Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

Journal of High Energy Physics, Nov 2017

We propose an optimization procedure for Euclidean path-integrals that evaluate CFT wave functionals in arbitrary dimensions. The optimization is performed by minimizing certain functional, which can be interpreted as a measure of computational complexity, with respect to background metrics for the path-integrals. In two dimensional CFTs, this functional is given by the Liouville action. We also formulate the optimization for higher dimensional CFTs and, in various examples, find that the optimized hyperbolic metrics coincide with the time slices of expected gravity duals. Moreover, if we optimize a reduced density matrix, the geometry becomes two copies of the entanglement wedge and reproduces the holographic entanglement entropy. Our approach resembles a continuous tensor network renormalization and provides a concrete realization of the proposed interpretation of AdS/CFT as tensor networks. The present paper is an extended version of our earlier report arXiv:​1703.​00456 and includes many new results such as evaluations of complexity functionals, energy stress tensor, higher dimensional extensions and time evolutions of thermofield double states.

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Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

HJE Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT Pawel Caputa 0 1 2 4 Nilay Kundu 0 1 2 4 Masamichi Miyaji 0 1 2 4 Tadashi Takayanagi 0 1 2 3 4 Kento Watanabe 0 1 2 4 0 Kashiwano-ha , Kashiwa, Chiba 277-8582 , Japan 1 Kitashirakawa Oiwakecho , Sakyo-ku, Kyoto 606-8502 , Japan 2 Kyoto University 3 Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo 4 Center for Gravitational Physics, Yukawa Institute for Theoretical Physics , YITP We propose an optimization procedure for Euclidean path-integrals that evaluate CFT wave functionals in arbitrary dimensions. The optimization is performed by minimizing certain functional, which can be interpreted as a measure of computational complexity, with respect to background metrics for the path-integrals. In two dimensional CFTs, this functional is given by the Liouville action. We also formulate the optimization for higher dimensional CFTs and, in various examples, nd that the optimized hyperbolic metrics coincide with the time slices of expected gravity duals. Moreover, if we optimize a reduced density matrix, the geometry becomes two copies of the entanglement wedge and reproduces the holographic entanglement entropy. Our approach resembles a continuous tensor network renormalization and provides a concrete realization of the proposed interpretation of AdS/CFT as tensor networks. The present paper is an extended version of our earlier report arXiv:1703.00456 and includes many new results such as evaluations of complexity functionals, energy stress tensor, higher dimensional extensions and time evolutions of thermo eld double states. AdS-CFT Correspondence; Anomalies in Field and String Theories; Confor- 1 Introduction 2 Formulation of the path-integral optimization General formulation Connection to computational complexity Optimization of vacuum states in 2D CFTs Tensor network renormalization and optimization 3 Optimizing various states in 2D CFTs Finite temperature states CFT on a cylinder and primary states Liouville equation and 3D AdS gravity 4 Reduced density matrices and EE Optimizing reduced density matrices Entanglement entropy Subregion complexity 5 6 7 8 2.1 2.2 2.3 2.4 3.1 3.2 3.3 4.1 4.2 4.3 6.1 6.2 6.3 6.4 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 Energy momentum tensor in 2D CFTs Evaluation of SL in 2D CFTs General properties of the Liouville action Vacuum states Primary states Finite temperature state Application to NAdS2=CFT1 Applications to higher dimensional CFTs Our formulation AdSd+1 from optimization Excitations in global AdSd+1 Holographic entanglement entropy Evaluation of complexity functional Comparison with earlier conjectures Relation to \complexity = action" proposal Higher derivative terms and anomalies { i { 9.1 9.2 9.3 Holographic complexity in the literature Higher derivatives in complexity functional and anomalies C.1 Poincare AdS5 with higher derivatives C.2 Global AdS5 with higher derivatives C.3 Excitation in global AdS5 with higher derivatives D Entanglement entropy and Liouville eld 36 38 of emergent spacetimes from tensor networks, as rst conjectured by Swingle [2], for the description of CFT states in terms of MERA (multi-scale entanglement renormalization ansatz) [3, 4].1 One strong evidence for this correspondence between holography and tensor networks, apart from the symmetry considerations, is the fact that the holographic entanglement entropy formula [15, 16] can naturally be explained in this approach by counting the number of entangling links in the networks. However, up to now, most arguments in these directions have been limited to studies of