Conformal Boundary Conditions from Cutoff AdS 3
aa r X i v : . [ h e p - t h ] O c t Prepared for submission to JHEP
Conformal Boundary Conditions from Cutoff AdS Evan Coleman, a Vasudev Shyam a,b a Stanford Institute for Theoretical Physics and Department of Physics, Stanford University, Stanford,CA 94305, USA b Perimeter Institute for Theoretical Physics, 31 Caroline St. N, N2L 2Y5, Waterloo ON, Canada
Abstract:
We construct a particular flow in the space of 2D Euclidean QFTs on a torus,which we argue is dual to a class of solutions in 3D Euclidean gravity with conformal boundaryconditions. This new flow comes from a Legendre transform of the kernel which implementsthe T ¯ T deformation, and is motivated by the need for boundary conditions in Euclideangravity to be elliptic, i.e. that they have well-defined propagators for metric fluctuations.We demonstrate equivalence between our flow equation and variants of the Wheeler de-Wittequation for a torus universe in the so-called Constant Mean Curvature (CMC) slicing. Wederive a kernel for the flow, and we compute the corresponding ground state energy in the low-temperature limit. Once deformation parameters are fixed, the existence of the ground stateis independent of the initial data, provided the seed theory is a CFT. The high-temperaturedensity of states has Cardy-like behavior, rather than the Hagedorn growth characteristic of T ¯ T -deformed theories. ontents T partition function of T ¯ T deformed CFT
33 Rewriting the flow equation 4
The T ¯ T deformation [1], [2] defines a flow on the space of two dimensional Euclidean quantumfield theories. This flow, parametrized by µ , defines a differential equation in terms of thepartition function Z that reads ∂ µ Z = Z h T ¯ T ( x ) i , (1.1)where h T ¯ T i is the expectation value of the so called T ¯ T operator. Written covariantly, itreads T ¯ T ( x ) = T µν T µν − ( T µµ ) . (1.2)If the theory at the origin of this flow, i.e. the undeformed theory is a conformal field theory,then there is a holographically dual description of this flow in terms of gravity in AdS witha finite radial cutoff surface, on which Dirichlet boundary conditions are imposed.If we deform a CFT, the only energy scale in the problem is the one introduced by the T ¯ T deformation and so the T ¯ T flow equation is the Callan–Symanzik equation.An interesting fact about the holographically dual description to this flow is that it co-incides with the radial development generated by the gravitational constraint equations inthe bulk. In particular, in the limit of large central charge of the undeformed CFT, theflow equation for the partition function Z [ g ] can be mapped on to the radial Hamiltonianconstraint equation (for the radial wavefunction ψ ( g )) in the bulk [3]. Away from this limit,the expectation [4], [5] is that the T ¯ T flow equation maps to the bulk radial Wheeler-de Witt– 1 –quation (WdW), which is a quantization of the aforementioned constraint. The diffeomor-phism constraint equations are simply a rewriting of the covariant conservation of the energymomentum tensor. In other words, 1 /c is roughly playing the role of the Planck’s constantin the bulk. This is the case of interest in the work to follow.Despite the myriad lessons learned from the case where Dirichlet boundary conditionsare imposed on this cutoff surface [3], these boundary conditions are not elliptic. This lack ofellipticity prevents one from calculating quantities such as the propagator for metric fluctua-tions or the one loop determinant in the bulk theory. It turns out that conformal boundaryconditions, where the trace of the extrinsic curvature of the cutoff surface and the conformalpart of the metric adapted to it are fixed, are elliptic [7] and would generically allow us toobtain the aforementioned quantities. We refer the readers to [6] for a more detailed expla-nation of this property. In this article, we will investigate the flow on the space of quantumfield theories that AdS gravity with these boundary conditions is dual to.Fixing Dirichlet boundary conditions in the bulk maps the on-shell action to the gener-ating functional log Z [ g ], which can be