On Gauss-bonnet gravity and boundary conditions in Lorentzian path-integral quantization
aa r X i v : . [ g r- q c ] J a n On Gauss-bonnet gravity and boundary conditions in Lorentzianpath-integral quantization
Gaurav Narain a ∗ a Center for Gravitational Physics, Department of Space Science,Beihang University, Beijing 100191, China.
Abstract
Recently there has been a surge of interest in studying Lorentzian quantum cosmology usingPicard-Lefschetz methods. The present paper aims to explore the Lorentzian path-integral ofGauss-Bonnet gravity in four spacetime dimensions with metric as the field variable. We employmini-superspace approximation and study the variational problem exploring different boundaryconditions. It is seen that for mixed boundary conditions non-trivial effects arise from Gauss-Bonnet sector of gravity leading to additional saddle points for lapse in some case. As an appli-cation of this we consider the No-boundary proposal of the Universe with two different settings ofboundary conditions, and compute the transition amplitude using Picard-Lefschetz formalism. Infirst case the transition amplitude is a superposition of a Lorentzian and a Euclidean geometricalconfiguration leading to interference incorporating non-perturbative effects coming from Gauss-Bonnet sector of gravity. In the second case involving complex initial momentum we note thatthe transition amplitude is an analogue of Hartle-Hawking wave-function with non-perturbativecorrection coming from Gauss-Bonnet sector of gravity. ∗ [email protected] . INTRODUCTION General relativity (GR) is a wonderful theory which is great at offering explanations for avariety of physical phenomenon ranging from astrophysical to cosmological scales. However,this model of gravity is not complete. It is one of the simplest gravity theory which tellshow the spacetime geometry curves in presence of matter and how the matter behaves incurved spacetime. But the theory starts to give erroneous results at short distance and losesreliability at such scales [1–7]. Similarly at ultra-large scales the theory predictions don’tagree with the experimental observations, and one has to invoke notions of dark-matter anddark energy to make an attempt at explaining observational data. Numerous model andmodifications of gravity has been proposed over the years to explain such deviations at bothends of energy scales.For example at ultra short length scales, motivated by lack of renormalizabilty of GR(which has only two time derivatives of the metric field) proposals have been made to incor-porate higher-time derivatives of the metric field. Such modifications are collectively referredto as higher-derivative theories of gravity. It has been noticed that incorporating higher-derivatives although addresses issues of renormalizability in four spacetime dimensions butthe theory lacks unitarity [8–10]. Some efforts have been made to tackle this unwantedproblem [11–14], in asymptotic safety approach [15, 16] and ‘
Agravity ’ [17].Overtime a need arose of having an amendment to GR which consist of higher-derivativesof the metric field, but when contributions from all such terms are summed over then thehighest order of time derivative is two. The Gauss-Bonnet (GB) gravity in four spacetimedimensions is one such simple modification of the GR. Here the dynamical evolution equa-tions of field remains unaffected. In four spacetime dimension GB sector of gravity actionis also topological and doesn’t play any role in dynamical evolution of spacetime metric.However, they play a key role in path-integral quantization of gravity where it is used toclassify topologies and has an important role to play at boundaries. The Gauss-Bonnetgravity action is following S = 116 πG Z d D x √− g (cid:20) −
2Λ + R + α (cid:18) R µνρσ R µνρσ − R µν R µν + R (cid:19)(cid:21) , (1)where G is the Newton’s gravitational constant, Λ is the cosmological constant term, α isthe Gauss-Bonnet coupling and D is spacetime dimensionality. The mass dimensions ofvarious couplings are: [ G ] = M − D , [Λ] = M and [ α ] = M − . This action falls in the classof lovelock gravity theories [18–20], and are a special class of higher-derivative gravity whereequation of motion remains second order in time.My interest in this paper is to explore the quantum properties of the GB gravity in thecontext of cosmology and the affect it has on path-integral quantization depending on variouskind of boundary conditions [21–25]. We start by considering a generic metric respectingspatial homogeneity and isotropicity in D spacetime dimensions. It is the FLRW metric inarbitrary spacetime dimension with dimensionality D . In polar co-ordinates { t p , r, θ, · · · } the FLRW metric can be expressed asd s = − N p ( t p )d t p + a ( t p ) (cid:20) d r − kr + r dΩ D − (cid:21) . (2)It consists of two unknown time-dependent functions: lapse N p ( t p ) and scale-factor a ( t p ).Here k = (0 , ±
1) is the curvature, and dΩ D − is the metric corresponding to unit sphere in D − G [bd , bd ] = Z bd bd D g µν e iS/ ~ ⇒ Z bd bd D N p ( t p ) D a ( t p ) e iS [ a,N p ] / ~ , (3)where bd and bd refers to field configuration at initial and final boundary, g µν is the fullmetric on left of ‘ ⇒ ’, incorporating all degree of freedom of the metric. While on the rightof ‘ ⇒ ’ we have reduced degree of freedom corresponding to mini-superspace approximation.Such transitional amplitudes depend crucially on the boundary conditions, and differentboundary conditions may or may not give rise to non-trivial features coming from the Gauss-Bonnet term in the gravitational action.Generically, to study such quantum transitions one investigates the behaviour of thecorresponding Euclidean functional integral given by G [bd , bd ] = Z bd bd D g µν exp ( − I [ g µν ]) . (4)Here g µν is the metric whose corresponding Euclidean action appears in the exponentialwhich is given by I [ g µν ]. This is the standard process of Euclideanization of functionalintegral which was originally Lorentzian in nature. It is obtained by performing a Wick-rotation of the time co-ordinate. This is done as the original Lorentzian functional integral ishighly oscillatory, and a Wick-rotation leads to a corresponding convergent and well-behavedfunctional integral. This whole process of Wick-rotation is standard procedure of defining awell-defined and convergent functional integrals in flat spacetime.Flat spacetime has a meaningful time co-ordinate and enjoys the properties of globalsymmetries to cast Lorentz group in to a compact rotation group under a transformationof the time co-ordinate. Such a beauty is not present in a generic curved spacetime. Thisimplies that the standard methodology of Wick-rotation used for defining sensible quantumfield theory (QFT) on flat spacetime is difficult to generalise reliably in a generic curvedspacetime where ‘time’ is just a parameter. The Feynman + iǫ -prescription in flat space-time QFT is a systematic way to choose a convergent integration contour for an otherwisehighly oscillatory integral. It naturally implements causality in path-integral in a system-atic manner by requiring that the euclideanised version of two-point function must satisfyOsterwalder-Schrader positivity. Such benefits exist only in flat spacetime and don’t getautomatically inherited to generic Lorentzian spacetime. The situation gets even more cum-bersome when spacetime becomes dynamical due to gravity and/or gravitational field is alsoquantized. Some attempts to incorporate Wick-rotation sensibly in curved spacetime havebeen made in [26–29]. However, more work needs to be done for it to mature.Picard-Lefschetz theory offers a methodology to carefully handle such oscillatory path-integrals, and is an extension of the Wick-rotation prescription to define convergent contourintegral on a generic curved spacetime. In this framework one uniquely finds contoursin the complexified plane along which the integrand is well-behaved. By definition theoriginal oscillatory integral then transforms to a convergent integral along such contours.Such contour-lines are termed Lefschetz thimbles . This framework has been used in pastin the context of Euclidean quantum gravity [30, 31]. Its recent usage has been in contextof Lorentzian quantum cosmology [32–34], where the authors studied gravitational path-integral in the mini-superspace approximation for Einstein-Hilbert gravity. Earlier attempts3sing this framework but in Euclidean quantum cosmology goes back to early 1980s [35–38]when issues regarding initial conditions were explored.Such investigations lead to tunnelling proposal [35–37] and no-boundary proposal [30,31, 38]. Euclidean gravitational path-integral (which is unbounded from below [39] dueto famous conformal factor problem [40]) needs a sensible choice of initial conditions anda contour of integration [41–43] to have it well-defined. Using the framework of Picard-Lefschetz theory one can directly find the contour of integration in Lorentzian spacetimeand study scenarios involving various boundary conditions in a systematic manner [32–34].In this paper we make use of Picard-Lefschetz theory to study the gravitational path-integral for the Gauss-Bonnet gravity in the mini-superspace approximation. We start byvarying the action with respect to field and study the nature of surface terms. We consid-ers three different type of boundary conditions: Dirichlet, Neumann and mixed boundaryconditions. It is seen that mixed boundary conditions may lead to non-trivial features inthe path-integral coming from Gauss-Bonnet sector of gravity action, while other bound-ary conditions don’t give any non-trivial features coming from Gauss-Bonnet sector. Mixedboundary conditions (MBC) are interesting and have also been previously explored in thecontext of Einstein-Hilbert gravity [44, 45] in relation with no-boundary proposal of theUniverse. We explore MBC in the context of gravitational path-integral of Gauss-Bonnetgravity, and find some non-trivial contribution coming from Gauss-Bonnet sector. As a spe-cial case we consider the no-boundary proposal of the Universe and find interesting featuresarising from the Gauss-Bonnet sector of gravity action.The outline of paper is follows: in section I we motivates our interest in studying thisproblem. In section II we discuss the mini-superspace approximation and compute themini-superspace action of theory. In section section III we discuss the action variationand study the various boundary conditions. In section IV we consider the path-integral ofgravity in mini-superspace approximation and start to compute the transition probability insaddle point approximation. Section V studies the integration over lapse in complex spacevia Picard-Lefschetz. In section VI we study the no-boundary proposal of Universe withmixed boundary conditions. In section VII we analyse the Hartle-Hawking wave-functionusing Lorentzian path-integral and the corrections it receive due to Gauss-Bonnet sector ofgravity. We finish off by presenting a conclusion and outlook in section VIII.
