Jackiw-Teitelboim Gravity in the Second Order Formalism
PPrepared for submission to JHEP
TIFR/TH/20-53
Jackiw-Teitelboim Gravity in the Second OrderFormalism
Upamanyu Moitra, a Sunil Kumar Sake, a Sandip P. Trivedi, a a Department of Theoretical Physics, Tata Institute of Fundamental Research,Colaba, Mumbai, India, 400005
Abstract:
We formulate the path integral for Jackiw-Teitelboim gravity in the secondorder formalism working directly with the metric and the dilaton. We consider the theoryboth in Anti-de Sitter(AdS) and de Sitter space(dS) and analyze the path integral for thedisk topology and the “double trumpet” topology with two boundaries. We also consider itsbehavior in the presence of conformal matter. In the dS case the path integral evaluates thewavefunction of the universe which arises in the no-boundary proposal. In the asymptoticAdS or dS limit without matter we get agreement with the first order formalism. Moregenerally, away from this limit, the path integral is more complicated due to the presenceof modes from the gravity- dilaton sector and also matter sector with short wavelengthsalong the boundary that are smaller than the AdS or dS scales. In the double trumpet case,for both AdS and dS, we find that bosonic matter gives rise to a diverging contributionin the moduli space integral rendering the path integral ill-defined. The divergence occurswhen the size of the wormhole neck vanishes and is related to the Casimir effect. Forfermions this divergence can be avoided by imposing suitable boundary conditions. In thiscase, in dS space the resulting path integral gives a finite contribution for two disconnecteduniverses to be produced by quantum tunneling. a r X i v : . [ h e p - t h ] J a n ontents A.1 Euclidean AdS disk 52A.2 Euclidean AdS double trumpet 53A.3 de Sitter 53
B Extrinsic Trace and Schwarzian action 54
B.1 Euclidean AdS disk 54B.2 Euclidean AdS double trumpet 56B.3 de Sitter 57
C Zeta-function regularization 59D Conformal Killing Vectors in the Euclidean AdS Disk 60E Estimate of the inner product of metric perturbations arising from largeand small diffeomorphism 62 – i –
Estimation of various determinants in Euclidean AdS disk 67
F.1 Asymptotic AdS limit case 72
G Matter coupling to the time reparametrization modes in AdS 77H de Sitter wavefunction using Euclidean AdS contour 80I More on AdS double trumpet calculations 82
I.1 Measure for large diffeomorphisms and Schwarzian action 82I.2 Matter in double trumpet calculations 88I.3 Coleman method for computing determinants in AdS double trumpet 90
J dS double trumpet determinants 97
J.1 Coleman method computation 97J.2 Direct calculation of scalar laplacian 99– 1 –
Introduction
Jackiw-Teitelboim (JT) gravity is a theory of two-dimensional gravity which has receivedconsiderable attention recently [1–29, 29–37, 37–106] In this paper we carry out a pathintegral quantization of the theory in the second order formalism which involves the metricand the dilaton. This is to be contrasted with the first order formalism, used in much of therecent discussion[3, 25, 107–109], which involves the spin connection and Vierbein alongwith the dilaton. Some of the motivation for our work comes from wanting to comparethe results we get from the second order formalism with those obtained in the first orderformalism. The second order formalism allows matter to be added in a direct way, and thispermits us to generalize our study of JT gravity to also include matter. Finally, one mighthope to glean some lessons about quantizing gravity in higher dimensions from the secondorder formalism.We study both JT gravity in Anti-de Sitter (AdS) space and in de Sitter (dS) spacehere. Our analysis includes the path integral for spacetimes with the topology of a diskwhich have Euler character χ = 1, with one boundary, and also spacetimes of the “doubletrumpet” kind with χ = 0 and two boundaries.For the disk topology we find in the asymptotic AdS or dS limit, obtained by takingthe dilaton and length of the boundary to diverge while keeping their ratio fixed, thatthe results of the second order path integral quantization agree with those obtained fromthe first order formalism. In particular, we find like in the first order formalism thatthe dynamics of the gravity-dilaton system is given by the reparametrization modes of theboundary (called time reparametrization modes) which are governed by an action involvingthe Schwarzian derivative. However away from the this limit, for example, even whenworking with a boundary of large but fixed length, we find that the path integral is morecomplicated to evaluate. This happens because the measure for diffeomorphisms is morecomplicated in general, due to the mixing of small and large diffeomorphisms with eachother, and also because the determinants which now arise have a complicated dependenceon the large diffeomorphisms. The underlying reason for all this is that away from theasymptotic limit there are modes with wavelengths shorter than the AdS or dS scale whichneed to be included in the path integral and their dynamics is not simple. In particularone needs to include an arbitrarily large number of higher derivative terms, beyond theSchwarzian, valued in Diff( S ) /SL (2 , R ) in order to incorporate their coupling with thelarge diffeomorphisms.The matter we include is conformally invariant - mostly free bosons or fermions, al-though some of our results are more general. In the presence of matter, for the disk, wefind again that in the asymptotic AdS or dS limit the path integral can be carried out butaway from this limit the matter determinant from quantum fluctuations has a dependenceon the large diffeomorphisms which requires us to go beyond the Schwarzian action andinclude the higher derivative terms mentioned above. It is worth mentioning that simpledimensional counting shows that the quantum effects of matter only arise away from theasymptotic limit when one is working at finite boundary length, and including them in asystematic manner along with the quantum effects from the gravity-dilaton sector is quite– 2 –on-trivial. Such an analysis would need to be carried out to go beyond the semi-classicallimit which has been analyzed in considerable detail recently where the number of matterfields N → ∞ , and the gravity-dilaton is treated as being classical.In AdS space the path integral for a single boundary has the interpretation of com-puting the partition function of the boundary theory at finite temperature. In dS spacethe path integral we carry out evaluates the wavefunction of the universe obtained fromthe no-boundary proposal first put forward by Hartle and Hawking [110]. The contour inthis case is more complicated and goes over spacetime regions with different signatures.We consider both the Hartle-Hawking (HH) contour, which involves initially a Euclideanregion with signature (2 ,
0) and then de Sitter space of signature (1 , , − AdS , followed by a region of dS space. We find agreement betweenboth contours and discuss how to carry out the path integral by analytically continuingfluctuations across regions of different signature. The short distance modes we were refer-ring to above, which render the calculations more complicated away from the asymptoticlimit, in the dS case refer to modes which are still dynamical and have not yet exited thehorizon. It is important to mention that especially in the dS case one would like to obtainthe wavefunction when the universe has finite size (and the dilaton takes a finite value) andit is therefore important to try and go beyond the asymptotic limit after including thesemodes, although we will not be able to make much progress in this direction here.The contribution of the double trumpet geometry is suppressed compared to the disktopology, since it has higher Euler character, by a factor of e Φ / G where Φ can be thoughtof as the horizon value of the volume of the internal space which gives rise to the JT AdSor dS theory after dimensional reduction. The double trumpet can be thought of as awormhole joining the two boundaries. Most of the comments above about agreement withthe first order formalism in the absence of matter and in the asymptotic AdS or dS limitsapply to the double trumpet geometry as well. The path integral now involves two setsof large diffeomorphisms which act independently at the two boundaries and also moduli,including one which corresponds to the size of the “neck” of the wormhole (called b below).We show how the correct measure for summing over these diffeomorphisms and moduliarise in the second order formalism.Once matter is added we find that its quantum effects give rise to a contribution inthe path integral which diverges when the neck goes to zero size. The quantum effectscan be thought of as giving rise to a Casimir effect which diverges when the neck becomesvanishingly small. We show that this happens both for bosons and also for fermions.The bosons have periodic while the fermions have anti-periodic boundary conditions alongthe time or temperature direction. This divergence is analogous to the tachyon divergencewhich arises on the world sheet for Bosonic string theory. Alternatively, we can also considerperiodic boundary conditions along this direction for the fermions, as would be appropriatefor example if one is evaluating an index Tr(( − F e − βH ) instead of the partition function .In this case we find that the quantum effects from fermions do not diverge when b → We are grateful to Shiraz Minwalla for emphasizing these alternate boundary conditions. – 3 –he path integral is well behaved.In dS space the double trumpet gives rise to an amplitude for two universes to ariseafter quantum tunnelling from “nothing”. The divergence in the b → dS spacetime.Finally we end with conclusions in section 7. Appendices A-J contain important additionaldetails. In this section we will consider JT gravity in Euclidean AdS space. The path integral forthe system is given by Z JT = (cid:90) D [ φ ] D [ g µν ]Vol(Ω) exp {− S JT } (2.1)where S JT , the action for Jackiw-Teitelboim gravity, involves the metric and a scalar, thedilaton. In Euclidean signature the action is given by S JT = − πG (cid:18)(cid:90) d x √ gφ ( R + 2) + 2 (cid:90) dx √ γφ ( K − (cid:19) , (2.2)where we have set the AdS length R AdS = 1.Note that the sum in eq.(2.1) is over all metric and dilaton configurations and ourmain task here will be to make this precise. This problem has invited considerable atten-tion recently, [3, 13]. In general, one must sum over all topologies subject to the boundaryconditions that are imposed. Our approach will be to work directly in the second orderformalism which involves a sum of metric configurations and not with the first order for-malism which has been used in much of the previous literature. This will also allow us toinclude matter easily, as we will see later. We will restrict ourselves in this section to therelatively simple case of the disk topology with one boundary.Before proceeding let us note that the action of JT gravity actually includes one addi-tional term which is topological, S top = − Φ πG (cid:18)(cid:90) √ gR + 2 (cid:90) √ γK (cid:19) = − Φ G χ (2.3)– 4 –here χ is the Euler characteristic of the manifold, related to the number of handles, H and boundaries, B by χ = 2 − H − B (2.4)Such a topological term arises for example when one constructs the JT action by dimen-sionally reducing from higher dimensions in the near horizon region of a near extremalblack hole and in that case it accounts for the ground state entropy of the extremal blackhole. We will mostly ignore S top for now, since we will be working on the disk topologywith fixed χ = 1, and work with the action eq. (2.2).We will formulate the path integral for a boundary of fixed length l with the dilatontaking a fixed value φ B at this boundary. An important limit in which the path integraleq.(2.2) has been studied is the asymptotic AdS limit. In this limit we introduce a cut-off (cid:15) to regulate the theory and take the limit (cid:15) →
0, with the dilaton and length of theboundary scaling like φ → J (cid:15) (2.5) l → β(cid:15) (2.6)with J, β fixed. By rescaling (cid:15) we can set β = 1 so there is actually only one dimensionlessparameter specifying the limit given by βJ . So, we take the boundary conditions for φ and l in this section as follows. φ → πJ β(cid:15) (2.7) l → π(cid:15) (2.8)We will see below when dealing with determinants that there is also the issue of takingthe cut-offs, introduced to regulate the determinants, to infinity; the asymptotic AdS limitthen needs to be defined more precisely keeping track of the correct order of limits.In this asymptotic AdS limit we will find complete agreement between the path integralin the second order and first order formalisms. In particular, we will show below how thesum over large diffeomorphisms, which correspond to fluctuations of the boundary, arises,with the correct measure, in the second order formalism as well. In the more generalcase where the dilaton takes a fixed value φ B at the boundary of length l we will showhow the path integral can be defined quite precisely, but will not be able to carry out theevaluation till the very end. Interestingly, we will find that the more general case differsfrom the asymptotic AdS one in important ways, even when φ B , l (cid:29)
1. This more generalcase will also be of interest when we turn to de Sitter space later in the paper.We have not specified yet what the Vol(Ω) factor in eq.(2.1) refers to. In definingthe path integral for any gauge theory one would only like to sum over physically distinctconfigurations. This can be achieved by summing over all configurations and then dividingby the volume of the gauge group. For our case we would therefore divide by the volumeof all diffeomorphisms which leave the geometry - along with the boundary - unchanged.– 5 –hese diffeomorphisms which will be defined more precisely below be will referred to as“small diffeomorphisms” and Vol(Ω), in eq.(2.1), then refers to the volume of these dif-feomorphisms, . In contrast, there will also be a set of “large diffeomorphisms”, these arephysically distinct configurations corresponding to different boundaries and we will sumover them without treating them as gauge transformations.
We first consider in our discussions below the general case of a disk with a boundary oflength l where the dilaton takes value φ B . As a limiting case we will then turn to theasymptotic AdS boundary conditions, eq.(2.7) and (2.8).Let us begin by specifying the measure for the sum over metrics more carefully. Thestarting point is as follows. We consider the space of metrics satisfying the required bound-ary conditions itself to be a Riemannian manifold and denote this space as R . A point inthis space is a metric g ab on a manifold with disk topology and boundary of length l . Thetangent space of all metric deformations at any particular point in R , T g R , correspondsto small deformations δg ab . This space is endowed with an ultra local inner product whichtakes the form, (cid:104) δ g, δ g (cid:105) = (cid:90) d x √ gg ac g bd δ g ab δ g cd (2.9)for two deformations, δ g, δ g . The inner product then defines a metric on R and themeasure for summing over different metrics is defined using the volume element whichfollows from this metric.In two dimensions things become especially simple because a general metric, g ab , aftera coordinate transformation can always be locally written in terms of a conformal factor σ as g ab = e σ ˆ g ab (2.10)where ˆ g ab is a fiducial metric. For the disk topology, the manifold can be covered by asingle coordinate chart and we can take ˆ g ab to be a constant negative curvature metric withcurvature ˆ R = − δg ab = δσg ab ⊕ range P ⊕ Ker P † . (2.11)Note that this is an orthogonal decomposition with respect to the inner product, eq.(2.9)In eq.(2.11) δσ is a perturbation in the conformal factor, P is an operator acting on vectorfields, V , as ( P V ) ab = ∇ a V b + ∇ b V a − ∇ · V g ab (2.12)and P † , which is the adjoint of P , acts on traceless metric perturbations as( P † δg ) b = − ∇ a δg ab (2.13)It is well known that the kernel of P † in general corresponds to moduli, which together with– 6 –he conformal factor then determine the metric, upto coordinate transformations. In factEq.(2.11) is the statement that any perturbation around a given metric can be written as acombination of an infinitesimal conformal transformation, an infinitesimal diffeomorphismand an infinitesimal change in the moduli. The Kernel of P † vanishes for the disk since ithas no moduli.The orthogonal decomposition in eq.(2.11 ) means that the measure for summing overmetrics can then be written as Dg ab = D [ σ ] D [ P V ] (2.14)Here D [ σ ] involves the volume element in the space of conformal deformations which arisesfrom the inner product, eq.(2.9). For deformations, δ σ, δ σ eq.(2.9), this takes the form,( δ σ, δ σ ) = (cid:90) d x √ gδ σδ σ (2.15)Similarly the measure D [ P V ] includes the volume element which arises from eq.(2.9) forthe metric perturbations of the form δg ab = ( P V ) ab .We now come to the main new element in this problem. The set of diffeomorphismswe sum over, whose measure we schematically denoted as D [ P V ] above, includes both“small” and “large” diffeomorphisms as mentioned above. Small diffeomorphisms leave theboundary unchanged and roughly speaking “fall off fast enough” towards the boundary.Large diffeomorphisms in contrast do not leave the boundary unchanged, in fact they canbe thought of as modes which describe the fluctuations of the boundary.Before proceeding, let us note that in general the space of vector fields on the disk alsohas a natural inner product given by (cid:104) δ V, δ V (cid:105) = (cid:90) d x √ gg ab δ V a δ V b (2.16)For P † to be the adjoint of P it is easy to see that a boundary term BT = 2 (cid:90) d x ∇ c ( √ gg ac g bd δg ab V d ) (2.17)which arises during the manipulation (cid:104) δg ab , P V (cid:105) = 2 (cid:90) d x ∇ c ( √ gg ac g bd δg ab V d ) + (cid:104) ( P † δg ) a , V b (cid:105) (2.18)must vanish.For the boundary term eq.(2.17) to vanish, the vector field V a must satisfy appropriateboundary conditions. We choose the small diffeomorphisms to correspond to vector fieldswhich satisfy the following two boundary conditions, n a V a = 0 ,t a n b P V ab = 0 (2.19)– 7 –here t a , n b are the tangent and normal vector to the boundary respectively. It is easyto see that the first condition ensures that the boundary remains unchanged and togetherthe two boundary conditions ensure that the the boundary term vanishes for δg = P V in eq.(2.17). Acting on the space of all such small diffeomorphisms P † P is therefore anadjoint operator.As was mentioned above, the small diffeomorphisms, which we have now defined pre-cisely, correspond to the gauge transformations and therefore Vol(Ω), in eq.(2.1) is givenby Vol(Ω) = Vol(sDiffeo) (2.20)where Vol(sDiffeo) denotes the volume of the group generated by the small diffeomorphisms.The additional large diffeomorphisms we would like to include arise from zero modes of P † P . We turn to describing them next. For now then putting together all the informationwe have acquired so far the partition function for the disk topology is given by Z JT = (cid:90) D [ φ ] D [ σ ] D [ P V ]Vol(sDiffeo) e − S JT (2.21)where D [ P V ] together refer to the measure for the sum over the small and large diffeomor-phisms.
Physically, as has already been noted [7], one can think of the large diffeomorphisms asfollows. Consider carrying out the path integral by first fixing a metric, summing overall configurations of the dilaton for this metric, and then summing over all metrics. Asthe dilaton varies, due to the boundary condition that φ = φ B on the boundary, theboundary must also fluctuate. The diffeomorphisms we are including correspond to thesefluctuations of the boundary and they can be thought of as different ways of cutting out asingle connected component, meeting our boundary conditions, from a given disk geometry.In particular, we will consider such diffeomorphisms which preserve the boundary lengthto be l .We will see in the next subsection that on carrying out the path integral for the dilatonfirst, along the contour we choose, we obtain a delta function constraint that localizes themetric path integral to geometries with constant curvature R = −
2. We restrict ourselvesto describing the large diffeomorphisms for such a geometry here.In general, any vector field on the disk can be written in terms of two scalar fields ξ, ψ as V = dξ + ∗ dψ (2.22)For a constant curvature metric with R = − P ,and therefore of P † P , arise from scalars ψ, φ which satisfy the equation ∇ ψ = 2 ψ, ∇ ξ = 2 ξ, (2.23)– 8 –he large diffeomorphisms arise from modes where ξ = 0, with ψ satisfying eq.(2.23).To be more explicit, take the metric for AdS in “polar coordinates” given by, ds = dr ( r −
1) + ( r − dθ (2.24)The θ coordinate is periodic θ ∈ [0 , π ] and can be thought of as the Euclidean timedirection. Solutions to eq.(2.23) in this coordinate system, with ψ ∼ e imθ , which areregular at the origin, r = 1 take the form ψ m = ˆ c m e imθ ( r + | m | ) (cid:18) r − r + 1 (cid:19) | m | (2.25)The modes with m = 0 , , − SL (2 , R )isometries of AdS .For other values | m | > P † P which correspond to the largediffeomorphisms of interest. The corresponding vector field for ψ in eq.(2.25) is given by,for V aL,m = ( V rL,m , V θL,m ) by V aL,m =ˆ c m e imθ (cid:18) r − r + 1 (cid:19) | m | (cid:32) im ( | m | + r ) , − (cid:0) | m | ( | m | + r ) + r − (cid:1) r − (cid:33) (2.26)where the subscript L, m denotes that it is a large diffeomorphism with mode number m .The resulting metric perturbations are δ ˆ g rr = 2 i ˆ c m m (cid:0) m − (cid:1) e imθ ( r − (cid:18) r − r + 1 (cid:19) | m | δ ˆ g rθ = − c m (cid:0) m − (cid:1) | m | e imθ r − (cid:18) r − r + 1 (cid:19) | m | δ ˆ g θθ = − i ˆ c m m (cid:0) m − (cid:1) e imθ (cid:18) r − r + 1 (cid:19) | m | (2.27)Note that these large diffeomorphisms do not satisfy the boundary conditions eq.(2.19) ingeneral.For r → ∞ the vector field takes the form, V aL,m = (cid:18) i ˆ c m mre imθ + O (cid:18) r (cid:19) , − ˆ c m e imθ + O (cid:18) r (cid:19)(cid:19) (2.28)In particular, since the θ coordinate transforms as θ → θ + V θ ; the vector field generatesreparametrizations of the θ coordinate (Euclidean time) in this limit. Also note that the– 9 –etric in the large r limit is given by δ ˆ g rr = 2 i ˆ c m m (cid:0) m − (cid:1) e imθ r + O (cid:18) r (cid:19) δ ˆ g rθ = − c m (cid:0) m − (cid:1) | m | e imθ r + O (cid:18) r (cid:19) δ ˆ g θθ = − i ˆ c m m (cid:0) m − (cid:1) e imθ + O (cid:18) r (cid:19) (2.29)and we see that δ ˆ g θr and the fractional change in the components δ ˆ g rr ˆ g rr , δ ˆ g θθ ˆ g θθ vanish inthis limit. As a result these diffeomorphisms give rise to asymptotic isometries, in thelimit r → ∞ . Note that the requirement that the vector field and the associated metricperturbations be real gives the conditionˆ c − m = ˆ c ∗ m (2.30)The action for the metric perturbations generated by the large diffeomorphisms staysfinite even in the asymptotic limit when the dilaton and total length scaling like eq.(2.8), asis well known and as we will also see below in section 2.5. The physical reason for this is thelow-dimension of spacetime we are working in here and the fact that the diffeomorphisms areasymptotic isometries in this limit. Due to their finite action these large diffeomorphismsneed to be included in the path integral.To be very explicit, for the metric, eq.(2.24), we note that the boundary of length l islocated at r = r B , where (cid:113) r B − l π (2.31)when l (cid:29) r B (cid:39) l π (cid:29)
1. Once a diffeomorphism is turned on we go to newcoordinates ˜ r = r + V r , ˜ θ = θ + V θ , where ( V r , V θ ) is the vector field leading to thediffeomorphism. The boundary will now be located at ˜ r = r B and so r = r B − V r (2.32)at the boundary. For large diffeomorphisms where V · n does not vanish, unlike for smalldiffeomorphism, eq.(2.19), the boundary will change. Let us also note that for the diffeo-morphisms eq.(2.28) with | m | > c m showing that these give rise to length preserving diffeomorphisms at the boundary. Inthe asymptotic AdS limit, eq.(2.7) and (2.8) we have that r B (cid:39) (cid:15) (2.33)and we see that r B → ∞ .More generally, away from the asymptotic AdS limit, when we consider a boundaryof finite length l and finite boundary value of the dilaton φ B , the large diffeomorphismscontinue to be give rise to physically distinct geometries and we need to include these– 10 –odes in the path integral in the general case as well.The resulting measure in the space of small and large diffeomorphisms is actually quitecomplicated in general. This is because the inner product which follows from eq.(2.9) isnot orthogonal between the small and large diffeomorphisms, and as a result the metricin the space of diffeomorphisms has off-diagonal components between the large and smalldiffeomorphisms. In the asymptotic AdS limit though these off-diagonal elements vanish,in a precise manner which we estimate below. As a result the measure simplifies allowingthe path integral to be explicitly carried out. More generally, for fixed φ B , l carrying outthe path integral is more challenging.To estimate how the off-diagonal components in the space of diffeomorphisms vanishesin the limit when r B → ∞ let us first consider the diagonal components of the metric.Starting from eq.(2.12) for two small diffeomorphisms V s , V s meeting boundary conditionseq.(2.19) we get that (cid:104) P V s , P V s (cid:105) = (cid:104) V s , P † P V s (cid:105) . (2.34)The subscript s in the diffeomorphisms is to indicate that it is a small diffeomorphism.Note that the inner product on the left is between two metric deformations, eq.(2.9), andon the right between two vector fields, eq.(2.16). The subscripts s i is to indicate that thevector fields correspond to small diffeomorphisms. On the other hand, the inner productbetween two large diffeomorphisms can be written as a boundary term, since they are zeromodes of P † P . With metric, eq.(2.24) and boundary at r = r B , this takes the form, (cid:104) P V
L,m P V