exponentiated to obtain the partition function Z [ g ].The functional dependence is on the boundary metric, which is the data that the Dirichletboundary condition specifies at the cutoff surface. On the torus, this partition function turnsout to depend only on the zero modes of the metric, namely the overall volume of the torus V and the real and imaginary parts of the modular parameter ( τ , τ ), i.e. Z T [ g ] = Z ( V, τ , τ ) . (1.3)In order to change boundary conditions, one needs to perform a canonical transformationon the phase space of the bulk theory. This canonical transformation induces a certain Laplacetransform of the T ¯ T deformed torus partition function, as a function of the volume:Γ( T, τ , τ ) = Z dV e − V T Z ( V, τ , τ ) . (1.4)The resulting object is a quantum effective action log Γ( T, τ , τ ) that depends on the tracemode of the energy momentum tensor, and behaves like the ordinary partition function interms of its dependence on the conformal modes of the metric. At the level of the free energy,it defines a Legendre transformation. The flow equation for Γ( T, τ , τ ) will be the quantityof interest in what follows, as will the thermodynamics of the new ensemble. Organization of the Article
This note is organized as follows. Section 2 reviews the state of the science. In Section 3,we rewrite the T ¯ T flow equation above, changing variables from µ to a volume scale V . Inthese variables, the flow equation becomes the Wheeler-DeWitt equation of 3D gravity. Tobetter demonstrate this correspondence from the gravity side, we study General Relativity in The caveat being the situation where the extrinsic curvature is positive or negative definite. In thatcase, even with Dirichlet boundary conditions, the differential operator appearing in the kinetic term of thelinearized theory is invertible [6]. – 2 –onstant Mean Curvature gauge. Then, in Section 4, we introduce the Legendre transformbetween V and its canonically conjugate variable T , related to the trace of the extrinsiccurvature that enters the boundary action. We show that the change in the boundary actiondue to the canonical transformation is exactly the right term to implement elliptic boundaryconditions starting from a bulk with Dirichlet boundary conditions. We go on to derive anddiscuss the thermodynamics of this new ensemble in the low- and high-temperature regimes. T partition function of T ¯ T deformed CFT In this article, we will parameterize the family of T ¯ T -deformed solutions by λ , the dimension-less T ¯ T coupling. The seed theory, i.e. the undeformed theory at λ = 0, is a conformal fieldtheory on the torus with central charge c . The modular parameter of the torus in question isdenoted τ = τ + iτ . The corresponding 2D line element reads: ds = | dx + τ dy | . (2.1)where x, y have period 2 πR . We would like to switch to units where we measure lengths interms of the coupling constant of the T ¯ T deformation, µ , that carries dimensions of lengthsquared. Then, we can introduce the dimensionless version of this coupling: λ = µR , (2.2)so that R = q µλ . The volume of the torus is now given by: V = Z d x √ g = 4 π µ τ λ . (2.3)In this language, the T ¯ T flow equation reads [8]: ∂ λ Z ( λ, τ , τ ) = (cid:18) τ (cid:0) ∂ τ + ∂ τ (cid:1) + λ ∂ λ ∂ τ − τ λ∂ λ (cid:19) Z ( λ, τ , τ ) . (2.4)This equation is analogous to a diffusion equation, and the partition function Z ( λ, τ , τ )admits a representation in terms of an analogue of the heat kernel: Z ( λ, τ , τ ) = τ πλ Z H d σσ e − λσ | σ − τ | Z CF T ( σ , σ ) . (2.5)Here, Z CF T ( σ , σ ) in the integrand is the initial condition Z ( λ = 0 , τ , τ ) = Z CF T ( τ , τ ),i.e. the flow originates from a CFT. It has been shown that the solutions to this differentialequation are invariant under modular transformations, like the seed CFT [8], [9]. Also notethat the domain of integration is the upper half plane H . This expression appeared in [10],[11], [12] and [4]. Note that it can also be derived from the prescription of [13], meaning that– 3 –t can be seen as an expression for the path integral of 2d ghost-free massive gravity coupledto a conformal field theory. Z CF T can then