II. MINI-SUPERSPACE ACTION
The FLRW metric given in eq. (2) is conformally related to flat metric and hence itsWeyl-tensor C µνρσ = 0. The non-zero entries of the Riemann tensor are [46–48] R i j = − (cid:18) a ′′ a − a ′ N ′ p aN p (cid:19) g ij ,R ijkl = (cid:18) ka + a ′ N p a (cid:19) ( g ik g jl − g il g jk ) , (5)where g ij is the spatial part of the FLRW metric and ( ′ ) denotes derivative with respect to t p . For the Ricci-tensor the non-zero components are R = − ( D − (cid:18) a ′′ a − a ′ N ′ p aN p (cid:19) , ij = (cid:20) ( D − kN p + a ′ ) N p a + a ′′ N p − a ′ N ′ p aN p (cid:21) g ij , (6)while the Ricci-scalar for FLRW is given by R = 2( D − (cid:20) a ′′ N p − a ′ N ′ p aN p + ( D − kN p + a ′ )2 N p a (cid:21) . (7)In the case of Weyl-flat metrics one can express Riemann tensor in terms of Ricci-tensorand Ricci scalar. R µνρσ = R µρ g νσ − R µσ g νρ + R νσ g µρ − R νρ g µσ D − − R ( g µρ g νσ − g µσ g νρ )( D − D − . (8)Due to this identity we have R µνρσ R µνρσ = 4 D − R µν R µν − R ( D − D − . (9)It allow us to simplify our Gauss-Bonnet gravity action for Weyl-flat metrics. Z d D x √− g (cid:0) R µνρσ R µνρσ − R µν R µν + R (cid:1) = D − D − Z d D x √− g (cid:18) − R µν R µν + DR D − (cid:19) . (10)On plugging the FLRW metric of eq. (2) in the action in eq. (1) we get an action for a ( t p )and N p ( t p ). S = V D − πG Z d t p (cid:20) a D − N p (cid:26) ( D − D − kN − a N p − D − aa ′ N ′ p +( D − D − a ′ N p + 2( D − N p aa ′′ (cid:27) + ( D − D − D − α (cid:26) a D − ( D − N p × ( kN p + a ′ ) + 4 a D − dd t p (cid:18) ka ′ N p + a ′ N p (cid:19)(cid:27)(cid:21) , (11)where V D − is the volume of D − V D − = Γ(1 / D/ (cid:16) πk (cid:17) ( D − / . (12)In D = 4 we notice that in the GB-sector terms proportional α either vanish or are totaltime-derivatives. The mini-superspace gravitational action becomes following in D = 4 S = V πG Z d t p (cid:20) kaN p − a N p − a a ′ N ′ p N p + 6 aa ′ N p + 6 a ′′ a N p + 24 α dd t p (cid:18) ka ′ N p + a ′ N p (cid:19)(cid:21) , (13)This action can be recast in to a more appealing form by a rescaling of lapse and scale factor. N p ( t p )d t p = N ( t ) a ( t ) d t , q ( t ) = a ( t ) . (14)5his set of transformation changes our original metric in eq. (2) into followingd s = − N q ( t ) d t + q ( t ) (cid:20) d r − kr + r dΩ D − (cid:21) , (15)and our action in D = 4 given in eq. (13) changes to following simple form S = V πG Z d t (cid:20) (6 k − q ) N + 3 ˙ q N + 3 q dd t (cid:18) ˙ qN (cid:19) + 24 α dd t (cid:18) ˙ q N + ˙ q N (cid:19) (cid:21) . (16)Here (˙) here represent derivative with respect to time t . It is worth noting the GB-part ofaction appears as a total derivative term. It will later be seen that this part plays a crucialrole in the action for the lapse N and will result in additional saddle points. In the path-integral this term will play crucial role as it will in some sense be incorporating topologicalcorrections. III. BOUNDARY ACTION AND BOUNDARY CONDITIONS
To find the boundary action and the relevant set of boundary conditions we start byvarying the action in eq. (16) with respect to q ( t ). From now on we work in the ADM gauge˙ N = 0, which implies that N ( t ) = N c (constant). We write q ( t ) = ¯ q ( t ) + ǫδq ( t ) (17)where ¯ q ( t ) satisfy the equation of motion, δq ( t ) is the fluctuation around it and ǫ is parameterused to keep a track of the order of fluctuation terms. Plugging this in eq. (16) and expandingto first order in ǫ we have δS = ǫV πG Z d t (cid:20)(cid:18) − N c + 3¨ qN c (cid:19) δq + 3 N c dd t ( qδ ˙ q ) + 24 α dd t (cid:26)(cid:18) N c + ˙ q N c (cid:19) δ ˙ q (cid:27) (cid:21) . (18)There will also be second order terms, but for the purpose of having a sensible boundaryvalue problem for the equation of motion this is sufficient. We notice that there are two totaltime-derivative pieces in the above equation which will be responsible for fixing appropriateboundary conditions. The term proportional to δq gives the equation of motion for q ¨ q = 23 Λ N c . (19)This is easy to solve and its general solution is q ( t ) = Λ N c t + c t + c , (20)where c , are constants and will be determined based on the boundary conditions. Thetotal-derivative terms in the above will result in a collection of boundary terms S bdy = ǫV πG (cid:20) N c ( q δ ˙ q − q δ ˙ q ) + 24 α (cid:26)(cid:18) δ ˙ q N c + ˙ q δ ˙ q N c (cid:19) − (cid:18) δ ˙ q N c + ˙ q δ ˙ q N c (cid:19)(cid:27)(cid:21) , (21)where q = q ( t = 0) , q = q ( t = 1) , ˙ q = ˙ q ( t = 0) , ˙ q = ˙ q ( t = 1) . (22)6 . Neumann Boundary condition (NBC) If we impose Neumann boundary condition (NBC) which is fixing ˙ q at both the ends ofthe q -trajectory [23, 44]. Then we notice that the surface term in eq. (21) vanish completely.˙ q , | NBC = fixed ⇒ δ ˙ q , | NBC = 0 , (23)where the | NBC refers to imposing Neumann boundary condition. However, it is soon realisedthat with this boundary condition the constant c , appearing in the solution to equationof motion (20) cannot be fixed uniquely. In particular c is left undetermined while c willhave two different values. This implies that it is not a well-posed problem as it leads toinconsistencies. This boundary condition cannot be imposed even though the surface termin eq. (21) vanishes entirely and one doesn’t have to incorporate any additional boundaryaction. B. Dirichlet Boundary condition (DBC)
In this boundary condition we fix the value of q at the two end points. This means wehave q , | DBC = fixed ⇒ δq , | DBC = 0 , (24)where the | DBC refers to imposing Dirichlet boundary condition. Our surface contribution ineq. (21) doesn’t vanish under the imposition of this boundary condition. In the case when α = 0 (only Einstein-Hilbert gravity), then in order to have a sensible Dirichlet boundaryvalue problem one has to add an extra boundary action. This is the well known Gibbon-Hawking-York term [21, 22, 40], which in mini-superspace reduces to S GHY = − V πG q ˙ qN c (cid:12)(cid:12)(cid:12)(cid:12) = − V πG (cid:18) q ˙ q N c − q ˙ q N c (cid:19) . (25)On varying the S GHY action it is noticed that it cancels the δ ˙ q terms at the boundary ineq. (21) for α = 0. It therefore sets up a successful imposition dirichlet boundary condition,atleast for the Einstein-Hilbert gravity part of theory, thereby leading to a consistent solutionto equation of motion.But the same thing can not be implemented for the Gauss-Bonnet sector of gravitationalsurface terms. They will be proportional to f ( ˙ q ) δ ˙ q , where f ( ˙ q ) = (1 / N c + ˙ q / N c ). Inprinciple one can construct a possible surface term for the Gauss-Bonnet sector. S GB | bdy = F ( q, ˙ q ) | . (26)During the process variation of action with respect to q to compute equation of motion, thissurface term on variation will lead to ǫ (cid:18) ∂F∂q δq + ∂F∂ ˙ q δ ˙ q (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (27)Then in order to cancel the surface contribution proportional to α in eq. (21), we noticethat implies ǫ ∂F∂ ˙ q δ ˙ q (cid:12)(cid:12)(cid:12)(cid:12) + ǫV πG α (cid:26)(cid:18) N c + ˙ q N c (cid:19) δ ˙ q (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , ⇒ F ( q, ˙ q ) = − V πG α (cid:18) ˙ q N c + ˙ q N c (cid:19) + g ( q ) . (28)7s the Gauss-Bonnet surface part in eq. (21) doesn’t have any term proportional to δq or δq , so this implies that g ′ ( q ) = 0, which can be fixed to zero. Then the total boundary actionis a summation of Gibbon-Hawking term from eq. (25) and Gauss-Bonnet part coming fromeq. (28). S surface = S GHY − V πG α (cid:18) ˙ q N c + ˙ q N c (cid:19) . (29)This when added to the boundary contributions coming from varying the bulk action resultsin complete cancelation of the terms proportional to α . As a result it doesn’t lead toany non-trivial contributions coming from Gauss-Bonnet sector. However, for Dirichletboundary conditions the equation of motion can still be solved without any inconsistencies,but the gravitational path-integral will not have non-trivial features coming from Gauss-Bonnet sector of gravitational action. In a sense if our motivation is look for situationswhere Gauss-Bonnet piece of gravitational action contribute non-trivially then DBC doesn’tfall in the category. C. Mixed Boundary condition (MBC)
