounded as function of α (figures 8,9,20,27,28,29); and as function of boundary geometry τ when the boundary topology is T , they both tend to + ∞ . Therefore no remarkable implication can be drawnfrom these results concerning the normalizability property of the Hartle-Hawking wave function.For the actions of the non-fundamental saddle points with time contours not circling singularities in the Riemannsurface of the scalar field (blue contours with k (cid:54) = 0 in figures 7 and 18), but containing non-minimum length ofEuclidean history, it turns out that I and ˜ I can differ violently from each other. For all such cases studied, theimaginary part of bulk action I is bounded while the boundary counterpart ˜ I diverges towards ±∞ as a functionof scalar deformation α (figure 10,11,21). For boundary topology T , it is also shown that the “high temperature”( τ → ∞ ) behavior is the same as what the scalar perturbation results tell: I and ˜ I tend simultaneously to + ∞ or −∞ . Thus this class of non-fundamental saddle points can probably be responsible for both the scalar divergencesand the high temperature divergences of the Hartle-Hawking wave function (in case they contribute). However weshould notice that these divergences do not look quite the same as what is found in [22].There is another category of non-fundamental saddle points studied whose complex time contours circle aroundsingular points in the Riemann surface of the scalar field (green contours in Fig.7 and Fig.18). This part is morechallenging due to the difficulty in adjusting the contours. Still some results are obtained just enough to claimthat for boundary topology T , different ways of circling around singularities lead to different saddle points as inFig.12; and for boundary topology S , while the perturbative analysis shows that letting time contours circling thesingularities of Φ does not result in new saddle points, numerical results as presented in Fig.19 show that it is notthe case.Intriguing connection has been empirically noticed between the intrinsic characteristics of the saddle points andthe scalar divergence of their contribution to the Hartle-Hawking wave function. It seems that the divergence of58 I is always accompanied by a singular point in the Riemann surface of the scalar field, which with the increaseof α , moves the further and further in the positive direction of Lorentzian time, and which tend to collide intothe integral contour of the equations of motion (see figures 16,25,26,31). A closer look shows that this singularityseems to be connected to another singularity by a branch cut, and that with the increase of α this branch cut isstretched longer and longer. It is not clear especially, if this is just a mathematical feature or it can have somephysical meaning.The perturbative results obtained in Sec.3 and Sec.5.1 give the correct leading order approximation of thenumerical results. This is seen in the many figures (not density plots) where the red dotted lines well overlap solidlines when α ∼
0. However the drastic difference between I and ˜ I for non-fundamental saddle points cannot berevealed by perturbative results.The formal holographic renormalizations Eqs (42) and (149) have passed the test of “one-point function”generation, in that Eq.(46) has been verified numerically for all the saddle points covered by numerical calculation. Issues which need further discussion
In the paper, all the claims of scalar/“temperature” divergence of the Hartle-Hawking wave function are drawn fromthe investigation of individual saddle-point contributions. The problem is that, whereas plenty of saddles are found,we do not have any clue which ones are actually picked up by the Hartle-Hawking path integral. Considerableamount of studies on this subject have been done in mini-superspace cosmological models of empty spacetime[37, 57], showing that which saddle points contribute depends on the initial choice of path integral contour, andthat there can be a natural choice of initial contour, which ends up picking up only part of saddles. However itis not clear to which degree of precision the result from mini-superspace formalism can provide guideline in thesituation where full gravitational degrees of freedom should be taken into account. Especially the situation becomesquite complicated with the presence of a scalar field. Therefore this paper simply assumes that every saddle pointmay contribute and studies what the consequence can be if a certain saddle point eventually contributes. Maybein general, due to the difficulty in directly operating the path integrals of gravity, a plausible alternative is to letthe boundary field theory result tell which bulk saddles really contribute to the Hartle-Hawking wave function.Furthermore, the drastic differences between I and ˜ I for certain saddle points seems to imply that in the passagefrom bulk to boundary, or from Ψ to ˜Ψ, the holographic renormalization changes the normalization property ofthe wave functions. This sounds contradictory to quantum mechanics, because the holographic renormalizationcorresponds to a representation change of the quantum system as discussed by the end of Sec.2.3, and thus shouldnot change the normalization property of the wave function required by the conservation of probability. However weshould remember that in the context of no-boundary quantum cosmology, we are dealing with generally covariantsystems where the probabilist interpretation of the wave functions is subtle. In the bulk it has been established eitherwith respect to classical histories [60] keeping the general covariance manifest, or with respect to the configurationof the universe at a given moment where the general covariance freedom is gauge-fixed to render the wave functionsunitary [61, 62]. However it is not clear how to consistently establish the boundary probabilist interpretation. Atleast if we take | ˜Ψ | as the boundary probability density, very likely it is inconsistent with that established in thebulk, since the integral transform relating Ψ and ˜Ψ is manifestly non-unitary with its kernel S ct complex.Finally an important technical limitation of the work is that the numerics work only for scalar mass aboveabout 0 .