be written as a partition sum Z CF T = X n e − τ E n + iτ J n , (2.6)where E n = ∆ n + ¯∆ n − c , J n = ∆ n − ¯∆ n . (2.7)Here ∆ n and ¯∆ n are the left and right conformal dimensions of the conformal field theoryand c is its central charge. We can find the T ¯ T -deformed partition function by assuming thata similar form will hold: Z ( λ, τ , τ ) = X n e − τ E n ( λ )+ iτ J n . (2.8)The reason why this form of the deformed partition function is justified (in particular, why theterm involving J n remains unmodified) is tied to the fact that the T ¯ T deformation preservestranslation invariance. Then, (2.4) leads to the following equation for the deformed energylevels E n : 2 λ E n ∂ λ E n + 4 ∂ λ E n + E n = J n . (2.9)This equation can be solved to obtain the deformed energy levels, given by: E n ( λ ) = − p λE n + λ J n λ . (2.10)The branch of the square root is selected so that E n ( λ →
0) = E n . Our fellow T ¯ T aficionadosshould note that this expression lacks the traditional factors of R as in Eq. (1.9) of [3]. Inorder to reinstate the R dependence, we take λ µR , so we find E n ( R ) = 2 R µ − r µ J n R + µE n R ! = R ˜ E n . (2.11)In other words, E is the product of R and what would normally be considered as the deformedenergy levels. In the analysis which follows, we derive expressions in terms of E n ( R ) ratherthan ˜ E n , to keep expressions simple. We can rewrite the flow equation in terms of the volume V = 4 π µ τ λ . τ (cid:0) ∂ τ + ∂ τ (cid:1) Z ( V, τ , τ ) + V (cid:18) π µ ∂ V − ∂ V (cid:19) Z ( V, τ , τ ) = 0 . (3.1)This form of the equation will prove to be useful in making a connection to the Wheeler-deWitt equation of three dimensional quantum gravity. In particular, if we look at the object: ψ ( V, τ , τ ) = e − V π µ Z ( V, τ , τ ) , (3.2)– 4 –he equation it satisfies: τ ( ∂ τ + ∂ τ ) ψ ( V, τ , τ ) + V (cid:18) π µ − ∂ V (cid:19) ψ ( V, τ , τ ) = 0 , (3.3)is identical to the Wheeler-de Witt equation in three dimensions with negative cosmologicalconstant, in constant mean curvature gauge. We now defineΛ ≡ π µ . (3.4)Henceforth, we work interchangeably with Λ and µ in order to keep equations simple anddraw important connections to the literature.Note that this exercise is the finite- c equivalent of deriving the trace flow equation. The Arnowitt–Deser–Misner (ADM) Hamiltonian and momentum constraints are [14] H = 1 √ g g ij g kl (cid:16) π ik π jl − π ij π kl (cid:17) − √ g ( R − H i = − ∇ j π ji = 0 . (3.6)At the level of an action, we can implement these constraints with the term H T ot = Z d D x (cid:0) N ( x ) H ( x ) + ξ i ( x ) H i ( x ) (cid:1) = H ( N ) + H i ( ξ i ) . (3.7)where the spacetime dimension of the bulk is D + 1.We would like to fix the mean curvature of the hypersurface to be constant. To do thatwe start by splitting the conjugate momentum π ij into traceless and trace components, anddefine the metric g ij as a conformal rescaling of a constant-curvature counterpart ¯ g ij viadilaton φ ( x ): π ij = σ ij + 1 D tr πg ij , g ij = e φ ( x ) ¯ g ij . (3.8)The gauge fixing condition imposes the constancy of T defined as: T = 2 D tr π √ g , ∇ i T = 0 (3.9)Specialising the the case of three dimensional gravity, in this gauge, the Hamiltonian con-straint becomes: H CMC = − √ ¯ ge φ (cid:0) T − (cid:1) + √ ¯ ge − φ σ ij σ ij + 2 √ ¯ g (cid:20) ¯∆ φ −
12 ¯ R (cid:21) = 0 , (3.10)where “barred” quantities are defined in terms of ¯ g ij . Integrating, we find: h CMC = Z d D x H CMC = − V ( T − τ δ ab p aτ p bτ = 0 . (3.11)– 5 –ote that classically, the condition for the existence of solutions is T ≥ On the reduced phase space, the dynamics of the theory is finite dimensional. It can be quan-tized as such, and a quantum mechanical theory is obtained. The wavefunctions of interestdepend on the modular parameters of the torus, as well as the volume. The correspondingmomenta act as derivatives with respect to the conjugate variables:ˆ