After not being able to have a consistent boundary value problem with Neumann bound-ary conditions and lack of obtaining non-trivial effects in the case of Dirichlet boundaryconditions, we next consider the situation with mixed boundary conditions where we spec-ify q at one end and ˙ q at another end. Similar mixed boundary conditions have also beeninvestigated in [25, 44, 45, 49], here inspired by their work we consider applying them incase of Gauss-Bonnet gravity.In this case there are two possibilities:Case (a) : Specify q and ˙ q ⇒ δq = δ ˙ q = 0 , Case (b) : Specify ˙ q and q ⇒ δq = δ ˙ q = 0 . (30)We will consider each of this cases individually in more detail later in paper. But first westudy the boundary action that is needed for each of these. The surface action for each ofthese is S ( a )surface = V πG (cid:20) q ˙ q N c + 24 α (cid:18) ˙ q N c + ˙ q N c (cid:19)(cid:21) , (31) S ( b )surface = − V πG (cid:20) q ˙ q N c + 24 α (cid:18) ˙ q N c + ˙ q N c (cid:19)(cid:21) . (32)During the computation of equation of motion, each of them can be varied and added tothe boundary action in eq. (21). This will result in S bdy + δS ( a )surface = ǫV πG (cid:20) N c ( q δ ˙ q + ˙ q δq ) + 24 α (cid:18) δ ˙ q N c + ˙ q δ ˙ q N c (cid:19)(cid:21) , (33) S bdy + δS ( b )surface = − ǫV πG (cid:20) N c ( ˙ q δq + ˙ q δq ) + 24 α (cid:18) δ ˙ q N c + ˙ q δ ˙ q N c (cid:19)(cid:21) . (34)From this one immediately notices that in former case ( a ) if we fix q and ˙ q then RHS ofeq. (33) vanishes. Similarly in the later case ( b ) if we fix q and ˙ q then the RHS of the eq.834) vanishes. In this way the boundary value problem is well-posed. Moreover the totalaction of theory is S ( a )tot = S + S ( a )surface = V πG Z d t (cid:20) (6 k − q ) N c + 3 ˙ q N c + 3 q ¨ qN c (cid:21) + V πG (cid:20) q ˙ q N c + 24 α (cid:18) ˙ q N c + ˙ q N c (cid:19)(cid:21) , (35) S ( b )tot = S + S ( b )surface = V πG Z d t (cid:20) (6 k − q ) N c + 3 ˙ q N c + 3 q ¨ qN c (cid:21) − V πG (cid:20) q ˙ q N c + 24 α (cid:18) ˙ q N c + ˙ q N c (cid:19)(cid:21) . (36)In each of these cases one can compute the momentum corresponding to field variable q ( t )by varying the bulk Lagrangian with respect to ˙ q . This is given by π = δ L δ ˙ q = 3 ˙ qN c . (37)It should be noted the bulk momentum in both the cases is same.The variational problem in the two cases is well-posed resulting in equation of motionwhose solution can be found consistently. The solution to equation of motion in each ofthese cases is given by q ( a ) ( t ) = Λ N c t + (cid:18) ˙ q − N c (cid:19) t + q , (38) q ( b ) ( t ) = Λ N c t + ˙ q ( t −
1) + (cid:18) q − Λ N c (cid:19) . (39)These solution can be plugged back in corresponding action of theory in eq. (35 & 36) toobtain action for the lapse N c . The lapse action for the two cases is given by, S ( a )tot = V πG (cid:20) kN c + 3 ˙ q (2 q + ˙ q )2 N c − (2 q + ˙ q ) N c Λ + 2 N c Λ α ˙ q N c (cid:18)
12 + ˙ q N c (cid:19)(cid:21) , (40) S ( b )tot = V πG (cid:20) kN c + 3 ˙ q ( ˙ q − q )2 N c + ( ˙ q − q ) N c Λ + 2 N c Λ − α ˙ q N c (cid:18)
12 + ˙ q N c (cid:19)(cid:21) . (41)The lapse action include non-trivial features coming from the Gauss-Bonnet sector of grav-itational action, which arise in the case of MBC. In the following we will study these twocases in more detail. IV. TRANSITION PROBABILITY
Generically once the action of a theory is known at the classical level then it can be usedin the path-integral to study the behaviour of the corresponding quantum system. In thecase investigated in present paper the well-known classical action of Einstein-Hilbert gravityis modified by inclusion of Gauss-Bonnet gravity terms which is topological in four spacetimedimensions. Although such topological extensions doesn’t affect dynamical evolution of fields9t the classical level as has been noted in the previous section, but their presence play acrucial role in dictating the choice of boundary conditions.In the case of gravity one can simplify the process of studying gravitational path-integralby restricting oneself to mini-superspace approximation. Within this approximation one canprecisely ask the following question: what is the transition amplitude from one 3-geometryto another? Is it possible to address this directly in Lorentzian signature without doing astandard Wick-rotation of time? and what is the role played by boundary conditions in thecomputation of this transition amplitude given by path-integral? The relevant quantity thatwe are interested in can be expressed in mini-superspace approximation [32, 41] as follows G [bd , bd ] = Z C D N c Z D q ( t ) exp (cid:18) i ~ S tot (cid:19) , (42)where bd and bd are initial and final boundary configurations respectively. The path-integral over q ( t ) is performed such that it respects those boundary conditions. For ourpresent case the above path-integral will be analysed with mixed boundary conditions asdiscussed in eq. (30) in previous section. S tot is the total action incorporating the appropriateboundary condition as given in (35 and 36), ‘ C ’ is the contour of integration for N c whichwill be chosen later (in the next section) using Picard-Lefschetz theory.Analysing this in mini-superspace approximation is still cumbersome as the usual issueof defining measure, having convergence, un-controllable oscillations on integrand are stillpresent. In standard path-integrals usually the highly oscillatory integrand is tamed by do-ing a Wick-rotation following Feynman + iǫ -prescription and considering the correspondingEuclidean path-integral. Here we make use of Picard-Lefschetz methods to obtain a contourof integration along which the integral is absolutely convergent.We start by considering the fluctuations around the solution to equation of motion, whichhas been obtained previously respecting the boundary conditions. q ( t ) = ¯ q ( a,b ) ( t ) + ǫ ′ √ πGQ ( t ) , (43)where ¯ q ( a,b ) ( t ) is the solution to equation of motion given in eq. (38 & 39), Q ( t ) is thefluctuation around the background ¯ q ( a,b ) ( t ), and ǫ ′ is the parameter to keep track of orderof terms. This decomposition can be plugged back in total action given in (35 and 36)and expanded to second-order in ǫ ′ . Q ( t ) obeys similar set of boundary conditions as thebackground ¯ q ( a,b ) ( t ): Case (a) : Specify Q and ˙ Q ⇒ Q = ˙ Q = 0 , Case (b) : Specify ˙ Q and Q ⇒ Q = ˙ Q = 0 . (44)After imposing these boundary conditions on Q and performing the expansion in powersof ǫ ′ we notice that first order terms in ǫ ′ vanish as ¯ q ( a,b ) ( t ) satisfies equation of motion.The second order terms are non-vanishing. The series in ǫ ′ stops at second order. The fullexpansion can be written as S ( a,b ) = S ( a,b )tot − ǫ ′ V N c Z d t ˙ Q , (45)where S ( a,b )tot is given in eq. (40 & 41). In the path-integral measure such a decompositionwill imply Z D q ( t ) ⇒ Z D Q ( t ) . (46)10s the action in eq. (45) separates into a part independent of Q and part quadratic in Q , therefore the path-integral over Q can be performed independently of the rest. Thispath-integral over Q is F ( N c ) = Z Q ′ [1]=0 Q [0]=0 D Q ( t ) Case (a) OR Z Q [1]=0 Q ′ [0]=0 D Q ( t ) Case (b) exp (cid:18) − iǫ ′ V ~ N c Z d t ˙ Q (cid:19) . (47)This path-integral is very similar to the path-integral for a free where the trajectories at endpoints are kept fixed. However, this one is slightly different as at one of the boundary weare fixing ˙ Q . A similar path-integral over mixed boundary conditions was encountered in[49] where the authors have computed it in appendix of the paper. Following the footstepsin [49] we note F ( N c ) = 1 √ πi . (48)The important point to note is that in case of mixed boundary conditions the above path-integrals leads a N c -independent numerical factor, unlike in case of Dirichlet boundary con-ditions where the above path-integral is proportional to N − / c .Then our transition amplitude G [bd , bd ] becomes G [bd , bd ] = 1 √ πi Z ∞ + d N c exp (cid:18) i ~ S ( a,b )tot (cid:19) , (49)where S ( a,b )tot is given in eq. (40 & 41). Now the task is reduced to performing the contourintegration over lapse N c . Here we will make use of complex analysis and Picard-Lefschetzformalism to analyse this integral. We start by studying the various saddle points of theaction S ( a,b )tot appearing in the exponent. A. Saddle points
The saddle points of the action can be found using ∂S ( a,b )tot ∂N c = 0 . (50)The important thing to note here is about the structure of S ( a,b )tot in terms of N c which canbe noticed from eq. (40 & 41). It has term proportional to N c , N c , 1 /N c and 1 /N c . Setting(8 πG ) = 1 the structural form for S ( a,b )tot can be written as S ( a,b )tot = V (cid:20) A ( a,b ) N c + B ( a,b ) N c + 2 N c Λ αC ( a,b ) N c (cid:21) , (51)where A ( a ) = 6 k − (2 q + ˙ q )Λ , A ( b ) = 6 k + ( ˙ q − q )Λ , (52) B ( a ) = 3 ˙ q (2 q + ˙ q )2 + 12 α ˙ q , B ( b ) = 3 ˙ q ( ˙ q − q )2 − α ˙ q (53) C ( a ) = ˙ q , C ( b ) = − ˙ q . (54)11his structure is largely same as in the case of Einstein-Hilbert gravity, except the emergenceof new additional term proportional to 1 /N c which is coming from the Gauss-Bonnet sector.The presence of this new term give rise to additional saddle points which are absent in thecase of Einstein-Hilbert gravity. ∂S ( a,b )tot ∂N c = 0 ⇒ A ( a,b ) − B ( a,b ) N c + 2 N c Λ − αC ( a,b ) N c = 0 . (55)It can be seen from this that the saddle point equation is cubic in N c , resulting in threepairs of roots. This cubic equation can be solved by the known methods of dealing withcubic polynomial equation. In particular if the cubic equation has real coefficients then thenature of roots can be determined by analysing the behaviour of the discriminant of cubicequation. Such a strategy is no longer valid if the coefficients are complex. The discriminantof cubic polynomial with real coefficients is given by∆ = (cid:0) A ( a,b ) (cid:1) (cid:0) B ( a,b ) (cid:1) + 8Λ (cid:0) B ( a,b ) (cid:1) + 12 α (cid:0) A ( a,b ) (cid:1) (cid:0) C ( a,b ) (cid:1) +36 α Λ (cid:0) A ( a,b ) (cid:1) (cid:0) B ( a,b ) (cid:1) (cid:0) C ( a,b ) (cid:1) − α Λ (cid:0) C ( a,b ) (cid:1) . (56)If ∆ > N c . If D < N c . By defining variables U = 34Λ (cid:0) A ( a,b ) (cid:1) + 32Λ B ( a,b ) ,V = 34Λ B ( a,b ) A ( a,b ) + 14Λ (cid:0) A ( a,b ) (cid:1) − α C ( a,b ) (57)one can write the roots as N ± = ± (cid:18) y + + y − − λ A ( a,b ) (cid:19) / ,N ± = ± (cid:18) y + ω + y − ω − λ A ( a,b ) (cid:19) / ,N ± = ± (cid:18) y + ω + y − ω − λ A ( a,b ) (cid:19) / , (58)where y ± = V ± r V − U ! / . (59)where 1, ω and ω are the three roots of unity. These are the six saddle points that arise inthis system.The boundary conditions decide the nature of A ( a,b ) , B ( a,b ) and C ( a,b ) . If they are realthen one can compute the discriminant of the cubic equation whose behaviour dictates thekind of roots for N c . We can collectively write the saddle point as N ± σ , where σ = 0, 1, and2. Corresponding to each of these saddle points we have a metric (cid:0) d s ( a,b ) σ (cid:1) = − N σ q ( a,b ) ( t ) d t + q ( a,b ) ( t ) (cid:20) d r − kr + r dΩ (cid:21) , (60)12here q ( a,b ) ( t ) is given by eq. (38 & 39). Note that it is N σ that enters the metric, whichimplies that the metric is same for each pair N ± σ of saddle points. As long as N σ is real andpositive, we are in Lorentzian signature. When it is real and negative then it is Euclideansignature, as in those cases N σ is imaginary. In cases when N σ is complex, the spacetimehas a mixed signature. Geometries become singular when q ( a,b ) ( t ) →