9, since otherwise the FindFit command fails to extract the value of Re β . This is an important reason why m = √ is not studied in the paper for 3d models. In fact it is interesting to extend the study to the saddle pointsof very low masses (even below √ ) and to work out the holographic renormalization. It is mostly because thenon-fundamental saddle points of small scalar mass show quite different features from those studied so far of scalar The same computations can be almost trivially done to the 3d models in this paper setting Φ = 0, and similar conclusions can bereached. .
93. I have not shown in the body of the paper, but actually the same calculation as in Sec.4.2 hasbeen done to some low mass models, but only in the bulk. For example when m = 0 .
25 for the non-fundamentalsaddle points of k = − , − , . . . , − k = 0) ones, and probably they have Im β that do not diverge, andhave I and ˜ I that are both bounded with the increase of α . If this is the case, it means that with the increase ofmass, there is a transition in the pattern of behavior of such non-fundamental saddle points. It will be interestingto find a way to study the saddle points of low mass to figure out what happens during this transition. What next
Since the formal formalisms obtained only requires the potential to take the form of Eq.(28), we can considercarrying further the study in Sec.6.1, trying different potentials to see whether novel physics can emerge. It willbe especially interesting if some exactly solvable potential can be investigated to work out everything analytically.To start with, maybe a good idea is to obtain such models by analytic continuation from black hole solutions withscalar hair, for example [43]. However it is even more desirable to find exactly solvable models with very low scalarmasses (at least lower than √ ). If possible, this will provide direct solution to the problem raised in the lastparagraph.The work in this paper has produced some indication of scalar divergence and “high temperature” divergencein the boundary Hartle-Hawking wave function. However the detail of the divergences do not have the same lookas those in [22, 23, 24]. For example, the temperature divergence in [24] arise at loop level, while in this paper, itappears at tree level with the participation of a scalar field. Also very obviously the Hartle-Hawking wave functionshere are even as function of scalar deformation α while in [22] it is obviously not the case. However there canstill be things to do to render the study more relevant to the work in [22]. One immediate possibility that canbe envisaged is to study the model of boundary topology S of scalar mass √
2. Moreover, using the formula inappendix A, the generalization to S × S maybe possible. If so we can work out its Hartle-Hawking wave functionto compare with the result of S boundary and examine the effects of different topologies. At the mean time, itwill be extremely interesting if some exactly solvable potential can be found. Acknowledgements
The initiation of the topic is attributed to Frederik Denef and Thomas Hertog. The many discussions with themhave been very beneficial. Ruben Monten and Yannick Vreys provided invaluable help on numerics without whicha large part of the work in this paper is not possible. In the starting phase of the work, Kristof Moors alsohelped L.Liu to understand the numerical schemes and some important conceptual issues. This work has alsobenefited from the discussions or communications with Alice Bernamonti, Adam Bzowski, Gabriele Conti, FedericoGalli, Herve Partouche and Hongbao Zhang. L.Liu especially thanks F.Denef for being constantly available andsupportive during the roughest time of the work, as well as for his comments on the manuscript. This work issupported by grants from the John Templeton Foundation and from the Odysseus Programme of the FlemishResearch Foundation.
Notes added