T ψ ( V, τ , τ ) = − ∂ V ψ ( V, τ , τ ) , (3.12)ˆ p τ a ψ ( V, τ , τ ) = − ∂ τ a ψ ( V, τ , τ ) . (3.13)With these conventions, the global Hamiltonian constraint equation reads:ˆ h CMC ψ ( V, τ , τ ) = (cid:2) τ ( ∂ τ + ∂ τ ) − V (cid:0) ∂ V − (cid:1)(cid:3) ψ ( V, τ , τ ) = 0 . (3.14)Note that the ordering of the V ˆ T and τ δ ab ˆ p a ˆ p b terms are picked automatically by therewriting of the T ¯ T flow equation (3.3). The relationship between radially quantized 3dquantum gravity and the T ¯ T deformation of CFTs was first noted in [5], and explainedfurther in [3], [13]. Such a connection is anticipated in higher dimensional generalizations ofthe T ¯ T flow as well, see [17–19]. Also, this equation involves only partial derivatives with respect to global quantities(
V, τ , τ ). In the bulk, this is a direct consequence of the CMC gauge condition. At large c ,the sphere partition function computed in [20], [17] also depends only on a global quantity,i.e. the radius of the sphere. At finite c , if we relate the radial wavefunction with the partitionfunction, then the CMC gauge fixing also leads to an ODE involving derivatives with respectto the radius only [21]. On the field theory side, this can be seen as a consequence of thecalculation in [10], and it is unclear as to why these two facts lead the same phenomenon.If we take the following ansatz for the wavefunction: ψ ( V, τ , τ ) = e − √ Λ V X n e − τ E n ( τ √ Λ V )+ iτ J n , (3.15)then we can recover the partition sum discussed in the previous sections, E n = V π µτ − r π µτ V E n + 4 π µ τ V J n ! . (3.16)This is simply a rewriting of the expressions we had above, recall the identification: λ = 4 π µ τ V . Note that this quantized flow equation is not an approximation of a functional differential equation, i.e. aminisuperspace approximation, but rather an exact expression because of the particular choice of gauge. – 6 – .3 Relationship to Jackiw–Teitelboim gravity
In this section, we will briefly note what happens when one of the cycles of the torus degener-ates. From the perspective of the wavefunction, this restriction is imposed as the condition: ∂ τ ψ JT = 0 , ψ JT ( V, τ ) = ψ ( V, τ = 0 , τ ) . (3.17)The constraint equation now reads: (cid:16) τ ∂ τ − V ( ∂ V − (cid:17) ψ JT ( V, τ ) = 0 . (3.18)Just as the wavefunction in the three dimensional theory can be related to a partition function,the same is true in the dimensionally reduced case: ψ JT ( V, τ ) = e − √ Λ V Z JT ( V, τ ) . (3.19)This object satisifes an equation identical to the one in appendix B of [22]: V (cid:16) √ Λ ∂ V − ∂ V (cid:17) Z JT + τ ∂ τ Z JT = 0 . (3.20)Further, if we take the ansatz: Z JT ( V, τ ) = X n g (cid:18) τ √ Λ V (cid:19) e − τ E n ( τ √ Λ V ) , g (cid:18) τ √ Λ V (cid:19) = 1 , (3.21)we find the energy levels obtained in (1.2) of [22]: E ± n ( V, τ ) = V π µτ (cid:18) ∓ r − τ V E n (cid:19) . (3.22)If the spectrum of the undeformed theory is continuous, we can write the general solution as: Z JT ( V, τ ) = Z ∞ dE ρ + ( E ) e − τ E + ( V,τ ) + Z ∞−∞ dE ρ − ( E ) e − τ E − ( V,τ ) . (3.23)One can find a density of states which accommodates both branches of E , i.e. we can choose ρ + ( E ) = sinh (cid:16) π √ CE (cid:17) , ρ − ( E ) = ( − sinh (cid:16) π √ CE (cid:17) , < E < λ b ρ ( E ) , E < C is related to the boundary value of the dilaton, and ˆ ρ ( E ) is an arbitrary function.A more in-depth analysis of this solution is available in [22].– 7 – Implementing conformal boundary conditions
For a review on boundary conditions in Euclidean gravity, see [6]. We will be interested inpolarized boundary conditions. These are imposed by specifying a Lagrangian submanifoldin the theory’s phase space. On such a submanifold, the symplectic form Ω vanishes, i.e.Ω = Z d D x (cid:0) δπ ij ∧ δg ij (cid:1) , Ω | L = 0 . (4.1)The Lagrangian submanifold corresponding to the Dirichlet boundary conditions is spec-ified as follows: L Dirichlet = (cid:26) ( g ij ; π ij ) : π ij ( g ) = δS [ g ] δg ij (cid:27) . (4.2)In this picture, the conjugate momentum to the metric is the quasilocal stress-energy tensorof Brown and York. In order to do perturbation theory, we will need to compute the prop-agator. This exercise involves inverting the second order differential operator appearing inthe kinetic term of the action for fluctuations of the metric to quadratic order. Ellipticity isthe requirement that the space of zero modes