0. In this case thespacetime volume goes to zero.For each of these saddle points one has a corresponding on-shell action. As the saddlepoints will generically be complex in nature therefore their corresponding on-shell actionwill have a real and an imaginary part. The momentum at the saddles can be computedusing eq. (37). π ( a,b ) = 3 ˙ q ( a,b ) N ± σ . (61)By making use of solution to equation of motion given in (38 & 39) one can compute themomentum at the end points. π ( a )0 = 3 N ( a ) ± σ " ˙ q − N ( a )2 σ , π ( a )1 = 3 ˙ q N ( a ) ± σ ,π ( b )0 = 3 ˙ q N ( b ) ± σ , π ( b )1 = 3 N ( b ) ± σ N ( b )2 σ q ! . (62)The crucial point to note here is that momentum at the boundaries can be complex if thesaddle point N σ is complex. This is interesting as it carries characteristics of tunneling phenomena. V. N c -INTEGRATION VIA PICARD-LEFSCHETZ We then go forth to compute the N c -integration. We will make use of Picard-Lefschetz(PL) theory to analyse the behavior of the integrand in the complex plane [50–53]. Alongwith PL theory we make use of WKB methods to compute the integral. For this we needthe set of saddle points and collection of steepest descent/ascent paths associated with eachsaddle point. A saddle point is termed ‘ relevant ’ if the steepest ascent path emanatingfrom it intersects the original contour integration. The original integration contour canthen be distorted to lie along the steepest descent paths passing through relevant saddlepoints. Instead of using the prescription of Wick-rotation to deform the contour, we followthe methods of PL-theory to choose a contour of integration uniquely, along which theintegrand is absolutely convergent.The problem of performing path-integration is reduced to a task of computing thimbles(steepest descent paths) on a complex plane. In the following we will give a review of Picard-Lefschetz formalism. We start by considering the path-integral in the following manner I = Z D z ( t ) e i S ( z ) / ~ , (63)where the exponent is functional of z ( t ). In general the integrand can be quite oscillatoryand hence not an easy task to compute the integral. In flat spacetime the global symme-tries of spacetime allow one to cast Lorentz group in to a compact rotation group under13 transformation of time co-ordinate. This privilege doesn’t exist in non-flat spacetimes.Such a transformation of time co-ordinate in flat spacetime leads to exponential damping ofabove integrand. In PL theory one analytically continues both z ( t ) and S ( z ) in to complexplane, and interprets S as an holomorphic functional of z ( t ). This implies that S satisfies afunctional form of Cauchy-Riemann conditions δ S δ ¯ z = 0 ⇒ ( δ Re S δx = δ Im S δy , δ Re S δy = − δ Im S δx . (64) A. Flow equations
On writing the complex exponential as I = i S / ~ = h + iH and writing z ( t ) = x ( t ) + ix ( t ), the downward flow is defined asd x i d λ = − g ij ∂h∂x j , (65)where g ij is a metric defined on the complex manifold, λ is flow parameter and ( − ) signrefers to downward flow. The steepest descent flow lines follow a trajectory dictated byabove equation. They are also knowns as thimbles (can be denoted by J σ ). Steepest ascentflow lines are defined with a plus sign in front of g ij in the eq. (65), and are denoted as K σ .Here σ refers to the saddle point to which these flow-lines are attached. The definition offlow lines immediately implies that the real part h (also called Morse function) decreasesmonotonically as one moves away from the critical point along the steepest descent curves.This can be seen by computingd h d λ = g ij d x i d λ ∂h∂x j = − (cid:18) d x i d λ d x i d λ (cid:19) ≤ . (66)This generically holds for any Riemannian metric. However, in this paper for simplicity weassume g z,z = g ¯ z, ¯ z = 0 and g z, ¯ z = g ¯ z,z = 1 /
2. This leads to a simplified version of flowequations d z d λ = ± ∂ ¯ I ∂ ¯ z , d¯ z d λ = ± ∂ I ∂z . (67)Using them it is easy to notice that the imaginary part of Im I = H is constant along theflow lines. d H d λ = 12 i d( I − ¯ I )d λ = 12 i (cid:18) ∂ I ∂z d z d λ − ∂ ¯ I ∂ ¯ z d¯ z d λ (cid:19) = 0 . (68)This is a wonderful feature of flow-lines and can be used to determine the structure of flow-lines in the complex N c -plane. It is seen that the oscillatory integral becomes convergent andwell-behaved along any of the steepest descent lines (thimbles). This motivates one checkif it is possible to analytically deform the original integration contour to integration alongeither one thimble or a sum of thimbles. This is a true generalization of Wick-rotation.In the complex N c -plane the flow equations corresponding to steepest descent (ascent)becomes the following in cartesian co-ordinatesDescent ⇒ d x d λ = − ∂ Re I ∂x , d x d λ = − ∂ Re I ∂x , (69a)Ascent ⇒ d x d λ = ∂ Re I ∂x , d x d λ = ∂ Re I ∂x . (69b)14t is noticed that the Im I doesn’t enter the flow equations as Im I = const . along the flowlines. Each saddle point has two steepest descent lines and two steepest ascent lines attachedto it. The boundary conditions and the parameter values dictate the location of the saddleson the complex N c -plane. Solving these flow equations can be sometimes hard as I can becomplicated. However, it is possible to deal with them numerically. One can bypass solvingthem entirely by making use of knowledge that H is constant along them. This determinesall the flow-lines. But to find out about the nature of flow lines one has to compute thegradient of first derivative (second order derivative of action at the saddle points). B. Choice of contour
Once the set of saddle points along with the set of steepest ascent/descent flow-linesassociated with each saddle point are known, one can begin to find the new contour ofintegration to which the original integration contour will be deformed. The integral incomplex N c -plane is absolutely convergent along this new contour (for more detail see [32,51, 52]).In the complex N c plane the behavior of h and H determines the ‘allowed’ regions (regionwhere integral is well-behaved) and ‘forbidden’ region (region where integral diverges). Welabel the former by J σ while later is denoted by K σ , and as mentioned previously σ refersthe saddle point. These regions have h ( J σ ) < h ( N σ ), while h ( K σ ) > h ( N σ ). h goes to −∞ along the steepest descent lines and ends in a singularity, while along the steepest ascentcontours h → + ∞ . These lines usually intersect at only one point where they are bothwell-defined. With a suitable choice of orientation one can writeInt ( J σ , K σ ′ ) = δ σσ ′ . (70)The purpose is to write the integral over the original contour as an integral along the newcontour which is sum of integrations done along Lefschetz thimbles. Schematically this canbe expressed as D = (0 + , ∞ ) ⇒ C = X σ n σ J σ , (71)in a homological sense for some integers n σ which will take value 0 or ± n σ = Int( C , K σ ) =Int( D , K σ ). As the intersection number is topological and doesn’t change if we deform thecontour, therefore the necessary and sufficient condition for a thimble J σ to be relevant isthat the steepest ascent curve from the corresponding saddle point intersects the originalintegration domain D . The integration contour is chosen to lie in the region J σ (which is the‘allowed’ region) and follow the contour trajectory dictated by the steepest descent paths[32]. In this circumstance there is no hindrance in smoothly sliding the intersection pointalong the K σ to the relevant saddle point.Once the original integration contour is deformed to a sum over integration done alongvarious relevant thimbles then we have I = Z C d z ( t ) e iS [ z ] / ~ = X σ n σ Z J σ d z ( t ) e iS [ z ] / ~ . (72)It is common that in such process more than one thimble contributes to integration, resultingin interference of contributions coming from various thimbles. This is feature of performing15omplex integration via Picard-Lefschetz methodology. The integration along each of thethimbles is absolutely convergent if (cid:12)(cid:12)(cid:12)(cid:12)Z J σ d z ( t ) e iS [ z ] / ~ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z J σ | d z ( t ) || e iS [ z ] / ~ | = Z J σ | d z ( t ) | e h ( z ) < ∞ . (73)If we denote the length along the contour path as l = R | d z ( t ) | , then convergence of aboveintegral require that e h ∼ /l as l → ∞ . The original integration hence can be analyticallydeformed into a sum of absolutely convergent integrals along various Lefschtez thimblespassing through relevant saddle points. If one does an expansion in ~ then to leading orderwe get the following I = Z C d z ( t ) e iS [ z ] / ~ = X σ n σ e iH ( N σ ) Z J σ d z ( t ) e h ≈ X σ n σ e iS [ N σ ] / ~ [ A σ + O ( ~ )] , (74)where A σ is the contribution coming after performing a gaussian integration around thesaddle point N σ . C. Flow directions