discretized lattice models so that we can apply the idea of tensor networks directly. Therefore, they at most serve as toy models of AdS/CFT as they do not describe the genuine CFTs which are dual to the AdS gravity (though they provide us with deep insights of holographic principle such as quantum error corrections [10, 17, 18]). Clearly, it is then very important to develop a continuous analogue of tensor networks related to AdS/CFT. There already exists a formulation called cMERA (continuous MERA) [5], whose connection to 1For recent developments we would like to ask readers to refer to e.g. [5{14]. { 1 { AdS/CFT has been explored in [7{9, 12, 14, 19]. Nevertheless, explicit formulations of cMERA are so far only available for free eld theories [5] (see [7, 8, 20, 21] for various studies) which is the opposite regime from the strongly interacting CFTs which possess gravity duals, the so-called holographic CFTs. A formal construction of cMERA for general CFTs can be found in [14, 19]. The main aim of this work is to introduce and explore a new approach which realizes a continuous limit of tensor networks and allows for eld theoretic computations. In our preceding letter version [22], we gave a short summary of our idea and its application to two dimensional (2D) CFTs. Essentially, we reformulate the conjectured relation between tensor networks and AdS/CFT from the viewpoint of Euclidean path-integrals. Indeed, HJEP1(207)9 the method called tensor network renormalization (TNR) [23, 24] shows that an Euclidean path-integral computation of a ground state wave function can be regarded as a tensor network description of MERA. In this argument, one rst discretizes the path-integral into a lattice version and rewrites it as a tensor network. Then, an optimization by contracting tensors and removing unnecessary lattice sites, nally yields the MERA network. The `optimization' here refers to some e cient numerical algorithm. In our approach we will reformulate this idea, but in such a way that we remain working with the Euclidean path-integral. More precisely, we perform the optimization by changing the structure (or geometry) of lattice regularization. The rst attempt in this direction was made in [14] by introducing a position dependent UV cut o . In this work, we present a systematic formulation of optimization by introducing a metric on which we perform the path-integral. The scaling down of this metric corresponds to the optimization assuming that there is a lattice site on a unit area cell. To evaluate the amount of optimization we made, we consider a functional I of the metric for each quantum state j i . This functional, which might appropriately be called \Path-integral Complexity", describes the size of our path-integration and corresponds to the computational complexity in the equivalent tensor network description.2 In 2D CFTs, we can identify this functional I with the Liouville action. The optimization procedure is then completed by minimizing this complexity functional I , and we argue that the minimum value of I is a candidate for complexity of a quantum state in CFTs. Below, we will perform a systematic analysis of our complexity functional for various states in 2D CFTs, lower dimensional example of NAdS2=CFT1 (SYK) as well as in higher dimensions, where we will nd an interesting connection to the gravity action proposal [32, 33]. Our new path-integral approach has a number of advantages. Firstly, we can directly deal with any CFTs, including holographic ones, as opposed to tensor network approaches which rely on lattice models of quantum spins. Secondly, in the tensor network description there is a subtle issue that the MERA network can also be interpreted as a de Sitter space [6, 11], while the re ned tensor networks given in [ 10, 13 ] are argued to describe Euclidean 2The relevance of computational complexity in holography was recently pointed out and holographic complexity was conjectured to be the volume of maximal time slice in gravity duals [25, 26] (for recent progresses 42]). We would also like to mention that for CFTs, the behavior of the complexity is very similar to the quantum information metric under marginal deformations as pointed out in [43] (refer to [27, 31, 44] for recent developments), where the metric is argued to be well approximated by the volume of maximal time slice in AdS. { 2 { hyperbolic spaces. In our Euclidean approach we can avoid this issue and explicitly verify that the emergent space coincides with a hyperbolic space, i.e. the time slice of AdS. This paper is organized as follows: in section 2, we present our formulation of an optimization of Euclidean path-integrals in CFTs and relate to the analysis of computational complexity and tensor network renormalization. We will also start with an explicit example for a vacuum of a 2D CFT. In section 3, we will investigate the optimization procedure in 2D CFTs for more general states such as nite temperature states