of this operator is at most finite dimensional,and it ensures (among other properties) that the leading-in-momentum component of the ki-netic operator for metric fluctuations is invertible. Dirichlet boundary conditions genericallyrun afowl of this requirement. Alternatively, we can use the so-called “conformal” boundaryconditions [7], where we specify the following Lagrangian submanifold: L Conformal = (cid:26) ( g ij ; π ij ) = (cid:0) ¯ g ij ( τ a ) , V ; σ ij ( p a ) , T (cid:1) : σ ij ( τ a ) = δSδ ¯ g ij , V ( T ) = δSδT (cid:27) , (4.3)where S = S ( T, τ , τ ). This choice of boundary conditions is elliptic and thus has a well-defined propagator. In switching from Dirichlet to conformal boundary conditions, we notethe following: Z d D x π ij δg ij = Z d D x (cid:18) σ ij + 1 D g ij tr π (cid:19) δg ij . (4.4)The trace component in Constant Mean Curvature gauge simplifies to1 D Z d D x (cid:0) tr π g ij δg ij (cid:1) = 2 D Z d D x (cid:18) tr π √ g (cid:19) δ √ g CMC = Z d D x T δV, (4.5)and the tracelessness of σ ij implies Z d D x σ ij δg ij = Z d D x ( σ ij δ ¯ g ij ) , (4.6)where ¯ g ij is as defined in (3.8). Then:Ω = Z d D x ( δ ¯ g ij ∧ δσ ij ) + δV ∧ δT = δ (cid:18)Z d D x ( σ ij δ ¯ g ij ) + T δV (cid:19) (4.7)= δ (cid:18)Z d D x ( σ ij δ ¯ g ij ) + T δV − δ ( V T ) (cid:19) = δ ( p a δτ a − V δT ) (4.8)– 8 –his is equivalent to a canonical transformation generated by the following shift in the action : G = V T. (4.9)More specifically, we are arguing that S GHY − S Conf = G, S
Conf = 1
D S
GHY . (4.10)We will now show this explicitly. In CMC gauge, the standard Gibbons-Hawking-York bound-ary term in the action takes the form: S GHY = − Z d D x √ g tr K = − − D Z d D x tr π. If we now plug in the form of G it becomes clear that S GHY − G = − (cid:18) − D + 1 D (cid:19) Z d D x tr π = − D (1 − D ) Z tr π, and since − D (1 − D ) Z tr π = − D Z tr K = 1 D S
GHY , (4.11)we thus conclude that the relevant boundary term for the new ensemble is exactly the oneprescribed by conformal boundary conditions, i.e. S Conf . Note that the change of boundaryconditions involving a Legendre transformation is much in keeping with the lesson of [24].We should rewrite our “wavefunction” ψ by making the Laplace transform explicit:Ψ( T, τ , τ ) = Z d V e − V T ψ ( V, τ , τ ) , (4.12)Γ( T, τ , τ ) = Z d V e − V T Z ( V, τ , τ ) (4.13)where we define Γ( T, τ , τ ) to be the partition function for this new ensemble. We cancompute the correlation functions of the Weyl mode from taking successive T derivatives ofthis object. In this way, it is similar to the dilaton effective action that features in [26].The Legendre transform of the Hamiltonian constraint yields modified flow equations forΨ and Γ:ˆ h CMC Ψ( T, τ , τ ) = (cid:18) − T ∂ T − ∂ T T − ( T − ∂ T + τ (cid:0) ∂ τ + ∂ τ (cid:1) (cid:19) Ψ = 0 . (4.14)Similarly, the partition function Γ must satisfy (cid:16) − (4 √ Λ + T ) T ∂ T − √ Λ + T ) ∂ T − τ ( ∂ τ + ∂ τ ) (cid:17) Γ = 0 . (4.15)The two equations above are the key results presented in this article. Note that this canonical transformation is identical to the one used in the symmetry trading map of [23]. Note that this expression looks very similar to the one appearing in [25], except we only integrate over the(zero mode of the) Weyl factor of the metric, and as such our Laplace transform is only partial. Our aims arealso different from those of the authors of [25]. – 9 – onnection to Schr¨odinger equation with York curvature time
As should be expected in the context of quantization, there is an ordering ambiguity implicitin the definition of the constraint equation (4.14). We fix the ambiguity by integrating byparts within the Laplace transform. This is identical to the prescription of [27, 28]. However,if one relaxes this ordering, it is possible to rearrange V and T to find an interesting variant:ˆ h CMC Ψ( T, τ , τ ) = (cid:16) − T ∂ T − ( T − ∂ T + τ ( ∂ τ + ∂ τ ) (cid:17) Ψ = 0 . (4.16)This version of the flow equation has appeared in the literature in a different form, see e.g.