L,m (cid:105) = δ m , − m π ˆ c m ˆ c − m | m | (cid:0) m − (cid:1) (cid:0)(cid:0) m + r B − (cid:1) + 2 | m | r B (cid:1) r B − (cid:18) r B − r B + 1 (cid:19) | m | (2.35)where V L,m is given in eq.(2.26) and δ m , − m is the Kronecker delta symbol.A general small diffeomorphism can be decomposed in a basis of vector fields goinglike ∼ e imθ which are eigenmodes of − i∂ θ and also eigenmodes of P † P with eigenvalues, λ .Denoting such a basis element as V s,m,λ , the inner product between a large diffeomorphism V L,m and V s,λ, − m can also be expressed as a boundary term and is given by (cid:104) P V s,λ, − m , P V L,m (cid:105) = √ γ n r V θs,λ, − m ( P V
L,m ) θr (cid:12)(cid:12) ∂ (2.36)where √ γ = (cid:113) r B − n r is the unit normal and | ∂ denotesthe boundary values at r = r B . It is easy to see that the expression above does not vanishin general.As discussed in the appendix E in the asymptotic AdS limit, when r B → ∞ , one getsthat the ratio (cid:104) P V s,λ, − m , P V L,m (cid:105) (cid:112) (cid:104)
P V s,λ, − m , P V s,λ,m (cid:105)(cid:104) P V
L,m , P V L, − m (cid:105) ∼ r / B (2.37)It is this ratio which determines the importance of the off-diagonal terms compared to thediagonal ones in the volume element for the sum over all diffeomorphisms. Since it vanisheswe learn that the off-diagonal terms can be neglected when r B → ∞ and the measure for– 11 –he small and large diffeomorphisms decouple.We learn that (cid:90) D [ P V ]Vol(sDiffeo) = (cid:90) D [ P V s ] D [ P V L ]Vol(sDiffeo) = (cid:90) (cid:113) det (cid:48) ( P † P ) D [ P V L ] . (2.38)where in the last equality we have carried out the integral over the small diffeomorphismssince the action is independent of them. The prime in det (cid:48) ( P † P ) indicates that the zeromodes have been removed. In fact, for the disk, with the boundary conditions, eq.(2.19),there are no zero modes, which we we explain in appendix D .The path integral in the asymptotic AdS limit then becomes, Z JT = (cid:90) D [ φ ] D [ σ ] D [ P V L ] (cid:113) det (cid:48) ( P † P ) e − S JT (2.39)From eq.(2.35) we also learn that when r B → ∞(cid:104) P V
L,m , P V L, − m (cid:105) = 8 π ˆ c m ˆ c − m | m | (cid:0) m − (cid:1) (2.40)leading to the measure D [ P V L ] = (cid:89) m ≥ d ˆ c m d ˆ c ∗ m π | m | (cid:0) m − (cid:1) . (2.41)where we used eq.(2.30) in obtaining the above form of the measure.Before closing this subsection let us introduce a variable u which is the rescaled properlength along the boundary. Using a small diffeomorphism, we can bring the boundary lineelement to the form, ds (cid:12)(cid:12)(cid:12)(cid:12) ∂ = du (cid:15) , u ∈ [0 , π ] (2.42)For the metric eq.(2.24), when the boundary length l → ∞ , we have near the boundary,where r (cid:29) ds = du (cid:15) (cid:39) dr r + r dθ (2.43)Using the form of the vectors fields specifying a large diffeomorphism V rL , V θL in eq.(2.28)it is easy to then show that u = ˜ θ . For infinitesimal transformations we can therefore write V rL = − rδθ (cid:48) ( u ) , V θL = δθ ( u ) (2.44)where δθ ( u ) = − (cid:88) | m | > ˆ c m e imθ ( u ) (2.45)On comparing we find that the measure obtained in eq.(5.40) agrees with that whicharises in the first order formalism as discussed in [3] and [13]. Let us also note, as wasmentioned above, that the θ direction can be thought of as the Euclidean time direction.From eq.(2.44) we also see that large diffeomorphisms act as reparametrizations of θ , when– 12 – → ∞ . For this reason we will sometimes refer to the large diffeomorphisms as timereparametrizations below. Having described the diffeomorphisms, small and large, in some detail, and the measurefor summing over them we now come back to a systematic evaluation of the path integral.Our starting point is eq.(2.21). We will first carry out the integral over the dilaton, thenover the conformal factor σ and finally turn to the sum over the diffeomorphisms. Also,to begin here we will consider the case of a general boundary of length l with the dilatontaking a value φ = φ B on the boundary, and consider the asymptotic AdS limit as a specialcase in subsection 2.5.We carry out the dilaton path integral in the background of a general metric g ab givenby eq.(2.10). To begin we write a general dilaton field as φ = φ ( r ) + δφ ( r, θ ) (2.46)where φ ( r ) is a fixed function of r given by φ ( r ) = Ar (2.47)The constant A is fixed to take the value A = φ B /r B (2.48)so that for the metric in eq.(2.24) the locus r = r B also has the required boundary valueof the dilaton φ = φ B with δφ = 0.Let us note in passing that φ ( r ) satisfies the classical equations of motion in JTgravity.The path integral for the dilaton requires us to sum over various values for δφ . Tocarry out this sum we adopt the prescription, now being commonly adopted, [3], of rotatingthe contour for δφ to lie along the imaginary axis [ − i ∞ , i ∞ ] so that after taking δφ → iδφ , δφ has the range [ −∞ , ∞ ]. Next we expand δφ into eigenmodes of the operator ( ˆ ∇ − ∇ is the scalar laplacian obtained from a metric ˆ g ab of constant negative curvatureˆ R = −
2. We require that δφ vanishes at the boundary (at r = r B ) so that the dilaton,eq.(2.46) takes the value φ B at the boundary as required.With this prescription the path integral over the dilaton gives (cid:90) D [ φ ] e − S JT = δ ( R [ σ ] + 2) e − S JT,∂ (2.49)where R [ σ ] = − e − σ (2 + 2 ˆ ∇ σ ) (2.50)is the Ricci scalar which arises from the metric eq.(2.10), δ ( R [ σ ] + 2), denotes a deltafunction which has non-trivial support only when the curvature R [ σ ] = −
2, and S JT,∂ is– 13 –he boundary part of the
J T action, eq.(2.2), which with φ = φ B takes the form, S JT,∂ = − πG (cid:90) ∂ dx √ γφ B ( K −
1) (2.51)After carrying out the dilaton path integral the partition function therefore takes the form, Z JT = (cid:90) D [ σ ] D [ P V ]Vol(sdiffeo) δ ( − e − σ (2 + 2 ˆ ∇ σ ) + 2) e − S JT,∂ (2.52)Let us note before proceeding that one could have considered another contour for doingthe dilaton integral. In fact from the higher dimensional point of view it is perhaps morenatural to consider a contour where the dilaton is real with a range [ − Φ , ∞ ], where Φ is the prefactor of the topological term, eq.(2.3), since this ensures that the volume of theinternal space does not become negative. We will not have anything further to say aboutsuch a contour here and leave it for future consideration.We have glossed over one subtlety above. The path integral as we mentioned at thebeginning is being done for a general metric of the form eq.(2.10) and the measure for thedilaton integral will therefore depend on σ the conformal factor. This measure arises froman inner product which, to begin with for two dilaton perturbations δ φ, δ φ , takes theform (cid:104) δ φ, δ φ (cid:105) = (cid:90) d x √ gδ φδ φ (2.53)and g ab is the metric including the conformal factor. The dependence of the measure on theconformal factor is the same as for a scalar field satisfying Dirichlet boundary conditions,and can be obtained from the conformal anomaly, as is discussed in appendix F. Since, aswe will see shortly below, satisfying the delta function in eq.(2.52) results in setting σ = 0,this dependence results at best in a constant multiplying the partition function. We cantherefore ignore this subtlety, since we are not keeping track of the overall multiplicativeconstant in the partition function. Next, we turn to the path integral over the Liouville mode. The delta function we obtainedin eq.(2.52) makes this easy. The argument of the delta function manifestly vanishes when σ = 0. Linearizing around it we get δ ( e − σ ( − − ∇ σ ) + 2) = δ (( − ∇ − δσ ) (2.54)It is easy to see that in conformal gauge the action, eq.(2.2), gives rise to a well-definedvariational principle with δσ vanishing on the boundary (where φ b , l are fixed). This is truebecause we have included the Gibbons-Hawking boundary term in the action. As a resultwe will sum over all Liouville mode fluctuations in the path integral subject to the conditionthat δσ vanishes on the boundary.Carrying out the integral over the non-zero modes of ( ˆ ∇ −
2) then gives rise to afactor of (det( − ˆ ∇ + 2)) − in the partition function. The zero mode is fixed by requiring– 14 –hat the boundary has length l .We note that, as for the dilaton above, the measure D [ σ ] in general has a non-trivialdependence on σ , since the inner product eq.(2.15) involves the full metric, eq.(2.10).However, again, this dependence which is the same as for a massless scalar and can beobtained from the conformal anomaly only gives rise to an overall multiplicative factor inthe partition function once we set σ = 0. The resulting determinant det( − ˆ ∇ + 2) thenonly depends on the metric ˆ g ab .Putting all this together gives Z JT = (cid:90) D [ P V ]Vol(sdiffeo) 1det( − ˆ ∇ + 2) e − S JT,∂ (2.55)Note that the determinant which appears above depends on the metric ˆ g ab and thereforeon the large diffeomorphisms. So far we have been considering the general case of a boundary of finite length. Thereare two complications in going further with the evaluation of the partition function whenthe boundary has a finite length, even for the case when the length is large. First, aswas discussed in section 2.2 the metric perturbations generated by the small and largediffeomorphisms are not orthogonal and as a result the measure for summing over them isquite complicated to obtain. Second, the dependence of the the determinant det( − ˆ ∇ + 2)on large diffeomorphisms is also not easy to obtain.To proceed, we will therefore take the asymptotic AdS limit where we take the lengthto go to infinity, while also taking the dilaton to diverge at the boundary, as given in eq.(2.7)and (2.8). Actually this limit has to be defined more precisely in the path integral wherewe are dealing determinants of various operators. These determinant are formally infiniteand need to be regulated. We will regulate the determinants by first keeping only thecontributions of eigenmodes with finite eigenvalues, take the asymptotic AdS limit, where l → ∞ , eq.(2.8), and then finally take the cut-off on the eigenvalues to go to infinity. Thisorder of limits is part of our definition of the asymptotic AdS limit. It will be responsiblefor some of the simplification which occurs.In the more general case where l is finite we need to regulate the determinants andthen take the cut-off introduced for regulating the determinants to zero keeping l fixed.This makes the evaluation of the determinants more complicated. To explain some of theresulting complications consider evaluating the determinant det( − ˆ ∇ + 2). We cannot useconformal invariance for evaluating this determinant, unlike det( − ˆ ∇ ) which arises for amassless scalar for which at least some information can be obtained, as we will see inthe next subsection. A direct evaluation of det( − ˆ ∇ + 2) is also not easy. For example,consider evaluating this determinant in the metric eq.(2.24). The eigenmodes of ( − ˆ ∇ + 2)can also be simultaneously chosen to be eigenmodes of ∂ θ . Denoting these modes by φ λ,m , Here we are thinking of the cut-off in the position space. In the momentum space, the cut-off would betaken to infinity. – 15 –e have that φ λ,m ∼ e imθ and ( ˆ ∇ − φ λ,m = − λφ λ,m . In the asymptotic AdS limit, asmentioned above, we first take the the boundary r B → ∞ , keeping m, λ fixed, and thentake m, λ → ∞ . This means that we are including modes whose wavelength along the θ direction Λ = r B /m (cid:29)
1. Reinstating the radius of AdS, R AdS , in this relation we see thatin the asymptotic AdS limit we are only including modes withΛ (cid:29) R AdS . (2.56)For such modes the asymptotic form for φ λ,m can be used and this considerably simplifiesthe analysis. One can then show that the determinant in this limit is independent of thelarge diffeomorphisms as discussed in appendix F.In the more general case when the length is finite, there are modes with m ≥ r B whosewavelength Λ ≤ R AdS (2.57)and the contributions of these modes also need to be included. This is harder to do since weneed to include terms with arbitrary number of derivatives beyond the Schwarzian term.For example these can arise in the action due to the expansion of the extrinsic trace toobtain the analog of eq.(B.38) in the Euclidean AdS disk.Similarly, there are high wavenumber modes, for both the large and small diffeomor-phisms, also with Λ < R
AdS whose contribution would need to be included at any finitevalue of l . This is again complicated due to two reasons. First, our estimate that theratio in eq.(2.37), for the inner product of normalized small and large diffeomorphisms issuppressed at large r B is valid only for modes with fixed mode number m , as r B → ∞ , asdiscussed further in appendix E. And the mixing between large and small diffeomorphismsdiscussed in subsection 2.2 above therefore does not vanish for modes with wavelengthΛ < R AdS . Second, because obtaining the contribution due to such modes, even after ne-glecting this mixing, is not straightforward, since terms beyond the Schwarzian derivativefor the large diffeomorphisms would need to be included, for example for the determinantdet (cid:48) ( P † P ) which arises from the small diffeomorphisms, see discussion above eq.(F.1).As a toy model in appendix H we show how these high wavenumber modes couldpotentially have a significant effect on the behaviour of the partition function when l , theboundary length becomes very big. In the cosmological context which we study beloweq.(2.57) is replaced by Λ ≤ H − (2.58)where H is the Hubble constant, and this condition therefore corresponds to modes whichhave not yet exited the horizon and “frozen out”. Our analysis shows that such modes cansignificantly affect the wavefunction. For all these reasons, here after in this subsection we only consider the asymptotic AdSlimit. Since the small and large diffeomorphisms become orthogonal in this limit the path– 16 –ntegral eq.(2.55) gives Z JT = (cid:90) D [ P V L ] (cid:112) det (cid:48) ( P † P )det( − ˆ ∇ + 2) e − S JT,∂ (2.59)The measure for the large diffeomorphisms is given in eq.(5.40). The prime in det (cid:48) ( P † P ) isto indicate that the zero modes in the space of small diffeomorphisms of the operator P † P are to be excluded in calculating the determinant. There is in fact one zero mode for theoperator P † P satisfying the boundary conditions eq.(2.19). More discussion on this zeromode is contained in the appendix F.1 above eq.(F.41).In the asymptotic AdS limit as we have defined it above one can show that bothdet (cid:48) ( P † P ) and det( − ˆ ∇ + 2) become independent of the large diffeomorphisms as is dis-cussed in appendix F. Upto a multiplicative constant which we are not keeping track of wethen get Z JT = (cid:90) D [ P V L ] e − S JT,∂ (2.60)This final expression agrees completely with what has been obtained from the first orderformalism, [3].For completeness let us carry out the remaining integral over V L here. As is wellknown fact the boundary action S JT,∂ gives rise to the Schwarzian term involving thetime reparametrization generated by the large diffeomorphism as follows. As discussed inappendix B the extrinsic curvature for a general boundary curve specified as ( r ( u ) , θ ( u ))is given by K = r ( u ) − (cid:15) r (cid:48)(cid:48) ( u )( r ( u ) − (cid:15)θ (cid:48) ( u ) (2.61)where r ( u ) is the radial coordinate along the boundary as a function of proper time u andprime denotes a derivative with respect to u . From eq.(2.43) we get that upto correctionssub-leading in r B dθdu = 1 (cid:15) r ( u ) (2.62)where r ( u ) is given in eq.(2.32) with V rL ( u ) = ir (cid:88) m ˆ c m me imθ ( u ) (2.63)Substituting gives, K = 1 + (cid:0) − θ (cid:48)(cid:48) + θ (cid:48) + 2 θ (3) θ (cid:48) (cid:1) θ (cid:48) (cid:15) + O (cid:0) (cid:15) (cid:1) (2.64)– 17 –he net result is the path integral eq.(2.60) with action S JT,∂ = − φ B (cid:15) πG (cid:90) π du Sch (cid:18) tan (cid:18) θ ( u )2 (cid:19) , u (cid:19) = − π GJ β (cid:88) m ≥ ( m − m )ˆ c m ˆ c − m (2.65)and measure eq.(5.40). This agrees with the result obtained earlier. In particular notethat the measure we have obtained from the second order formalism above agrees with themeasure obtained in [13] see also [3].The integral over the modes ˆ c m is in fact one-loop exact [13]. Using the measureeq.(5.40), the action eq.(2.65) and noting eq.(2.30), we have the path integral as Z JT = e π (4 GJβ ) − (cid:90) (cid:89) m ≥ dc m dc ∗ m π | m | ( m −
1) exp π GJ β (cid:88) m ≥ ( m − m )ˆ c m ˆ c ∗ m = e π (4 GJβ ) − (cid:89) m ≥ πGJ βm = e π (4 GJβ ) − (32 πGJ β ) − / √ π (2.66)Adding the topological term eq.(2.3) for completeness gives the partition function inthe asymptotic AdS limit to be Z JT = exp (cid:20) Φ G + π GJ β (cid:21) (32 πGJ β ) − / √ π (2.67)Let us conclude this section with a remark. As mentioned above for the boundarylocated at a finite value for the length l with dilaton taking value φ B , the path integralwe have defined is still quite explicit, eq.(2.21), but much harder to fully evaluate. This istrue even when the boundary length l (cid:29)
1, which one might expect is simpler than thatof the general case. We hope to return to this issue and also to the analogous one in dSJT gravity, where it is related to computing the wavefunction at late but finite time afterincluding modes which have not yet exited the horizon, in the future.
We shall next extend the analysis of the previous section to include additional scalarmassless matter fields. The path integral is given by Z JT + M = (cid:90) D [ φ ] D [ g µν ]Vol(sdiffeo) (cid:32) N (cid:89) i =1 D [ ϕ i ] (cid:33) exp {− S JT − S M } (3.1)– 18 –here S JT is the same action for the JT theory as before, the measure Dφ, Dg µν and thevolume of small diffeomorphisms, Vol(sdiffeo) are the same as above and S M is the actionfor the minimally coupled massless scalar fields, ϕ i , given by S M = N (cid:88) i =1 (cid:90) d x √ g ( ∂ϕ i ) (3.2)where N is the number of matter fields. As can be seen from the above action, the matterfields do not directly couple to the dilaton. We will carry out the path integral for fixedboundary values for the scalar fields, ϕ i (cid:12)(cid:12) ∂ = ˆ ϕ i ( u ) (3.3)with u being, upto a multiplicative constant, the proper length along the boundary,eq.(2.42). The resulting partition function is a functional of ˆ ϕ i ( u ), besides being a functionof the length l and the boundary value of the dilaton φ B , as before. We discuss the generalcase of finite l, φ B first and then turn to the asymptotic AdS limit below.Working in conformal gauge we can carry out the integral over the dilaton and theLiouville modes. Since the matter fields do not couple to the dilaton directly the dilatonintegral will localize the path integral to constant negative curvature metrics as before andallow us to set the the Liouville mode σ in eq.(2.10) to vanish. After the Liouville modeintegral is done we are then left with the integral over diffeomorphisms and the matterfields, giving Z JT + M = (cid:90) D [ P V ] (cid:16)(cid:81) Ni =1 D [ ϕ i ] (cid:17) Vol(sdiffeo) 1det( − ˆ ∇ + 2) e − S JT,∂ − S M (3.4)The measure for the scalar fields in this path integral is to be evaluated using a metric ˆ g ab ,with curvature ˆ R = −
2. This measure follows from the standard ultra local inner productfor two scalar perturbations given by( δ ϕ i , δ ϕ i ) = (cid:90) d x (cid:112) ˆ g δ ϕ i δ ϕ i (3.5)Thus the background geometry for the scalar path integral is hyperbolic space with aboundary determined by the large diffeomorphisms.To perform the path integral over the fields ϕ i , we first expand it around the classicalsolution obtained by solving the scalar laplacian equation with the boundary conditionspecified by eq.(3.3) and also demanding that the solution is regular everywhere in theinterior.Let us denote the resulting solution to be ϕ (0) i ( r, θ ). Expanding the fields ϕ i aroundthis solution as ϕ i = ϕ (0) i + δϕ i (3.6)– 19 –he boundary condition eq.(3.3) translates to the Dirichlet condition, δϕ i | ∂ = 0 (3.7)We can write the path integral for the matter fields then as Z M = (cid:90) (cid:32) N (cid:89) i =1 D [ ϕ i ] (cid:33) e − S M = e − S M,cl (cid:90) (cid:32) N (cid:89) i =1 D [ δϕ i ] e (cid:82) d x √ ˆ g ˆ g ab ∂ a δϕ i ∂ b δϕ i (cid:33) (3.8)where S M,cl , the classical contribution resulting from ϕ (0) i , is given after using the equationsof motion, by a boundary term, S M,cl = (cid:88) i (cid:90) ∂ ds √ γϕ (0) i ∂ n ϕ (0) i (3.9)with ∂ n being the normal derivative at the boundary. Note that the Laplacian ˆ ∇ has nozero modes for the Dirichlet boundary conditions satisfied by δϕ i . Thus the path integralover δϕ i is straightforward and gives, Z M = (cid:90) (cid:32) N (cid:89) i =1 D [ δϕ i ] (cid:33) e − S M = e − S M,cl (cid:16) det (cid:16) − ˆ ∇ (cid:17)(cid:17) N (3.10)In much of the discussion later in this paper, we will drop the factor of that appearsin the determinant in the above expression as it will only change the overall numericalcoefficient of the path integral which we are not keeping track of. Both S M,cl and thedeterminant on the RHS depend on the large diffeomorphisms. This dependence is noteasy to obtain, for the general case of a finite boundary of length l , as is discussed inappendix F. The reason, related to the discussion towards the end of the subsection 2.5, isthe presence of high wave number modes with wavelength less than the boundary length,Λ < R AdS . The Schwarzian action is no longer sufficient to describe the dependence onthe large diffeomorphisms for such modes. In addition, as is also discussed in subsection2.2, the subsequent step involving the integral over the diffeomorphisms is also not easy tocarry out in this case.Keeping these points in mind we again restrict ourselves to the asymptotic AdS limit forthe subsequent evaluation of the path integral. As discussed in appendix F the dependenceof the large diffeomorphisms in det( − ˆ ∇ ) vanishes in the asymptotic AdS limit when r B →∞ , after a suitable length dependent counter term is added. If the boundary values of thescalars ϕ (0) i vanish the path integral is therefore unchanged (upto an overall temperatureindependent prefactor) by the presence of the matter in the asymptotic AdS limit. And thethermodynamics essentially does not change, other than a possible change in the groundstate entropy.When the boundary values ϕ (0) i are non-zero, the matter sector does couple to thelarge diffeomorphisms. In appendix G, eq.(G.13), we show that the dependence of S M,cl is– 20 –iven, for r B → ∞ , by S M,cl = 12 (cid:90) du du θ (cid:48) ( u ) θ (cid:48) ( u ) (cid:88) i ˆ ϕ i ( u ) ˆ ϕ i ( u ) F ( θ ( u ) , θ ( u )) (3.11)Here θ ( u ) specifies the time reparametrization as a function of the boundary proper length u , and θ (cid:48) = dθdu . ˆ ϕ i ( u ) is the value of the scalars along the boundary and the function F isdefined in eq.(G.7). At linear order in δθ ( u ) this becomes, S M,cl = (cid:90) ∂ du du ˆ ϕ ( u ) ˆ ϕ ( u )( δθ (cid:48) ( u ) F ( u , u ) + δθ ( u ) ∂ u F ( u , u )) (3.12)Upto an overall constant the path integral then takes the form, Z JT + M = (cid:90) D [ P V L ] e − S JT,∂ e − S M,cl (3.13)The measure D [ P V L ] is defined in eq.(5.40), and the action S JT,∂ involves the Schwarzianderivative of θ ( u ), eq.(2.65). The path integral can then be done by integrating out thelarge diffeomorphisms perturbatively, including the self interactions from the Schwarzianterm and the interactions with the matter fields, to obtain Z JT + M as a function of theboundary values of the scalar fields ˆ ϕ i ( u ), β and J . We will not go into the details here.These calculations are also discussed in [7]. Let us end this section with some comments. We have seen that the path integral at finitevalues of the boundary length l is difficult to calculate even when l (cid:29)
1. Some of thereasons for this were mentioned above. On the other hand the quantum effects of mattervanish when l → ∞ since the matter determinant does not couple to large diffeomorphismsin this limit anymore as was also mentioned above.One way to obtain a tractable situation where quantum effects due to matter can beincorporated is to consider a semi-classical limit by taking G , the gravitational constantwhich appears in front of the JT action eq.(2.2), to vanish, G →
0, with the number ofscalar fields, N → ∞ keeping GN fixed, [18]. In this limit the measure for the diffeo-morphisms is not important since gravity is classical and quantum fluctuations over thesediffeomorphisms can be ignored, similarly the dependence of det( − ˆ ∇ + 2) on the largediffeomorphisms can be neglected. However the quantum effects of matter remain. Thislimit has received considerable attention recently, [5, 8, 18, 27]. The saddle point equationsin this limit for the system we are considering were obtained in [18]. It was found thatthey can typically be solved only in slowly varying situations where the excited modes havewavelengths Λ (cid:29) R AdS as discussed in [18].More generally one could consider a system away from the semi-classical limit, with afinite number of matter fields, where we are interested in the response to slowly varyingsources provided for example by the boundary values ˆ ϕ (0) i . The sources are varying at– 21 –avelength Λ (cid:29) R AdS . (3.14)In this case one can consider constructing a Wilsonian effective action which will containthe sources coupled to the large diffeomorphisms by integrating out the other degrees offreedom. The determinants which arise must be valued in Diff( S ) /SL (2 , R ) and can beexpanded in a derivative expansion. The leading term in this expansion which dependson the large diffeomorphisms is the Schwarzian term, other terms involve more derivativesand would be suppressed when eq.(3.14) is met.The resulting effective action, after adding a suitable counter term to cancel a boundarylength dependent term, is then given by Z JT + M [ β, J, ˆ ϕ i ( u )] = (cid:90) D [ P V L ] e − S (3.15)where the action, see appendix F, is S = (cid:15) πG C (cid:90) du Sch(tan( θ ( u ) / , u ) + S M,cl . (3.16) S M,cl above arises from the quadratic action of the scalar fields and depends on theboundary values ˆ ϕ i ( u ) and time reparametrization θ ( u ); to leading order it is is givenin eq.(3.11), a correction at O ( (cid:15) ) can also be similarly obtained. The measure in eq.(3.15)is the Diff( S ) /SL (2 , R ) invariant measure given in eq.(5.40) above. The coefficient C infront of the Schwarzian action to begin with, before the short wavelength modes have beenintegrated out, is given by C = − φ B + G ( N −