and primary states. In section 4, we apply our optimization procedure to reduced density matrices. We show that the holographic entanglement entropy and entanglement wedge naturally arise from this computation. In section 5, we will study the energy stress tensor of our 2D CFTs in a di erence of Liouville action, which corresponds to a relative complexity. In section 7, we apply our optimization to one dimensional nearly conformal quantum mechanics like SYK models. In section 8, propose and provide various support for generalization of our optimization to higher dimensional CFTs. We also compare our results with existing literature of holographic complexity. In section 9, we discuss the time evolution of thermo- eld dynamics in 2D CFTs as an example of time-dependent states. Finally, in section 10 we summarize our ndings and conclude. In appendix A, we comment on the connection of our approach to an earlier work on the relation between the Liouville theory and 3D gravity. In appendix B, we give a brief summary of the results on holographic complexity in literature, focusing on CFT vacuum states. In appendix C, we study the properties of complexity functional in the presence of higher derivatives and in appendix D, we discuss connections between entanglement entropy and Liouville eld. 2 Formulation of the path-integral optimization Here we introduce our idea of optimization of Euclidean path-integrals, which was rst presented in our short letter [22]. We consider a discretized version of Euclidean pathintegral which produces a quantum wave functional in QFTs, having in mind a numerical computation of path-integrals. The UV cut o (lattice constant) is written as throughout this paper. The optimization here means the most e cient procedure to perform the pathintegral in its discretized form.3 In other words, it is the most e cient algorithm to numerically perform the path-integrals which leads to the correct wave functional. 2.1 General formulation We can express the ground state wave functional in a d dimensional QFT on Rd in terms of a Euclidean path-integral as follows: 0['~(x)] = Z 0 D'(z; x)A e SQF T (') ('( ; x) '~(x)): (2.1) 3Please distinguish our optimization from other totally di erent procedures such as the optimization changes the tensor network structures as in tensor network renormalization [23, 24]. 1 dimensional space coordinate of Rd 1. We set z = at the nal time when the path-integral is completed for our convenience. However, we can shift this value as we like without changing our results as is clear from the time translational invariance. Now we perform our discretization of path-integral in terms of the lattice constant . We start with the square lattice discretization as depicted in the left picture of gure 1. To optimize the path-integral we can omit any unnecessary lattice sites from our evaluation. Since only the low energy mode k 1= survives after the path-integral for the period , we can estimate that we can combine O( = ) lattice sites into one site without losing so much accuracy. It is then clear that the optimization via this coarse-graining procedure leads to the middle picture in gure 1, which coincides with the hyperbolic plane. One useful way to systematically quantify such coarse-graining procedures is to introduce a metric on the d dimensional space (z; x) (on which the path integration is performed) such that we arrange one lattice site for a unit area. In this rule, we can write the original at space metric before the optimization as follows: ds2 = 1 2 d 1 i=1 dz2 + X dxidxi : ! Consider now the optimization procedure in this metric formulation. The basic rule is to require that the optimized wave functional opt is proportional to the correct ground state wave function (i.e. the one (2.1) for the metric (2.2) ) even after the optimization i.e. opt['(x)] / background metric for the path-integration The optimization can then be realized by modifying the ds2 = gzz(z; x)dz2 + gij (z; x)dxidxj + 2gzj (z; x)dzdxj ; gzz(z = ; x) = 2 ; gij (z = ; x) = ij 2 ; giz(z = ; x) = 0; (2.2) (2.3) where the last constraints argue that the UV regularization agrees with the original { 4 { one (2.2) at the end of the path-integration (as we need to reproduce the correct wave functional after the optimization). In conformal eld theories, because there are no coupling RG ows, we should be able to complete the optimization only changing the background metric as in (2.3). However, in non-conformal eld theories, actually we need to modify external elds J (such as mass parameter or other couplings of various interactions) in a position dependent way J (z; x). The same is true for CFT states in the presence of external elds. To nalize the optimization procedure, we should provide a su cient condition for the metric to be \maximally" optimized. Thus, we assume that for each quantum state j i there exists a functional I [gab(z; x)] whose minimization with respect to the metric gab gives such maximal optimization.4 In this way, once we know the functional I , we can nalize our optimization procedure. As we will see shortly, in 2D CFTs we can explicit identify this functional I [gab(z; x)]. 