(3.4) in [16] as well as [29]. In the reduced phase space quantization of 3D gravity in a torusuniverse, with T parametrizing the York curvature time slice, the analog of the Schr¨odingerequation is: ∂ Ψ ∂T = − √ T − q τ ( ∂ τ + ∂ τ ) Ψ . (4.17)We refer to this expression as the “York-Schr¨odinger equation.” Now, note that( ∂ τ + ∂ τ ) τ = 0 · τ . Since τ is an eigenfunction of this linear differential operator with eigenvalue 0, it is aneigenfunction of the square root of that operator with eigenvalue 0. We can therefore rewritethe York-Schr¨odinger equation as follows: − p T − ∂ T (cid:16)q ∂ τ + ∂ τ Ψ (cid:17)| {z } − √ T − τ ∂ T Ψ = τ ( ∂ τ + ∂ τ )Ψ . (4.18)Rearranging, we find the same flow in (4.16). Γ and exact solutions for ΨWriting out (4.13) with V in terms of λ allows us to do the Laplace transform explicitly:Γ = Z ∞ d (cid:18) τ √ Λ λ (cid:19) e − τ √ Λ λ T τ πλ Z H d σσ e − λσ | σ − τ | Z CF T ( σ , σ ) (4.19)One can perform the λ integral to find:Γ = √ Λ πT Z H d σσ Z CF T ( σ , σ ) (cid:16) − √ Λ τ σ T | σ − τ | (cid:17) . (4.20)This kernel can be shown to satisfy the flow in (4.15). A condition on the integrand whicharises as a requirement for the convergence of the λ integral is T (cid:18) T √ Λ − (cid:19) τ < √ Λ( σ − τ ) + √ Λ (cid:18) σ − (cid:18) − T √ Λ (cid:19) τ (cid:19) , (4.21)– 10 –hich determines a circle or an annulus in the ( σ , σ ) plane, depending on the magnitudesand signs of different parameters. To derive the above constraint, one must separately consider √ Λ ≷
0. When that square root is negative, one should write the Laplace transform overnegative values of λ , to keep V positive. The overall sign of the argument to the exponentialbeing integrated must be negative, which gives the stated result.This constraint makes the convergence properties of Γ somewhat subtle, as we have toredefine the integration over d σ within only a subregion of the upper half-plane. However,because the integrand itself solves the differential equation (4.15), we can restrict the domainof integration however we like and it will not affect the actual flow.The solutions to the V, τ , τ flow equation for ψ in (3.3) have been studied extensivelyin the literature, see [28] for a helpful review. Separation of variables yields a complete setof solutions in that case. We can find solutions to our Ψ flow equation (4.14) by a similarmethod. Set Ψ = α ( T ) β ( τ ) γ ( τ ), and divide by Ψ on both sides to find2 + 4 T α ′ ( T ) α ( T ) + ( T − α ′′ ( T ) α ( T ) = τ (cid:18) β ′′ ( τ ) β ( τ ) + γ ′′ ( τ ) γ ( τ ) (cid:19) ≡ − P (4.22)Where P is dimensionless and constant in all parameters. We can rearrange the expressionon the RHS to see further that β ′′ ( τ ) β ( τ ) = Pτ − γ ′′ ( τ ) γ ( τ ) ≡ − (2 πJ ) (4.23)Where J is also constant in all parameters. This gives us 3 separate equations which we cansolve for α , β , and γ . In fact, since the zeroth-order Maass form determines the form of the τ and τ dependence in both the T and V flows, our β and γ will take the same form asin [28]. In particular, modular invariance requires J ∈ Z , and we obtain: α ( T ) = α (1) 2 F (cid:18)
14 (3 − ν ) ,
14 (3 + ν ); 12 ; T (cid:19) ( ν = √ − P )+ α (2) T √ Λ F (cid:18)
14 (5 − ν ) ,
14 (5 + ν ); 32 ; T (cid:19) β ( τ ) = β (1) e πiJτ + β (2) e − πiJτ (4.24) γ ( τ ) = γ (1) √ τ K ν (2 π | J | τ )In the absence of a specific surface on which the CFT “lives,” it is difficult to impose anyfurther constraints on the form of this solution. Nevertheless we can comment on its variousproperties, for example convergence. The form of α ( T ) is simply the appropriate solutionto the hypergeometric differential equation. The defining series for F around T /
4Λ = 0happens to converge only for T < T >