The direction of flow lines either emanating from the saddles or going into it can bedetermined analytically (to some extent) by expanding the N c -action of theory given in eq.(40 & 41) around the saddle points given in eq. (58). If we write N c = N σ + δN (where N σ is any saddle point of action), then the action has a power series expansion in δN . S (0) = S (0) σ + d S (0) d N c (cid:12)(cid:12)(cid:12)(cid:12) N = N σ δN + 12 d S (0) d N c (cid:12)(cid:12)(cid:12)(cid:12) N = N σ ( δN ) + · · · . (75)The first order terms will vanish identically by definition.The second order terms can be computed directly from the action in eq. (40 & 41), byjust taking double-derivative with respect to N c . From this the direction of flow-lines can bedetermined. One should recall that the imaginary part of exponential iS (or H ) is constantalong the flow lines. This implies that Im [ iS − iS ( N s )] = 0. The second variation at thesaddle point can be written as d S (0) / d N c = re iρ , where r and ρ depends on boundaryconditions. Near the saddle point the change in H will go like∆( H ) ∝ i (cid:18) d S (0) d N (cid:12)(cid:12)(cid:12)(cid:12) N σ (cid:19) ( δN c ) ∼ n σ e i ( π/ θ σ + ρ σ ) , (76)where we write δN = n σ e iθ σ and θ σ is the direction of flow lines at the corresponding saddlepoint. Given that the imaginary part H remains constant along the flow lines, so this means θ σ = (2 k − π − ρ σ , (77)where k ∈ Z .For the steepest descent and ascent flow lines, their corresponding θ des / aes σ is such thatthe phase for ∆ H correspond to e i ( π/ θ σ + ρ σ ) = ∓
1. This implies θ des σ = kπ + π − ρ σ , θ aes σ = kπ − π − ρ σ . (78)These angles can be computed numerically for the given boundary conditions and for grav-itational actions. 16 . Saddle-point approximation Once the set of saddle points, flow directions and steepest descent/ascent paths associatedwith them (denoted by J σ / K σ respectively) are known, it is then easy to find the relevant saddle points. A saddle point is termed relevant if the steepest ascent path emanatingfrom it intersect with the original contour of integration. In the current case the originalintegration contour is (0 + , ∞ ). The original integration contour then becomes sum over thecontribution coming from all the Lefschetz thimbles passing through relevant saddle points.We can then do saddle-point-approximation to compute the transition amplitude in eq. (49).In the ~ → G [bd , bd ] ≈ √ πi X σ n σ exp (cid:20) i ~ S ( a,b )tot ( N σ ) (cid:21) Z J σ d N c exp (cid:20) i ~ (cid:16) S ( a,b )tot (cid:17) N c N c ( N c − N σ ) (cid:21) , (79)where we consider only the leading order term in ~ . Here N σ are the relevant saddle pointsfor the various boundary conditions given in eq. (58), S ( a,b )tot ( N σ ) is the on-shell action whichcan be computed from eq. (40 & 41) for the relevant saddles. (cid:16) S ( a,b )tot (cid:17) N c N c is the secondvariation of the action with respect to N c computed at the relevant saddle points.On writing N − N σ = ne iθ σ , where θ σ is the angle the Lefschetz thimble make with thereal N -axis while σ corresponds to relevant saddle point. Then the above integration canbe performed easily. It gives the following G [ q , q ] = 1 √ πi X σ n σ | (cid:16) S ( a,b )tot (cid:17) N c N c | − / exp (cid:20) iθ σ + i ~ S ( a,b )tot ( N σ ) (cid:21) . (80)In the next section we will make use of it and apply it to the case of no-boundary proposalof Universe to compute the transition amplitude. VI. NO-BOUNDARY UNIVERSE
This is special boundary condition where the Universe start from nothing. In the currentsituation this implies q = 0, implying that Universe started with a zero scale factor a [30, 32–34, 44, 45]. In case ( a ) where we specify field q at one end point while its first derivative ˙ q at another end point this immediately leads to a simplified N c action. However, at the finalboundary, following the solution to equation of motion from eq. (38) we also have a relation˙ q ( a )1 = q ( a )1 + Λ N c . (81)This allow us to express the derivative of field in terms of field value at the final boundary.This is useful as one can study the problem by doing the analysis in terms of q wherewe require that for physical reasons q >
0. A real and positive q immediately implies apossibly complex ˙ q ( a )1 if N c is complex. On the other hand this also means that if q has afixed real positive value and the number of relevant saddle points are more that one, then itwill imply that at the final boundary ˙ q ( a )1 will have multiple values. This is contradictory toour initial requirement that in case ( a ) ˙ q ( a )1 is fixed at final boundary, and implies that theUniverse at final time has multiple values of ˙ q ( a )1 .17n the case ( b ) on other hand the boundary conditions require fixing ˙ q ( b )0 and q ( b )1 . Thisimplies that at t = 0 by following the solution 39, if the Universe started from nothing( q ( b ) ( t = 0) = 0), then it leads to a relationship between ˙ q ( b )0 and q ( b )1 ˙ q ( b )0 = q ( b )1 − Λ N c . (82)Here if at final boundary q ( b )1 > q ( b )0 could be complexwhen N c is complex. Moreover, when the number of relevant saddle points are more thanone, then the final geometry is seen to arising from superposition multiple initial geometries.This is quite possible and doesn’t lead to a contradiction unlike in case ( a ). In the followingwe will study this particular scenario in more detail. We can plug the relation in eq. (82)in the action for N c for case ( b ) in eq. (41) to obtain the N c -action for the no-boundaryproposal. S ( b )tot = V πG (cid:20) kN c + { ( α Λ − q + 12 α } N c Λ3 + (9 + 2 α Λ) q + 24 αq N c + (3 + 2 α Λ)Λ N c
54 + αq N c (cid:21) . (83)It should be noted that if we set α = 0 then we get the action for the no-boundary Universein case of pure Einstein-Hilbert gravity. We note that this residual action is bit differentfrom the action that one obtains in the case of dirichlet boundary conditions [32–34]. This isbecause we used mixed boundary conditions to arrive at the action in eq. (83). The saddlepoint equation correspondingly isCase (b) : Λ (3 + 2 α Λ)18 N c + (cid:26) k − q Λ + α Λ(12 + q Λ)3 (cid:27) N c − (cid:26) αq + (cid:18)
92 + α Λ (cid:19) q (cid:27) N c − αq = 0 . (84)The interesting thing to note here is that in case ( b ) there exist a q for which the coefficientof N c in eq. (84) can vanishe. This will offer some simplification in the expressions forsaddle points.The saddle-point equation is cubic in N c with real coefficients. Its nature of roots canbe decided based by analysing the behaviour of its discriminant in the parameter space ofcouplings and boundary value q . It is seen that for positive k , Λ, and α the discriminantis always positive for q ≥
0. This is interesting as it quickly implies that the saddle pointequation has three distinct real roots for N c . Also as α and q are positive, the zeroth-orderterm in N c in the saddle-point equation is positive. This means that the product of threeroots has to be positive. It leads to two possibilities: either all roots for N c are positiveor one is positive and other two are negative. However, as the coefficient of N c is negativeso this immediately implies the later case with one positive root and two negative roots for N c . This means that we have two saddle points lying on real-axis in complex N c -plane (onepositive and one negative); while four saddle points lie on imaginary axis in complex N c -plane (two of them in positive imaginary axis, while other two in negative imaginary axis).It is worth stating here that saddle point where N c > N c < relevant saddle points. The second variation is given by (cid:0) S ( b ) (cid:1) ,N c N c = V πG (cid:20) (3 + 2 α Λ)Λ N c α Λ) q + 24 αq N c + 12 αq N c (cid:21) . (85)At real saddle points the second variation is also real, while when N c is complex then thesecond variation will also be complex.The nature of relevance of each of these saddle-points depends on the parameter valuesand whether the steepest ascent path emanating from them intersects the original integrationcontour. In principle this seems like a well-defined way of finding out the relevance of saddlepoints. However, in practice often the action has large amount of symmetry. Due to thisthere is degeneracy between steepest ascent and steepest descent curves. It means that thesteepest ascent curve from one saddle point overlaps with the steepest descent curve fromanother saddle point. To lift these degeneracy one can add a small perturbation in the N c action which helps in breaking symmetry. Lifting this degeneracy also aid us to correctlylocate the relevant saddle points.We will consider a numerical example to investigate the state of art once parameters arefixed to some value. For numerical analysis and to lift degeneracy of the system we consideradding a small perturbation to the N c action S pert = iV ǫ ′′ N c πG , (86)where ǫ ′′ is a small parameter. Notice that the perturbation is imaginary in nature and it issomewhat reminiscent to + iǫ -prescription in standard flat spacetime field theory. In a sensewe are inspired by the Feynman’s + iǫ -prescription to choose this perturbation.For purpose of better understanding the system, we pick up an example. We considerthe value of parameters: k = 1, Λ = 3, α = 2 and ǫ ′′ = 10 − (we have set 8 πG = 1). For k = 1 the volume V = π which can be computed using eq. (12). We compute the saddlepoints following the eq. (84). As discussed previously it is seen that the cubic equation in N c has three distinct roots: N , N and N . As expected the Re( N ) >