26 + q ) . (3.17)The first term on the RHS, φ B , is from the classical JT action. The matter fields contributethe second term, GN , eq.(F.16); the third term, − G , comes from det (cid:48) ( P † P ), eq.(F.23)and the term with coefficient q which we have not been able to determine and shouldbe of order unity, arises from det( − ˆ ∇ + 2), see eq.(F.24). The factor of (cid:15) multiplyingthe Schwarzian shows that the effect of the matter determinant etc vanishes when (cid:15) → C will be renormalized from this starting value though, once weintegrate out the short wavelength modes.This Wilsonian effective action can then be used for calculating the long-wavelengthproperties of the system, including computing loop effects from modes meeting eq.(3.14).If necessary a renormalization procedure can be carried out to make such calculationsoptimal. We leave a further analysis along these lines for the future. In this section we will extend our discussion to consider the path integral over connectedgeometries with two boundaries in Euclidean AdS. These spaces have Euler character χ = 0.Such a spacetime is often referred to as the double trumpet geometry. The path integral,– 22 –enoted as Z DT reads Z DT = (cid:90) D [ φ ] D [ g µν ]Vol(Ω) exp {− S JT } (4.1)The action for the path integral is given as in eq.(2.2) above, with boundary terms atboth boundaries. Note that in this case the boundary contribution to the path integralwill arise from both boundaries. The non-trivial part of the calculation, like for the disk,is to correctly identify the metric configurations which need to be included in the pathintegral and obtain a measure for summing over them. We discuss this issue first here andthereafter in the next subsection will carry out the path integral in a systematic manner,analogous to section 2, by first summing over the dilaton, the conformal mode and thenthe diffeomorphisms.The boundary conditions we impose are that φ takes values φ B, , φ B, , at the twoboundaries which are taken to have lengths l , l respectively. It will be convenient tokeep in mind the following background metric for the double trumpet, which has curvature R = −
2, with two boundaries, at r → ∞ , r → −∞ (henceforth also referred to right andleft boundaries respectively): ds = dr r + 1 + ( r + 1) dθ , θ ∼ θ + b (4.2)Note that this metric can be obtained from eq.(2.24) by the analytic continuation, r → − ir, θ → iθ, (4.3)however now the periodicity of θ is a free parameter b . This parameter actually correspondsto a modulus and we will integrate over it in the path integral, as we will see below.While we first set up the path integral with general boundary conditions, as in the caseof the disk, it will turn out to be difficult to carry out the calculations all the way throughin this general case. As a result, at some point in the discussion below we will specialize tothe asymptotic AdS limit. This limit is defined for two boundaries by taking the length ofboth boundaries to go to infinity and also taking φ B → ∞ , while keeping the ratio φ B /l tobe fixed at each boundary. The ratio φ B /l takes an independent value at both boundaries,i.e., we take (cid:15) → φ B, = 2 πJ β (cid:15) , l = 2 π(cid:15) φ B, = 2 πJ β (cid:15) , l = 2 π(cid:15) (4.4)The resulting answer will then depend on both the parameters Jβ and Jβ . The asymptoticlimit corresponds to taking (cid:15) , (cid:15) → (cid:15) , (cid:15) we cantake the renormalized lengths of the two boundaries in the double trumpet to be β , β – 23 –o that the path integral for the double trumpet geometry can be interpreted as giving acontribution to the connected two point function of the partition functions (cid:104) Z ( β ) Z ( β ) (cid:105) for two boundary theories. We will have more to say about this interpretation in section4.2 below.In general metric perturbations about a metric g ab can be decomposed, similar toeq.(2.11) in subsection 2.1, as δg ab = δσg ab ⊕ δ ˜ g ab (4.5)Here δ ˜ g ab are traceless metric perturbations which will include, for the double trumpet,perturbations produced by small diffeomorphisms, large diffeomorphisms and moduli. Wedescribe all three types of perturbations below.Perturbations produced by small diffeomorphisms are generated by vector fields V s andof the form δ ˜ g ab = P V s (4.6)where the operator P is given in eq.(2.12) . The vector fields V s satisfy the boundary condi-tions, eq.(2.19) at both the boundaries. These perturbations describe the same spacetimeafter a coordinate transformation and therefore give rise to gauge transformations. Theirvolume Vol(sdiffeo) is the factor Vol(Ω) in the denominator of eq.(4.1).The perturbations produced by large diffeomorphisms describe fluctuations of bothboundaries in double trumpet case. They correspond, as in the disk, to zero modes ofthe operator P † P , with the operator P † as defined in eq.(2.13) acting on traceless metricperturbations. Denoting a vector field which generates such a transformation by V L thecondition V L satisfies is P † P V L = 0 (4.7)and the metric perturbation it produces is δ ˜ g ab = ( P V L ) ab . Taking a cue from the diskcase, in identifying these we look for diffeomorphisms which reduce to reparametrizationsof the two boundary circles, in the limit where both boundaries have large length. Suchdiffeomorphisms can be obtained by setting V L = ∗ dψ (4.8)with ψ being a scalar field satisfying the eq.(2.23) .Solutions to this equation for a background metric eq.(4.2) can be obtained fromeq.(2.25) after noting the analytic continuation eq.(4.3) and are given by ψ ( r, θ ) = (cid:88) | m | > e i ˜ mθ (cid:32) A m ( r + ˜ m ) (cid:18) r + ir − i (cid:19) i ˜ m + B m ( r − ˜ m ) (cid:18) r + ir − i (cid:19) − i ˜ m (cid:33) (4.9)where ˜ m = 2 πmb (4.10)Note that we have twice the number of solutions compared to the disk case, since there are– 24 –wo modes for every value of m . And unlike the disk, there is no condition of regularity inthe interior which cuts down the number of solutions since the coordinate system in whichthe metric in eq.(4.2) is written is non-singular everywhere. We should also mention thatthe sum in eq.(4.9) does not include an m = 0 mode. We will have more to say about thissector shortly.Before proceeding let us note that the solution in eq.(4.9) has functions involving ( r + i )and ( r − i ) raised to various powers. These are defined, for a general exponent a , as follows( r + i ) ia = exp[ ia log( r + i )] , ( r − i ) ia = exp[ ia log( r − i )] (4.11)with the log function in both cases being defined to have a branch cut along the negativereal axis, i.e., ln z = ln | z | + i Arg( z ) , Arg( z ) ∈ [ − π, π ] (4.12)Also in our definition, (cid:18) r + ir − i (cid:19) i ˜ m = ( r + i ) i ˜ m ( r − i ) i ˜ m = exp (cid:20) i (cid:18) ˜ m r + i ) − ˜ m r − i ) (cid:19)(cid:21) = exp (cid:20) − ˜ m r + i ) − Arg( r − i )) (cid:21) (4.13)With these definitions we see that the reality of ψ imposes the condition A ∗− m = B m (4.14)on the coefficients in eq.(4.9).We will sometimes find it convenient to work with linear combinations of the basiselements used in the expansion in eq.(4.9), and rewrite ψ as follows, ψ ( r, θ ) = (cid:88) | m | > e i ˜ mθ γ m (cid:32) ( r + ˜ m ) (cid:18) r + ir − i (cid:19) i ˜ m − ( r − ˜ m ) (cid:18) r + ir − i (cid:19) − i ˜ m (cid:33) + (cid:88) | m | > e i ˜ mθ δ m (cid:32) ( r + ˜ m ) (cid:18) r + ir − i (cid:19) i ˜ m e ˜ mπ − ( r − ˜ m ) (cid:18) r + ir − i (cid:19) − i ˜ m e − ˜ mπ (cid:33) (4.15)The coefficients γ m , δ m are related to A m , B m by γ m + e ˜ mπ δ m = A m , γ m + e − ˜ mπ δ m = − B m . (4.16)It is easy to see that the radial functions multiplying γ m , δ m vanish respectively as r → ∞ and r → −∞ respectively, to leading order . In this basis it is therefore manifestly clear thatthe expansion eq.(4.9) includes independent perturbations at the two ends. Using eq.(4.8)it is also easy to see that the resulting diffeomorphisms become independent reparametriza-tions of the θ direction at r ± ∞ .We now turn to the m = 0 sector. There are two solutions to eq.(2.23) in this sector,– 25 –hese are independent of θ and are given by ψ = s r (4.17) ψ = t π (cid:18) ir log (cid:18) r + ir − i (cid:19)(cid:19) (4.18)where t, s are arbitrary coefficients. It is easy to see that ψ corresponds to the U (1)isometry under which θ → θ − s . Since it keeps the metric and boundary unchanged itdoes not correspond a distinct spacetime, and we must not sum over it in the path integral.On the other hand ψ give rises to the diffeomorphism V tw = ∗ dψ . The subscript m is toindicate that it is a modulus and tw is to denote that this vector field introduces a relativetwist between the two boundaries. It is easy to see that V rtw = 0 and V θtw = t π (cid:20) − rr + 1 − i log (cid:18) r + ir − i (cid:19)(cid:21) (4.19)As r → ∞ , V θtw →
0, while as r → −∞ , V θtw → t . Thus this diffeomorphism produces arelative twist between the θ variables parametrizing the circles at the two boundaries inthe limit when the boundaries have length l → ∞ . It is in fact one of the two moduliassociated with this geometry. The corresponding metric perturbation is given by( P V tw ) µν = − tπ ( r + 1) (cid:32) (cid:33) (4.20)The other modulus for this space is related to the parameter b which is the size of the θ circle, eq.(4.2). Consider a vector field V µb = δbb ( − r, θ ) (4.21)under which θ → θ + δbb θ , so that the periodicity of θ changes. The subscript b is to denotethat this is a vector field corresponding to the modulus parameter b . This vector field is notsingle valued on the circle, however the metric perturbation it gives rise to, is well-definedand single valued, ( P V b ) µν = 2 δbb (cid:32) − r )
00 1 (cid:33) . (4.22)We will also include this metric perturbation in the sum over all configurations in the pathintegral. Note that both V b , V tw satisfy P † P V b = 0 = P † P V tw (4.23)To summarize the discussion so far then we have argued that general metric deforma-tions which we sum over include changes in the conformal factor and deformations asso-ciated with small diffeomorphisms which preserve the boundaries, large diffeomorphisms– 26 –hich changes the boundaries, and two moduli.Let us now turn to describe the measure in the space of all metric deformations.This measure arises from the inner product in the space of metric perturbations, eq.(2.9)as described in subsection 2. The decomposition in eq.(4.5) is an orthogonal one withrespect to the inner product eq.(2.9). Thus the measure in eq.(2.1) can be written as D [ g ab ] = D [ σ ] D [˜ g ab ] where D [ σ ] is the measure for the sum over conformal factors followingfrom the inner product, eq.(2.15).As discussed in appendix E in general the large and small diffeomorphisms included in D [˜ g ab ] are not orthogonal to each other and the resulting measure is hard to obtain, evenwhen the lengths l , l of the two boundaries are large but finite. This is because modes ofwave number ˜ m coming from small and large diffeomorphisms, meeting the condition˜ m/l ≥ O (1) (4.24)can mix with each other. The resulting complications for the disk topology are discussedin subsection 2.5 and there are similar issues in the double trumpet as well, see discussionafter eq.(E.40).In order to avoid these complications we will therefore finally have to resort to theasymptotic AdS limit as described above. In this limit there are no modes which meetthe condition eq.(4.24) (since the l → ∞ limit is taken while keeping the mode number ˜ m fixed). The small and large diffeomorphisms, and moduli are all orthogonal to each otherin this limit, see discussion after eq.(I.22), and the measure then splits up into a product D [˜ g ab ] = D [ P V s ] D [ P V L ] D [ P V mod ] (4.25)where the three terms on the RHS denote the measures for summing over the small andlarge diffeomorphisms and the two moduli with V mod denoting V tw , V b .We will describe these three measures in more detail next. The definition of D [ P V s ] isthe same as in the disk case and we get in an analogous way, after carrying out the integralover the small diffeomorphisms that (cid:90) D [ P V s ]Vol(sdiffeos) = (cid:113) det (cid:48) ( P † P ) (4.26)Note that there is a zero mode of P † P which corresponds to the U (1) isometry of eq.(4.2).The measure in the space of large diffeomorphisms is described in appendix I.1. Ex-pressing the complex modes γ m , δ m appearing in eq.(4.15) in terms of the real variables p m , q m , r m , s m , as γ m = p m + iq m , δ m = r m + is m (4.27)– 27 –e find that the measure in terms of the coefficients above is given by eq.(I.13) (cid:90) D [ P V L ] = (cid:90) (cid:89) m ≥ dp m dq m dr m ds m (16 b (cid:0) ˜ m + ˜ m (cid:1) sinh ( π ˜ m )) (4.28)Finally for the moduli, from eq.(4.20) and eq.(4.22), it is straightforward to evaluate theinner products using eq.(2.9) and we find (cid:104) P V tw , P V tw (cid:105) = 4 t bπ ; (cid:104) P V b , P V b (cid:105) = 4 π ( δb ) b ; (cid:104) P V tw , P V b (cid:105) = 0 (4.29)the measure for moduli after integrating over the twist modulus turns out to be (cid:90) D [ P V tw ] D [ P V b ] = 4 (cid:90) dt db = 4 (cid:90) bdb (4.30)Here we have used the fact that the range of t is [0 , b ], since a twist between the two endswhich is bigger than b in magnitude can always be brought to lie in this range using theperiodicity θ (cid:39) θ + b . We now turn to a systematic evaluation of the path integral eq.4.1. As for the disk, wewill first carry out the dilaton path integral, then the integral over the Liouville mode andfinally, after taking the asymptotic AdS limit, the integral over the diffeomorphisms andmoduli.It is convenient, but not essential, to expand the dilaton about a background φ , φ = φ + δφ (4.31)where the background φ = αr + iβ (cid:20) r log (cid:18) r + ir − i (cid:19) − i (cid:21) (4.32)The coefficients α, β can be adjusted so that φ takes the values φ B , φ B at the twoboundaries in the metric, eq.(4.2) which are located at r B , r B . For the boundary lengths l , l (cid:29) r B (cid:39) − l b , r B (cid:39) l b (4.33)The fluctuation δφ about the background then satisfies Dirichlet boundary conditions atthe two ends. Let us also note that there is no solution to the equations of motion wherethe metric has the required form eq.(4.2) so as to meet the condition R = −
2, and thedilaton takes the values φ B , φ B at both ends with φ B , > δφ is purely imagi-nary, as in the disk topology. This yields, Z DT = (cid:90) D [ σ ] D [˜ g ab ]Vol(sdiffeo) δ ( R + 2) e − S JT,∂ − S JT,∂ (4.34)– 28 –as in the disk we will not keep track of any overall constants in Z from here on carefully.)Note that the delta function imposes the constraint R = − σ we write the metric as g ab = e σ ˆ g ab , (4.35)where ˆ g ab is a metric obtained from eq.(4.2) after carrying out changes due to small andlarge diffeomorphisms as well as the moduli. We will impose Dirichlet boundary conditionson σ at the two ends. This condition is needed to obtain a well defined variational principlein the presence of the Gibbons-Hawking boundary terms at boundaries.This gives Z DT = (cid:90) D [˜ g ab ]Vol(sdiffeo)det( − ˆ ∇ + 2) e − S JT,∂ − S JT,∂ (4.36)Note that with the Dirichlet boundary conditions we are imposing on the Liouville modedet( − ˆ ∇ + 2) has no zero modes.At this point to simplify the measure and also deal with det( − ˆ ∇ +2), which in generaldepends on the large diffeomorphisms and also moduli, in a tractable manner, we take theasymptotic AdS limit described earlier, eq.(4.4). The measure D [˜ g ab ] then becomes aproduct, eq.(4.25) and proceeding as discussed at the end of the previous subsection weget Z DT = (cid:90) bdb D [ P V L ] (cid:112) det (cid:48) ( P † P )det( − ˆ ∇ + 2) e − S JT,∂ − S JT,∂ (4.37)As argued in appendix F both (cid:112) det (cid:48) ( P † P ) and det( − ˆ ∇ + 2) become independent ofthe large diffeomorphisms in this limit. Furthermore their ratio has important cancellations.In particular, the exponential divergences that we shall discuss shortly, cancel in this ratio,see appendix I.3. The action, obtained in eq.(I.22), is given by S JT,∂ + S JT,∂ = b πGJ (cid:18) β + 1 β (cid:19) + (cid:88) m ≥ πG b J ˜ m ( ˜ m +1) sinh ( ˜ mπ ) (cid:18) p m + q m β + r m + s m β (cid:19) (4.38)where p m , q m , r m , s m are related to the modes γ m , δ m appearing in eq.(4.15) by the relationeq.(4.27). The measure for summing over the large diffeomorphisms is given in eq.(4.28).All this then leads to Z DT = (cid:90) bdb (cid:89) m ≥ dp m dq m b ˜ m ( ˜ m + 1)csch ( π ˜ m ) exp − b πGJ β (cid:88) m ≥ m ( ˜ m + 1)csch ( π ˜ m ) (cid:0) p m + q m (cid:1) × (cid:89) m ≥ dr m ds m b ˜ m ( ˜ m + 1)csch ( π ˜ m ) exp − b πGJ β (cid:88) m ≥ m ( ˜ m + 1)csch ( π ˜ m ) (cid:0) r m + s m (cid:1) (4.39)– 29 –hich agrees with eq.(127) of [3]. Note that we have cancelled terms in the action whichare proportional to length that arise from the various determinants by adding countertermwith suitably chosen coefficient, see discussion around eq.(F.25).Doing the integrals over p m , q m , r m , s m in eq.(4.39) we get Z DT = (cid:90) bdb e − b πGJ (cid:16) β + β (cid:17) π GJ √ β β (4.40)which further yields Z DT = 1 π √ β β ( β + β ) (4.41)as the final result in agreement with eq.(135) of [3] (we have not been careful about theoverall numerical factor as discussed above).We note that away from the asymptotic AdS limit the additional modes, meetingcondition eq.(4.24) would enter in the calculation and one would have determine to theirdependence in both det (cid:48) ( P † P ) and det( − ˆ ∇ + 2). This dependence is not easy to obtainand would involve an infinite number of higher derivative terms beyond the Schwarzian.Similarly the measure for summing over such modes is not easy to calculate. For all thesereasons we will not attempt a calculation of Z DT in this more general case here. Next we turn to adding matter to the theory and consider its effect in the path integralwhile summing over connected geometries with two boundaries. To begin we take a freebosonic massless scalar field. It’s action is given by S M = 12 (cid:90) d x √ g ( ∂ϕ ) (4.42)and is the same as eq.(5.47) with N = 1. We will also consider fermionic matter subse-quently.The matter field does not couple to the dilaton and we can carry out the integral over φ and thereafter over the conformal factor σ as before, leading to the partition function, Z DT + M being given by Z DT + M = (cid:90) D [˜ g ab ] D [ ϕ ]Vol(sdiffeo)det( − ˆ ∇ + 2) e − ( S JT,∂ + S JT,∂ + S M ) (4.43)where S JT,∂ , is given by eq.(B.17) at each of the boundaries.First let us consider the case where the matter vanishes at the boundaries. Carryingout the path integral over ϕ then gives, Z DT + M = (cid:90) D [˜ g ab ]Vol(sdiffeo)det( − ˆ ∇ + 2) (det( − ˆ ∇ )) / e − S JT,∂ − S JT,∂ (4.44)– 30 –he factor (det( − ˆ ∇ )) / in the denominator arose from the integral over the matter fieldand with the matter field vanishing at both boundaries is obtained from the product ofeigenvalues of the laplacian ˆ ∇ with Dirichlet boundary conditions. It is easy to see thatwith these boundary conditions the operator has no zero modes. In general (det( − ˆ ∇ )) / will depend both on the moduli, see appendix I.2, and the large diffeomorphisms as dis-cussed in appendix F.To proceed we now take the asymptotic AdS limit, eq.(4.4). In this limit the depen-dence on the large diffeomorphisms of (det( − ˆ ∇ )) / vanishes, as discussed in appendixF. However there is still an important dependence on the modulus b as we discuss shortlybelow and in appendix I.2. Also the measure breaks up into a measure over the small andlarge diffeomorphisms and moduli as mentioned in eq.(4.25). Carrying out the integralover the small diffeomorphisms then gives, Z DT + M = (cid:90) bdb D [ P V L ] (cid:112) det (cid:48) ( P † P )det( − ˆ ∇ + 2)(det( − ˆ ∇ )) / e − S JT,∂ − S JT,∂ (4.45)Here the measure for summing over the large diffeomorphisms D [ P V L ] is given in eq.(4.28)and the two boundary actions, S JT,∂ , S JT,∂ are given in eq.(B.18). Note that in theasymptotic AdS limit both (cid:112) det (cid:48) ( P † P ) and det( − ˆ ∇ + 2) are independent of the largediffeomorphisms. Again, as mentioned earlier, their ratio has crucial cancellations as dis-cussed in appendix I.3.The matter determinant depends on the modulus b and this dependence is given by Z M [ b ] = 1(det( − ˆ ∇ )) / = e − b η ( ib π ) (4.46) η ( τ ) on the RHS is the Dedekind eta function.Keeping all these facts in mind and carrying out the integral over the large diffeomor-phisms then gives, Z DT + M = (cid:90) bdbe − b πG (cid:16) β + β (cid:17) π GJ √ β β Z M [ b ] (4.47)Now we come to a rather interesting consequence of having added the matter. Using thewell known properties of η ( τ ) under modular transformations it is easy to see, as discussedin appendix I.2, eq.(I.50), that as b → Z M [ b ] → (cid:114) b π e π b (4.48)As a result the integral over the modulus b diverges as b → and the partition function Z DT + M is in fact not well defined. To examine the behaviour of the wavefunction as b →∞ , we note from the results for determinants evaluated in appendix I.2,I.3, see eq.(I.52), This divergence is more general, see for example eq.(I.75), and arises in (det( − ˆ ∇ + m )) − for anynon-negative m . – 31 –I.70), that in this limit, the contribution from various determinants can atmost go as e xb , x >
0. However the boundary terms of the JT theory, after integrating over the largediffeomorphisms has the behaviour e − yb , y >
0, see eq.(4.47). Thus the wavefunction isconvergent at the other end of the b -integral as b → ∞ .Why does the divergence as b → b → θ direction with minimum length b and the length of thisgeodesic goes to zero when b vanishes. The divergence is related to the quantum stresstensor of matter giving a negative contribution due to the Casimir effect which blows upas the size of the neck vanishes.The result for the double trumpet partition function in the absence of matter, eq.(4.41)can be interpreted as a two point correlation between the partition functions of bothboundary theories (cid:104) Z ( β ) Z ( β ) (cid:105) , [3], which could arise for example in a boundary theorywith random couplings. The divergence, once bosonic matter is added, suggests that thedominant contribution in the sum over geometries will arise when the neck goes to zero sizeresulting in the two ends not being connected at all and the connected two point functionfor the partition function vanishing. This suggests that in the presence of bosonic matterone is describing a more conventional system without random couplings. To put it anotherway, the theory with matter is ill-defined due to the divergence above. To make it well-defined, one possibility could be to take the result from the double trumpet which peaksat b = 0 as a clue and simply disallow all topologies except the disk.However, it could well be that this is not the only possibility, it is certainly not a veryelegant one. Instead, perhaps further study will show that the path integral can be madewell defined in various ways and the resulting dynamics would then determine whetherwormholes are allowed or not depending on how the divergence is tamed . We leave amore detailed investigation along these lines for the future.We can also consider what happens if fermionic matter is added instead of the bosonicmatter we considered above. Let us take as an example one complex free fermion field ψ with central charge c = 1 and action S M,f = (cid:90) d x √ g ¯ ψγ µ ∂ µ ψ (4.49)where the subscript f in S M,f in to indicate the fermionic nature of matter. Since we arethinking of θ direction as the Euclidean time direction, or as the temperature directions,we impose anti-boundary conditions along it. In addition let us also impose anti-periodicconditions, i.e. NS boundary conditions, in the radial direction.The partition function, Z M,f as a function of the moduli b can then be easily writtendown and is given by Z M,f = Tr ( NS ) e − bH (4.50) We are grateful to the members of the TIFR String Theory group for a lively discussion on this questionresulting in the more nuanced comments presented above! – 32 –here H the Hamiltonian is given by H = (cid:88) r = n +1 / rc †− r c r −
124 (4.51)In the notation used in [112] eq.(10.7.8a), for the boundary conditions above, Z M,f = Z ( τ ) (4.52)where τ = ib π . Z M,f will replaces the factor Z M [ b ] in eq.(4.47). To understand the b → Z ( τ ) = Z ( − /τ ), (eq.(10.7.14) of [112]),we learn that Z M,f → e π b . (4.53)As a result, once again the integral over b diverges.We could also consider imposing periodic (Ramond) instead of anti-periodic boundaryconditions in the radial direction at the two ends of the double trumpet, while still keepingthe boundary conditions along the temperature direction to be anti-periodic. This gives Z M,f = Tr ( R ) e − bH , which in the b → Z M,f = Tr ( NS ) e − bH − Tr ( R ) e − bH (4.54)and now the leading divergence at small b would cancel. From eq.(10.7.14) of [112] we seethat Tr ( R ) e − bH → Z ( − /τ ) under the modular transformation, τ → − τ . Thus, we getthat after this modular transformation Z M,f = Z ( − /τ ) − Z ( − /τ ) = T r ( NS ) [(1 − ( − F )˜ q H ] (4.55)where ˜ q = e − π b , and F denotes the fermion number operator, under which the NS vacuumhas charge 0 and the ψ, ψ † operators have charge ± F = 1. Thesehave H = − = , so that as b → Z M,f (cid:39) e − π b → Z M,f now decays very rapidly as b → b convergent in the region where b → θ direction . This would correspond to calculating not the partition function butan index Tr[( − F e − βH ]. For the disk topology it would not be possible to impose thisboundary condition since the θ direction shrinks to zero size and going around it is a 2 π rotation under which the fermion must be anti-periodic. But we can do so for the doubletrumpet since the θ circle has a finite size everywhere in the geometry. Imposing NSboundary conditions along the radial direction in the periodic case would give Z M,f = Z ( τ )which after a modular transformation becomes Z ( − /τ ). In the limit b → Z M,f ∼ e − π b . (4.57)We see that this now vanishes as b →