4Λ isnevertheless accessible. This is because F ( a, b ; c ; z ) can be written as a linear combinationof F ’s with the argument z replaced with one of the other 5 cross-ratios involving z and 1.– 11 –n addition, γ ( τ ) is well-behaved for all P even though the order becomes pure imaginaryfor P > ; this follows from standard properties of the modified Bessel functions.Regarding α , one should note the difference with the analogous solution to the York-Schr¨odinger equation: α Y S ( T ) = α (1) cos (cid:18) √ P arctanh (cid:18) T √ T − (cid:19)(cid:19) + α (2) sin (cid:18) √ P arctanh (cid:18) T √ T − (cid:19)(cid:19) (4.25)The behavior of β and γ is again unaffected. In addition, the apparent singular behavior in α ( T ) at T = 4Λ is a removable discontinuity. We ultimately wish to study the thermodynamics of this new system, specifically the energylevels. However, a na¨ıve approach will not work, because the energies in the Z ( V, τ , τ )ensemble depend nontrivially on τ and V . The flow equation for Γ can be solved by separationof variables just as Ψ can. In analogy with the expressions in (4.24), we find the ratherunenlightening solution: α ( T ) = α (1) 2 F (cid:18)
12 (3 − ν ) ,
12 (3 + ν ); 2; − T √ Λ (cid:19) + α (2) G − T √ Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (1 + ν ) , − (1 − ν ) − , ! , (4.26)where ν , β ( τ ), and γ ( τ ) are the same as for Ψ. To gain some insight into the spectrum, wemight instead consider the discrete version of the Laplace transform:Γ( T, τ , τ ) = X n e − nV T Z ( nV, τ , τ ) . (4.27)Since the left-hand side is independent of V , the Hamilton-Jacobi equations tell us that ∂ V Γ = 0 = X n ne − nV T (cid:16) − T + ∂ nV Z ( nV, τ , τ ) (cid:17) , = ⇒ T = P n ne − nV T ∂ nV Z ( nV, τ , τ ) P n ne − nV T . (4.28)However, inverting the sums to study V is not a tractable approach. Taking a direct ansatzfor the energy levels fails similarly, as it requires solving nonlinear second-order differentialequations. We are thus restricted to studying the thermodynamics of this ensemble in variouslimits.For τ ≫ τ = 0, which can be viewed as the low-temperature limit, the Legendretransform can be performed explicitly. The inverse temperature is given by β = τ R where R is the radius of the torus. We will nevertheless proceed with our present conventions and– 12 –reat τ as the inverse temperature. We will follow a line of reasoning similar to that in [9]which was aimed at extracting the density of states from Z ( λ, τ , τ ). The leading term inthe low temperature expansion of the partition function is: Z ( τ , V ) | τ ≫ ∼ e V π µτ (cid:16) ∓ √ cτ V (cid:17) . . (4.29)We now apply the Hamilton-Jacobi equation, i.e.0 ≡ ∂ log Γ ∂V = ∂ log Z∂V − T, (4.30)to find V ( T, τ ): V ± ( T, τ ) = cτ √ Λ ± s T ( T − √ Λ) ! , (4.31)which is real provided T √ Λ < T √ Λ >
4. Then, by plugging this back into the low temper-ature limit, we obtain the asymptotic form of Γ:Γ(
T, τ ) | τ ≫ = e − τ E o , (4.32)where E ± o ( X ) = − c (cid:18) − X (cid:19) ± s X ( X − ! − s X ( X − . (4.33)Here, we write the effective ground state energy in terms of the dimensionless quantity X = T √ Λ . We note that this quantity is real whenever V ( T, τ ) is real. Moreover, it does not dependon τ , unlike E o and V . This means that E o can be properly interpreted as an energy. Notealso that Γ is invariant under modular transformations. To understand this, we restate (4.13),Γ( T, τ , τ ) = Z dV e − V T Z ( V, τ , τ ) , and note that T, V , and Z ( V, τ , τ ) are all modular invariant. Thus we can apply an Stransformation that takes τ → τ and obtain the high temperature limit of Γ( T, τ ):Γ (cid:18) T, τ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) τ ≫ = e E oτ . (4.34) One might reasonably be concerned that the constraint in (4.21) ruins this argument, because Γ can nolonger be written in terms of Z ( V, τ , τ ). In this article, we consider (4.13) to be the definition of Γ. Whetherit converges only under a reduced class of conditions is a question we relegate to future work. – 13 –e can take the inverse Laplace transform of this quantity to obtain the density of states: ρ ( ǫ ) = Z i ∞− i ∞ dτ e τ ǫ + E oτ . (4.35)Using the saddle point approximation, we see Cardy-like growth at high energies: ρ ( ǫ ) ∼ e √ E o ǫ . (4.36)This is the regime that the high temperature expansion can accurately shed light on. Notethat our findings are quite different from the case with Dirichlet boundary conditions, whereone finds Hagedorn behaviour. Also note that E o ∼ c , so the above result also reflects thebehaviour of the density of states in that limit. This is much akin to what happens in CFTs. In this note, we have derived a modification to the flow equation of the T ¯ T deformationwhich implements conformal boundary conditions in the bulk dual, rather than the Dirichletboundary conditions of lore. By rewriting the flow in terms of a characteristic volume scale V ,we identified it with the Wheeler-DeWitt equation of AdS gravity. Then, starting from thebulk gravitational action with Dirichlet boundary conditions (i.e. with the Gibbons-Hawking-York boundary term), we change the variational problem to impose conformal boundaryconditions. We show that in Constant Mean Curvature gauge, the term needed to shiftbetween these pictures can be interpreted as a Legendre transform of the torus path integral.The resulting system has a ground state whose existence is independent of the CFT data oncethe deformation parameters are fixed.One may reasonably ask why we were interested in this problem in the first place, giventhat 3D gravity has no gravitons. In fact, if it can be well-defined, the graviton propagator willstill contribute to any perturbative expansion in G N , but will only enter as e.g. an internalline in a Feynman diagram. Holographically, the avatar of such an expansion is large- c perturbation theory, which was the regime of interest here and in [3]. We wonder whetherthe lack of a µ -dependent breakdown of the ground state of the Legendre transformed theoryis due to the elliptic nature of the boundary conditions.We see many paths forward left to explore. Exact results for the full spectrum and den-sity of states may be within reach, despite the troubling nonlinearities which generically arisein deriving E n ( X ) directly. In this vein, one option may be to integrate the kernel in (4.20)exactly, perhaps by taking advantage of its similarity to the structure of the Feynman propa-gator. Generalizations of this flow are also immediately available for the many generalizationsof T ¯ T . We are particularly interested in applying our procedure to dS holography using theflow prescribed in [30].It would also be valuable to construct a string-theoretic analog of the flow in T , in thespirit of [31, 32]. In those works, a deformation of the worldsheet sigma model implements asingle-trace variant of the T ¯ T deformation in the putative 2D boundary CFT of the AdS -like– 14 –pacetime. For the “holographic” sign of the deformation, their procedure yields spacetimeswith naked singularities. We are curious whether a similar procedure is possible for ourproposed flow, and if so, what the salient features are of the corresponding geometries.There is one overarching question which we hope to address in future work: where doesthe analogy with the BTZ phase space enter in this new picture? The answer is unclear,given the results presented. However, there are some rough hints of a correspondence worthmentioning at the level of discussion. In our case, we started by deforming a unitary CFTwithout a gravitational anomaly and thus E o = − c and J o = 0. However, if we start witha theory with a gravitational anomaly, and further assume that our expressions remain validin this setting, then the expression one obtains for the volume is: V ± ( T, τ ) = − E o τ √ Λ ± s − (cid:18) J o E o (cid:19) s − T ( T − √ Λ) . (5.1)In order for this quantity to be real, one must separately consider | J o E o | ≷
1, and then identifythe range of T √ Λ which keeps the argument to the square root nonnegative. This is eerilyreminiscent of the bounds on angular momentum which arise in the BTZ phase space, butwe will curtail our speculative commentary here. Acknowledgments
We thank Eva Silverstein, Shouvik Datta, Gonzalo Torroba, Jorrit Kruthoff, Ronak Soni,G. Joaquin Turiaci, Zhenbin Yang, and Victor Gorbenko for their insightful comments. Thework of EAC is supported by the US NSF Graduate Research Fellowship under Grant DGE-1656518. During early stages of this work, VS was supported in part by the PerimeterInstitute for theoretical Physics. Research at Perimeter Institute is supported in part by theGovernment of Canada through the Department of Innovation, Science and Industry Canadaand by the Province of Ontario through the Ministry of Colleges and Universities.
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