0, Re( N ) < N ) < q . Each of these roots for N c gets a small imaginary part due to theperturbation, which thereby give rise to a small deviation in the saddle points valueIm (cid:0) δN ± (cid:1) = ∓ ν , ν > , Re (cid:0) δN ± (cid:1) = ± ν , ν > , Re (cid:0) δN ± (cid:1) = ± ν , ν > . (87)In figure 1 we plot the real and imaginary part of these roots as a function of q (the valueof q at t = 1). To determine the nature of saddle points ( relevant or irrelevant ) one has tostudy the steepest descent/ascent flow lines corresponding to each saddle point. These flowlines can be drawn by exploiting the knowledge that along these lines H ( N c ) = H ( N σ ). Inthe absence of perturbation in (86) there will be some degeneracy in the sense that steepestdescent line of one saddle will overlap with the steepest ascent line from another saddle.The addition of perturbation helps in lifting this degeneracy. In order to find the relevance of saddle points it is also crucial to analyse the nature of Morse-function h at each of thesesaddle points. Picard-Lefschetz theory dictates that relevant saddle points must be reached19 (cid:2) (cid:3) (cid:4) (cid:5) (cid:6)(cid:1) - (cid:6)(cid:7) - (cid:6)(cid:1) - (cid:7)(cid:1)(cid:7)(cid:6)(cid:1) (cid:1) (cid:1) (cid:8)(cid:9) ( (cid:2) (cid:1) (cid:2) ) N N N (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6)(cid:1) (cid:1)(cid:7)(cid:1)(cid:1)(cid:7)(cid:8)(cid:6)(cid:7)(cid:1)(cid:6)(cid:7)(cid:8)(cid:2)(cid:7)(cid:1)(cid:2)(cid:7)(cid:8)(cid:9)(cid:7)(cid:1)(cid:9)(cid:7)(cid:8) (cid:1) (cid:1) | (cid:10)(cid:11) (cid:2) (cid:0) (cid:2) (cid:12)(cid:13) (cid:2) (cid:2) (cid:2) |* (cid:6)(cid:1) (cid:3) N N N FIG. 1. Here we analyse the real and imaginary part of the saddle points ( N , N and N ) forvarious values of q . In this numerical example we consider k = 1, Λ = 3, α = 2 and ǫ ′′ = 10 − .There is a small imaginary part which comes due to the perturbation added to the total N c -action.As discussed the real part of N remains positive, while real part of N and N remains alwaysnegative. In the plot on left we show the real part of various N σ , while on the right plot we see thebehavior of | Im( N σ ) / Re( N σ ) | as a function of q . For the plot on right we have scaled the valueby 10 to plotting purpose. by flowing down from the original integration contour via steepest ascent paths. This willimmediately imply that h < relevant saddle points. A complex action bypasses thisrigid constraint though. However, in our present case this is not possible.If we plot h ( N σ ) against q we notice that for some of saddle points h changes sign as q is varied. This is shown in figure. 2. For the present situation there are six saddle points. (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6)(cid:1) - (cid:1)(cid:7)(cid:6)(cid:8) - (cid:1)(cid:7)(cid:6)(cid:1) - (cid:1)(cid:7)(cid:1)(cid:8)(cid:1)(cid:7)(cid:1)(cid:1)(cid:1)(cid:7)(cid:1)(cid:8)(cid:1)(cid:7)(cid:6)(cid:1)(cid:1)(cid:7)(cid:6)(cid:8) (cid:1) (cid:1) (cid:9) ( (cid:2) σ ) h ( N - ) h ( N + ) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6)(cid:1) (cid:6)(cid:2) - (cid:2)(cid:1)(cid:1)(cid:1) - (cid:6)(cid:1)(cid:1)(cid:1)(cid:1)(cid:6)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:1) (cid:1) (cid:1) (cid:7) ( (cid:2) σ ) h ( N - ) h ( N + ) h ( N - ) h ( N + ) h ( N - ) h ( N + ) FIG. 2. Plotting Morse-function h for various saddle point against q . For this we considerparameter values k = 1, Λ = 3, α = 2 and ǫ ′′ = 10 − . From the plots we notice that only h ( N − )and h ( N − ) remains always negative. h ( N ± ) change sign after a certain threshold q th1 . Only those which can be reached by flowing downward along the steepest ascent lines fromthe original integration contour are relevant . The saddle points N +0 and N − lie in lower-rightand upper-left quadrant respectively. Only the former can be reached via steepest ascentlines from original integration contour and hence is relevant . The saddles N +1 and N − liein upper-right and lower-left quadrant respectively. Both these saddle-point can be reachedfrom original integration contour by flowing downward along the steepest ascent flow lines.However only the later lie in allowed region with corresponding h < N − is therefore relevant . The saddle points N +2 and N − lie in lower-right and upper-leftquadrant respectively. Both are irrelevant : the former can’t be reached via a steepest ascentpath from original integration contour while later has h >
0. So out of six saddle-points20nly two of them are relevant : N +0 and N − . In figure 3 we consider an example of the above ■■⊗⊗ ⊗⊗▲▲ ⊗⊗ ⊗⊗ (cid:1)(cid:2) ( (cid:1) (cid:1) ) (cid:3)(cid:4) ( (cid:1) (cid:1) ) FIG. 3. We consider the case of no-boundary Universe where we choose parameter values: k = 1,Λ = 3, ǫ ′′ = 10 − and α = 2. We take q = 3. We plot on x -axis real-part of N c while the y -axisis imaginary part of N c . The red lines correspond to steepest descent lines (thimbles J σ ), whilethe thin black lines are steepest ascent lines and denoted by K σ . The various saddle points N σ areshown in blue. The blue cross-circle are irrelevant saddle points. The saddle point blue-square andblue-triangle are relevant . The steepest ascent line emanating from it can be connected to originalintegration contour. H remains constant along the red and black lines, and is equal to the value of H ( N σ ). The green region is allowed region with h < h ( N σ ) for all values of σ . The orange region(forbidden region) has h > h ( N σ ) for all σ . The light-green, light-orange and un-colored region hasintermediary values. The boundary of these region is depicted in brown lines. Along these line wehave h = h ( N σ ). The original contour of integration (0 , ∞ + ) is shown by thick black line. scenario. We plot the set of saddle points along with the collection of flow lines associatedwith each saddle. The coloring of graph is done obeying the relation between the values ofMorse-function at various saddle points. The region where h ( N c ) < h ( N − ) is colored green.The region where h ( N c ) > h ( N +1 ) is colored orange. The light-green and light-orange regionhas intermediary values. The boundary of these regions is depicted in brown lines. Thesteepest descent lines are shown in red while the steepest ascent lines are shown in black.The thick black-line depicts the original integration contour. The upward flow throughoriginal integration contour only hits the saddle points N +0 , N − and N − . However, only theformer two are relevant with corresponding h <
0, while N − is irrelevant with corresponding h becoming positive for q > q th1 (threshold value).The deformed contour of integration can be chosen such that it passes through all the relevant saddle points (depicted in blue-triangle and blue-square), and follow closely theLefschetz-thimbles passing though them. The saddles depicted by blue-triangle is predom-inantly imaginary, and hence correspond to a predominantly Euclidean geometry, while21he saddle point depicted in blue-square is predominantly real and hence correspond to apredominantly Lorentzian geometry.The deformed contour starts at blue-triangle then circles around following the Lefschetz-thimble (red-line) lying in lower-right quadrant. Then it approaches origin. Thereafternear the origin it turns back, hovers around in the green region following the steepest-descent line approaching the blue-square. Thereafter it follows the Lefschetz-thimble (redline) connecting blue-square in the upper-right quadrant. For the two relevant saddle thecorresponding ˙ q at the initial boundary can be computed using eq. (82). As the N c at thetwo saddle point is different, as a result the initial ˙ q is different, indicating that the finalgeometry is a super-position of two different initial configurations. | (cid:1) ( (cid:2)(cid:3) (cid:1) (cid:1) )| (cid:1) (cid:2)(cid:1) (cid:3)(cid:1) (cid:4)(cid:1) (cid:5)(cid:1) (cid:6)(cid:1)(cid:1)(cid:1)(cid:7)(cid:1)(cid:1)(cid:1)(cid:7)(cid:1)(cid:6)(cid:1)(cid:7)(cid:1)(cid:2) (cid:1) (cid:7)(cid:1)(cid:8)(cid:1) (cid:7)(cid:1)(cid:3) (cid:1) (cid:1) FIG. 4. We consider the case of no-boundary Universe where we choose parameter values: k = 1,Λ = 3, ǫ ′′ = 10 − and α = 2. Here we plot the transition amplitude G (0 , q ) as q is varied. We numerically compute the transition amplitude in this particular case and plot it infigure 4 as a function of q . For this situation under consideration we have θ N = 0 and θ (cid:4) = π/