0, and the divergence goes away. The periodic bound-ary conditions have reversed the sign of the Casimir energy and there is no obstructionto a wormhole connecting the two ends now. If we impose periodic boundary conditionsalong the radial direction as well as along the θ direction, the partition function continuesto behave like eq.(4.57) as b → b should then converge, the double trumpet geometry connecting the two endswould contribute to the path integral and the dual theory would involve averaging overcoupling constants in some way. In this section we consider JT gravity in dS space. This corresponds to the action forgravity and the dilaton S JT,dS = − i πG (cid:18)(cid:90) d x √− gφ ( R − − (cid:90) ∂ dx √ γφK (cid:19) (5.1)In comparison with eq.(2.2) we see that the cosmological constant is positive here and weare working in units where the Hubble constant H is given by H = 1 (5.2)Note also that the boundary term above differs from the corresponding one in AdS. In theAdS case there is a term proportional to the length of the boundary, going like (cid:82) dx √ γ ,which is absent here. For the AdS case this term can be thought of as a counter termwhich is added, with a suitable coefficient, to remove a divergence which arises in the pathintegral when we take the limit where the length of the boundary diverges. However, in the We are grateful to Shiraz Minwalla for bringing this example to our attention and for emphasizing itsimportance. – 34 –S case the dependence of the wavefunction on the length of the boundary is of physicalsignificance and we should not be adding such a term .We also note that the path integral in our conventions is given by Z JT,dS = (cid:90) D [ g µν ] D [ φ ]Vol(Ω) e − S JT,dS (5.3)the measure etc. which appears above will be discussed in more detail below. Matter canalso be added to the system. Later on we will consider conformal matter, specifically scalarfields with action eq.(5.47) or fermionic fields with action eq.(5.48).We will study the wavefunction of the universe as given by the no boundary proposal.This wavefunction gives the probability amplitude for a universe which has length l whenthe dilaton takes the value φ B and it is given by the partition functionΨ dS [ φ B , l ] = Z JT,dS = (cid:90) D [ g µν ] D [ φ ]Vol(Ω) e − S JT,dS (5.4)For a single connected universe this partition function needs to be calculated over geome-tries which have one boundary with length l where φ = φ B . One can think of φ as providinga clock for the universe and the wavefunction as giving the amplitude for the universe tohave different lengths at time φ B .A key new element in the calculation, in comparison to the AdS case with disk topology,is that the path integral involves metrics of different spacetime signatures. There are twocontours which have been suggested to calculate the no-boundary wavefunction. In theconventional Hartle- Hawking proposal, [110], the contour studied involves Euclidean dSwhich is a sphere, S , with metric of signature (2 , S , to Minkowski dS with signature (1 , S . We willrefer to this as the Hartle-Hawking (HH) contour below. In contrast, in the Maldacenacontour, [24], we start at the north pole but evolve along − AdS which is a Euclideangeometry of signature (0 , ,
0) or (0 ,
2) the action, denoted by S JT,edS is given by S JT,edS = − πG (cid:18)(cid:90) d x √ gφ ( R + 2) + 2 (cid:90) dx √ γφ ( K ) (cid:19) , (5.5) In addition with our choice of conventions the relative sign between the bulk and boundary terms isopposite to that in eq.(2.2), for a space-like boundary. – 35 –or a contour which passes through regions of different signature we will calculate piece-wisethe contribution to S JT and add them to get the full result, keeping track of boundarieswhich arise when the different signature pieces are glued together.Note that there is actually an additional topological term in the action, eq.(2.3). Thisterm is also present in the de Sitter case we are considering here. For (2 ,
0) and (0 , S top, = − Φ πG (cid:18)(cid:90) (cid:112) | g | R + 2 (cid:90) ∂ (cid:112) | γ | K (cid:19) (5.6)and for (1 ,
1) signature by S top, = − i Φ πG (cid:18)(cid:90) (cid:112) | g | R − (cid:90) ∂ (cid:112) | γ | K (cid:19) (5.7)where Φ is a parameter which suppresses topological fluctuations. When dS JT gravityarises from higher dimensions it is related to the volume of the extra dimensions and thetopological term gives a contribution proportional to the higher dimensional dS entropy.For a contour which passes through regions of different signature we will again calculatepieces wise the contribution to S top and add them.In this section we focus on the case with a single boundary. In this case S top = − Φ G withthe contribution coming from the part of the contour which has (2 ,
0) or (0 ,
2) signature.This yields e − S top = e Φ04 G (5.8)Note that the higher dimensional de Sitter entropy is given by S dS = Φ G (5.9)and is twice in magnitude compared to S top .The three spacetimes mentioned above can be described with the metric ds = dr (1 − r ) + (1 − r ) dθ (5.10)the region r < S , and r > − AdS . Taking r → ± ir (5.11)gives dS space with signature (1 ,
1) which can be written as ds = − dr r + 1 + ( r + 1) dθ (5.12). We consider below a value of the length of the boundary l > π which is classicallyallowed for dS spacetime with H = 1. Using the ( r, θ ) coordinate system mentioned above,– 36 – − PQO S T
Figure 1 . Analytic continuations in the complex r-plane for the no-boundary wavefunction we can understand the HH and Maldacena contours in Fig.1. The branch cut in the figurecorresponds to the branch points at r = ± , −∞ of the associated Legendre function P µν ( r )that arises as a solution to the eigenvalue equation of the operator ( − ˆ ∇ + 2) in EuclideanAdS, see eq.(F.37),(I.58). The HH contour, shown in blue corresponds to starting at r = 1( P ), going till r = 0( O ) and then continuing till either Q or S depending on whetherwe started at r = 1 just above or below the real axis respectively. The Maldacena contour,shown in green, starts at r = 1( P ) in the − AdS metric and proceeds along real axis to r (cid:29) T ) and then is analytically continued to Q or S .The steps to calculate the path integral have many similarities to the AdS case con-sidered previously and we will mainly emphasize some of the key new points below.We first find a classical solution meeting our boundary conditions, obtain its contri-bution to the action, then expand around it and compute the contributions due to thequantum fluctuations.For the first contour, the classical equations are solved for the (2 ,
0) signature case bythe S with metric eq.(5.10) and dilaton, φ = iAr (5.13)– 37 –n continuing to (1 ,
1) signature using eq.(5.11) we get Minkowski dS space with φ = Ar. (5.14)Note that if the boundary value φ B is real, A must be real and therefore in the Euclideansignature region the dilaton is imaginary. To meet the boundary conditions we take theboundary to be at r = r B , where l = 2 πr B (5.15)and fix A to be A = φ B /r B (5.16)For the second contour, in the (0 ,
2) signature part the solution is also given by eq.(5.13)(but now for r >
1) and again continuing to (1 ,
1) signature gives the same result for thedilaton and the same values for r B , A as above.The resulting classical action gives rise to the saddle point contribution of the wave-function [24] S cl = iφ B G (cid:115)(cid:18) l π (cid:19) − l (cid:29)
1, the leading term denoted by ˆ S cl , is given byˆ S cl = iφ B l πG (5.18)The result for the action and wavefunction are the same in the two contours [24, 110].This is because the value of the on-shell classical action only depends on the data closeto the boundary ( φ B and the extrinsic curvature of the surface of length l ) and these aredetermined by the dS part of the contour.When l < π we are in the classically disallowed region. The action is given by S cl = φ B (cid:113) − (cid:0) l π (cid:1) G (5.19)Note that the wavefunction which in the WKB approximation is given byΨ dS ∼ e − S cl (5.20)is exponentially suppressed in the classically disallowed region, with the exponential sup-pression growing with time, φ B , and is classically oscillating in the allowed region. Next, expanding about the classical part we compute the contributions due to the quantumfluctuations in the path integral. The calculation is closely related to that in section 2.3.We write the dilaton as φ = φ + δφ (5.21)– 38 –here δφ is the fluctuation which we will integrate over. And we work in conformal gaugeby writing the metric as g ab = ˆ g ab e σ (5.22)where ˆ g ab is a conformally flat metric of appropriate signature with curvature ˆ R = 2.The Liouville mode fluctuations can be expanded in eigenmodes of the operator ˆ ∇ + 2. δσ = (cid:88) c λ φ λ ( ˆ ∇ + 2) φ λ = − λφ λ (5.23)where the modes φ λ are regular at the north pole and the coefficients c λ , φ λ are chosen sothat δσ is real for the (2 ,
0) or (0 ,
2) signature parts of the contour. For the Minkowski(1 ,
1) part we take the modes to be given by analytic continuation from the value it takesin the (2 ,
0) or (0 ,
2) segments of the geometry, depending on whether we are consideringthe HH or Maldacena contours and impose Dirichlet boundary conditions, φ λ (cid:12)(cid:12) ∂ = 0 (5.24)at the boundary r = r B . This ensures δσ = 0 at the boundary. For the dilaton fluctuationwe do a similar expansion δφ = (cid:88) d λ φ λ (5.25)in terms of the same mode functions φ λ , except that the expansion coefficients d λ arechosen to be “phase mismatched” compared to c λ so that in the (2 ,
0) or (0 ,
2) parts thedilaton fluctuation is purely imaginary. Again by analytic continuation we obtain δφ inthe (1 ,
1) region and impose Dirichlet boundary conditions on it as well.The dilaton and Liouville mode path integrals can then be done in a way very similarto the AdS case, leading toΨ dS [ φ B , l ] = e − ˆ S cl (cid:90) D [ P V ]det (cid:16) ˆ ∇ + 2 (cid:17) Vol(sdiffeo) e − S JT,∂ (5.26)where S JT,∂ , which arises entirely from the boundary term in eq.(5.1) and ˆ S cl is given ineq.(5.18). Note that when the boundary length l (cid:29) r B , eq.(5.15), meets thecondition, r B (cid:29)
1, we get, as discussed in appendix B.3, that the boundary action is givenby S JT,∂ = − iφ B (cid:15) πG (cid:90) du Sch (cid:18) tan (cid:18) θ ( u )2 (cid:19) , u (cid:19) (5.27)where the line element along the boundary is ds = du (cid:15) (cid:39) − dr r + r dθ (5.28)– 39 –ith (cid:15) ∼ /r B (cid:28) ∇ + 2) (similarly comments alsoapply to to ( P † P ) and ˆ ∇ which will appear below). We do this by analytic continuationas follows. Suppose we are working in the metric eq.(5.10) and its analytic continuationeq.(5.12). We promote the radial variable which appears in the metric and in the operator( ˆ ∇ + 2) to be a complex coordinate and analytically continue the operator as we go alongthe contour. The eigenmodes φ λ are constructed to be regular at the north pole or theorigin of the disk for the (2 ,
0) or (0 ,
2) signature regions respectively and then analyticallycontinued to dS spacetime. And the eigenvalues are then determined by imposing Dirichletboundary conditions at the boundary. Let us also note that the fluctuations φ λ will notbe real everywhere along the two contours. We will take them to be real in the (2 ,
0) or(0 ,
2) regions, this will result in them being complex, in general, in the dS region of thepath integral.The vector fields V which appear in eq.(5.26) include large and small diffeomorphisms, V L and V s , respectively as in the AdS case. The small diffeomorphisms V s , which satisfythe boundary conditions eq.(2.19), generate the gauge transformations whose volume is inthe denominator in eq.(5.26). The large diffeomorphisms V L are zero modes of P † P andcorrespond to different ways in which the boundary wiggles with φ = φ B on the boundary.The operators P, P † are given in eq.(2.12) and (2.13). These vector fields can be definedin the (2 , ,
2) signature regions as in the AdS case and are also then analyticallycontinued to the (1 ,
1) region.More specifically, V L can be written as V L = ∗ dψ where ψ is a zero mode of ( ˆ ∇ + 2),satisfying ( ˆ ∇ + 2) ψ = 0 . (5.29)In the (2 ,
0) or (0 ,
2) segments of the contour ψ is given from eq.(2.25) to be ψ = (cid:88) | m | > ˆ c m e imθ ( r + | m | ) (cid:18) r − r + 1 (cid:19) | m | (5.30)the m = 0 , − ,
1) is then obtained by analytic continuation after taking ( r → − ir, θ → θ ),and becomes, ψ = (cid:88) | m | > ˆ c m e imθ ( − ir + | m | ) (cid:18) r − ir + i (cid:19) | m | (5.31)with the coefficients ˆ c m being chosen so that ψ is real in the region r (cid:29) c − m = − ˆ c ∗ m . (5.32)In general the vector field V L obtained from ψ is then complex in this region. It is given– 40 –y V L,m = ˆ c m e iθm (cid:18) r − ir + i (cid:19) | m | (cid:32) m ( r + i | m | ) , i (cid:0) ir | m | − m + r + 1 (cid:1) r + 1 (cid:33) (5.33)and the corresponding metric perturbations are given by( P V L ) ab = 2ˆ c m e iθm (cid:0) m − (cid:1) ( r + 1) (cid:18) r − ir + i (cid:19) | m | (cid:32) m ( r + 1) − | m || m | m ( r + 1) (cid:33) (5.34)Now we come to a complication similar to what we found in the AdS case. While thepath integral is quite clearly defined as we have seen, evaluating it explicitly, for a fixed φ B and l is difficult even when l is large. This is because the metric perturbations resultingfrom the small and large diffeomorphisms are not orthogonal resulting in a measure thatis difficult to calculate and also because evaluating the determinant ( ˆ ∇ + 2) is non-trivial.To simplify things we therefore consider the asymptotic limit, in which φ B → πβJ (cid:15) , l → π(cid:15) . (5.35)with (cid:15) → dS = e − ˆ S cl (cid:90) D [ P V L ] (cid:112) det (cid:48) ( P † P )det (cid:16) ˆ ∇ + 2 (cid:17) e − S JT,∂ (5.37)Moreover both det (cid:48) ( P † P ) and det (cid:16) ˆ ∇ + 2 (cid:17) do not depend on V L in this limit and cantherefore be taken out of the integral over V L - the arguments leading to this conclusionare analogous to the AdS case, see also appendix F. The measure for summing over thediffeomorphisms is obtained from the measure for metric perturbations. This is given ineq.(2.9) with δg ab being given by eq.(2.27) for the (2 ,
0) or (0 ,
2) segment of the contourand obtained in the (1 ,
1) segment by continuation. The resulting inner product is thengiven by (cid:104)
P V
L,m P V
L,m (cid:105) = δ m , − m − iπ ˆ c m ˆ c m | m | (cid:0) m − (cid:1) r B + 1 (cid:18) r B − ir B + i (cid:19) | m | (cid:0) i | m | r B − m + r B + 1 (cid:1) (5.38)which in the r B → ∞ limit becomes, (cid:104) P V
L,m
P V L, − m (cid:105) (cid:39) − iπ ˆ c m ˆ c − m (cid:0) m − (cid:1) | m | (5.39)– 41 –hus the integral over the large diffeomorphisms reduces to the integral over the modes ˆ c m with the standard well known measure [3, 13], D [ P V L ] = (cid:89) | m |≥ d ˆ c m d ˆ c ∗ m π | m | (cid:0) m − (cid:1) . (5.40)The action in terms of the modes ˆ c m is obtained from eq.(5.27) by noting that θ ( u ) isspecified by eq.(2.32) and eq.(2.62). So we see that θ ( u ) (cid:39) u + (cid:80) m ≥ m ˆ c m e imu . The actionthen becomes S JT,∂ = − iφ B (cid:15) G (cid:88) m ≥ ˆ c m ˆ c − m ( m − m ) (5.41)The integral of the Schwarzian action with the above measure is shown to be one-loopexact [13], leading to the wavefunction in this limit beingΨ dS = exp (cid:20) − iφ B l πG + iπφ B Gl + φ G (cid:21) ( − iπGJ β ) − / √ π (cid:112) det (cid:48) ( P † P )det( ˆ ∇ + 2) (5.42)where we have also added the contribution from S top , and ˆ S cl , eq.(5.8) and (5.17). Puttingin eq.(5.35) givesΨ dS = exp (cid:20) − iπ GβJ (cid:15) + iπ GβJ + φ G (cid:21) ( − iπGJ β ) − / √ π (cid:112) det (cid:48) ( P † P )det( ˆ ∇ + 2) (5.43)More correctly, this is the value of Ψ upto an overall coefficient which we have not fixed.Note that in the limit we are considering the first term in the exponent which arises fromeq.(5.18) S cl = iπ GβJ (cid:15) → ∞ (5.44)and thus the wavefunction has very rapid fluctuations in its phase.It is worth mentioning that the two determinants on the RHS above in general dependon l and can also give rise to a term diverging linearly like l in the exponent of Ψ, as isdiscussed especially for the AdS case in appendix F.1 in considerable detail. There aresome subtle issues which arise in this context having to do with how the determinants areregulated in the UV, and related to the order of limits involved while taking the asymptoticdS limit. This is also connected to the discussion below.It is worth emphasizing that while we have considered the asymptotic limit eq.(5.35),eq.(5.36) since it is analogous to the asymptotic AdS limit which was also tractable, in thecontext of cosmology one really wants to obtain Ψ for fixed values of l, φ B . The l → ∞ then is the the limit of Ψ obtained first for such fixed values.The case with l fixed is considerably more complicated, as was emphasized in the AdSspacetime, and we unfortunately have to postpone such an investigation for the future.It is however worth noting that the different order of limits required when we work at l – 42 –xed and take the cut-off on the eigenvalues, which regulates the determinants, to infinityfirst can yield a significantly different result. In this limit modes of the form e imθ withmode number m ≥ l have a physical wavelength which lies within the universe and suchmodes can play an important role in determining the behaviour of the determinants andthe resulting behaviour of the wavefunction. In contrast in the asymptotic limit, since l → ∞ first, all modes which are kept in the determinants have a diverging physical wavelength. To illustrate how the behaviour at fixed and large l might be different we evaluatethe integral over the large diffeomorphisms with a measure obtained from the inner producteq.(5.38). We find that the behaviour of Ψ changes quite dramatically at large l and beginsto decay exponentially going like Ψ dS ∼ e − l (5.45)due to the presence of the extra modes with m ≥ l (5.46)in the path integral, see eq.(H.16) in appendix H. We hasten to reiterate here that thiscalculation is not really self consistent because these modes mix with small diffeomorphisms,since we are at finite l (or non-zero (cid:15) ) , eq.(2.37), and this mixing needs to be included inobtaining the correct measure while integrating over them. Our purpose in presenting thediscussion of appendix H is mainly to emphasize that a different result can be potentiallyobtained with the different order of limits, due to such modes once they are correctlyincluded in the path integral. We end this section by making some comments about the case with matter. We againconsider N free bosonic scalar fields, as in the AdS case with the action S M = i N (cid:88) i =1 (cid:90) d x √− g ( ∂ϕ i ) (5.47)Although we will only discuss about bosonic fields for now, we mention the action for afree fermionic field for completeness, which is given by S M,f = i (cid:90) d x √− g ¯ ψγ µ ∂ µ ψ (5.48)At late times for r B (cid:29) S M,cl = − (cid:90) du du θ (cid:48) ( u ) θ (cid:48) ( u ) ˆ ϕ ( u ) ˆ ϕ ( u ) F ( θ ( u ) , θ ( u )) (5.49)where u is the rescaled proper length along the boundary, eq.(5.28), and ˆ ϕ i ( u ) is the latetime value of ϕ i . The details of the above result can be found in appendix G, see eq.(G.17)– 43 –he quantity F in eq.(5.49) is given by eq.(G.7) of appendix G. To obtain the behaviour ofthe wavefunction in the asymptotic dS limit we would couple the Schwarzian action to thematter action above and integrate the large diffeomorphisms to obtain the wavefunction asa functional of the boundary values ˆ ϕ i ( u ). This was studied in considerable detail in [24].We will not pursue this line of investigation further here.One can also include quantum corrections due to the matter fields. The quantumcorrections come from (det( − ˆ ∇ )) − N/ which arises when one integrates out the matterfields. The dependence on the large diffeomorphisms of this determinant is suppressed atlarge l going like O (1 /r B ) ∼ O ( (cid:15) ) analogous to the case of AdS disk, see discussion aftereq.(F.16). The resulting term to quadratic order in the diffeomorphisms vanishes in theasymptotic dS limit. One can include the quantum effects of matter and neglect thosedue to the other degrees of freedom, which are difficult to obtain at finite (cid:15) , as mentionedabove by working in the semi-classical limit where we take N → ∞ and G → GN fixed. Solving the resulting saddle point equations now with the additional quantumeffective action to O ( (cid:15) ) yields the wavefunction as a function of φ B for large values of l inthe theory. We leave further investigation of this interesting limit for the future.One can also try to go beyond the semi-classical limit and include the quantum effectsof matter as well as the gravitational degrees by working at fixed and large l . However, nowmodes within the horizon meeting the condition eq.(5.46) will need to be included and thisis more challenging as discussed above. We also leave an investigation of this interestingcase for the future.Before ending let us give a few more details on the matter determinant calculation. Tocalculate the matter path integral (cid:90) Dϕe − S M (5.50)with the scalar ϕ being subject to Dirichlet boundary conditions at the dS boundary weproceed as follows. We consider the operator ˆ ∇ which is obtained in different regions ofspacetime along the contour by analytic continuation as per our discussion of the operator( ˆ ∇ + 2) above. And expand φ in terms of the complete set of eigenmodes of this operatorwhich satisfy the equation ˆ ∇ ϕ λ = − λϕ λ . The eigenmodes are analytically continued fromthe (2 ,
0) or (0 ,
2) regions to the (1 ,
1) region. Specifically in the (2 ,
0) or (0 ,
2) regionsthese modes, which satisfy the regularity condition, at r = 1 ( r being the radial coordinatein eq.(5.10)) are given by P −| m | v − / ( r ) e imθ where P −| m | v − / ( r ) are associated Legendre functions,with eigenvalue λ = v − . After analytic continuation to dS they become P −| m | v − / ( ± ir ) e imθ .The eigenvalues are obtained by imposing Dirichlet boundary conditions on these modes.In the complex r plane shown in Fig.1, P −| m | v − / ( r ) has singularities at ±
1. The contoursshown in Fig.3 illustrates how the analytic continuations we have in mind are to be carriedout. It also shows why the HH and Maldacena contours will agree since both avoid anysingularities and the solutions along these contours can be analytically continued to eachother. For completeness we should also mention that the inner product eq.(2.9) which goesinto defining the measure of the path integral, (5.40), should be analytically continued aswell along the contour. – 44 – de Sitter double trumpet
Let us now turn to discussing the analogue of the double trumpet spacetime in the contextof de Sitter space. More specifically, as in section 5 we will consider the no-boundaryproposal for calculating the wavefunction but now we ask about the amplitude for twodisconnected universes of length l , l to arises when the dilaton takes values φ B , φ B respectively. The result for this amplitude is suppressed by a factor of e S dS / where S dS is given by eq.(5.9), compared to the amplitude for producing one universe. We will findthat the amplitude to produce two disconnected universes is non-zero in pure JT gravity.Once matter is included the result for the double trumpet space can be finite, or have adivergence of the kind we found in the AdS case which arises when the neck of the wormholeshrinks to zero size due to quantum effects of the matter stress-tensor.We will start by considering pure dS JT gravity and then add matter.Let us note that the pure JT theory does not have a classical solution with the doubletrumpet topology and the dilaton meeting its boundary conditions. For carrying out thepath integral in this case we have to use the Maldacena contour. Along this contour thegeometry has a segment with − AdS metric of signature (0 ,