In the past years in studying the scalar-deformed dS no-boundary cosmology, R.Monten and Y.Vreys have tremen-dously improved the numerical tools initially developed in [26, 27] for computing the Hartle-Hawking wave functions.60hen still working on the present paper, L.Liu had many discussions with them, and thereby could apply the newlydeveloped numerical tools to the models in the present paper. The work by Monten and Vreys is not yet published,while their paper is in preparation [35] which will include the important findings in 4d among other results. L.Liuhereby emphasizes the importance of [35] to the present paper and the authorship of the numerical algorithms ofMonten and Vreys, although the present paper is finalized earlier. A Action principles for mini-superspace no-boundary saddle points ofboundary topology S d × S d This appendix presents the action principles for the mini-superspace no-boundary saddle points minimally coupledto a scalar field. The boundary topology is set to be the product of two spheres: S d × S d where d + d ≥ d and d to appropriate values. Action
Let the mini-superspace time coordinate be χ , and let radii of S d and S d be respectively a and b , which arefunctions of χ only. The spacetime metric is (cid:96) − ds = − N ( χ ) dχ + a ( χ ) d Ω d + b ( χ ) d Ω d , (153)where (cid:96) is the dS radius related to the cosmological constant as Λ = ( d + d )( d + d − (cid:96) , and d Ω d , are the lineelements of the spheres S d , respectively. Let the minimally coupled scalar field be Φ = Φ( χ ) with potential V (Φ).It a function of χ only due to the mini-superspace formalism. Then we assume, for simplicity, that the kineticterms in the Lagrangian are normalized like L ∝ R + ˙Φ − V (Φ) where R is the Ricci scalar. Thus the total actionfor the no-boundary saddle points, which sums up the Einstein-Hilbert term, the Gibbons-Hawking term and thescalar field term, is2 κ S = ˆ χ ∗ χ o dχ a d b d (cid:34) d ( d − (cid:18) − ˙ a a + 1 a (cid:19) + d ( d − (cid:32) − ˙ b b + 1 b (cid:33) − d d ˙ a ˙ bab − ( d + d )( d + d −
1) + ˙Φ − V (Φ) (cid:35) − a d o b d o (cid:32) d ˙ a o a o + d ˙ b o b o (cid:33) . (154)Here κ = πG(cid:96) D − Ω Ω with Ω , the surface area of the unit spheres S d , . The lapse is set to unit for simplicity. Thesouth pole is at χ = χ o and the spacetime boundary of the saddle point is at χ = χ ∗ . a o and b o stand for a ( χ o )and b ( χ o ) respectively. 61 amiltonian constraint and equations of motion The equations derived from varying the action (154) are d ( d − (cid:18) ˙ a a + 1 a (cid:19) + d ( d − (cid:32) ˙ b b + 1 b (cid:33) + 2 d d ˙ aa ˙ bb − ( d + d )( d + d − − ˙Φ − V (Φ) = 0 , (155)2( d −
1) ¨ aa + 2 d ¨ bb + ( d − d − (cid:18) ˙ a a + 1 a (cid:19) + d ( d − (cid:32) ˙ b b + 1 b (cid:33) +2 d ( d −
1) ˙ aa ˙ bb − ( d + d )( d + d −
1) + ˙Φ − V (Φ) = 0 , (156)2( d − bb + 2 d ¨ aa + ( d − d − (cid:32) ˙ b b + 1 b (cid:33) + d ( d − (cid:18) ˙ a a + 1 a (cid:19) +2 d ( d −
1) ˙ aa ˙ bb − ( d + d )( d + d −
1) + ˙Φ − V (Φ) = 0 , (157)¨Φ + d ˙ aa ˙Φ + d ˙ bb ˙Φ + V (cid:48) (Φ) = 0 . (158)We can simplify the second order equations (156) and (157) using the Hamiltonian constraint. This leads to¨ aa + ( d − (cid:18) ˙ a a + 1 a (cid:19) + d ˙ aa ˙ bb − ( d + d ) − V (Φ) d + d − , (159)¨ bb + ( d − (cid:32) ˙ b b + 1 b (cid:33) + d ˙ aa ˙ bb − ( d + d ) − V (Φ) d + d − . (160) South pole conditions
Without loss of generality, we let the first sphere (the a -sphere) shrink smoothly to zero size at the south pole χ o .Thus we can expand the metric and the scalar field near the south pole as follows: a ( χ o + (cid:15) ) = i (cid:15) + a (cid:15) + O ( (cid:15) ) , (161) b ( χ o + (cid:15) ) = b o + b (cid:15) + b (cid:15) + O ( (cid:15) ) , (162)Φ( χ o + (cid:15) ) = Φ o + c (cid:15) + c (cid:15) + O ( (cid:15) ) . (163)In order to determine the south pole conditions, arising from the no-boundary proposal, we substitute theseexpansions into the equations (155)–(158) and read off the coefficients of different orders of (cid:15) . Solving the resultingalgebraic equations which we get b = a = c = 0 , (164) c = − V (cid:48) (Φ o )2( d + 1) , (165) b = (cid:2) ( d + d )( d + d −
1) + 2 V (Φ o ) (cid:3) b o − ( d + d − d − d + 1)( d + d − b o . (166)62hus the initial conditions for numerical calculation are, in terms of the Lorentzian time, a ( χ o + (cid:15) ) = i(cid:15), ˙ a ( χ o + (cid:15) ) = i ; (167) b ( χ o + (cid:15) ) = b o , ˙ b ( χ o + (cid:15) ) = (cid:2) ( d + d )( d + d −
1) + 2 V (Φ ) (cid:3) b − ( d + d − d − d + 1)( d + d − b o (cid:15) ; (168)Φ( χ o + (cid:15) ) = Φ o , ˙Φ( χ o + (cid:15) ) = − V (cid:48) (Φ o ) d + 1 (cid:15) , (169)where Φ o and b o are complex, whose phases are to be adjusted such that the solution fit the boundary conditionsassigned at χ = χ ∗ . Saddle-point action for direct numerical computation