2.2 Connection to computational complexity At an intuitive level, the optimization corresponds to minimizing the number of pathintegral operations in the discretized description. As we will explain in subsection 2.4, we can map this discretized Euclidean path-integration into a tensor network computation. Tensor networks are a graphical description of wave functionals in quantum many-body systems in terms of networks of quantum entanglement (see e.g. [45, 46]). The optimization of tensor network was introduced in [23, 24], called tensor network renormalization. We are now considering a path-integral counterpart of the same optimization here. In the tensor network description, the optimization corresponds to minimizing the number of tensors. We can naturally identify this minimized number as a computational complexity of the quantum state we are looking at. Let us brie y review the relevant facts about the computational complexity of a quantum state (for example, see [47{50]). In quantum information theory, a quantum state made of qubits can be constructed by a sequence of simple unitary operations acting on a simple reference state. The sequence is called a quantum circuit and the unitary operations are called quantum gates. As a simple choice, we use 2-qubit gates for simple unitary operations and a direct product state for a simple state which has no real space entanglement ( gure 2). The quantum circuit (gate) complexity of a quantum state is then de ned as a minimal number of the quantum gates needed to create the state starting from a reference state. Because the quantum circuit is a model of quantum computation, here we refer to the complexity as the computational complexity.5 Based on the above considerations as well as the evidence provided in the following section, we are naturally lead to a conjecture that a computational complexity C of a state j i is obtained from the functional introduced before by a minimization: C = Mingab(z;x) [I [gab(z; x)]] : (2.4) 4In non-conformal eld theories or in the presence of external elds in CFTs, this functional depends on gauge elds for global currents and scalar elds etc. as I [gab(z; x); Aa(z; x); J(z; x); : : :]. 5The relevance of computational complexity in AdS/CFT was recently pointed out and holographic computations of complexity have been proposed in [25, 26, 32, 33]. { 5 { state j i can be constructed by simple local (2-qubit) unitary operations from a simple reference state, for example, a product state j0ij0ij0i . In other words, the functional I [gab(z; x)] for any gab(z; x) estimates the amount of complexity for that network corresponding to the (partially optimized) path-integral on the space with the speci ed metric. Understanding of the properties of this complexity functional I , which might appropriately be called \Path-integral Complexity", is the central aim of this work. As we will soon see, this functional will be closely connected to the mechanism of emergent space in the AdS/CFT. 2.3 Optimization of vacuum states in 2D CFTs Let us rst see how the optimization procedure works for vacuum states in 2D CFTs. We will study more general states later in later sections. In 2D CFTs, we can always make the general metric into the diagonal form via a coordinate transformation. Thus the optimization is performed in the following ansatz: ds2 = e2 (z;x)(dz2 + dx2); e2 (z= ;x) = 1= 2; where the second condition speci es the boundary condition so that the discretization is ne-grained when we read o the wave function after the full path-integration. Obviously this is a special example of the ansatz (2.3). Thus the metric is characterized by the Weyl scaling function (z; x). Remarkably, in 2D CFTs, we know how the wave function changes under such a local Weyl transformation. Keeping the universal UV cut o , the measure of the pathintegrations of quantum elds in the CFT changes under the Weyl rescaling [51]: where SL[ ] is the Liouville action6 [52] (see also [51, 53]) [D']gab=e2 ab = eSL[ ] SL[0] [D']gab= ab ; SL[ ] = 24 c Z 1 Z 1 1 dx 2 i : Liouville action for a more general reference metric. 