2) which then connects to dSspace ending in two boundaries as shown in Fig.2. The − AdS segment is now describedby the double trumpet geometry with signature (0 , ds → − ds for − AdS space, and incorporating fluctuations about this fiducial metric. To join this spacetime todS space with two disconnected boundaries we continue the two ends of the double trumpettaking r → ± ir . This gives at each end the metric ds ≡ ˆ g ab dx a dx b = − dr r − r − dθ , θ ∼ θ + b (6.1)which can be easily seen to be a metric with curvature ˆ R = 2 describing the “Milne” regionof dS space. By taking the boundary to lie at r = r B , and choosing r B , suitably at thetwo ends we can impose the condition that the two universes have lengths l , l . To carryout the path integral for the dilaton we expand it about a background value, φ , which isthe analogue of eq.(4.32), and which takes values, φ B , φ B at the two ends of de Sitterspace. The fluctuation about this background is then given by δφ and both φ and δφ , areanalytically continued across the different regions of the spacetime with signatures (0 , , δσ are also defined across thetwo regions with different signature by analytic continuation. Both δφ, δσ are expanded interms of the eigenmodes of the operator ( ˆ ∇ + 2), with a relative factor of i between theirexpansion coefficients, as for the dS integral with disk topology.The result of the path integral over the dilaton and the conformal factor then gives forthe wavefunction, Ψ DDT , when l , l (cid:29) DDT = e − i (cid:16) φB l πG + φB l πG (cid:17) (cid:90) D [˜ g ab ]Vol(sdiffeo)det( ˆ ∇ + 2) e − S JT,∂ − S JT,∂ (6.2)– 45 – → ± ir r → ± ir Figure 2 . dS double trumpet from AdS double trumpet where S JT,∂ , = − iφ B (cid:15) πG (cid:90) ∂ , du Sch (cid:18) tan (cid:18) θ ( u )2 (cid:19) , u (cid:19) (6.3)and D [˜ g ab ] is the measure for summing over traceless metric deformations which includessmall, large diffeomorphisms and the moduli.As discussed in section 5 earlier in the dS case, the case of physical interest is one wherethe lengths l , l are finite. However this is a difficult situation in which to make progresssince there are modes meeting the condition, eq.(5.46), which have not yet exited the hori-zon. The dynamics of such short distance modes is difficult to evaluate. To make progresswe therefore consider the asymptotic limit where eq.(4.4) is met. In this asymptotic limitall modes have exited the horizon and are frozen out by the exponential expansion of theuniverse. It could be, as suggested in [24] eq.(2.12), that in fact this asymptotic limit is themore fundamental quantity in dS space and the finite length case should be thought of asarising from it by integrating back in some of the modes which are yet to exit the horizon.In the asymptotic limit, reasoning analogous to the AdS case in subsection 2.5, andfor the disk topology in dS, subsection 5.2, then leads to the conclusion thatΨ DDT = exp (cid:20) − iφ B l + iφ B l πG (cid:21) Z ∂,DDT (6.4)where Z ∂,DDT = (cid:90) bdb e − ib πGJ (cid:16) β + β (cid:17) × (cid:89) m ≥ dp m dq m b ˜ m ( ˜ m + 1)csch ( π ˜ m ) exp − ib πGJ β (cid:88) m ≥ ˜ m ( ˜ m + 1)csch ( π ˜ m ) (cid:0) p m + q m (cid:1) × (cid:89) m ≥ dr m ds m b ˜ m ( ˜ m + 1)csch ( π ˜ m ) exp − ib πGJ β (cid:88) m ≥ ˜ m ( ˜ m + 1)csch ( π ˜ m ) (cid:0) r m + s m (cid:1) (6.5)– 46 –oing the integral over p m , q m etc in eq.(6.6) gives, Z ∂,DDT = (cid:90) bdbe − ib πGJ (cid:16) β + β (cid:17) iπ GJ √ β β (6.6)which in turn gives, Ψ DDT = exp (cid:20) − iφ B l + iφ B l πG (cid:21) π √ β β ( β + β ) (6.7)We can also consider adding matter. Consider a massless scalar field with actioneq.(5.47). We impose vanishing boundary conditions for the scalars at the two ends. Otherboundary conditions can also be similarly dealt with by a straightforward extension of themethods discussed in this paper, but we will not do so here. These cases would give rise tothe wavefunction for two disconnected universes with the scalar field taking some specifiedvalues in these universes.The path integral can be carried out along the lines described above for this case too.It follows from the previous section that in the asymptotic limit discussed above Z DDT + M = (cid:90) bdb e − ib πGJ (cid:16) β + β (cid:17) iπ GJ √ β β Z M,s [ b ] (6.8)where Z M,s [ b ] = 1(det( − ˆ ∇ )) / (6.9)As discussed in appendix J when b → Z M,s diverges like eq.(4.48). As in the AdScase this divergence arises due to the Casimir effect for matter which results in a divergingstress tensor when the size of the neck in the wormhole goes to zero.Now let us turn to the fermionic case with action (5.48). For AdS the anti-periodicboundary condition along the θ direction is needed for computing the thermal partitionfunction, however here it is upto us to specify whether the fermion satisfies periodic oranti-periodic boundary conditions along the θ direction. When the boundary conditionis anti-periodic the disk topology path integral corresponding to producing one universegives a well-defined result but the double trumpet has a divergence of the form eq.(4.53).When the boundary condition is periodic (for both universes) the disk topology does notcontribute, since the θ circle shrinks to zero size and therefore the fermion must necessarilyhaving anti-periodic boundary conditions along it. The leading contribution then arisesfrom the double trumpet topology for two connected universes. In this case one cannotproduce one universe from nothing through quantum tunnelling they must come in at leasta pair!Let us end with one comment. The continuation from the − AdS to dS can be doneby taking r → ± ir . We have not specified, either in the case of the disk or the doubletrumpet, which of the two analytic continuations needs to be carried out. In the disktopology, as far as we can tell, it does not make a difference and both are allowed. For thedouble trumpet there are a total of four possibilities, see Fig.3, since we have two choices– 47 – − A B CDE FG
Figure 3 . dS double trumpet continuations from AdS double trumpet at either end. More precisely, as can be seen from the figure, we can start at r = −∞ fromeither above the real axis at A (green contour) or below the real axis at E (blue contour).In each case there are again two possibilities after going along the imaginary axis. Forexample, for the blue contour, at B , we can continue to either A or F . All of these seem tobe allowed, as far as we can tell. When dealing with eigenmodes for various determinants,e.g. a scalar laplacian, the modes need to be analytically continued, we find it is possibleto do so for all four continuations. A proper understanding of this issue is also left for thefuture. In this paper we have formulated the path integral for JT gravity in the second orderformalism working directly with the metric and the dilaton. This allows one to incorporatematter easily as well. It also allows one to investigate whether spacetimes with differenttopologies can contribute to the path integral. We considered both AdS and dS spacetimesin our analysis. For matter, we have mostly considered free bosons or fermions, but many ofour conclusions extend more generally to conformal matter and some even more generallyto non-conformal matter.Many questions remain to be followed up.– 48 –e found agreement with the first order formalism for the pure JT theory in theasymptotic AdS or dS limits. Away from this limit the path integral is more complicateddue to the presence of modes whose wavelength along the boundary is short and in par-ticular smaller than the radius of AdS or dS. Understanding the dynamics of these modesand carrying out the path integral more generally, is an important open question. Onemight hope that some of the determinants which arise can be made more tractable bya general analysis, of the kind that leads to a simplification in the ratio of determinants √ det (cid:48) ( P † P )det( − ˆ ∇ +2) for compact manifolds ([109, 113, 114]), and such simplifications might helpwith the problem.In dS space the path integral is more non-trivial to define since the no-boundaryproposal, which is what we explored here, requires one to include regions of spacetimewith different signatures along the contour of the integral. In our formulation we continuemodes analytically between these regions while carrying out the path integral. While thismeans the general metric, dilaton and matter configurations we sum over are complex,the resulting path integral is well defined, as best as we can tell. However, this needs tobe investigated more carefully further. Another issue for the dS case is how to deal withthe divergences which arise when we take the asymptotic limit. In AdS space these aredealt with by the standard procedure of holographic re-normalization after adding suitablecounter-terms which are local on the boundary. But it is less clear if such a procedure isthe correct one to adopt in dS space. It would be worth understanding this issue better aswell. Calculating the path integral away from the asymptotic limit is especially importantin the dS case, since we are interested in the wavefunction for a universe of finite size andat finite values of the dilaton. More generally, it will be worth establishing firmly whethera precise and sensible formulation of JT gravity can be given in dS space, as a start evenfor spacetimes with the topology of the disk.Adding matter introduces some interesting new facets. Most important, as we haveseen above, is the divergence which arises in the integral over moduli space while computingthe path integral for the double trumpet topology in the presence of matter. This divergenceis due to the Casimir effect leading to a negative stress tensor which diverges when theneck of the wormhole goes to zero size and is the analogue of the tachyon divergence inBosonic string theory. While the precise result we get is simply that the path integral isill-defined, the divergence suggests that perhaps the presence of matter would cause thespacetime to disconnect into two pieces each of disk topology with one boundary. In fact,taking a cue from the divergence one could simply posit that to get a well-defined theoryone should disallow higher topologies and only keep spacetimes with the topology of thedisk. However this is clearly too premature. Rather, further analysis is needed to see whatare the possible ways to make the path integral well-defined and whether the divergencecan be cured while still allowing the two boundaries to stay connected. It could well bethat the fate of the wormhole depends on the details of how the divergence is tamed. Thisis an important issue which we hope to investigate more fully in later workThe double trumpet geometry in AdS can be thought of as a contribution to theconnected two point function of the partition functions of the two boundary theories,– 49 – Z ( β ) Z ( β ) (cid:105) . If the double trumpet and more generally wormholes survive in the pathintegral it would suggest that the boundary theories dual to JT gravity involve some uncon-ventional features, for example random couplings over which one needs to sum, resultingin this connected correlation. JT gravity can be obtained by dimensional reduction fromhigher dimensional near-extremal black holes, [16][17] and one expects that the higher di-mensional systems which arise for example in string theory are more conventional with aHamiltonian with fixed coupling constants. The dimensional reduction gives rise typicallyto a lot of extra matter. It would be satisfying if the presence of this matter itself causeswormholes to pinch off and the two boundaries to disconnect. However, as was mentionedabove, this needs to be investigated further. It could also be that the dimensional reduc-tion to two dimensions removes essential degrees of freedom in the system and thereafterwormholes are allowed in the two dimensional theory .It is also worth drawing attention to the fact that the divergence mentioned above doesnot always arise. For example, in the double trumpet if one consider fermions instead ofbosons with periodic boundary conditions along the two boundaries then the Casimir effectreverses its sign and the divergence goes away. The periodic boundary conditions in theAdS context would be appropriate for computing an index Tr(( − F e − βH ) instead of thepartition function Tr( e − βH ). Investigating the behavior of the divergence as we vary thekind of matter and the boundary conditions we impose on it is another important directionto pursue.A similar divergence for the double trumpet also arises in the dS case. And as inthe AdS case with fermions, by imposing periodic boundary conditions along the spatialboundary the divergence goes away. This suggests that for appropriate matter and bound-ary conditions the wavefunction of the universe in the no-boundary proposal can have anamplitude to create multiple universes. It is clearly important to understand this moredeeply. In this context one would also like to study the “pants diagram” which corre-sponds to one universe tunnelling into two, and more generally study the role of highertopologies. If wormhole do arise, the proper setting for quantum cosmology would be thethird quantized one, where one is dealing with a multiverse.We have focussed on the no-boundary proposal in dS space here. There are otherpossibilities that are also worth investigating [115–117] . One would also like to add aninflaton to the theory and study the resulting dynamics of the system, including how itbehaves in the presence of a potential for the inflaton, with metastable minima. It wouldalso be interesting to explore the role of “bra-ket” wormholes and their contribution todensity matrices[58, 118, 119].Finally, we have not explored the Lorentzian AdS theory here. The fact that thequantum effects of matter only arise when we are away from the asymptotic AdS limit,with a boundary of finite length, is true for the Lorentzian case as well, as was discussedin [18]. Thus, for discussing the effects of Hawking radiation by coupling the JT system toexternal radiation, as has been done quite extensively in recent literature,[8, 18, 27, 28, 34], Another possibility is that the higher dimensional theory itself has wormholes due to averaging overthe various ground states of the extremal system. We are grateful to Ashoke Sen for emphasizing thispossibility to us. – 50 –ne needs to work away from the asymptotic limit. To incorporate corrections beyond theleading semi-classical analysis (obtained with N matter species by taking N → ∞ ) onewould then also need to include the effects of the short wave length modes mentioned in thefirst few paragraphs above. The dynamics of these modes might in fact play a key role inthe recovery of information during the evaporation process. A discussion of the Lorentziantheory will also be of interest from the point of view of potentially taming moduli spacedivergences that were discussed above, see [120].Clearly two dimensional gravity is a rich and fascinating playground. While resultsobtained in lower dimensional settings might not always apply to higher dimensions onecan hope to gain some important insights from them. We look forward eagerly to exploringsome of the questions mentioned above in the future. Acknowledgments
We thank Onkar Parrikar, Ashoke Sen and the TIFR String theory group including AbhijitGadde, Gautam Mandal and Shiraz Minwalla, for useful discussions. We acknowledge thesupport of the Government of India, Department of Atomic Energy, under Project No. 12-R&D-TFR-5.02-0200 and support from the Infosys Foundation in form of the Endowmentfor the Study of the Quantum Structure of Spacetime. S. P. T. acknowledges support froma J. C. Bose Fellowship, Department of Science and Technology, Government of India.Most of all, we are grateful to the people of India for generously supporting research inString Theory. – 51 –
Coordinate transformations
A.1 Euclidean AdS disk
The metric for the Euclidean
AdS disk geometry is given by ds = ( r − dt + dr ( r −
1) (A.1)Redefining the coordinate t, r as r = cosh( ρ ) , t = θ (A.2)we get ds = dρ + sinh ( ρ ) dθ (A.3)Defining the coordinate r ∗ as r ∗ = log (cid:16) tanh (cid:16) ρ (cid:17)(cid:17) (A.4)The metric then becomes ds = dθ + dr ∗ sinh ( r ∗ ) (A.5)Writing in term of the complex coordinates defined as ζ = θ − ir ∗ ds = − dζ d ¯ ζ sin ( ζ − ¯ ζ ) (A.7)Futher doing a coordinate transformation ζ = arctan( x ) , ¯ ζ = arctan(¯ x ) ⇒ x = tan( ζ ) , ¯ x = tan ¯ ζ (A.8)it is easy to see that the metric becomes ds = − dxd ¯ x ( x − ¯ x ) = dt + dz z where x = t + iz r defined byˆ r = exp( r ∗ ) = (cid:114) r − r + 1 (A.10)as ds = 4 d ˆ r + ˆ r dθ (1 − ˆ r ) (A.11)– 52 – .2 Euclidean AdS double trumpet The line element for this geometry is given by ds = ( r + 1) dt + dr ( r + 1) (A.12)The two boundaries correspond to the limits r → ∞ and r → −∞ . Performing thecoordinate transformations r = sinh( ρ ) , t = θ, (A.13)we find that the metric is given by ds = dρ + cosh ( ρ ) dθ (A.14)It has to be noted that the θ direction is periodic with period 2 π . Defining r ∗ coordinateas dr ∗ = dρ cosh( ρ ) → r ∗ = 2 arctan (cid:16) tanh (cid:16) ρ (cid:17)(cid:17) ⇒ tan( r ∗ ) = sinh( ρ ) (A.15)In term of the r ∗ coordinates the metric becomes ds = dr ∗ + dθ cos ( r ∗ ) (A.16)This can be written in complex coordinates as ζ = t − ir ∗ , ¯ ζ = t + ir ∗ ds = 4 dζ d ¯ ζ cosh ( ζ − ¯ ζ ) (A.18)To get it to the Poincare form, consider the further coordinate transformation x = coth( ζ ) , ¯ x = tanh ¯ ζ (A.19)and hence the metric becomes ds = − dxd ¯ x ( x − ¯ x ) (A.20) A.3 de Sitter
The metric for the de Sitter spacetime in 2 dimensions, for Lorentzian signature, is givenin the global coordinates as ds = − dτ + cosh ( τ ) dθ (A.21)– 53 –s before, we define the coordinate r ∗ as dr ∗ = dτ cosh( τ ) = 2 arctan (cid:16) tanh (cid:16) τ (cid:17)(cid:17) ⇒ cosh τ cos r ∗ = 1 (A.22)and the metric in the r ∗ becomes ds = dθ − dr ∗ cos ( r ∗ ) (A.23)Doing the transformation r ∗ = arctan( r ) (A.24)gives us ds = (1 + r ) dθ − dr r + 1 (A.25)From eq.(A.22) and (A.24), we find r = sinh τ (A.26)Defining the null coordinates ζ ± = θ ± r ∗ ds = 4 dζ + dζ − cos ( ζ + − ζ − ) (A.28)The coordinate transformation x + p = cot ζ + , x − p = − tan ζ − (A.29)The line element eq.(A.28) becomes ds = 4 dx + p dx − p ( x + p − x − p ) (A.30) B Extrinsic Trace and Schwarzian action
B.1 Euclidean AdS disk
Consider the metric ds = dr r − r − dθ (B.1)The boundary is located in the region where r → ∞ . The general boundary curve isspecified by ( r ( u ) , θ ( u )) where u is the proper boundary time. The line element on the– 54 –oundary is ds (cid:12)(cid:12)(cid:12)(cid:12) ∂ = du (cid:15) (B.2)where (cid:15) (cid:28)
1. The tangent vector to the boundary curve is given by V ≡ ∂ u = r (cid:48) ( u ) ∂ r + θ (cid:48) ( u ) ∂ θ (B.3)The unit normalized normal vector is then given by n r = ( r − θ (cid:48) (cid:112) θ (cid:48) ( r − + r (cid:48) n θ = − r (cid:48) (cid:112) ( r − (cid:112) θ (cid:48) ( r − + r (cid:48) (B.4)The extrinsic trace is given by K = ∇ µ n µ = ∂ r n r = ∂ u n r r (cid:48) = √ r − (cid:16)(cid:0) r − (cid:1) r (cid:48) θ (cid:48)(cid:48) ( u ) + θ (cid:48) (cid:16) r (cid:48)(cid:48) + r (cid:16) − rr (cid:48)(cid:48) + 3 r (cid:48) + (cid:0) r − (cid:1) θ (cid:48) (cid:17)(cid:17)(cid:17)(cid:16) r (cid:48) + ( r − θ (cid:48) (cid:17) / (B.5)where it is understood that r, θ are functions of u . Differentiating the line element relationgives 2 (cid:18) rr (cid:48) θ (cid:48) − rr (cid:48) ( r − + r (cid:48) r (cid:48)(cid:48) r − (cid:0) r − (cid:1) θ (cid:48) θ (cid:48)(cid:48) ( u ) (cid:19) = 0 (B.6)Using this to simplify the eq.(B.5), we get K = r − (cid:15) r (cid:48)(cid:48) ( r − (cid:15)θ (cid:48) (B.7)Noting from eq.(B.1) and eq.(B.2) that to leading order r θ (cid:48) (cid:39) (cid:15) − ⇒ r (cid:39) (cid:15)θ (cid:48) (B.8)The above relation need to be extended to one higher to obtain the leading schwarzianterm in the extrinsic trace. Doing so gives r (cid:39) (cid:15)θ (cid:48) ( u ) + (cid:15) (cid:18) θ (cid:48) ( u )2 − θ (cid:48)(cid:48) ( u ) θ (cid:48) ( u ) (cid:19) (B.9)– 55 –ith this relation between θ, r , expanding the extrinsic trace to quadratic order in (cid:15) , weget K = 1 + (cid:15) (cid:0) − θ (cid:48)(cid:48) ( u ) + θ (cid:48) + 2 θ (3) ( u ) θ (cid:48) (cid:1) θ (cid:48) + O (cid:0) (cid:15) (cid:1) = 1 + (cid:15) (cid:18) Sch (cid:18) tan (cid:18) θ ( u )2 (cid:19) , u (cid:19)(cid:19) + O ( (cid:15) ) (B.10)We will now provide some formulae that will be useful later on in appendix H. B.2 Euclidean AdS double trumpet
The line element is given by eq.(4.2). The geometry now has two boundaries in the asymp-totic region, i.e near r → ±∞ . We need to be a bit careful when evaluating the boundaryterms in the J T action as the relative signs play a crucial role in the final result of the pathintegral. Consider a curve given by ( r ( u ) , θ ( u )) where u is proportional to the boundaryproper time. We have ∂ u = r (cid:48) ( u ) ∂ r + θ (cid:48) ( u ) ∂ θ (B.11)The unit normalized normal vector is then given by n r = ± ( r + 1) θ (cid:48) (cid:112) θ (cid:48) ( r + 1) + r (cid:48) n θ = ∓ r (cid:48) (cid:112) ( r + 1) (cid:112) θ (cid:48) ( r + 1) + r (cid:48) (B.12)where the upper sign corresponds to the right boundary and the lower sign to the leftboundary. The extrinsic trace is given by K = ∇ µ n µ = ∂ r n r = ∂ u n r r (cid:48) = √ r + 1 (cid:16)(cid:0) r + 1 (cid:1) r (cid:48) θ (cid:48)(cid:48) + θ (cid:48) (cid:16) r (cid:16) r (cid:48) + (cid:0) r + 1 (cid:1) θ (cid:48) (cid:17) − (cid:0) r + 1 (cid:1) r (cid:48)(cid:48) (cid:17)(cid:17)(cid:16) r (cid:48) + ( r + 1) θ (cid:48) (cid:17) / (B.13)where it is understood that r, θ are functions of u . Consider the situation when both theboundaries of the double trumpet geometry have the line element as given in eq.(B.2) withthe same parameter (cid:15) . The coordinate u is chosen such that its range is same as the rangeof the θ coordinate. From eq.(4.2), we have to leading order r θ (cid:48) (cid:39) (cid:15) ⇒ r (cid:39) ± (cid:15)θ (cid:48) (B.14)where we need to use + sign at the right boundary and − at the left boundary. Moregenerally, the parameter (cid:15) need not be same at both boundaries. The above relation needto be extended to one higher to obtain the leading Schwarzian term in the extrinsic trace.– 56 –oing so gives r (cid:39) ± (cid:18) (cid:15)θ (cid:48) ( u ) − (cid:15) (cid:18) θ (cid:48) ( u )2 + θ (cid:48)(cid:48) ( u ) θ (cid:48) ( u ) (cid:19)(cid:19) (B.15)With this relation between θ, r , expanding the extrinsic trace to quadratic order in (cid:15) , weget K = 1 − (cid:15) (cid:0) θ (cid:48)(cid:48) + θ (cid:48) − θ (3) θ (cid:48) (cid:1) θ (cid:48) + O (cid:0) (cid:15) (cid:1) = 1 + (cid:15) Sch (cid:18) tanh (cid:18) θ ( u )2 (cid:19) , u (cid:19) + O ( (cid:15) ) (B.16)We get the above action for both the signs in eq.(B.15), or in other words, at both theboundaries. This relative plus sign in between the boundary terms at both the bound-aries is important because of the dependence on the moduli of the integral over the largediffeomorphisms as we shall see later in appendix I. The boundary term of the J T actioneq.(2.51), at either boundary, then becomes S JT,∂ = φ B (cid:15) πG (cid:90) b du Sch (cid:18) tanh (cid:18) θ ( u )2 (cid:19) , u (cid:19) (B.17)Denoting the boundaries ∂ , ∂ and writing the two boundary terms explicitly we have S JT,∂ = S JT,∂ + S JT,∂ = φ B (cid:15) πG (cid:18)(cid:90) ∂ du Sch (cid:18) tanh (cid:18) θ ( u )2 (cid:19) , u (cid:19) + (cid:90) ∂ du Sch (cid:18) tanh (cid:18) θ ( u )2 (cid:19) , u (cid:19)(cid:19) (B.18) B.3 de Sitter
Consider the metric in global coordinates given by ds = − dτ + cosh τ dθ (B.19)The general boundary curve is specified by ( τ ( u ) , θ ( u )) where u is the proper boundarytime. The line element on the boundary is given by ds (cid:12)(cid:12)(cid:12)(cid:12) ∂ = du (cid:15) (B.20)The tangent vector to the boundary curve is given by V ≡ ∂ u = τ (cid:48) ( u ) ∂ τ + θ (cid:48) ( u ) ∂ θ (B.21)The unit normalized normal vector is then given by n τ = − θ (cid:48) ( u ) cosh τ (cid:113) θ (cid:48) ( u ) cosh τ − τ (cid:48) ( u ) n θ = − τ (cid:48) ( u )cosh τ (cid:113) θ (cid:48) ( u ) cosh τ − τ (cid:48) ( u ) (B.22)– 57 –he extrinsic trace is given by K = − θ (cid:48)(cid:48) τ (cid:48) cosh τ + θ (cid:48) (cid:0) τ (cid:48)(cid:48) cosh τ − τ (cid:48) sinh τ (cid:1) + θ (cid:48) sinh τ cosh τ (cid:0) θ (cid:48) cosh τ − τ (cid:48) (cid:1) / (B.23)where it is understood that τ, θ are functions of u . Noting from eq.(B.19) and eq.(B.20)that to leading order e τ θ (cid:48) ( u ) (cid:39) (cid:15) − ⇒ τ ( u ) (cid:39) − log (cid:18) (cid:15)θ (cid:48) (cid:19) (B.24)Correcting this relation to one higher order in (cid:15) , we have τ ( u ) = − ln (cid:18) (cid:15)θ (cid:48) (cid:19) + (cid:15) (cid:18) θ (cid:48)(cid:48) θ (cid:48) − θ (cid:48) (cid:19) (B.25)and expanding to quadratic order in (cid:15) , we get K = 1 − (cid:15) (cid:0) − θ (cid:48)(cid:48) + θ (cid:48) + 2 θ (cid:48)(cid:48)(cid:48) θ (cid:48) (cid:1) θ (cid:48) + O (cid:0) (cid:15) (cid:1) = 1 − (cid:15) Sch (cid:18) tan (cid:18) θ ( u )2 (cid:19) , u (cid:19) + O ( (cid:15) ) (B.26)Also, we can simplify the extrinsic trace formula eq.(B.23) as follows using eq.(B.19) andeq.(B.20). Doing so, we get K = τ (cid:48)(cid:48) + ( (cid:15) − + τ (cid:48) ) tanh τ(cid:15) − √ (cid:15) − + τ (cid:48) (B.27)We now provide some useful formulae of the same calculations in the metric eq.(5.12).The unit normal vector components are given by n r = ( √ r + 1) θ (cid:48) (cid:113) ( r + 1) θ (cid:48) − r (cid:48) , n θ = r (cid:48) √ r + 1 (cid:113) ( r + 1) θ (cid:48) − r (cid:48) (B.28)The extrinsic trace is then given by K = √ r + 1 (cid:16) θ (cid:48) (cid:16)(cid:0) r + 1 (cid:1) r (cid:48)(cid:48) + r (cid:16)(cid:0) r + 1 (cid:1) θ (cid:48) − r (cid:48) (cid:17)(cid:17) − (cid:0) r + 1 (cid:1) r (cid:48) θ (cid:48)(cid:48) (cid:17)(cid:16) ( r + 1) θ (cid:48) − r (cid:48) (cid:17) / (B.29)Expanding r ( u ) in terms of θ ( u ) as r ( u ) = 1 (cid:15)θ (cid:48) + (cid:15) θ (cid:48) ( u ) (cid:18) θ (cid:48)(cid:48) ( u ) θ (cid:48) ( u ) (cid:19) − (cid:15)θ (cid:48) ( u ) + O ( (cid:15) ) (B.30)we get the extrinsic trace as in eq.(B.26). We now derive some formulae that will be usedin appendix H. Taking τ → τ − i π ds = − ( dτ + sinh τ dθ ) (B.32)which is negative of the metric for global AdS . Using eq.(B.31) then becomes K = − τ (cid:48) sinh( τ ) θ (cid:48)(cid:48) + θ (cid:48) (cid:0) cosh( τ ) (cid:0) τ (cid:48) + sinh ( τ ) θ (cid:48) (cid:1) − τ (cid:48)(cid:48) sinh( τ ) (cid:1)(cid:0) τ (cid:48) + sinh ( τ ) θ (cid:48) (cid:1) / (B.33)Now, taking the line element on the boundary to be ds = − sinh τ du (B.34)where τ is an arbitrary fixed value of τ . The boundary relation then reads − τ (cid:48) − sinh τ θ = − sinh τ (B.35)where τ is an arbitrary value. Expanding τ and θ as τ = τ + δτ, θ = u + δτ (B.36)and solving for δτ iteratively to quadratic order in δθ , we get δτ ( u ) = − tanh τ δθ (cid:48) ( u ) + 14 tanh τ sech τ (cid:0) (cosh(2 τ ) + 3) δθ (cid:48) ( u ) − δθ (cid:48)(cid:48) ( u ) (cid:1) + O (cid:0) κ (cid:1) (B.37)Using eq.(B.37) to expand the extrinsic trace eq.(B.33) to quadratic order in δθ , we find K = − coth τ − csch τ sech τ (cid:16) δθ (3) ( u ) + δθ (cid:48) ( u ) (cid:17) −
14 csch τ cosh τ (cid:16) δθ (cid:48)(cid:48)(cid:48) + 4 δθ (4) δθ (cid:48)(cid:48) − (3 cosh(2 τ ) + 5) δθ (cid:48)(cid:48) + 2 sinh τ δθ (cid:48) − τ ) + 3) δθ (3) δθ (cid:48) (cid:17) (B.38) C Zeta-function regularization
In this appendix, we mention some useful formulae pertaining to Zeta-function regular-ization that are used in this work. The Riemann-zeta function, denoted by ζ ( s ) is, givenby ζ ( s ) = (cid:88) m =1 m s , (C.1)and it has the specific values ζ (0) = − , ζ (cid:48) (0) = − ln √ π. (C.2)– 59 –he generalized Zeta function, ζ ( s, m ) is given by ζ ( s, m ) = (cid:88) m =0 m + m ) s , ζ (cid:48) (0 , m ) = ln Γ( m ) √ π (C.3)Consider the sum (cid:80) m> ln (cid:0) αm (cid:1) . Defining ζ A ( s ) as ζ A ( s ) = (cid:88) m =1 λ sm ⇒ (cid:88) m =2 log λ m = − ζ (cid:48) A (0) − ln λ (C.4)and using λ m = αm in the above, we get that ζ A ( s ) = (cid:88) m =1 m s α s = ζ ( − s ) α s ⇒ ζ (cid:48) A (0) = − ζ (cid:48) (0) − ζ (0) ln α = ln (cid:16) √ πα (cid:17) (C.5)Using the result eq.(C.5) in eq.(C.4), we get (cid:88) m =2 ln (cid:16) αm (cid:17) = − ln √ πα − ln α = − ln √ πα (C.6)Now consider the sum (cid:80) m =2 ln( m − m ). Defining ζ B ( s, m ) as ζ B ( s, m ) = (cid:88) m =0 λ sm ⇒ (cid:88) m =2 log λ m = − ζ (cid:48) B (0 , m ) − ln λ − ln λ (C.7)Using the value of λ m = m − m , we see that ζ B ( s, m ) = (cid:88) m =0 m − m ) s = ζ ( s, − m ) ⇒ ζ (cid:48) B (0 , m ) = ζ (cid:48) (0 , − m ) = ln Γ( − m ) √ π (C.8)Using the result of eq.(C.8) in eq.(C.7), we get (cid:88) m =2 ln( m − m ) = − ln Γ( − m ) √ π − ln(1 − m ) − ln( − m ) = − ln (cid:18) Γ(2 − m ) √ π (cid:19) (C.9)Generalizing the above results, we note here a general formula for the zeta function-regularized product, (cid:89) m ≥ αm ( m − m ) . . . ( m − m p )( m − ˜ m ) . . . ( m − ˜ m q ) → (2 π ) p − q √ πα Γ(2 − ˜ m ) . . . Γ(2 − ˜ m q )Γ(2 − m ) . . . Γ(2 − m p ) (C.10) D Conformal Killing Vectors in the Euclidean AdS Disk