To compute the saddle-point action, we can simplify the expression (154) using the hamiltonian constraint (155),which gives κ S = ˆ χ ∗ χ o dt a d b d (cid:34) d ( d − a + d ( d − b − ( d + d )( d + d − − V (Φ) (cid:35) − a d o b d o (cid:32) d ˙ a o a o + d ˙ b o b o (cid:33) . (170)The convention adopted in this paper is to use plain letters for any action in general and calligraphic counterpartsfor their on-shell values. This expression is useful for models of dimension higher than 3. In 3d models where d = d = 1 or d = 2 and d = 0 or d = 0 and d = 2, more efficient ways of evaluation are available which areworked out in Sec.2.3 and Sec.5.2. B Asymptotic expansions near future boundary of asymptotic dS
When performing the holographic renormalization, it is important to know the leading behaviors of the metric andscalar field near the dS boundaries. This appendix presents these asymptotic expansions for the models studied inthe paper. Let the coordinate time be χ and the asymptotic boundary be Re χ → ∞ . To study the asymptoticbehaviors, we can mimic the procedure in [12], putting e − χ = u and expanding the metric components and thescalar field in series of u . Then inserting the series into the equations of motion we can work out the coefficientsorder by order in principle, and we only need to conserve the leading orders.Below to be presented are the results for the two types of boundary topologies covered in the paper: T and S d . The scalar potential is assumed to have the form V (Φ) = (cid:96) − m Φ + O (Φ ), and the mass range consideredis d − < m < d , where d = 2 , , . . . is the spatial dimension.63 oundary topology T The equation of constraint and the equations of motion are derived from Eqs (155)–(158) with d = d = 1:2 u a u b u − ab − ab ( u Φ u + m Φ ) + O (Φ ) = 0 , (171)2 u a uu + 2 ua u − a + a ( u Φ u − m Φ ) + O (Φ ) = 0 , (172)2 u b uu + 2 ub u − b + b ( u Φ u − m Φ ) + O (Φ ) = 0 , (173) ab ( u Φ uu + u Φ u ) + u ( a u b + ab u )Φ u + m ab Φ + O (Φ ) = 0 . (174)Then we can expand the fields as power series of u , inserting them into the equations to determine the coefficientsof the leading orders. First we need to set Φ = 0 and obtain the leading order of a and b , and then plug theseleading orders into Eq.(174) to find the leading order of Φ, and then plug the leading order of Φ back into Eqs(171)–(173) to find the sub-leading orders of a and b . By this procedure, we find the following asymptotic behaviorsΦ( u ) = u ∆ − (ˆ α + . . . ) + u ∆ + ( ˆ β + . . . ) , (175) a ( u ) = C a u (cid:32) − ˆ α u − + a u − ˆ β u + + . . . (cid:33) , (176) b ( u ) = C b u (cid:32) − ˆ α u − + b u − ˆ β u + + . . . (cid:33) , (177)where dots stand for higher orders in u ; C a , C b , ˆ α , ˆ β are constants, and ∆ ± = 1 ± √ − m , and a , b are complexconstants which satisfy a + b + m ˆ α ˆ β = 0 . (178)Note that due to the range of mass d − < m < d ( d = 2), we have < ∆ − < > ∆ + > Boundary topology S d The equations of motion for spatial topology S d are d ( d − u a u + 1) − d ( d − a − a ( u Φ u + m Φ ) + O (Φ ) = 0 , (179) d ( d − u a uu + ua u − a ) + a (cid:2) ( d − u Φ u − m Φ (cid:3) + O (Φ ) = 0 , (180) a ( u Φ uu + u Φ u ) + d u a u Φ u + m a Φ + O (Φ ) = 0 . (181)Following the same scheme in obtaining Eqs (175)–(177), we find the asymptotic behaviors to beΦ = u ¯∆ − (ˆ α + . . . ) + u ¯∆ + ( ˆ β + . . . ) , (182) a = C a u (cid:34) u C a − ˆ α u − − m ˆ α ˆ βd ( d − u d − ˆ β u + + . . . (cid:35) , (183)where C a , ˆ α , and ˆ β are complex constants, and ¯∆ ± = (cid:0) d ± √ d − m (cid:1) . Due to the mass range chosen, we have d − < ¯∆ − < d and d +12 > ¯∆ + > d . Puting d = 2 we obtain the results for the case of topology S that we study64n detail in the paper: Φ = u ∆ − (ˆ α + . . . ) + u ∆ + ( ˆ β + . . . ) , (184) a = C a u (cid:34) − ˆ α u − + (cid:18) C a − m ˆ α ˆ β (cid:19) u − ˆ β u + + . . . (cid:35) , (185)where C a , ˆ α , and ˆ β are complex constants, and ∆ ± = 1 ± √ − m . References [1] A. Strominger, “The dS / CFT correspondence,” JHEP (2001) 034 [hep-th/0106113].[2] A. Strominger, “Inflation and the dS / CFT correspondence,” JHEP (2001) 049 [hep-th/0110087].[3] J. M. 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