4. Both saddle-points contribute in exponentially suppressed manner as h < q . At each q the weight of blue-square saddle point is more than the weight of blue-triangle saddle favouring a Lorentziangeometry. VII. COMPLEX INITIAL MOMENTUM
In this section we consider a very simple model of no-boundary proposal where we directlyfix the initial field derivative ˙ q . Certainly, this scenario fall in case ( b ) category as discussedin subsection III C. To properly motivate the choice of ˙ q we start by considering deSitter(dS) geometry which will be solution to bulk equation of motion. This means that for22 = 3 λ > d = 4 we have for the spacetime dS metric in eq. (2) N p = 1 , a ( t p ) = 1 λ cosh ( λt p ) . (88)dS can be embedded in 5-dimensions where in closed slicing it can pictured as hyperboloidhaving a minimum spatial extent at t p = 0. The intuition behind the no-boundary proposalis that the geometry is rounded off, so as to have no boundary in the beginning of time.This can be achieved by analytically continuing the original dS metric to Euclidean time,starting exactly at the waist of hyperboloid at t p = 0. This means t p = ∓ i (cid:16) τ − π λ (cid:17) , ≤ τ ≤ π λ . (89)This means that along the Euclidean section the dS metric transforms in to that of a 4-sphered s = d τ + 1 λ sin ( λτ ) dΩ . (90)This geometry has no boundary at τ = 0 and smoothly closes off.It should be emphasised that there are two possibilities of the time rotation to Euclideantime above, corresponding to the sign appearing in eq. (89). Each of these choices correspondto a different Wick rotation. The upper sign correspond to the standard Wick rotation whichis also used in the flat spacetime QFT. It is also the sign chosen in the work of Hartle andHawking [31, 54]. For this sign the perturbations around the geometry are stable andsuppressed. The lower sign in eq. (89) correspond to Vilenkin’s tunneling geometry wheresmall perturbation around the geometry are unsuppressed [33, 42]. The process of Wickrotation can also be thought of the lapse N p changing its value from N p = 1 to N p = ∓ i ,thereby implying that the total time T p = R N p d t p becoming complex valued.This can be translated into the language of metric in eq. (15) and will thereby implysinh ( λt p ) = λ N HH t + i , (91)where N HH will turn out to the saddle-point value of of the lapse integral corresponding toHartle-Hawking geometry [31, 54]. It is given by N HH = p λ q − λ − iλ , (92)where q = q ( t = 1). The HH-geometry fall in case ( b ) of the mixed boundary conditions.On comparing it with eq. (39) one has q ( t = 0) = 0, while the q ( t ) is given by˙ q = q − λ N , ⇒ q HH ( t ) = λ N t + (cid:0) q − λ N (cid:1) t , (93)where 0 ≤ t ≤
1. From this we can immediately notice the complex nature of saddle pointvalue N HH encodes the direction of Wick rotation. This can be seen by computing themomentum using eq. (37) at t = 0 ˙ q HH N HH (cid:12)(cid:12)(cid:12)(cid:12) t =0 = + i . (94)23otivated by the Hartle-Hawking geometry [31, 54] where we notice that the initial momen-tum is complex and appears with positive sign resulting in a stable and suppressed behaviourof fluctuations, we can consider appling this boundary condition in the case ( b ) scenario thatis considered in this paper. More clearly motivated by the works of Hartle-Hawking [31, 54],we choose the following mixed boundary condition in the case ( b )˙ q ( b )0 = +2 iN c , q ( b ) ( t = 1) = q . (95)Plugging this special condition in eq. (39) and (41) we get q ( b ) ( t ) = λ N c t + 2 iN c t + q − iN c − λ N c , (96)and the corresponding action for lapse N c is given by S HHtot = V πG (cid:20) λ N c + 6 iλ N c − q λ N c − i (3 q + 8 α ) (cid:21) (97)respectively. There are few crucial things to note here for this special mixed boundarycondition: (1) the action for lapse N c is complex (2) the action is no longer singular at N c = 0. The former is a direct consequence of the imposition of complex initial momentumwhich subsequently leads to complex geometries. A complex action will mean that even forreal values of lapse N c there will be a non-zero weighting of the corresponding geometricalconfiguration.The later point about the lack of N c = 0 singularity can be understood by realising thatas we are fixing the initial momentum (and not the initial size of geometry). As resultwe are summing over all possible initial 3-geometry size and their transition to 3-geometryof size q . This will also include a transition from q → q . Such a transition can occurinstantaneously i.e. with N c = 0. This means that there is nothing singular happening at N c = 0.An interesting third observation is that the saddle point equation following from actionin eq. (97) is quadratic in N c .d S HHtot d N c = 0 ⇒ λ N c + 2 iλ N c − q λ = 0 . (98)This quadratic equation has only two saddle point solution, unlike the scenarios studied inprevious section where there were six saddle points. In the present case the saddle points arealso independent of the Gauss-Bonnet coupling α . These saddle points have a very simpleexpression N ± = − i ± p q λ − λ . (99)It should be noted that N + is same as the saddle point considered in the work of Hartle-Hawking [31, 54].At this point our interest is to compute eq. (49) for the case ( b ) for the boundary conditionmentioned in eq. (95). The N c -action is given in eq. (97). As the integrand is not singularfor N c = 0, so one can extend the range of the N c -integration from −∞ to ∞ . Then wehave G [ ˙ q = 2 iN c , q ] = 12 √ πi Z ∞−∞ d N c exp (cid:18) i ~ S HHtot (cid:19) . (100)24his can be performed using the Picard-Lefschetz and WKB methods. Once the saddlepoints for the action S HHtot are known, one can compute the steepest ascent/descent flow linescorresponding to each of the saddle point. A saddle point is termed relevant if the steepestascent path emanating from it hits the original integration contour which is ( −∞ , + ∞ ). Ifthe action is real then it implies that the relevant saddle points will have their correspondingMorse-function h <
0. However, in the case when action is complex this obstruction can beevaded. ■■ ●● (cid:1)(cid:2) ( (cid:1) (cid:1) ) (cid:3)(cid:4) ( (cid:1) (cid:1) ) FIG. 5. We consider the case of no-boundary Universe where we impose the mixed boundarycondition: with Euclidean momentum at t = 0 and fixed final size at t = 1. The lapse action givenin eq. (98) is complex. We take ˙ q = +2 iN c motivated by work of Hartle-Hawking [31, 54]. For thepurpose of this numerical example we take λ = 1, α = 2. We choose final boundary condition tobe q = 3. We plot on x -axis real-part of N c while the y -axis is imaginary part of N c . The red linescorrespond to steepest descent lines (thimbles J σ ), while the thin black lines are steepest ascentlines and denoted by K σ . Both the saddle points are depicted in blue: N − (blue-square) and N + (blue-circle). Both saddle points are relevant . The steepest ascent lines emanating from both ofthem intersects the original integration contour ( −∞ , + ∞ ) which is shown by thick-black line. TheMorse-function h is same for both saddle points: h ( N ± ) > H remains constant along the redand thin-black lines, and is equal to the value of H ( N σ ). The light-green region is allowed regionwith h < h ( N σ ) for all values of σ . The light-orange region (forbidden region) has h > h ( N σ ) for all σ . The boundary of these region is depicted in brown lines. Along these line we have h = h ( N σ ). The analyse the nature of Morse-function at each saddle point we first compute the on-shell action, which is obtained by plugging the saddle point solution given in eq. (99) backin the action given in eq. (97). The on-shell action at the two saddle points is given by, S HH ± = 2 π " − i (cid:18) λ + 4 α (cid:19) ∓ ( q λ − / λ . (101)25t should be emphasised here that only the imaginary part of the action gets correctionfrom the Gauss-Bonnet sector of gravity while the real parts remains unaffected and issame as for pure Einstein-Hilbert gravity. This immediately implies that for q > /λ , theMorse-function for the two saddle points is h ( N ± ) = 2 π ~ (cid:18) λ + 4 α (cid:19) . (102)It is real-positive and independent of q . However, it receives a correction from the Gauss-Bonnet sector of gravity action. By analysing the steepest ascent flow lines emanating fromboth the saddle points it is realised that both of them are relevant . Even though for bothof them h ( N ± ) > − i ∞ . The second part of contour starts − i ∞ in lower-right quadrant, follows the red-line, crosses the positive real-axis, then goesto the upper-right quadrant following the red-line. The Picard-Lefschetz theory then givesthe transition amplitude in the saddle point approximation as G [ ˙ q = 2 iN c , q ] = 12 √ πi (cid:20) exp (cid:18) iS HHtot ( N − ) ~ (cid:19) + exp (cid:18) iS HHtot ( N + ) ~ (cid:19)(cid:21) = e − iπ/ √ π exp (cid:20) π ~ (cid:18) λ + 4 α (cid:19)(cid:21) cos " π λ ~ (cid:18) q − λ (cid:19) / . (103)We notice that we get a non-perturbative correction from the Gauss-Bonnet sector of gravityto the Hartle-Hawking wave-function from a Lorentzian path-integral. This transitionalamplitude is fully non-perturbative and incorporates the non-trivial features coming fromthe Gauss-Bonnet coupling. VIII. CONCLUSION