In this appendix, we shall explicitly evaluate the conformal Killing vectors for the EuclideanAdS disk topology with the metric given in polar coordinates as in eq.(2.24). First, notethat conformal Killing vectors satisfy the condition
P V = 0. This immediately impliesthat P † P V = 0 for a CKV. Thus, we only look for CKVs in the sector of zero modes of– 60 –he operator P † P . For a general zero mode of this operator, we can write the vector fieldas V am = k ∇ a ψ m + k (cid:15) ab ∇ b ψ m (D.1)where ψ m is given in eq.(2.25) and the condition that the vector field be real means that k , k are real. The metric components are then given by( P V m ) rr = 2ˆ c m (cid:0) m − (cid:1) e iθm (cid:16) r − r +1 (cid:17) | m | ( k | m | + ik m )( r − ( P V m ) rθ = 2 i ˆ c m (cid:0) m − (cid:1) e iθm (cid:16) r − r +1 (cid:17) | m | ( k m + ik | m | ) r − P V m ) θθ = − i ˆ c m (cid:0) m − (cid:1) e iθm (cid:18) r − r + 1 (cid:19) | m | ( k m − ik | m | ) (D.2)It can be seen from the above that if all the metric components δg ab = ( P V m ) ab were tovanish, the possibilities are m = 0 , , − , k (cid:54) = 0 , k = 0 (D.3)or k = ik sign( m ) (D.4)The diffeomorphisms corresponding to k (cid:54) = 0 , k = 0 in eq.(D.3) are exact isometries of AdS . This is straightforward to see since if m = 0 ± P V = 0,and with k = 0 is follows that ∇ · V = 0, leading to the conclusion that ∇ a V b + ∇ b V a = 0.Among the set of CKVs those given by eq.(D.3) correspond to vector fields of the form V µ = q ∇ µ ˆ ψ + q ∇ µ ˆ ψ + q ∇ µ ˆ ψ (D.5)where ˆ ψ = r, ˆ ψ = (cid:112) r − θ, ˆ ψ = (cid:112) r − θ (D.6)and q i are arbitrary real constants. These give rise to an SL (2 , R ) algebra. The abovefunctions ˆ ψ i are in fact linear combinations of the solutions appearing in eq.(2.25) for m = 1 , − , P † P has no zero modes.– 61 – Estimate of the inner product of metric perturbations arising fromlarge and small diffeomorphism
We are interested in calculating the inner product (cid:104)
P V s , P V L (cid:105) (cid:112) (cid:104) P V s , P V s (cid:105)(cid:104) P V L , P V L (cid:105) (E.1)Let us calculate each of the terms in the above expression. In terms of scalar field, ψ , ξ λ , ψ λ , the vector fields corresponding to large and small diffeomorphisms are given by V aL = (cid:15) ab ∇ b ψ (E.2)and V as,λ,m = ∇ a ξ λ,m + (cid:15) ab ∇ b ψ λ,m (E.3)respectively.The large diffeomorphism V aL being a zero mode of the P † P translates into the condi-tion that ∇ ψ = 2 ψ (E.4)and the small diffeomorphism being an eigenmode of P † P with eigenvalue λ , i.e P † P V s = λV s translates into ∇ ψ λ,m = (2 + λ ) ψ λ,m ∇ ξ λ,m = (2 + λ ) ξ λ,m (E.5)The first of the boundary conditions for the small diffeomorphisms in eq.(2.19) just becomes V · n = 0 ⇒ V rs,λ,m = 0 ⇒ g rr ∂ r ξ λ,m + ∂ θ ψ λ,m = 0 (E.6)which gives g rr ∂ r ξ λ,m + imψ λ,m = 0 (E.7)at the boundary r = r B . The second condition becomes t a n b ( P V s,λ,m ) ab = 0 ⇒ ( P V s,λ,m ) θr = 0 ⇒ ∇ θ ( V s,λ,m ) r + ∇ r ( V s,λ,m ) θ = 0 ⇒ ∂ r V θs,λ,m = 0 (E.8), at r = r B , where the last line is obtained by using the first condition in eq(E.7). Expressed– 62 –n terms of the ξ λ,m , ψ λ,m , this condition becomes ∂ r V θs,λ,m = 0 ⇒ im∂ r ( g θθ ξ λ,m ) − ∂ r ψ λ,m = 0 (E.9)To understand how these conditions can be met in the asymptotic AdS limit when r B → ∞ more clearly, let us first examine the scalar field equation for ψ λ,m carefully. The generalsolution is given by ψ λ,m = e imθ (cid:16) c P mv − ( r ) + c Q mv − ( r ) (cid:17) (E.10)where v = 12 √ λ (E.11)and P mα , Q mα are the associated Legendre functions of the first and second kind respectively.Regularity at the origin of the above solution forces us to choose c = 0. Then behaviourof this solution for r (cid:29) ψ λ,m ( r ) = c ( r − − v f ( λ, m ) + r − + v g ( λ, m )) (E.12)where f ( λ, m ) , g ( λ, m ) are some specific expressions which can be read off from the asymp-totic behaviour of the associated Legendre functions and the θ dependence is not explicitlyshown. From the above, it is clear that if v is imaginary, the expression has the functionalform ψ λ,m ( r ) = c √ r F cos( w log r − β ) (E.13)where w = − iv and F, β are given by F = 2 (cid:112) f g , tan β = i ( g − f ) g + f (E.14)It is clear from the expression eq.(E.13) that the magnitude of the scalar field solution ψ λ,m ∼ c √ r . The same analysis holds for ξ λ,m , albeit with a different constant instead of c , say d . It is now clear that one way to satisfy the conditions eq.(E.7) and eq.(E.9) isto choose the constants c , d such that d ∼ c r B so that in eq.(E.7), the two terms arecomparable and cancel each other whereas in eq.(E.9), the second term dominates, givingrise to the condition ∂ r ψ λ,m (cid:12)(cid:12)(cid:12)(cid:12) r = r B (cid:39) λ . The alternate way is to choose the constants c , d suchthat c ∼ d r B so that in eq.(E.9), the terms are comparable and cancel each other whereas– 63 –n eq.(E.7), the first term dominates giving rise to the condition ∂ r ξ λ,m (cid:12)(cid:12)(cid:12)(cid:12) r = r B (cid:39) λ . These two ways of meeting the conditions eq.(E.7) andeq.(E.9) give rise to two sets of eigenvalues and in fact exhaust all the possibilities.We now proceed to evaluate the various inner products. We shall first evaluate theexpressions in general and then take the asymptotic AdS limit to get the estimates. Con-sider the inner product of two metric perturbations, one with a large diffeomorphism andone with a small diffeomorphism. (cid:104) P V s,λ,m , P V L, − m (cid:105) = (cid:104) V s,λ,m , P † P V L, − m (cid:105) + (cid:90) ∂ dsn a V bs,λ,m ( P V L, − m ) ab = (cid:90) r = r B dθ √ γn r V θs,λ,m ( P V L, − m ) rθ = 2 πr B V θs,λ,m ( P V L, − m ) rθ (cid:12)(cid:12)(cid:12)(cid:12) r = r B (E.17)In the large r B limit, from eq.(2.27), we have( P V L, − m ) r,θ = − c − m | m | ( m − r B (E.18)From the eq.(E.2), we get V θs,λ,m = g θθ imξ λ,m − ∂ r ψ λ,m (cid:39) imr ξ λ,m − ∂ r ψ λ,m (E.19)which for either set of eigenvalues determined by eq.(E.15) or eq.(E.16) becomes, using theequations of motion, V θs,λ,m (cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:39) − ( λ + 2) ψ λ,m ( r B )2 r B (E.20)Then, using eq.(E.18) and eq.(E.20) in eq.(E.17), we get (cid:104) P V s,λ,m , P V L, − m (cid:105) = 2 π ( λ + 2) ψ λ,m ( r B )ˆ c − m | m | ( m − r B (E.21)The inner product of two metric deformations both corresponding to large diffeomorphismshas already been obtained in eq.(2.35) which in the asymptotic AdS limit becomes (cid:104) P V
L,m , P V L, − m (cid:105) (cid:39) π ˆ c m ˆ c − m | m | ( m −
1) (E.22)We now are left to calculate inner product of two metric perturbations corresponding to– 64 –mall diffeomorphisms. This is given by (cid:104)
P V s,λ ,m , P V s,λ ,m (cid:105) = λ δ λ ,λ δ m , − m (cid:104) V s,λ ,m , V s,λ ,m (cid:105) (E.23)The inner product of two small diffeomorphisms can be manipulated as follows. (cid:104) V s,λ,m , V s,λ, − m (cid:105) = (cid:90) d x √ g ( ∇ a ξ λ,m ( V s,λ, − m ) a + (cid:15) ab ∇ b ψ λ,m ( V s,λ, − m ) a )= − (cid:90) d x √ g ( ξ λ,m g ab + (cid:15) ab ψ λ,m ) ∇ b ( V s,λ, − m ) a + (cid:90) ∂ ds n b ( g ab ξ λ,m + (cid:15) ab ψ λ,m )( V s,λ, − m ) a = − (cid:90) d x √ g ( ξ λ,m ∇ ξ λ, − m + ψ λ,m ∇ ψ λ, − m ) + (cid:90) ∂ dθ r B n b ( (cid:15) ab ψ λ,m ( V s,λ, − m ) a )= − ( λ + 2) (cid:18)(cid:90) d x √ g ( ξ λ,m ξ λ, − m + ψ λ,m ψ λ, − m ) (cid:19) + 2 π r B ψ λ,m ( V s,λ, − m ) θ (cid:12)(cid:12)(cid:12)(cid:12) ∂ = − ( λ + 2) (cid:18) π r B ψ λ,m ψ λ, − m (cid:12)(cid:12)(cid:12)(cid:12) ∂ + (cid:90) d x √ g ( ξ λ,m ξ λ, − m + ψ λ,m ψ λ, − m ) (cid:19) (E.24)For either set of eigenvalues, only one of the term in the bulk integral dominates. Since,the bulk integral is positive definite we see that |(cid:104) P V s,λ,m , P V s,λ, − m (cid:105)| ≥ | ( λ + 2) r B ψ λ,m ( r B ) ψ λ, − m ( r B ) | (E.25)Putting together eq.(E.21),(E.22) and eq.(E.25), we find in the asymptotic AdS limit, that (cid:104) P V s , P V L (cid:105) (cid:112) (cid:104) P V s , P V s (cid:105)(cid:104) P V L , P V L (cid:105) ≤ O ( r − B ) (E.26)Let us now consider the case when there are some modes such that m ∼ r B (cid:29) . (E.27)This modes will need to be included when we are considering the general case with a bound-ary of finite length. Our estimates need to be revised for such modes. We will examinebelow the case where the eigenvalue λ (cid:28) m while m becomes big meeting eq.(E.27).We start with the analysis of the scalar field solution. For r ∼ O (1), and λ such that m (cid:29) λ , the equation eq.(E.5) for the scalar field ψ λ,m , with θ dependence being e imθ , is ∂ r (( r − ∂ r ψ λ,m ) − m r − ψ λ,m = 0 (E.28)the solution for which we take to be ψ λ,m = (cid:18) r − r + 1 (cid:19) | m | (E.29)– 65 –his solution can be extended till the region where r is such that1 (cid:28) r (cid:28) m λ (E.30)So, in this region, the solution eq.(E.29) becomes ψ λ,m = exp (cid:18) − | m | r (cid:19) (E.31)The scalar field equation, eq.(E.5), for r (cid:29) ∂ r ( r ∂ r ψ λ,m ) − m r ψ λ,m = (2 + λ ) ψ λ,m (E.32)the solution for which is given by ψ λ,m = (cid:114) | m | r ( c Γ(1 + v ) I v (cid:18) | m | r (cid:19) + c Γ(1 − v ) I − v (cid:18) | m | r (cid:19) ) (E.33)where I a is the modified Bessel function of the first kind and v is as defined in eq.(E.11).Matching this with the solution eq.(E.31) in the region 1 (cid:28) r (cid:28) m , we find c = Γ( v ) √ π , c = − c Γ(1 − v )Γ(1 + v ) (E.34)and so the scalar solution becomes ψ λ,m ( r ) = (cid:114) | m | π r cosec( πv ) (cid:18) I − v (cid:18) | m | r (cid:19) − I v (cid:18) | m | r (cid:19)(cid:19) = (cid:114) | m | πr K v (cid:18) | m | r (cid:19) (E.35)where K v is the modified Bessel function of the second kind. The scalar ξ λ.m will alsobehave in the same way. We also see from eq.(E.35) that the scalar field solution ψ λ,m isa function of the combination (cid:0) mr (cid:1) ∼ O (1) near the boundary. We also see from eq.(E.15),eq.(E.16) that the boundary conditions can be met when the relative coefficient between ψ λ,m , ξ λ,m is order unity.We can now estimate the magnitudes of the required quantities. First, from eq.(E.18)and (E.22), we find that ( P V L, − m ) rθ (cid:112) (cid:104) P V
L,m , P V L, − m (cid:105) ∼ O (cid:18) √ m (cid:19) (E.36)To estimate the magnitude of V s,λ,m , we note as mentioned above that scalar fields ψ λ,m , ξ λ,m in eq.(E.35) are a function of the combination (cid:0) mr (cid:1) ∼ O (1) near the boundary. This gives, V θs,λ,m = (cid:32) imr ξ λ,m r − ψ (cid:48) λ,m m (cid:33) ∼ O (cid:18) m (cid:19) (E.37)– 66 –here the prime is a derivative with respect to the quantity (cid:0) rm (cid:1) . To estimate the value of (cid:104) P V s,λ,m , P V s,λ, − m (cid:105) , we show using eq.(E.24) that this quantity is of O ( √ m ). To see this,consider the bulk integral in eq.(E.24) (cid:90) d x √ gψ λ,m ψ λ, − m = 2 π (cid:90) r c (cid:18) r − r + 1 (cid:19) | m | + (cid:90) r B r c ψ λ,m ψ λ, − m (cid:16) mr (cid:17) dr (cid:39) πm (cid:90) rBmrcm dxψ λ,m ( x ) ψ λ, − m ( x ) (cid:39) O ( m ) (E.38)where r c is such that 1 (cid:28) r c (cid:28) m . It is also easy to see, by noting eq.(E.37) that theboundary term is also of the same order as above and so (cid:113) (cid:104) P V s,λ,m , P V s,λ, − m (cid:105) (cid:39) O ( √ m ) (E.39)Thus putting together eq.(E.39), eq.(E.37) and eq.(E.36) and noting eq.(E.17), we findthat (cid:104) P V s , P V L (cid:105) (cid:112) (cid:104) P V s , P V s (cid:105)(cid:104) P V L , P V L (cid:105) (cid:39) O (1) (E.40)We can easily extend the above analysis for the case of Double trumpet topologyin Euclidean AdS spacetime discussed in 4. To evaluate the quantity in eq.(E.1), wewould proceed as before. The discussion till eq.(E.16) continues to hold true except thatthe eigenvalues are now determined by imposing either eq.(E.15) or eq.(E.16) at boththe boundaries. Taking the left and right boundaries to be located at r = − r B and r = r B respectively. Since the vector fields for small and large diffeomorphisms can bechosen so that the modes corresponding to left and right boundaries can be decoupled, thecorresponding boundary terms in eq.(E.17) will be independent of each other. The innerproduct of two large diffeomorphisms in the basis in eq.(4.15) is calculated in detail inappendix I.1, the final result appearing in eq.(I.11). The calculation of inner product ofsmall diffeomorphisms is again analogous to the disk case with the expression in eq.(E.24)interpreted as having two boundary terms which again are independent of each other. Thusit immediately follows that we will have the result analogous to eq.(E.26) in the presentcase of double trumpet topology also. F Estimation of various determinants in Euclidean AdS disk
In this section we shall discuss the computation of various determinants in detail in AdSspacetime for the Euler characteristic χ = 1 corresponding to the disk topology. To beginwith, we will compute the determinant of the scalar Laplacian. This requires the specifica-tion of appropriate boundary conditions which we take to be Dirichlet boundary conditions.We will mostly consider the case when the boundary has large length l ∼ (cid:15) (cid:29) (cid:15) →
0, we will find that a length dependent counter-termneeds to be added to get a finite result, and that the dependence on large diffeomorphismsvanishes. We will make essential use of the conformal anomaly in the analysis.Similar results will also be obtained for det (cid:48) ( P † P ). For det( − ˆ ∇ + 2), on generalgrounds, upto O ( (cid:15) ), the dependence of the large diffeomorphisms will be shown to be ofthe form of the Schwarzian action with a coefficient which is linear in (cid:15) , but we will not beable to obtain the precise value of this coefficient.Coming back to the scalar case, we are interested in the dependence of det( − ˆ ∇ ) onthe large diffeomorphisms. On general grounds this dependence should be a functionalof Diff( S ) /SL (2 , R ), since it is easy to see that diffeomorphisms lying in the SL (2 , R )isometry group of AdS must leave the determinant unchanged . This imposes a strongrestriction on the kind of terms that appear in the final result for the determinant. Thesimplest such term that one can write is proportional to length of the boundary. Thenext term, which involves two derivative with respect to u , eq.(2.42) - the rescaled propertime along the boundary- is uniquely given by the Schwarzian action. On dimensionalgrounds its coefficient must go like (cid:15) , and by using the conformal anomaly we can obtainthe coefficient in front of this action as we show below. Beyond this, in general additionalterms will also be present - these will involve additional derivatives of u and correspondinglyadditional powers of (cid:15) . If we consider modes whose mode number m , eq.(2.26) is smallenough meeting the condition m(cid:15) (cid:28) , (F.1)so that their wavelength meets the condition, Λ (cid:29) R AdS , eq.(2.56), then these additionalterms will be suppressed. For modes of higher mode number where eq.(F.1) is not metthese higher order terms must all be retained and the resulting behaviour of the determi-nant is much more non-trivial to obtain. These arguments can also be applied to the doubletrumpet with two boundaries and de Sitter case when we calculate the no-boundary wave-function by analytic continuation from the (2 ,
0) or (0 ,
2) signature metrics as discussed in5.1. Let us now show how the conformal anomaly can be used to obtain the coefficient ofthe first two terms mentioned above, involving the boundary length and the Schwarzianderivative. We can expand the determinantln det( − ˆ ∇ ) = c (cid:90) ds + (cid:15) c (cid:90) du Sch (cid:18) tan (cid:18) θ ( u )2 (cid:19) , u (cid:19) + O ( (cid:15) ) (F.2)where u is renormalized boundary proper time, eq.(2.42), and θ is the coordinate appearingin the line element of the AdS metric in eq.(2.24). Our task now simplifies to evaluatingthe constants c and c . To fix the constants we consider a non-wavy boundary specified To be very explicit, suppose the initial metric is taken as eq.(2.24) with the boundary specified as r = r B . Let the new coordinates after the SL (2 , R ) transformation be ˜ r, ˜ θ . The boundary is now specifiedto be ˜ r = r B . Since neither the metric nor the specification of the boundary in terms of the coordinate haschanged, it naturally follows that the value of the determinant of the laplacian operator will not change. – 68 –y r = r B , and use the conformal transformation property of the determinant to fix itsdependence on the value r B . For this it is convenient to work in the coordinate system ˆ r, θ in which the line element is given by eq.(A.11). ds = 4(1 − ˆ r ) ( d ˆ r + ˆ r dθ ) , ˆ r ∈ [0 ,
1] (F.3)The boundary r = r B in the metric eq.(2.24) is specified in terms of ˆ r coordinate asˆ r = ˆ r B = (cid:114) r B − r B + 1 (F.4)Defining the coordinate ¯ ρ as ˆ r = ¯ ρ ˆ r B , we find that the boundary specification now becomes¯ ρ = 1. The line element becomes ds = ˆ g ab dx a dx b = 4ˆ r B (1 − ˆ r B ¯ ρ ) ( d ¯ ρ + ¯ ρ dθ ) ≡ e σ ¯ g ab dx a dx b (F.5)where ¯ g ab is defined as d ¯ s = ¯ g ab dx a dx b = d ¯ ρ + ¯ ρ dθ , ¯ ρ ∈ [0 , , θ ∈ [0 , π ] (F.6)where as mentioned earlier the boundary is located at ¯ ρ = 1. It is easy to see from eq.(F.5)that r B -dependence is entirely in the conformal factor with the flat metric ¯ g ab independentof r B . Now, we note that the conformal transformation property of the determinant of ascalar laplacian with Dirichlet boundary conditions for conformally related metricsˆ g ab = e σ ¯ g ab (F.7)is given by det( − ˆ ∇ ) ˆ g det( − ¯ ∇ ) ¯ g = exp {− S σ } (F.8)where S σ is given by S σ = 16 π (cid:20) (cid:90) d x √ ¯ g (¯ g ab ∂ a σ∂ b σ + ¯ Rσ ) + (cid:90) ∂ d ¯ s ¯ Kσ (cid:21) (F.9)From eq.(F.5) and eq.(F.6), we note that e σ = 4ˆ r B (1 − ˆ r B ¯ ρ ) , ¯ R = 0 (F.10)and normal vector to the boundary normalized with ¯ g ab and the corresponding extrinsic– 69 –race are ¯ n ¯ ρ = 1 , ¯ K = 1¯ ρ = 1 , ˆ K = e − σ ( ¯ K + ¯ n a ∂ a σ ) = 12ˆ r B + ˆ r B − ˆ ∇ ) ˆ g det( − ¯ ∇ ) ¯ g = exp (cid:26) − (cid:18) ˆ r B − ˆ r B (cid:19) −
13 ln 2ˆ r B (cid:27) (F.12)Now, take the case where r B = (cid:15) (cid:28)
1, so that the boundary length l (cid:29)
1. To be moreprecise, this (cid:15) defined by the boundary value of r B is the same as that in eq.(2.42) toleading order, there will be subleading corrections between the two variables. However, weshall work, for now, consistently with it being defined as defined by the value of r B . But,Using eq.(F.4) we get det( − ˆ ∇ ) ˆ g det( − ¯ ∇ ) ¯ g = exp (cid:18) − (cid:15) + 16 ln (cid:18) e (cid:19) + (cid:15) (cid:19) (F.13)where we used eq.(2.8) to obtain final equality. Note that det( − ¯ ∇ ) ¯ g is some constantindependent of r B and hence (cid:15) . From eq.(F.2), for the boundary at r B = (cid:15) − , we getdet( − ˆ ∇ ) = exp (cid:18) πc (cid:15) + π(cid:15) ( c − c ) (cid:19) (F.14)So comparing, we get c = − π , c = 14 π (F.15)and so we haveln det( − ˆ ∇ ) = − π (cid:90) ds + (cid:15) π (cid:90) du Sch (cid:18) tan (cid:18) θ ( u )2 (cid:19) , u (cid:19) + O ( (cid:15) ) (F.16)Note that the dimensional analysis we had mentioned above which fixes the powersof (cid:15) in each term in eq.(F.14) can be understood as follows. The line element eq.(2.42)is invariant under ( (cid:15), u ) → ( λ(cid:15), λu ). Under this rescaling (with tan( θ/
2) unchanged )the Schwarzian term, Sch(tan( θ/ , u ) → λ Sch(tan( θ/ , u ), while the line element ds isinvariant. This fixes the powers of (cid:15) appearing in the coefficients. Also note that afteradding a counter term to cancel the length dependent first term in eq.(F.16) which goeslike 1 /(cid:15) , we get are left with the Schwarzian term and additional subleading correctionswhich all vanish in the asymptotic AdS limit where (cid:15) → S OS for the J T gravity in the presence of matter fields satisfying vanishing Dirichlet boundaryconditions at finite temperature in the semi-classical limit, G → , N → ∞ with fixed GN – 70 –s given by S OS = S JT,∂ + S M,qm (F.17)where S JT,∂ is given by first line in eq.(2.65) and S M,qm is given by S M,qm = N − ˆ ∇ ) . (F.18)For the finite temperature case taking θ ( u ) = πβ u, φ B = J(cid:15) + GN , and introducing thecounterterm mentioned to cancel the first term in det( − ˆ ∇ ) which is length-dependent, wesee that the value of the on-shell action becomes S OS = − π GJ β (cid:18) − GN J (cid:15) (cid:19) (F.19)which indeed matches with the results in [18].We should alert the reader to an important issue connected to the above calculation.The formula relating the scalar laplacian determinant for conformally related metrics givenin eq.(F.9) is different from the one appearing in [121] by an extra term∆ S σ = − π (cid:90) ˆ Kds. (F.20)Indeed, in general, the bulk conformal anomaly and Wess-Zumino consistency conditions[122] fix the form of the action S σ completely upto the possibility of an additional term ofthis type. While [121] do report that such a term arises for the determinant with Dirichletboundary conditions we find that its presence leads to disagreement with the semi-classicalresults in [18], and have accordingly not included it here.We now extend the above considerations to compute the value of the determinant of theoperator P † P . This is in fact straightforward. Once again, we can expand the determinantin powers of (cid:15) , with the first two terms being,ln det (cid:48) ( P † P ) = k (cid:90) ds + k (cid:15) (cid:90) Sch (cid:18) tan (cid:18) θ ( u )2 (cid:19) , u (cid:19) (F.21)We will now use the same trick of considering a non-wavy boundary and use the conformaltransformation property of det (cid:48) P † P to compute its r B dependence and then match thecoefficients by expanding in (cid:15) where r B = (cid:15) . For conformally related metrics in eq.(F.7)the determinants of the operator P † P are related as [121](det (cid:48) P † P ) ˆ g (det (cid:48) P † P ) ¯ g = exp {− S σ } (F.22)We note that only the prefactor in the exponent in eq.(F.22) is different from the scalar casedue to the difference in the central charges . So again by comparing eq.(F.21) and eq.(F.22) We must mention though that we have not been too careful about the possible presence of a counter – 71 –or the geometry, eq.(2.24) with boundary at r B = (cid:15) , we get, upto an (cid:15) independentprefactor which we are not retaining,det (cid:48) ( P † P ) = exp (cid:18) − π (cid:90) ds + 264 π (cid:15) (cid:90) Sch (cid:18) tan (cid:18) θ ( u )2 (cid:19) , u (cid:19)(cid:19) (F.23)This is of the same form as in the scalar case and once the first term is removed by asuitable counter term again vanishes in the (cid:15) → − ˆ ∇ + 2) is more complicated. Since it arises after doing thepath integral for a massive scalar of mass 2, we cannot use the conformal anomaly to obtainuseful information about it . However, we can still argue from the requirement that thedeterminant is valued in Diff( S ) /SL (2 , R ) that it can be expanded asln det( − ˆ ∇ + 2) = q (cid:90) ds + q (cid:15) π (cid:90) du Sch (cid:18) tan (cid:18) θ ( u )2 (cid:19) , u (cid:19) (F.24)where q , q ∼ O (1) constants. Once again it then follows that the dependence on the largediffeomorphisms vanishes in the asymptotic AdS limit.To reiterate a point made earlier, note that the length-dependent terms in variousdeterminants eq.(F.16), eq.(F.23) and eq.(F.24) grow like (cid:15) , i.e. linearly in the length ofthe boundary, and thus diverge in the asymptotic AdS limit, (cid:15) →
0. To obtain a finiteresult in this limit we need to add a boundary term to the JT action, eq.(2.51), δS JT,∂ = A (cid:90) ∂ ds (F.25)and fix the constant A appropriately so as to cancel this divergence. F.1 Asymptotic AdS limit case
We now come to a subtlety having to do with the order in which various limits are beingtaken while evaluating the determinants. The computations above of the determinantsused the Weyl anomaly and are valid for an arbitrary boundary. We see above that theresult at leading order in the length l for the determinants go likedet ˆ O ∼ e ( Cl ) (F.26)where C is a constant which depends on the operator ˆ O . More precisely the determinantsabove are obtained for the case of a given geometry with a finite length boundary byintroducing a cut-off (for large eigenvalues ), regulating the product of eigenvalues of therelevant operators and then taking the cut-off to infinity in a manner which is consistentwith the Weyl anomaly. In contrast, in the asymptotic AdS limit, as was mentioned above,we are interested in first taking the limit when the boundary length l → ∞ , keeping the cut-off on eigenvalues fixed, and thereafter taking the limit where this cut-off goes to infinity term of the form eq.(F.20) in eq.(F.22). Also there could be some subtleties due to zero modes. – 72 – . Here we will show that this second order of limits can give a different result, andin particular the leading term, eq.(F.26), which is exponential in l can be absent in theasymptotic AdS limit. To carry out the calculations in this limit we will use a method firstdiscussed by Coleman [123].Let us illustrate this method for the Simple Harmonic Oscillator (SHO). Consider twosimple harmonic oscillators with frequencies w and w constrained to move between x = 0and x = L . Let ψ (1) λ be the solution to the equation( − ∂ x + w − λ ) ψ (1) λ ( x ) = 0 (F.27)with the appropriate boundary conditions. ψ (2) λ satisfies a similar equation with the fre-quency w . Suppose we impose Dirichlet boundary conditions at both ends. Coleman’sformula then states that det( − ∂ x + w − λ )det( − ∂ x + w − λ ) = ψ (1) λ ( L ) ψ (2) λ ( L ) (F.28)In the above formula the left and right hand sides are to be regarded as a function of thecomplex variable λ . The formula follows from noting that the zeros and poles of the leftand right hand slides are the same, and that both sides go to unity as λ → ∞ in anydirection except the real axis. It then also follows that upto a constant independent of ω det( − ∂ x + w ) = ψ ( λ =0) ( L ) (F.29)where the RHS is the value of the wavefunction obtained at x = L for the operator withfrequency ω by starting at x = 0 with the correct boundary conditions. One importantpoint to note here is that the normalization of the solution ψ λ ( x ) should be fixed in sucha way that when viewed as a function of λ , any spurious zeros or poles, other than thosecorresponding to the actual eigenvalues of the operator ( − ∂ x + ω ) in the solution ψ λ ( x )are cancelled and that the ratio of two solutions with different frequencies goes to unityas λ → ∞ in any direction other than along the real axis . The solution ψ λ =0 is thenobtained by taking the λ = 0 limit of this appropriately normalized ψ λ solution.A similar formula also follows for Neumann boundary conditions or mixed boundaryconditions where we set aψ + bψ (cid:48) = 0 (at say both x = 0 , L ) with eq.(F.29) being replacedby det( − ∂ x + w ) = aψ ( L ) + bψ (cid:48) ( L ) . (F.30)where the subscript refers to taking the λ = 0 solution as in eq.(F.29). The normalizationof the solution ψ is fixed as explained before.We will now adopt the same strategy to calculate the determinants det( − ˆ ∇ ), det( − ˆ ∇ + As we will see below after expanding modes along the θ direction in their Fourier components e imθ , thecut-off on the eigenvalues of relevance will be on the mode number m . In the present case we achieve this by taking ∂ x ψ ( x = 0) = 1. – 73 –) and det (cid:48) ( P † P ), in the case of the asymptotic AdS spacetime. In applying this method tothe AdS case we expand the modes for the operator in the basis of modes in the θ direction, e imθ , and then working for any fixed value of m obtain a one-dimensional problem in theradial direction. In this one -dimensional problem we take the boundary to go to infinity l → ∞ and then use the Coleman method to obtain the determinant of the radial operator.The full determinant is then be obtained by taking the product of contributions over allvalues of the mode number m , and then taking in this product | m | → ∞ . We see thereforethat in the calculations below while working in the asymptotic AdS limit with the order oflimits mentioned above, first taking l → ∞ and then taking mode number m → ∞ .Let us first compute det( − ˆ ∇ ) in this manner. The boundary condition at x = 0 inthe SHO is now replaced by the requirement of the regularity of the solution in the interior.The solution to the eigenvalue equation ˆ ∇ ψ λ = − λψ λ for fixed mode number m which isregular everywhere in the interior, chosen such that the ratio of two solutions with differentmode numbers goes to unity as | λ | → ∞ other than along the real axis, and without haveany spurious zeros, poles or branch points in the variable λ , is given by ψ λ,m = P −| m | v − ( r ) , m (cid:54) = 0 P v − ( r ) , m = 0 (F.31) v = 12 √ − λ and P βα is the associated Legendre function of the first kind. In fact, using the asymptoticform of the P −| m | v − ( r ) , P −| m | v − ( r ) = (cid:32) (2 r ) − v − Γ( − v ) √ π Γ (cid:0) − v + | m | + (cid:1) (cid:33) + O (cid:16) r − v − / (cid:17) + (cid:32) (2 r ) v − Γ( v ) √ π Γ (cid:0) v + | m | + (cid:1) (cid:33) + O (cid:16) r v − / (cid:17) (F.32)we see that the asymptotic form has v → − v symmetry. For the case of Dirichlet boundarycondition at r = r B , the eigenvalues are obtained by solving ψ λ,m ( r B ) = 0 (F.33)Let the eigenvalue be labelled by λ m,n . The subscript m in λ m,n denotes mode number m and the n labels the various eigenvalues for this particular mode number. To computethe determinant in the asymptotic AdS limit we take the asymptotic form of the solutioneq.(F.37), with λ = 0, which is given by ψ ,m = (cid:40) | m | +1) − | m | ) r + O ( r − ) m (cid:54) = 01 m = 0 (F.34) We have followed the conventions of [124] in obtaining the asymptotic forms here and elsewhere in thismanuscript. – 74 –sing this asymptotic form, the determinant computed using eq.(F.28) readsln det( − ˆ ∇ ) = − ∞ (cid:88) m = −∞ ,m (cid:54) =0 ln(Γ( | m | + 1)) (F.35)which is manifestly independent of r B .Next consider the determinant for the operator ( − ˆ ∇ + 2). The eigenvalue equationfor the operator reads ( ˆ ∇ − ψ λ = − λψ λ (F.36)which for the mode number m now has the regular solution ψ λ,m = P −| m | v − ( r ) , m (cid:54) = 0 P v − ( r ) , m = 0 (F.37) v = 12 √ − λ To compute the determinant in the asymptotic AdS limit we take the asymptotic form ofthe solution eq.(F.37), with λ = 0, which is given by ψ ,m = | m | +2) r − | m | r Γ( | m | ) + O (cid:0) r − (cid:1) m (cid:54) = 0 r m = 0 (F.38)We now use this to compute det (cid:16) − ˆ ∇ + 2 (cid:17) with the Dirichlet boundary condition eq.(F.33)using eq.(F.28). This givesln det( − ˆ ∇ + 2) = − ∞ (cid:88) m =1 ln(Γ(2 + | m | )) + m = ∞ (cid:88) m = −∞ ln r B (F.39)which is independent of r B since (cid:80) m = ∞ m = −∞ ln r B = 0.Thus we see that for both the operators considered above, we get no dependencegrowing exponentially as in eq.(F.26), in the asymptotic AdS limit. For det( − ˆ ∇ ) we sawin the previous subsections that such a dependence does arise when we consider a differentorder of limits.The computation of the determinant det (cid:48) P † P is very similar. Let us first study thecase when the index v in eq.(F.37) is imaginary. We shall later see that there exist oneeigenvalue when v is real. For v imaginary, the main difference in this computation wouldbe the boundary conditions on the scalar field. For a general vector field decomposed as ineq.(2.22) the boundary conditions eq.(2.19), in the asymptotic AdS limit gives two possible– 75 –onditions on the scalar fields.These are ∂ r ψ λ,m = 0 , ξ λ,m ∼ ψ λ,m r B ∂ r ξ λ,m = 0 , ψ λ,m ∼ ξ λ,m r B (F.40)The determinant of the eigenvalues determined by the second of the condition above, ∂ r ξ λ,m =0, is straightforward to compute. The appropriately normalized solution is again given byeq.(F.38) and so, the product of these eigenvalues is given by taking the derivative ofeq.(F.38) which still gives eq.(F.39).The contribution from the other set of eigenvalues ∂ r ψ λ,m = 0 are more complicated.First, let us note that even though the boundary condition is a second order equation,it can be understood as a mixed boundary condition of the form eq.(F.30) upon usingthe eigenvalue equation eq.(F.36) for ψ λ,m . The contribution from the modes m (cid:54) = 0 isstraightforward to obtain by taking the second derivative of the corresponding asymptoticexpression in eq.(F.38). However, for m = 0 mode, since the solution is just ψ , = r ,taking the double derivative gives zero, which using the analog of eq.(F.29) then showsthat the determinant for the m = 0 sector is zero. The zero mode is in fact ψ , itselfand it corresponds to the U (1) isometry of AdS under which θ → θ + c . We are actuallyinterested in evaluating det (cid:48) ( P † P ) and would therefore need to evaluate the determinantwithout the zero eigenmode.We have not been able to find a fully satisfactory way of dealing with this complication.One might hope to proceed as follows. We consider in the m = 0 sector the operatordet( − ˆ ∇ + 2 − λ ) for non-zero λ and then take the λ → λ thearguments above lead to the conclusion thatdet( − ˆ ∇ + 2 − λ ) λ = ∂ r ψ λ, λ (F.41)and so we get det( − ˆ ∇ + 2) = lim λ → ∂ r ψ λ, λ (F.42)Computing the solution ψ λ, in a perturbation series in λ near λ = 0 by imposing regularitynear the origin, we get the solution to O ( λ ) to be ψ λ, = r + λ − r ln(1 + r )) (F.43)and hence lim λ → ∂ r ψ λ, λ (cid:39) − r B (F.44)– 76 –o, the net value of the determinant det (cid:48) ( P † P ) becomesln det (cid:48) ( P † P ) = − ∞ (cid:88) m =1 ln(Γ(2 + | m | )) + 2 ∞ (cid:88) m =1 ln (cid:32) − | m | | m | ) r B (cid:33) + ln (cid:18) − r B (cid:19) (F.45)The resulting r B dependence, after doing the sum by zeta function regularization, is thengiven by ln det (cid:48) ( P † P ) = 2 ln r B (F.46)This is not a very satisfactory result though since in the r B → ∞ limit the resultingdivergence in the determinant cannot be removed by a local counter-term (unlike for aterm which is growing linearly with r B ). We leave a proper resolution of this puzzle forthe future.Let us end with some comments. It is easy to see that there exists one additionaldiscrete eigenvalue when v = corresponding to λ = 2. Consider the scalar fields ψ λ,m , ξ λ,m that satisfy the equation ˆ ∇ ψ λ,m = 0 = ˆ ∇ ξ λ,m with mode number m , the regular solutionsfor which are taken to be ψ λ,m = α m e imθ (cid:18) r − r + 1 (cid:19) | m | , ξ λ,m = β m e imθ (cid:18) r − r + 1 (cid:19) | m | (F.47)It is then easy to see that near the boundary r (cid:29)
1, with the choice of constants β m = i sgn( m ) α m , the boundary conditions (2.19) are satisfied thus showing that λ = 2 is a gen-uine eigenvalue, which does not belong to the either of the sets of eigenvalues in eq.(F.40).Also, note that this discrete eigenvalue exists only for m (cid:54) = 0, since when m = 0 the vec-tor field constructed out of these scalar fields vanishes identically everywhere. Further,there are no other eigenvalues apart from the ones we have obtained so far. Including thiseigenvalue of course does not change the r B dependence obtained above.The calculation of determinants discussed in this subsection can be easily extended tothe general case when the boundary is located at large but finite value of r B , and also tode Sitter spacetime. G Matter coupling to the time reparametrization modes in AdS
In this appendix we will describe in more detail the coupling of the matter to the timereparametrization modes in the classical action eq.(3.9). The result can be obtained fora general boundary of length l but for simplicity we will work out the case l (cid:29) r B given by eq.(5.15)in terms of l , so that r B (cid:29)
1. We next turn on a large diffeomorphism. Under such adiffeomorphism the new coordinates asymptotically close to the boundary are given by˜ θ = f ( θ ) , ˜ r = r/f (cid:48) ( θ ) (G.1)– 77 –ith the boundary lying at ˜ r = r B . f (cid:48) ( θ ) denotes a derivative of f ( θ ) with respect to θ .It is easy to see from our definition of the rescaled proper time u , eq.(2.42), and eq.(2.33)that u = ˜ θ = f ( θ ) (G.2)The infinitesimal version of these transformations follows from eq.(2.28) and is discussedin eq.(2.44) eq.(2.45). Note that the coordinate r varies along the resulting wavy boundaryas r ( θ ) = r B f (cid:48) ( θ ) (G.3)We will consider one massless scalar ϕ here. A general solution to the massless scalarequation ∇ ϕ = 0 is given by ϕ (0) ( r, θ ) = (cid:88) m p m e imθ (cid:18) r − r + 1 (cid:19) | m | (G.4)where p m are coefficients which are fixed by the form of ϕ at the boundary. Near theboundary at large r we get ϕ (0) = (cid:88) m p m e imθ (cid:18) − | m | r (cid:19) ≡ ϕ − ( θ ) − r ϕ + ( θ ) (G.5)where ϕ − ( θ ) = (cid:88) m p m e imθ , ϕ + = (cid:88) m | m | p m e imθ = (cid:90) dθ (cid:48) F ( θ, θ (cid:48) ) ϕ − ( θ (cid:48) ) (G.6)and F ( θ, θ (cid:48) ) = ∞ (cid:88) m = −∞ | m | e im ( θ − θ (cid:48) ) (G.7)If ϕ is given by the function ˆ ϕ ( u ) along the boundary, with u being the rescaled properlength as above, then we get (to leading order ) ϕ − ( θ ) = (cid:88) m p m e imθ = ˆ ϕ ( f ( θ )) (G.8)which determines the Fourier coefficients p m in terms of the functions ˆ ϕ and f . It is easyto see that the classical action for the scalar S = 12 (cid:90) √ gd x ( ∂ϕ ) (G.9)reduces on shell to a boundary term, S = 12 (cid:90) ds ϕn µ ∂ µ ϕ (G.10)– 78 –here ds is the line element along the boundary and n µ the unit normal. This gives S = 12 (cid:90) dθ dθ ϕ − ( θ ) ϕ − ( θ ) F ( θ , θ ) (G.11)where ϕ − ( θ ) is given in terms of the boundary function ˆ ϕ ( u ) and f ( u ) by eq.(G.8). Invertingeq.(G.2) we can express θ as a function of uθ ( u ) = f − ( u ) (G.12)which allows us to also express eq.(G.11) as S = 12 (cid:90) du du θ (cid:48) ( u ) θ (cid:48) ( u ) ˆ ϕ ( u ) ˆ ϕ ( u ) F ( θ ( u ) , θ ( u )) (G.13)This gives the classical action in terms of the boundary time reparametrizations specifiedby θ ( u ) and the boundary value of the scalar ˆ ϕ ( u ).At linear order in the diffeomorphisms we have ˜ θ given in terms of θ in eq.(2.44), (2.45),inserting this in eq.(G.13) gives S = (cid:90) ∂ du du ˆ ϕ ( u ) ˆ ϕ ( u )( δθ (cid:48) ( u ) F ( u , u ) + δθ ( u ) ∂ u F ( u , u )) (G.14)This result agrees with (3.12) in subsection 3, after being generalised to N scalar fields.Also, for the discussion in subsection (3.1) these formulas need to be extended to O ( (cid:15) )if we are to include the dependence on the large diffeomorphisms coming from the quantumpart, i.e. the scalar laplacian determinant in (3.10). This can be done in a straightforwardfashion along the lines above, but we spare the reader the details.Now, in the case of de Sitter spacetime, the matter coupling to the time reparametriza-tion modes can be obtained in an almost similar fashion. The massless scalar field equationin the coordinate system eq.(5.12) now has the solution ϕ ( r, θ ) = (cid:88) m p m (cid:18) r − ir + i (cid:19) | m | e imθ (G.15)This in the limit r (cid:29) ϕ ( r, θ ) = (cid:88) m p m e imθ (cid:18) − i | m | r (cid:19) = ϕ − ( θ ) − ir ϕ + ( θ ) (G.16)where ϕ − , ϕ + are as before in eq.(G.5). The matter coupling can then be obtained in amanner analogous to that in the AdS case above. Doing so, we get S M,cl = i (cid:90) dθ dθ ϕ − ( θ ) ϕ − ( θ ) F ( θ , θ ) (G.17)where F is as defined in eq.(G.7). So, comparing eq.(G.11) and (G.17), we see that the– 79 –xpressions upto the factor of i and so the linearized version in eq.(G.14) will also have anadditional factor of i . H de Sitter wavefunction using Euclidean AdS contour
In this appendix we calculate the wavefunction for the de Sitter spacetime in the non-asymptotic limit, by considering modes which have m > l where l is the length of theboundary. Although in such a case we need to carefully calculate various quantities suchas determinants, measure for large and small diffeomorphisms which does not decouple,we ignore all such subtleties and evaluate the measure for large diffeomorphisms and tryto do the path integral. To evaluate the wavefunction, we follow the Maldacena contour,described in subsection 5.1 and so we first do the computation in the negative AdS metricof signature (0,2) and then analytically continue to the (1,1) de Sitter spacetime. Considerthe metric given by ds = − ( dτ + sinh τ dθ ) (H.1)It is easy to compute the Ricci scalar for this metric which turns out to have the value R = 2. This metric is the negative of the AdS metric written in global coordinates. Asbefore, we find that the zero modes of the operator P † P are given by the vector field aseither the gradient or the curl of a scalar which satisfies the scalar Laplacian equation inthe background eq.(H.1). So, we get the vector field as V a = (cid:15) ab ∇ b ψ, ψ = − e imθ ˆ c m ( | m | + r ) (cid:18) r − r + 1 (cid:19) | m | , r = cosh τ (H.2)where (cid:15) = − | √ g | and ˆ c − m = ˆ c ∗ m so that the scalar field and the vector field constructedout of it is real. The components of the vector field written explicitly are V a = (cid:32) i ˆ c m me imθ ( | m | + cosh τ ) tanh | m | (cid:0) τ (cid:1) sinh τ , − ˆ c m e imθ (cid:0) | m | cosh τ + m + sinh τ (cid:1) tanh | m | (cid:0) τ (cid:1) sinh τ (cid:33) (H.3)The coefficients The corresponding metric perturbations are given by δg ττ = − i ˆ c m m ( m − τ tanh | m | (cid:16) τ (cid:17) e imθ δg τθ =2ˆ c m ( m − | m | csch τ tanh | m | (cid:16) τ (cid:17) e imθ δg θθ =2 i ˆ c m m ( m −
1) tanh | m | (cid:16) τ (cid:17) e imθ (H.4)– 80 –or an arbitrary τ = τ , we have the boundary term for the inner product of two metricperturbations analogous to that of the eq.(2.17) to be (cid:104) P V
L,m , P V L, − m (cid:105) =2 (cid:90) τ = τ dθ √ gg ττ ( V τ δg ττ + V θ δg τθ )= (cid:88) | m | > π ˆ c m ˆ c − m | m | ( m −
1) tanh | m | (cid:16) τ (cid:17) (cid:16) | m | csch ( τ ) + 1 + 2 | m | coth( τ )csch( τ ) (cid:17) (H.5)So the measure for the path integral over ˆ c m is given byˆ M = (cid:88) m> π ˆ c m ˆ c − m | m | ( m −
1) tanh | m | (cid:16) τ (cid:17) (cid:16) | m | csch ( τ ) + 1 + 2 | m | coth( τ )csch( τ ) (cid:17) (H.6)The extrinsic for the fluctuations around the saddle τ = τ is obtained in eq.(B.33), whichexpressed in terms of an expansion δθ ( u ) = (cid:88) m ˜ c m e imu (H.7)is given by δK (2) = − (cid:88) m ≥ ˜ c − m ˜ c m m (cid:0) m − (cid:1) tanh τ sech τ (H.8)where the superscript on δK is to indicate that this is the quadratic term in time reparametriza-tion modes ˜ c m . The relation between ˆ c m and ˆ c m is obtained by noting that δθ = V θ andis given by ˜ c m = − ˆ c m csch τ (cid:0) | m | cosh τ + sinh τ + m (cid:1) tanh | m | (cid:16) τ (cid:17) (H.9)Using this to find the quadratic action in terms of the variables ˆ c m , we obtain δK (2) = − (cid:88) m ≥ ˆ c m ˆ c − m | m | ( m − τ sech τ (cid:0) | m | cosh τ + sinh τ + m (cid:1) tanh | m | (cid:16) τ (cid:17) (H.10)The path integral over the modes ˆ c m is given by,Ψ nAdS ,Ψ nAdS = exp (cid:20) φ B cosh τ πG (cid:21) (cid:90) ˆ M d ˆ c m d ˆ c ∗ m exp (cid:26) ˜ γ (cid:90) π du φ B sinh τ δK (2) (cid:27) = exp (cid:20) φ B cosh τ πG (cid:21) (cid:88) m> π cosh τ (cid:0) sinh τ + m + | m | cosh τ (cid:1) ˜ γmφ B (cid:0) | m | cosh τ + m + sinh τ (cid:1) (H.11)where u is related to the proper time on the boundary and is defined through the relationeq.(B.34). Note that we have ignored the contribution from the topological term eq.(2.3)and the exponential prefactor above is the classical contribution coming from the leadingterm in the extrinsic trace eq.(B.33). The sum in eq.(H.11) can be regulated using zeta-– 81 –unction regularization. Defining the variables α, m , m , m , m , as α = 32 π cosh τ ˜ γφ B , ˜ γ = 18 πGm = 14 (cid:16) − τ − √ (cid:112) − cosh(2 τ ) (cid:17) , m = 14 (cid:16) √ (cid:112) − cosh(2 τ ) − τ (cid:17) m = 14 (cid:16) − τ − (cid:112) − τ ) (cid:17) , m = 14 (cid:16)(cid:112) − τ ) − τ (cid:17) (H.12)we get the regularized value to beΨ nAdS = exp[˜ γφ B cosh τ ] (cid:112) (2 πα ) Γ(2 − m ) Γ(2 − m ) Γ(2 − m )Γ(2 − m ) (H.13)For evaluating large τ (cid:29) πe τ = l , and so we get, using theStirling approximation for the Gamma functions,Ψ nAdS τ (cid:29) −−−→ l √ π (cid:114) (˜ γφ B ) l exp (cid:18) ˜ γφ B l π + l π ln (cid:18) lπ √ e (cid:19) + l (cid:18) − √ (cid:19)(cid:19) (H.14)Now doing a continuation to the Lorentzian de Sitter by taking τ → τ ± iπ ⇒ l → ± il (H.15)we getΨ dS = ± l πi (cid:115) (˜ γφ B ) ( ± il ) exp (cid:18) − l ± ˜ γφ B l π ± il π ln (cid:18) lπ √ e (cid:19) ± il (cid:18) − √ (cid:19)(cid:19) (H.16)We see that there is an exponential damping term for large l . The ± signs in the aboveexpression correspond to the ± signs in eq.(H.15) for the different ways of analytic continu-ation. As can be seen from eq.(H.16), we find that the exponential damping is independentof the choice of analytic continuation. Moreover this exponential damping cannot be re-moved by adding a length-dependent counterterm with a real coefficient as that would havean explicit factor of i as in the action eq.(5.1). I More on AdS double trumpet calculations
I.1 Measure for large diffeomorphisms and Schwarzian action
In this appendix we will elaborate more on the calculation of the measure for the largediffeomorphisms in the double trumpet topology and also show the calculation of theSchwarzian action in explicit detail. The line element is given by eq.(4.2) The solutions forthe scalar field ψ satisfying eq.(2.23) is given in eq.(4.15). We will use the form in eq.(4.9)to calculate the measure and the form in eq.(4.15) to evaluate the Schwarzian action andfinally relate them using eq.(4.16). We can now construct the vector field corresponding– 82 –o the large diffeomorphisms. In the disk topology, the modes m = ± , m = 0 mode. So, the largediffeomorphisms correspond to modes with | m | ≥
1. The components of the vector fieldeq.(4.8) computed in terms of the solution eq.(4.9) is given by V rL = ∂ θ ψ = i ˜ me i ˜ mθ (cid:18) r − ir + i (cid:19) − ( i ˜ m ) (cid:32) A m ( ˜ m + r ) + B m ( r − ˜ m ) (cid:18) r − ir + i (cid:19) i ˜ m (cid:33) V θL = − ∂ r ψ = − e i ˜ mθ r + 1 (cid:18) r − ir + i (cid:19) − ( i ˜ m ) (cid:32) A m (cid:0) ˜ m + ˜ mr + r + 1 (cid:1) + B m (cid:0) ˜ m − ˜ mr + r + 1 (cid:1) (cid:18) r − ir + i (cid:19) i ˜ m (cid:33) (I.1)The metric perturbations obtained by δg ab = ( P V ) ab is given by( P V L ) rr = 2 i (cid:0) ˜ m + ˜ m (cid:1) ( r + 1) e i ˜ mθ (cid:18) r − ir + i (cid:19) − ( i ˜ m ) (cid:32) A m + B m (cid:18) r − ir + i (cid:19) i ˜ m (cid:33) ( P V L ) rθ = − (cid:0) ˜ m + ˜ m (cid:1) r + 1 e i ˜ mθ (cid:18) r − ir + i (cid:19) − ( i ˜ m ) (cid:32) A m − B m (cid:18) r − ir + i (cid:19) i ˜ m (cid:33) ( P V L ) θθ = − i (cid:0) ˜ m + ˜ m (cid:1) e i ˜ mθ (cid:18) r − ir + i (cid:19) − ( i ˜ m ) (cid:32) A m + B m (cid:18) r − ir + i (cid:19) i ˜ m (cid:33) (I.2)It is now straightforward to compute the measure for the modes corresponding to thelarge diffeomorphisms. The measure is obtained by taking the inner product of two metricperturbations P V (1) L and P V (2) L using eq.(2.9), which just becomes the boundary term givenin eq.(2.17). We now have two boundary terms due to the two boundaries as r → ±∞ .The value of this boundary term at a single boundary is given by BT = ± (cid:88) | m | > b ˜ m ( ˜ m + 1) A m A − m − B m B − m + A m B − m (cid:0) m + 2 ˜ mr + r + 1 (cid:1) (cid:16) r − ir + i (cid:17) − i ˜ m r + 1 ∓ b ˜ m ( ˜ m + 1) A − m B m (cid:0) m − mr + r + 1 (cid:1) (cid:16) r − ir + i (cid:17) i ˜ m r + 1 (I.3)where the upper sign is to be used at the right boundary ( r → ∞ ) and the lower sign atthe left boundary ( r → −∞ ). The relative sign between the two boundaries arises dueto the change in the sign of the outward normal used to compute this boundary term ineq.(2.17). In the asymptotic limit, eq.(I.3) becomes, BT = ± b ˜ m ( ˜ m + 1) (cid:32) − B m B − m + A m A − m + A m B − m (cid:18) r − ir + i (cid:19) − i ˜ m − A − m B m (cid:18) r − ir + i (cid:19) i ˜ m (cid:33) (I.4)– 83 –e will now evaluate each of the boundary terms separately. The contribution to themeasure coming from the boundary term at r → ∞ , denoted M , can be evaluated byfollowing the conventions in eq.(4.12),(4.13). We see that as r → ∞ (cid:18) r − ir + i (cid:19) ± i ˜ m = exp (cid:18) ± i ˜ m (cid:18) r − ir + i (cid:19)(cid:19) (cid:39) M = (cid:88) | m |≥ b ˜ m ( ˜ m + 1) A m B − m (I.6)Noting that as r → −∞ , (cid:18) r − ir + i (cid:19) ± i ˜ m = exp (cid:18) ± i ˜ m (cid:18) r − ir + i (cid:19)(cid:19) (cid:39) exp( ± ˜ mπ ) (I.7)the contribution to the measure from the boundary term at r → −∞ , denoted M , is givenby M = − (cid:88) | m |≥ b ˜ m ( ˜ m + 1) A m B − m e − mπ (I.8)So the total measure, M , becomes M = M + M = (cid:88) | m |≥ b ˜ m ( ˜ m + 1) A m B − m (1 − e − mπ ‘ ) (I.9)Noting that δ m = − δ ∗− m , γ m = − γ ∗− m (I.10)which follow from the requirement that the solution eq.(4.15) be real, we can rewrite thisin term of the modes γ m , δ m using eq.(4.16) as M = (cid:88) | m |≥ b ˜ m ( ˜ m + 1)(( γ m γ ∗ m + δ m δ ∗ m ) sinh(2 ˜ mπ ) + 2( γ m δ ∗ m + δ m γ ∗ m ) sinh( ˜ mπ )) (I.11)Further expressing the complex variables γ m , δ m in terms of the real variables p m , q m , r m , s m as in eq.(4.27) we find that the measure becomes M = (cid:88) m ≥ b ˜ m ( ˜ m + 1) sinh( ˜ mπ ) (cid:0) ( p m + q m + r m + s m ) cosh( ˜ mπ ) + 2( p m r m + q m s m ) (cid:1) (I.12)– 84 –eading off the measure for the large diffeomorphism modes p m , q m , r m , s m , from the above,we have (cid:90) D [ P V L ] = (cid:90) (cid:89) m ≥ (cid:2) (16 b ˜ m (cid:0) ˜ m + 1 (cid:1) sinh( π ˜ m )) dp m dq m dr m ds m (cid:3) (I.13)We will now evaluate the action for the large diffeomorphisms. The action is given by theboundary term in the JT action eq.(2.51). Using eq.(I.5),(I.7), we see that the scalar fieldsolution eq.(4.15) as r → ∞ becomes ψ (cid:12)(cid:12) r →∞ (cid:39) (cid:88) m e i ˜ mθ δ m r sinh( ˜ mπ ) (I.14)and at r → −∞ , we get ψ (cid:12)(cid:12) r →−∞ (cid:39) − (cid:88) m e i ˜ mθ γ m r sinh( ˜ mπ ) (I.15)which shows that the the large diffeomorphism at the left and right boundaries are inde-pendent and so we can independently compute the action at each of the boundaries. Inthe asymptotic AdS limit, for the parametrization of θ ( u ) in terms of the diffeomorphismas θ ( u ) = b π u + V θL ( u ) (I.16)the boundary term in the JT action, eq.(B.17) to the quadratic order in large diffeomor-phisms, becomes S JT,∂ = b πGJ β + 116 πGJ β (cid:90) π du (cid:34) ( ∂ u V θL ) + (cid:18) πb (cid:19) ( ∂ u V θL ) (cid:35) (I.17)where we used the fact the dilaton is of the form eq.(2.7) at the boundary in the asymptoticAdS limit. Noting that near the boundary at r → ∞ , the large diffeomorphism is given by V θL ( u ) (cid:39) (cid:88) m e i ˜ mb π u δ m sinh( ˜ mπ ) (I.18)the term quadratic in the large diffeomorphism in the action is given by S JT,∂ = b πGJ β + 12 πG b J β (cid:88) m ≥ ˜ m ( ˜ m + 1) sinh ( ˜ mπ ) δ m δ ∗ m (I.19)where we have used eq.(I.10) and the first line in eq.(4.4) to obtain the above equation.Similarly the action for the boundary term near r → −∞ can be obtained by noting the– 85 –ector field corresponding to the large diffeomorphisms is given by V θL ( u ) (cid:39) − (cid:88) m e i ˜ mu γ m sinh( ˜ mπ ) (I.20)and the term quadratic in the large diffeomorphism in the action is given by S JT,∂ = b πGJ β + 12 πG b J β (cid:88) m ≥ ˜ m ( ˜ m + 1) sinh ( ˜ mπ ) γ m γ ∗ m (I.21)where again we have used eq.(I.10) and the second line in eq.(4.4) to obtain the aboveresult. Combining eq.(I.21) and eq.(I.19) and expressing in terms of p m , q m , r m , s m usingeq.(4.27), the net action then becomes S JT,∂ = b πGJ (cid:18) β + 1 β (cid:19) + (cid:88) m ≥ πG b J ˜ m ( ˜ m + 1) sinh ( ˜ mπ ) (cid:18) p m + q m β + r m + s m β (cid:19) (I.22)We will now elaborate on the orthogonality of different classes of metric perturbationsnamely, those corresponding to twist, b -modulus, small and large diffeomorphisms. Thediscussion regarding the inner product of small and large diffeomorphisms is presentedtowards the end of E which in the asymptotic AdS limit satisfies the inequality (cid:104) P V s , P V L (cid:105) (cid:112) (cid:104) P V s , P V s (cid:105)(cid:104) P V L , P V L (cid:105) ≤ O ( r − ) (I.23)with the notation being the same as in appendix E. Let us first consider the inner product ofthe metric perturbation corresponding to b -modulus with others. From eq.(4.22), eq.(4.20),noting the metric eq.(4.2) and using the definition eq.(2.9), it is straightforward to see that (cid:104) P V mod , P V tw (cid:105) = 0 (I.24)Further since there is not θ dependence in P V b where as the large diffeomorphisms havethe dependence on θ as e imθ , m ≥