In this paper we study the path-integral of gravitational theory where the gravitationaldynamics is governed by Einstein-Hilbert gravity with an addition of Gauss-Bonnet gravity.We study this setup in four spacetime dimensions directly in Lorentzian signature. Infour spacetime dimensions the Gauss-Bonnet sector of gravity action is also topological anddoesn’t contribute in the bulk dynamics. However it has a crucial role to play at boundaries.Depending on the nature of boundary conditions the Gauss-Bonnet modifications will affectthe study of path-integral. This paper aims to investigate these issues by considering thegravitational path-integral in a reduced setup of mini-superspace approximation.We start by considering the mini-superspace action of the theory and vary it with respectto field variables to study the dynamical equation of motion and the nature of boundaryterms. To have a consistent boundary value problem one has to incorporate additionalterms at the boundary. We notice that with Neuman boundary condition one ends upwith inconsistencies in fixing the free parameters in the solution to equation of motion.With Dirichlet boundary conditions on other hand it is seen that no non-trivial effects26rise from the Gauss-Bonnet sector. However, in the case of mixed boundary conditions(where one specifies q ( t ) at one end point and its derivative ˙ q ( t ) at another end point) wenotice that the Gauss-Bonnet start to play a non-trivial role. Although the solution to theequation of motion for q ( t ) doesn’t gets contribution from the Gauss-Bonnet sector, theaction for the lapse N c gets non-trivial additions due the non-vanishing boundary terms.Such non-trivialities coming due to the Gauss-Bonnet sector later leads to richer featureswhile evaluating the integration over lapse N c .The paper aims to study the transition amplitude from one 3-geometry to another andinvestigate the circumstances under which the Gauss-Bonnet sector starts to affects thisamplitude in a non-trivial manner. Such a transition amplitude is dictated by a path-integral over q ( t ) and a contour integration over lapse N c . The path-integral over q ( t ) canbe performed exactly as the Gauss-Bonnet part controls only he boundary while the bulkremains unaffected. The path-integral over q ( t ) is governed entirely by the Einstein-Hilbertpart of gravity action. Once the integral over q ( t ) respecting the boundary conditionsis performed, we are left with an contour integration over lapse N c with the integrandcontaining non-trivial features coming from the Gauss-Bonnet sector.We analyse this contour integration by lifting lapse N c to a complex plane and makinguse of Picard-Lefschetz theory to investigate the nature of integrand. We find the saddlepoints of the N c -action and realise that they occur in three pairs. This is a new featureof the Gauss-Bonnet gravity which is absent in the case of pure Einstein-Hilbert gravityhaving only two (or less) pairs of saddle points. In the mixed boundary conditions case theGauss-Bonnet sector contributes non-trivially and give rise to additional saddle points inthe complex N c plane. The three pairs of saddle points follow from the cubic saddle-pointequation in N c . Moreover, if the cubic polynomial equation has real coefficients then thenature of saddle points can be determined by analysing the discriminant ∆ of the cubicequation, which in turn depend on parameter values and boundary conditions.As an application of this we considered an example of no-boundary Universe, and analysethe transition amplitude in this setup. In this situation the initial q = 0. This has conse-quences: in case ( a ) ˙ q ( a )1 and q ( a )1 are related, while in case ( b ) ˙ q ( b )0 and q ( b )1 are related. Ineither case we have multiple relevant saddle points, so this implies that for real-positive q inthe case ( a ) we will have multiple possibilities for ˙ q ( a )1 . This is contradictory to our originalboundary condition requirement where ˙ q ( a )1 is supposed to be fixed at the final boundary.Case ( a ) is therefore ruled out. Such a contradiction doesn’t happen in case ( b ). In case( b ) for a fixed real-positive q ( b )1 there are multiple value for ˙ q ( b )0 for the corresponding rele-vant saddle-points. This is acceptable as the final geometry can be seen as arising from thesuperposition of multiple allowed initial configurations.In case ( b ) by making use of the relation in eq. (82) one can obtain the action for lapse N c entirely in terms of q . As q is positive, so the action for lapse is entirely real. The saddlepoint equation is cubic in N c with real coefficients. We realise that for positive k , Λ, and α there are three distinct real roots for N c : one positive and two negative. This implies thatone saddle point is always real-positive while its ‘twin’ is real-negative. There are four saddlepoints lying on imaginary axis: two on positive imaginary axis while their conjugate twinson negative imaginary axis. There are total of six saddle-points, which is new comparedto the pure Einstein-Hilbert gravity. Attached to each saddle-point there are two Lefschetzthimbles and two steepest ascent lines. Only three of the saddle point can be reachedby flowing downward along the steepest ascent lines starting from the original integrationcontour (0 + , ∞ ). Out of the three only two have their corresponding Morse-function h < relevant . One of the relevant saddle-point lies on negative imaginary axiswhile other lies on positive real-axis. The deformed contour of integration passes throughthese relevant saddles following the Lefschetz-thimbles. The deformed contour thereforeincorporate contributions from both saddles resulting in interference. The full amplitude isa superposition of the contribution coming from two configurations with the more weightassociated to Lorentzian saddle compared to Euclidean saddle.We consider another special case of no-boundary proposal with a complex initial mo-mentum. Here we are inspired by the past works of Hartle-Hawking [31, 54], where theauthors noticed that a particular choice of Wick-rotation leads to stable and suppressedperturbations. This choice of Wick-rotation eventually implies that the initial momentumin the cosmic evolution was complex. Inspired by their work we choose a special initialboundary condition to be ˙ q ( b )0 = +2 iN c . This particular scenario falls in the category ofcase ( b ) of mixed boundary conditions. We notice that in particular in this situation thelapse N c action is complex, and that the action is non-singular at N c = 0. Moreover, inthis case we have only two saddle-points and both are relevant . We compute the transitionamplitude from the initial to final configuration and obtain an analogue of Hartle-Hawkingwave-function having non-perturbative correction from the Gauss-Bonnet sector of gravitytheory.Certainly, more work needs to be done in this direction as many things are still unexplored.In the study in section VI we haven’t directly fixed the initial line ˙ q in the case ( b ), rathersort of derived it by imposing condition that q ( b )0 = 0 and q ( b )1 >
0. This two requirementseventually leads to two relevant saddle points. For each of these saddle point there is acorresponding fixed ˙ q ( b )0 . This is hinting at fact that the final geometry of Universe isarising due to superposition of the two very different initial configurations. Perhaps thereare two different copies of Universe initially whose evolution and interference results in thefinal geometry of Universe. Were these two Universe entangled in past and overtime thisentanglement grew stronger resulting in current Universe? This is hard to answer in presentmanuscript.Another crucial thing missing in this paper is an analysis about the behaviour of fluc-tuations, which are important to understand the stability of Universe. In past works onno-boundary Universe it was noticed that such models are unstable to fluctuations [33].This is a worrisome feature which if it exists make the model less reliable. Past attemptsto overcome these issues involved imposing different types of boundary conditions for back-ground and for fluctuations [44, 45]. It is worth asking this same question in the case ofthe Gauss-Bonnet gravity too. Does the Gauss-Bonnet modifications leads to a more stablebehaviour of fluctuations? If not then what kind of boundary conditions should be im-posed for the fluctuations? Moreover, in the case of HH-model the choice of Wick-rotationleads to stable and suppressed behaviour of fluctuations [31, 54]. Currently it is not clearwhether these fluctuations will remain suppressed when non-perturbative corrections fromGauss-Bonnet gravity are incorporated. We plan to address this in our future work. Acknowledgements
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