1, we have (cid:104)
P V b , P V L,m (cid:105) = 0 (I.25)Considering the inner product of a metric perturbation for small diffeomorphism with
P V mod we get (cid:104)
P V b , P V s,λ,m (cid:105) = (cid:104) P † P V b , V s,λ,m (cid:105) + (cid:90) ∂ V as,λ,m n b ( P V b ) ab = (cid:90) ∂ V θs,λ,m n r ( P V b ) θr = 0 (I.26)where in obtaining the second line we used eq.(4.23) and for a small diffeomorphism thateq.(E.6) is satisfied at the boundary. The vanishing of the second line then follows by noting– 86 –hat P V b does not have off-diagonal components, see eq.(4.20). Now, we shall consider theinner product of V tw with other metric perturbations. The inner product with P V b isalready obtained in eq.(I.24). The inner product of P V tw with large diffeomorphisms alsovanishes since as before the large diffeomorphisms have a non-trivial θ dependence whereasthe twist perturbation has no θ dependence as it corresponds to m = 0 sector, see eq.(4.20).The inner product of P V tw and P V s,λ,m , can be simplified following the steps in eq.(I.26)to obtain (cid:104)
P V tw , P V s,λ,m (cid:105) = (cid:90) ∂ V θs,λ,m n r ( P V tw ) θr (I.27)which is non-zero for m = 0 mode of V s,λ,m after noting eq.(4.20). So, to estimate it, weconsider the quantity, (cid:104) P V tw , P V s,λ,m (cid:105) (cid:112) (cid:104) P V tw , P V tw (cid:105)(cid:104) P V s,λ, − m , P V s,λ,m (cid:105) (I.28)We shall show that in the asymptotic AdS limit, the above quantity goes as O ( r − / ). Thequantity (cid:104) P V tw , P V tw (cid:105) has already been computed in eq.(4.29). For m = 0, using eq.(E.3),we have that V θs,λ, = ∂ r ψ λ, ∼ O ( r − / ) (I.29)It is also easy to see from line element eq.(4.2) that the normal vector at either of theboundaries r = − r B or r = r B , has the behaviour n r ∼ O ( r ) (I.30)The analog of the calculation leading to eq.(E.24) for the double trumpet shows that |(cid:104) P V s,λ, − m , P V s,λ,m (cid:105)| ≥ (cid:12)(cid:12)(cid:12)(cid:12) ( λ + 2) πrψ λ,m ψ λ, − m (cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:12)(cid:12)(cid:12)(cid:12) ∼ O ( r ) (I.31)where ∂ stands for both the boundaries. From eq.(4.20), we have( P V tw ) r,θ = − tπ ( r + 1) ∼ O ( r − ) (I.32)Further noting that (cid:104) P V tw , P V tw (cid:105) ∼ O ( r ) from eq.(4.29), we have, combining eq.(I.29),(I.30), (I.31) that (cid:104) P V tw , P V s,λ,m (cid:105) (cid:112) (cid:104) P V tw , P V tw (cid:105)(cid:104) P V s,λ,m , P V s,λ, − m (cid:105) ∼ O ( r − / ) (I.33)which vanishes in the asymptotic AdS limit. So, in total, noting all the above results,eq.(4.25) then follows immediately. – 87 – .2 Matter in double trumpet calculations In this appendix we shall elaborate on the details used in the discussion in subsection 4.2.We shall carefully evaluate the determinant of scalar laplacian, det( − ˆ ∇ ). We will considermassless scalar in the background of the double trumpet topology with the metric writtenin conformally flat coordinate system, eq.(A.16), as ds = dr ∗ + dθ cos r ∗ , r ∗ ∈ (cid:104) − π , π (cid:105) , θ ∈ [0 , b ] (I.34)We can compute the dependence on b by noting that the metric above is conformallyflat and so we can use the conformal anomaly to evaluate the contribution due to theconformal factor and then compute the contribution from the flat metric separately. The b dependence coming from the conformal anomaly can be evaluated using the conformalanomaly since the theory of a massless scalar field is a conformal field theory. The relationbetween determinants of conformally related metrics ˆ g ab = e σ ¯ g ab is given bydet( − ˆ ∇ )det( − ¯ ∇ ) = exp (cid:26) − π (cid:20) (cid:90) d x √ ¯ g (¯ g ab ∂ a σ∂ b σ + ¯ Rσ ) + (cid:90) ∂ d ¯ s ¯ Kσ (cid:21)(cid:27) (I.35)where quantities denoted by hats are calculated with respect to the metric ˆ g . In the caseat hand σ = − ln cos r ∗ (I.36)and ¯ R = 0. For a non-wavy boundary, the boundary can be specified by r ∗ = − r ∗ , r ∗ = r ∗ , r ∗ , > g , at the left and the rightboundaries is given by ¯ n µ = (1 , , ¯ n µ = ( − ,
0) (I.38)and the extrinsic curvature ¯ K = 0. So, we havedet( − ˆ ∇ )det( − ¯ ∇ ) = exp (cid:18) − b π (cid:90) r ∗ − r ∗ ( ∂ r ∗ σ ) (cid:19) = exp (cid:20) − b π (tan r ∗ + tan r ∗ − r ∗ − r ∗ ) (cid:21) (I.39)We shall now show the computation of the b dependence in the determinant coming fromthe flat metric given by ds = dr ∗ + dθ r ∗ ∈ [0 , π ] , θ ∈ [0 , b ] (I.40)– 88 –here we have shifted the range of r ∗ by π so as to simplify the computations. Taking θ circle as the time direction in the Euclidean scalar field theory, the periodicity of the θ circle determines the temperature at which we need to calculate the thermal partitionfunction. So, we have (cid:90) Dϕe − (cid:82) ( ˆ ∇ ϕ ) = Z M,flat [ b ] = Tr (cid:16) e − bH (cid:17) (I.41)the subscript f in Z f to denote that the computation is being done for the flat metric, H is the Hamiltonian given by H = (cid:90) dr ∗ ( π ϕ + ( ∂ r ∗ ϕ ) )) (I.42)where π ϕ = ∂ t ϕ, t = θ (I.43)The solution for the matter equation ¯ ∇ ϕ = 0 (I.44)with the boundary conditions ϕ (0 , t ) = 0 = ϕ ( π, t ) (I.45)is given by ϕ = ∞ (cid:88) n =0 sin( nr ∗ ) n √ π (cid:0) α n e − iω n t + α − n e iw n t (cid:1) , ω n = n (I.46)The Hamiltonian in terms of the modes is then given by H = 12 ∞ (cid:88) n =1 ( α n α † n + α † n α n ) = ∞ (cid:88) n =1 α † n α n −
124 (I.47)The partition function then becomes Z M,flat [ b ] = Tr( e − bH ) = ∞ (cid:89) n =1 e b − e − bn = 1 η ( τ ) (I.48)where η ( τ ) is the Dedekind eta function and τ is related to b by τ = ib π (I.49)– 89 –sing the modular transformation property of η ( τ ), we can study the behaviour of thepartition function near b = 0. So, we have Z M,flat [ b ] = 1 η ( τ ) = √− iτη ( − τ − ) = (cid:114) b π Z M,flat (cid:20) π b (cid:21) ⇒ lim b → Z M,flat [ b ] = (cid:114) b π lim b → Z M,flat (cid:20) π b (cid:21) = (cid:114) b π e π b (I.50)Putting together eq.(I.39) and eq.(I.48), we get the full dependence of det (cid:16) − ˆ ∇ (cid:17) asdet( − ˆ ∇ ) = (cid:18) η (cid:18) ib π (cid:19)(cid:19) exp (cid:20) − b π (tan r ∗ + tan r ∗ − r ∗ − r ∗ ) (cid:21) (I.51)The terms tan r ∗ , tan r ∗ in the exponent above diverges when r ∗ , r ∗ → π . This canbe avoided by adding a length-dependent term with the appropriate coefficient, followingwhich we have, in the limit r ∗ , r ∗ → π ,det( − ˆ ∇ ) = (cid:18) η (cid:18) ib π (cid:19)(cid:19) exp (cid:20) − b π (tan r ∗ − sec r ∗ + tan r ∗ − sec r ∗ − π ) (cid:21) (cid:39) (cid:18) η (cid:18) ib π (cid:19)(cid:19) e b (I.52)One can also compute the contribution to the determinant from the flat metric eq.(I.40)directly by noting that the eigenvalues in the geometry eq.(I.40) with Dirichlet boundaryconditions along the θ and r ∗ directions are given by λ = n + ˜ m (I.53)where n is the mode number along the r ∗ direction and ˜ m is related to the mode number m in the θ direction by eq.(4.10). I.3 Coleman method for computing determinants in AdS double trumpet
In this section, we shall show in detail how to evaluate the various determinants in theEuclidean AdS double trumpet geometry. The metric for the double trumpet geometry isgiven by ds = dr r + 1 + ( r + 1) dθ (I.54)The left and right boundaries are taken to be located at r = − r B and r = r B respectivelywhere r B , r B >
0. The solution for the wave equationˆ ∇ ψ = − λψ (I.55)– 90 –or the modes of the form e i ˜ mθ for the θ dependence is given by ψ λ,m = k P i ˜ mv − ( ir ) + k Q i ˜ mv − ( ir ) (I.56)where v = (cid:114) − λ, ˜ m = 2 πmb (I.57)For m (cid:54) = 0 an equivalently good independent basis of solutions is ψ λ,m = k P i ˜ mv − ( ir ) + k P − i ˜ mv − ( ir ) (I.58)We shall use this form of the general solution to compute the determinants of ( − ˆ ∇ ) and( − ˆ ∇ + 2) with Dirichlet boundary conditions. To begin with, let us compute the value ofthe determinant of ( − ˆ ∇ ) using Coleman Trick. To do this, we need to impose the requiredboundary conditions and find the appropriately normalized solution to get the correct ˜ m dependence. We note however, that the overall r B normalization of the solution, where r B > r B or r B is the location of the boundary, cannot be uniquelyfixed. We will discuss more about this point later. Consider first the case of ˜ m (cid:54) = 0 modes.For this, imposing the Dirichlet boundary conditions at the left boundary, say specified as r = − r B we get the relation between the constant k and k as k = − k P i ˜ mv − ( − ir B ) P − i ˜ mv − ( − ir B ) (I.59)and so the solution becomes ψ λ,m = k P − i ˜ mv − ( − ir B ) (cid:18) P i ˜ mv − ( ir ) P − i ˜ mv − ( − ir B ) − P − i ˜ mv − ( ir ) P i ˜ mv − ( − ir B ) (cid:19) (I.60)Choose k = kP − i ˜ mv − ( − ir B ) and so the solution becomes ψ λ,m = k (cid:18) P i ˜ mv − ( ir ) P − i ˜ mv − ( − ir B ) − P − i ˜ mv − ( ir ) P i ˜ mv − ( − ir B ) (cid:19) (I.61)Now, we need to fix the constant k based on the following requirements. Viewed as acomplex function of λ , the ratio of two solutions with different mode numbers m and m , should have no extra poles of zeroes other those corresponding to the eigenvalues ofthe operator ( − ˆ ∇ ). Also, the ratio of two solutions should approach unity as | λ | goesto infinity in any direction other than the positive real axis. This completely fixed the ˜ m dependence of the constant k although the r B , r B dependence is ambiguous. This can befixed by the requirement that the final answer should be independent of r B since otherwise,in the asymptotic AdS limit that would mean that we would get a ln r B divergence in thenet action, which cannot be cancelled away by a local counterterm. First, it is useful– 91 –o note the asymptotic expansions of the associated Legendre functions P ba ( ± ir B ). Fromeq.8.1.5 of [124], we see that for at the left boundary, P µν ( z ) = 2 − ν − Γ( − − ν ) z µ − − ν √ π ( z − µ Γ( − ν − µ ) F (cid:18) ν − µ , ν − µ , ν + 32 , z − (cid:19) ν Γ( + ν ) z µ + ν √ π ( z − µ Γ(1 + ν − µ ) F (cid:18) − ν − µ , − ν − µ , − ν + 12 , z − (cid:19) (I.62)Using this and the fact that F ( a, b, c, z ) (cid:39) O ( z ) for small z , we find that the asymptoticforms at − ir B and ir B are P i ˜ mv − ( ± ir B ) = e ∓ iπ (cid:32) e ∓ iπv (2 r B ) − v − √ π Γ( − v )Γ( − i ˜ m − v ) + e ± iπv (2 r B ) v − √ π Γ( v )Γ( − i ˜ m + v ) (cid:33) (I.63)However, there is a small subtlety which is as follows. After imposing the Dirichlet bound-ary conditions at the left boundary and relating the coefficients, the solution has to becontinued through r = 0 to the right boundary. In doing so, one has to use eq.8.1.4 of [124]and so our asymptotic expansions will have a relative factor of e ˜ mπ between z = − ir B and z = ir B due to the presence of the branch cut between (1 , −∞ ). The physics problem ofcontinuing the solution through r = 0 is unambiguous as it is perfectly well-defined pointin the double trumpet, where as the expansions in [124] are defined with a different choiceof branch which can be seen from the factor ( z − − µ in eq 8.1.4 and 8.1.5 . Thus, thesolution is not continuous across r = 0. So, to get rid of this problem, we multiply theasymptotic form at ir B by an extra factor of e π ˜ m . Doing so, we have P i ˜ mv − ( − ir B ) = e iπ (cid:32) e iπv (2 r B ) − v − √ π Γ( − v )Γ( − i ˜ m − v ) + e − iπv (2 r B ) v − √ π Γ( v )Γ( − i ˜ m + v ) (cid:33) P i ˜ mv − ( ir B ) = e π ˜ m e − iπ (cid:32) e − iπv (2 r B ) − v − √ π Γ( − v )Γ( − i ˜ m − v ) + e iπv (2 r B ) v − √ π Γ( v )Γ( − i ˜ m + v ) (cid:33) (I.64)Using these asymptotic forms, we can immediately calculate the solution at r = ir B to be ψ λ,m = k (cid:18) P i ˜ mv − ( ir B ) P − i ˜ mv − ( − ir B ) − P − i ˜ mv − ( ir B ) P i ˜ mv − ( − ir B ) (cid:19) = kπ (4 r B r B ) − (cid:32) mπ )(4 r B r B ) v Γ( v ) Γ( − i ˜ m + v )Γ( − i ˜ m + v ) − e πiv sinh(2 ˜ mπ ) v sin( πv ) (cid:18) r B r B (cid:19) v + ( v → − v ) (cid:33) (I.65)From the above we see that as | v | → ∞ , the second term and the corresponding v → − v are subleading and so we need to choose k such that ˜ m dependence in the remaining termscancels in this limit and that there are no extra poles or zeroes. So, in the limit | λ | → ∞ ,– 92 –e get ψ λ,m = 2 sinh( ˜ mπ ) kπ √ r B r B (cid:0) (4 r B r B ) v + (4 r B r B ) − v (cid:1) (I.66)From the above expression, we see that there will be no spurious poles or zeroes in theratio of two solutions. So we choose k as k = 12 sinh( ˜ mπ ) (I.67)The solution then becomes ψ λ,m = (4 r B r B ) − π sinh( ˜ mπ ) (cid:32) mπ )(4 r B r B ) v Γ( v ) Γ( − i ˜ m + v )Γ( + i ˜ m + v ) − e πiv sinh(2 ˜ mπ ) v sin( πv ) (cid:18) r B r B (cid:19) v + ( v → − v ) (cid:33) (I.68)Now, the final step in the Coleman method is to evaluate the solution at λ = 0 to get thevalue of the determinant. Doing so, we have v = , the expression in eq.(I.65) becomes toleading order ψ , ˜ m = 1Γ(1 − i ˜ m )Γ(1 + i ˜ m ) = sinh( ˜ mπ )˜ mπ (I.69)Now, it remains to evaluate the contribution from the ˜ m = 0 sector. The contributionfrom this sector can only be dependent on r B , r B . We shall choose the normalization soas to cancel this dependence. Thus, we can ignore the contribution from the ˜ m = 0 sector.So, the value of the determinant for the operator ( − ˆ ∇ ) is given byln det( − ˆ ∇ ) = m = ∞ (cid:88) m = −∞ ,m (cid:54) =0 ln sinh( ˜ mπ )˜ mπ = 2 ln η (cid:18) ib π (cid:19) − ln 2 π / (I.70)which in the limit b → − ˆ ∇ ) (cid:39) b e − π b (I.71)Comparing this with the answer obtained earlier for the determinant in eq.(I.51), we seethat there is a mismatch of the exponential term coming from the conformal part of themetric in the earlier calculation. We could not satisfactorily establish the reason for this butit could be due to the different order of limits that are being implemented in the Colemanmethod used above in calculating the determinant, as was also mentioned in eq.F.1. Now,we will evaluate det( − ˆ ∇ + 2) in the same manner for Dirichlet boundary conditions. Thesolution is the same as in eq.(I.58) except that the value of v is now given by v = (cid:114) − λ (I.72)– 93 –he arguments leading to eq.(I.68) remain the same and hence we get the same expressionas in eq.(I.68). In the final step of computing the determinant when we set λ = 0, we get v = and so the leading term in the solution becomes ψ , ˜ m = r B r B Γ(2 − i ˜ m )Γ(2 + i ˜ m ) = sinh( ˜ mπ ) π ˜ m (1 + ˜ m ) r B r B (I.73)To get rid of the r B , r B dependence, we can further normalize the solution by a factorof r B r B . The reasoning for this is same as in the calculation of ( − ˆ ∇ ), namely that theabsence of local counterterms to cancel ln r B divergent term in the determinant. We wouldalso get dependence on r B , r B from the ˜ m = 0 sector which can again be normalizedto unity and so we will ignore the contribution from the ˜ m = 0 sector. The value of thedeterminant of ( − ˆ ∇ + 2) is then given byln det( − ˆ ∇ + 2) = m = ∞ (cid:88) m = −∞ ,m (cid:54) =0 ln sinh( ˜ mπ )˜ mπ (1 + ˜ m ) = 2 ln η (cid:18) ib π (cid:19) − ln (cid:0) √ πb sinh( b/ (cid:1) (I.74)which in the limit b → − ˆ ∇ + 2) (cid:39) b e − π b (I.75)Now, we shall repeat the steps for the operator P † P . As in the case of the disk, we havetwo sets of eigenvalues corresponding to the boundary conditions ∂ r ξ = 0 and ∂ r ψ = 0,see eq.(E.15),(E.16). Let us first compute the contribution to the eigenvalues from ξ withthe boundary condition ∂ r ξ = 0 . (I.76)The solution for ξ is given ξ λ,m = k P i ˜ mv − ( ir ) + k P − i ˜ mv − ( ir ) (I.77)Now, imposing the boundary condition eq.(I.76), the constants k , k are related by k = − k ∂ r B P i ˜ mv − ( − ir B ) ∂ r B P − i ˜ mv − ( − ir B ) (I.78)Choosing k = k∂ r B P − i ˜ mv − ( − ir B ), we get the solution to be ξ λ,m = k (cid:18) P i ˜ mv − ( ir ) ∂ r B P − i ˜ mv − ( − ir B ) − P − i ˜ mv − ( ir ) ∂ r B P i ˜ mv − ( − ir B ) (cid:19) (I.79)– 94 –o, we have ∂ r ξ λ,m ( r ) (cid:12)(cid:12) r = r B = k (cid:18) ∂ r B P i ˜ mv − ( ir B ) ∂ r B P − i ˜ mv − ( − ir B ) − ∂ r B P − i ˜ mv − ( ir B ) ∂ r B P i ˜ mv − ( − ir B ) (cid:19) (I.80)which using eq.(I.64) becomes ∂ r ξ λ,m ( r ) (cid:12)(cid:12) r = r B = k (4 r B r B ) (cid:32) mπ )(2 v − Γ( v ) (4 r B r B ) v Γ( − i ˜ m + v )Γ( + i ˜ m + v ) − π sinh(2 ˜ mπ )(1 − v ) ve − πiv sin( πv ) (cid:18) r B r B (cid:19) v (cid:33) + ( v → − v ) (I.81)From the above expression it is easy to see that for large | λ | , we need to choose k as ineq.(I.67). Doing so and setting λ = 0 ⇒ v = in eq.(I.81), we get the leading term to be ∂ r B ξ ( r B ) = sinh( ˜ mπ ) π ˜ m (1 + ˜ m ) (I.82)We see that there is no r B , r B dependence in the above expression. Again, we ignorethe ˜ m = 0 sector with the understanding that it only gives an r B dependence which canbe cancelled away by an appropriate normalization. Let us now evaluate the value of thedeterminant coming from the eigenvalues corresponding to the boundary condition ∂ r ψ = 0 (I.83)at both the ends. The solution to begin with is given in eq.(I.58). Imposing the boundarycondition eq.(I.83) at the left end r = − r B gives the relation k = − k ∂ r B P i ˜ mv − ( − ir B ) ∂ r B P − i ˜ mv − ( − ir B ) (I.84)Choosing k = k∂ r B P − i ˜ mv − ( − ir B ), we get the solution to be ψ λ,m = k (cid:18) P i ˜ mv − ( ir ) ∂ r B P − i ˜ mv − ( − ir B ) − P − i ˜ mv − ( ir ) ∂ r B P i ˜ mv − ( − ir B ) (cid:19) (I.85)So, we have ∂ r ψ λ,m ( r ) (cid:12)(cid:12) r = r B = k (cid:18) ∂ r B P i ˜ mv − ( ir B ) ∂ r B P − i ˜ mv − ( − ir B ) − ∂ r B P − i ˜ mv − ( ir B ) ∂ r B P i ˜ mv − ( − ir B ) (cid:19) (I.86)– 95 –hich using eq.(I.64) becomes ∂ r ψ λ,m ( r ) (cid:12)(cid:12) r = r B = k (4 r B r B ) − (cid:32) mπ )(2 v − (2 v − Γ( v ) (4 r B r B ) v Γ( − i ˜ m + v )Γ( + i ˜ m + v ) (cid:33) + k (4 r B r B ) − (cid:18) − πe πiv sinh(2 ˜ mπ )(1 − v )(9 − v ) v sin( πv ) (cid:18) r B r B (cid:19) v (cid:19) + ( v → − v ) (I.87)Again noting that in the limit of | λ | (cid:29)
1, the first terms dominates and so we choose k as in eq.(I.67). With this choice of normalization in eq.(I.85), setting λ = 0, we get theleading term as ∂ r ψ λ,m ( r ) (cid:12)(cid:12) r = r B = ˜ m sinh( ˜ mπ ) πr B r B (I.88)In getting the above result, one has to keep the subleading terms arising from expandingthe hypergeometric functions appearing in eq.(I.62) when evaluated for ν = v − = 1.Again ignoring the r B , r B dependence with the understanding that they can be cancelledby an appropriate normalization and also the ˜ m = 0 sector, we get the total value of thedeterminant of P † P asln det (cid:48) P † P = m = ∞ (cid:88) m = −∞ ,m (cid:54) =0 (cid:18) ln (cid:18) sinh( ˜ mπ )˜ mπ (1 + ˜ m ) (cid:19) + ln (cid:18) ˜ m sinh( ˜ mπ ) π (cid:19)(cid:19) (I.89)The prime in the determinant is to indicate the exclusion of zero modes that appear in the m = 0 sector. Using eq.(I.89) and eq.(I.74), we getln √ det (cid:48) P † P det( − ˆ ∇ + 2) = m = ∞ (cid:88) m = −∞ ,m (cid:54) =0
12 ln (cid:0) ˜ m (1 + ˜ m ) (cid:1) b → −−→ b π (I.90)So, we find that there is a non-trivial b -dependence in the ratio √ det (cid:48) P † P det( − ˆ ∇ +2) . However, usingresults in string theory for the partition function of the ghost fields, it it straightforwardto compute the determinant of the operator P † P directly. To do so we use the conformallyflat form of the double trumpet geometry as in eq.(I.34). The contribution due to theconformal factor is same as in eq.(I.39) with a factor of −
26 multiplied in the exponent.The contribution from the flat part is then obtained by reading off the result from eq.7.4.1of [122], the string theory vacuum amplitude for open strings on a cylinder which gives (cid:113) det (cid:48) ( P † P ) (cid:39) (cid:32) η ( ib π ) b (cid:33) (I.91)which then has the correct b → dS double trumpet determinants J.1 Coleman method computation
In this section, we shall compute the determinants in the de Sitter double trumpet topology.We will find that the computation of the determinants is very similar to that in the caseof AdS double trumpet and so we shall only work out in detail the case of scalar laplaciandeterminant. To do so, we shall view the metric of the de Sitter double trumpet as arisingfrom the analytic continuation of the − AdS double trumpet which is given by ds = − dr r − (1 + r ) dθ (J.1)and continuing by r → ± ir (J.2)at both the ends of the AdS double trumpet, we get the metric of dS double trumpet as ds = − dr r − r − dθ , θ ∼ θ + b (J.3)The solutions for an eigenvector of the scalar laplacian in the AdS double trumpet, withmode number m , satisfying the equationˆ ∇ ψ = − λψ (J.4)whose solution is given in eq.(I.56). Continuing this solution by analytic continuation weget the eigenvector for the dS double trumpet topology as ψ λ,m = k P i ˜ mv − ( r ) + k Q i ˜ mv − ( r ) (J.5)where v = (cid:114)
14 + λ, ˜ m = 2 πmb (J.6)An equivalently good basis of solution for ˜ m (cid:54) = 0 is P ± ˜ mv − in which the solution abovebecomes ψ λ,m = k P i ˜ mv − ( r ) + k P − i ˜ mv − ( r ) (J.7)Now, to have te correct asymptotic expansions at the left and the right boundaries, weneed to look at the contour a bit more carefully. Due to the presence of the branchcut from ( − , ∞ ) in the complex r plane, we take the contour for the de Sitter doubletrumpet to start at r = −∞ just below the real axis and first rotate counterclockwise to r = − ir B , r B → ∞ . We then end up in the AdS double trumpet geometry in which– 97 –e go from the left end to the right end along the imaginary axis from which we pick arelative factor of e ˜ mπ , see discussion after eq.(I.63). From the point r = ir B , r B → ∞ ,we rotate clockwise to end up at r = ∞ . However, note that there are other equivalentlygood choices of contour, say rotating counterclockwise in the last step or beginning fromabove the cut at r = −∞ . In total, we have four possible choices of contours, which aredepicted in Fig.3. Our choice corresponds to the path EDCBF . Other choices will onlychange the phase factor e ± iπv and will end up giving the same final answer. Now, followingthe same steps as in the AdS double trumpet, imposing the Dirichlet boundary conditionsat one end of the boundary r = − r B , we get the form of the solution in eq.(I.61) as ψ λ,m = k (cid:18) P i ˜ mv − ( r ) P − i ˜ mv − ( − r B ) − P − i ˜ mv − ( r ) P i ˜ mv − ( − r B ) (cid:19) (J.8)With the choice of contour we made, the asymptotic forms of the associated Legendrefunctions now read P i ˜ mv − ( − r B ) = e iπ (cid:32) (2 r B ) − v − √ π Γ( − v ) e iπv Γ( − i ˜ m − v ) + (2 r B ) v − √ π Γ( v ) e − iπv Γ( − i ˜ m + v ) (cid:33) P i ˜ mv − ( r B ) = e π ˜ m (cid:32) (2 r B ) − v − √ π Γ( − v )Γ( − i ˜ m − v ) + (2 r B ) v − √ π Γ( v )Γ( − i ˜ m + v ) (cid:33) (J.9)and hence we get, at r = r B (cid:29)
1, the value of the scalar field solution as ψ λ, ˜ m = k (cid:18) P i ˜ mv − ( r B ) P − i ˜ mv − ( − r B ) − P − i ˜ mv − ( r B ) P i ˜ mv − ( − r B ) (cid:19) = kπ (4 r B r B ) − e iπ (cid:32) mπ )(4 r B r B ) v Γ( v ) Γ( − i ˜ m + v )Γ( + i ˜ m + v ) e − iπv − e iπv sinh(2 ˜ mπ ) v sin( πv ) (cid:18) r B r B (cid:19) v + ( v → − v ) (cid:33) (J.10)The same reasoning as in the case of AdS double trumpet, around eq.(I.66) shows that k should be chosen as in eq.(I.67) and so the wavefunction and consequently, the laplaciandeterminant are as given by eq.(I.70). It should be noted that the only difference in theassociated Legendre function expansions in the two cases as seen from eq.(I.64) and eq.(J.9)is only in the overall constant phases and factors of v -dependent exponentials, due to thefact that argument of the associated Legendre function is imaginary and real in the AdSand dS double trumpets respectively. However, this doesn’t make a difference in the valueof k and hence the value of the solution ψ λ,m for v = becomes ψ , ˜ m = sinh ˜ mπ ˜ mπ (J.11)which is the same as in eq.(I.69). It then immediately follows that the value of the scalarlaplacian determinant is same as before eq.(I.70) upto an irrelevant numerical constant . It can be argued easily that the other choices of contours in Fig.3 will only result in an overall unim-portant constant in the final value of the determinant. – 98 –he limit b → − ˆ ∇ + 2), we can use the same result in eq.(J.10) exceptthat the definition of v becomes v = (cid:114)
94 + λ (J.12)and hence we get the same result again for ( − ˆ ∇ + 2) as in eq.(I.74) upto irrelevantnumerical constant. The same arguments apply for the operator P † P and so we get theresult eq.(I.89) for this operator. J.2 Direct calculation of scalar laplacian
We shall now show an alternative but direct way of evaluating the scalar determinant withDirichlet boundary conditions at both ends. This calculations is analogous to the one inAdS double trumpet in appendix I.2 by writing the metric in conformally flat form andthen separately evaluating the conformal factor and flat metric contributions. We considerthe analogue of the Maldacena contour here where we consider the − AdS metric for thedouble trumpet and analytically continue it to the dS double trumpet. The − AdS doubletrumpet meric is given by ds = − r ∗ ,AdS ( dr ∗ ,AdS + dθ ) , r ∗ ,AdS ∈ [0 , π ] , θ ∈ [0 , b ] (J.13). Now we first do the analytic continuation at the left end of the above − AdS line elementby doing r ∗ ,AdS = ± ir ∗ ,dS (J.14)The double trumpet metric in the conformally flat form is given by ds = 1sinh r ∗ ,dS ( − dr ∗ ,dS + dθ ) (J.15)The contribution from the conformal factor can be obtained by using eq.(F.8) with theconformal factor given by e σ = − r ∗ ,AdS (J.16)for the part of the contour through AdS double trumpet. We are then left to evaluatethe contribution from flat part of the line element. Taking the solution for the eigenvalueequation of − ˆ ∇ with the mode number m , the equation for which readsˆ ∇ ψ = − λψ (J.17)– 99 –he solution satisfying the Dirichlet boundary conditions is given by ψ ∼ e i ˜ mθ sin (cid:16)(cid:112) λ − ˜ m r ∗ ,AdS (cid:17) (J.18)where ˜ m is related to m by eq.(J.6). Continuing this solution to the dS using eq.(J.14)gives ψ ∼ ± e i ˜ mθ sinh (cid:16)(cid:112) λ − ˜ m r ∗ ,AdS (cid:17) (J.19)which correctly satisfies the Dirichlet boundary conditions as r ∗ ,AdS →
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