6Here we take the reference metric is at ds2 = dz2 + dx2. Later in section (6), we will present the { 6 { (2.5) (2.6) (2.7) The constant c is the central charge of the 2D CFT we consider. The kinetic term in SL represents the conformal anomaly and the potential term arises the UV regularization which manifestly breaks the Weyl invariance. In our treatment, we simply set = 1 below by suitable shift of . Therefore, the wave functional gab=e2 ab ('~(x)) obtained from the Euclidean pathintegral for the metric (2.5) is proportional to the one gab= ab ('~(x)) for the at metric (2.2) thanks to the conformal invariance. The proportionality coe cient is given by the Liouville action as follows7 gab=e2 ab ('~(x)) = eSL[ ] SL[0] gab= ab ('~(x)): (2.8) can be identi ed as follows8 Let us turn to the optimization procedure. As proposed in [22], we argue that the optimization is equivalent to minimizing the normalization factor eSL[ ] of the wave functional, or equally the complexity functional I 0 for the vacuum state j 0i in 2D CFTs, I 0 [ (z; x)] = SL[ (z; x)]: The intuitive reason is that this factor is expected to be proportional to the number of repetition of the same operation (i.e. the path-integral in one site). In 2D CFTs, we believe this is only one quantity which we can come up with to measure the size of path-integration. Indeed it is proportional to the central charge, which characterizes the degrees of freedom. Thus the optimization can be completed by requiring the equation of motion of Liouville action SL and this reads With the boundary condition e2 (z= ;x) = 2, we can easily nd the suitable solution where we introduced w = z + ix and w = z to (2.10): which leads to the hyperbolic plane metric 4 e 2 = (w + w)2 = z 2; ds2 = dz2 + dx2 z2 : This justi es the heuristic argument to derive a hyperbolic plane H2 in gure 1. Indeed, this hyperbolic metric is the minimum of SL with the boundary condition. To see this, we rewrite SL = c Z 24 c Z 12 dx[e ]zz==1 cL 12 ; 7Here we compare the optimized metric gab = e2 ab with gab = ab. To be exact we need to take the latter to be the original one (2.2) i.e. gab = 2 ab. However the di erent is just a constant factor multiplication and does not a ect our arguments. So we simply ignore this. 8In two dimensional CFTs, as we will explain in section 6, due to the conformal anomaly we actually However this does not change out argument in this section. R dx is the length of space direction and we assume the IR behavior e2 (z=1;x) = 0. The nal inequality in (2.13) is saturated if and only if (2.14) and this leads to the solution (2.11). In this way, we observe that the time slice of AdS3 dual to the 2D CFT vacuum emerges after the optimization. We will see more evidences throughout this paper that geometries obtained from our optimization coincides with the time slice in AdS/CFT. This is consistent with the idea of tensor network description of AdS/CFT and can be regarded as its continuous version. We would like to emphasize that the above argument only depends on the central charge c of the 2D CFT we consider. Therefore this should be applied to both free and interacting CFTs including holographic ones. It is also interesting to note that the optimized value of SL, i.e. our complexity C 0 , scales linearly with respect to the momentum cut o 1 and the central charge c as C 0 = Min [SL[ ]] = cL 12 ; (2.15) and this qualitatively agrees with the behavior of the computational complexity [25, 26] of a CFT ground state and the quantum information metric [43] for the same state, both of which are given by the volume of time slice of AdS. In this relation, our minimization of SL nicely corresponds to the optimization of the quantum circuits which is needed to de ne the complexity. 2.4 Tensor network renormalization and optimization As argued in our preceding letter [22] (see also [14]), our identi cation of the Liouville action with a complexity i.e. (2.9) is partly motivated by an interesting connection between the tensor network renormalization (TNR) [23, 24] and our optimization procedure of Euclidean path-integral. This is because the number of tensors in TNR is an estimation of complexity and the Liouville action has a desired property in this sense, e.g. it is obvious that the Liouville potential term R e2 (i.e. the volume) measures the number of unitary tensors in TNR. Soon later this argument was sharpened in the quite recent paper [56] where the number of isometries is argued to explain the kinetic term R (@ )2 in Liouville theory. An Euclidean path-integral on a semi-in nite plane (or cylinder) with a boundary condition on the edge gives us a ground state wave functional in a quantum system. The path-integral can be approximately described by a tensor network which is a collection of tensors contracted with each other. Using the Suzuki-Torotter decomposition [54, 55] and the singular value decomposition of the tensors, we can rewrite the Euclidean path-integral into a tensor network on a square lattice ( gure 3). Tensor network renormalization (TNR) is a procedure to reorganize the tensors to ones on a coarser lattice by inserting projectors (isometries) and unitaries (disentanglers) with removing short-range entanglement.9 This is a step of TNR ( gure 4). Repeating this procedure, we can generate a RG ow properly 9Note that by adding a dummy or ancilla state j0i we can equivalently regard an isometry as a unitary. { 8 { approximately described by a tensor network on a square lattice. MERA (+IR tensors) UV bdy 1 step of TNR Repeat the steps network with removing short-range entanglement. From the UV boundary, isometries (coarsegraining) and unitaries (disentanglers) accumulate and the MERA network grows with the TNR steps. and end up with a tensor network at the IR xed point. For the ground state wave functional in a CFT, it ends up with a MERA (Multi-scale Entanglement Renormalization Ansatz) network made of isometries and disentanglers. The MERA network clearly contains smaller numbers of the tensors than ones in the tensor network on the original square lattice before the coarse-graing. In this sense, this MERA network is an optimal tensor network to approximately describe the Euclidean path-integral. Our optimization procedure is motivated by TNR. In our procedure ( gure 1), the tensor network on the square lattice corresponds to the Euclidean path-integral on at space with a UV cuto . Changing the tensor network with inserting isometries and entanglers corresponds to deforming the back-ground metric for the path-integral. And the MERA network, which is the tensor network after the TNR procedure, approximately corresponds to the optimized path-integral. Actually, it is not di cult to estimate the amount of complexity for each tensor network during the TNR optimization procedure, by identifying the complexity with the number of tensors, both isometries (coarse-graining) and unitaries (disentanglers). For simplicity, consider an Euclidean path-integral for the ground state wave function in a 2d CFT, which is performed on the upper half plane ( < z < 1; 1 < x < +1). First we consider { 9 { at a speci c layer. This also represents the one step (s-th) contribution in the process of tensor network renormalization, which nally reaches the MERA network. This corresponds to s th terms R22ss 1 dz( ) in (2.17). the original square lattice. Since we suppose that each tensor have unit area, the uniform metric is given by e2 (z) = 2 as in (2.2). Therefore, the total number of tensors, which are only unitaries, is estimated from the total volume: Z 1 1 dx Z 1 1 dz 2 = Z 1 Z 1 dx dze2 : 1 Then, performing the TNR procedure, the number of the tensors or the square lattice sites is reduced by the factor (1=2)2 per step. On the other hand, the isometries and disentanglers accumulate from the UV boundary [23, 24]. Refer to gure 4. At the k-th step of TNR, the total area changes into Z 1 1 dx Z 1 2k dz (2k )2 + Xk Z 1 1 s=1 1 dx Z 2s 2s 1 dz 1 (2s 1 ) (2s ) + 1 (2s )2 : (2.17) The rst term is the contribution from the tensors on the coarser lattice. The second term is the contribution from the MERA network. For the s-th layer of the MERA network, we have dxdz=((2s 1 ) (2s )) isometries and dxdz=(2s )2 per unit cell. This contribution is depicted in gure 5. This network corresponds to the metric MERA layer s-th layer (2.16) (2.18) HJEP1(207)9 e 2 = n (2k ) 2 (z 2k ): z 2 (z < 2k ): Obviously, the rst and third term in (2.17) are approximated by the Liouville potential integral R e2 [22]. The second term arises because of the non-zero gradient of and is 3 Optimizing various states in 2D CFTs Here we would like to explore optimizations in 2D CFTs for more general quantum states. First it is useful to remember that the general solutions to the Liouville equation (2.10) is Note that functions A(w) and B(w) describe the degrees of freedom of conformal mappings. For example, if we choose A(w) = w; B(w) = 1=w; then we reproduce the solution for vacuums states (2.11). nite temperature T = 1= . In the thermo eld double description [58], the wave functional is expressed by an Euclidean path-integral on a strip de ned by 4 ( z1) < z < 4 ( z2) in the Euclidean time direction, more explicitly 4 <z< 4 1 D'(z; x)CA e SCFT(') ' (z1; x) '~1(x) ' (z2; x) '~2(x) : where '~1(x) and '~2(x) are the boundary values for the elds of the CFT (i.e. '~(x)) at z = 4 respectively. Minimizing the Liouville action SL leads to the solution in (3.2) given by: well-known (see e.g. [51, 57]): e 2 = 4A0(w)B0(w) (1 which coincides with the time slice of eternal BTZ black hole (i.e. the Einstein-Rosen bridge) [58]. This leads to 2 iw A(w) = e ; B(w) = e 2 iw : e 2 = If we perform the following coordinate transformation then we obtain the metric z tan = tanh 2 ; ds2 = d 2 + (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) Thus the dependence of the wave function on looks like gab=e2 ab ('~) ' eSL e 2h (0) gab= ab ('~): This shows that the complexity function should be taken to be The equation of motion of I leads to where we set The solution can be found as which leads to the expression: I [ (w; w)] = SL[ (w; w)] 2h (0): e 2 + 2 (1 a) 2(w) = 0; a = 1 12h c : A(w) = wa; B(w) = wa; e 2 = 4a2 jwj2(1 a)(1 jwj2a)2 : Now consider 2D CFTs on a cylinder (with the circumference 2 ), where the wave functional is de ned on a circle jwj = 1 at a procedure, we obtain the geometry A(w) = w and B(w) = w given by xed Euclidean time. After the optimization which is precisely the Poincare disk and is the solution to (2.10). Then we consider an excited state given by a primary state j i. This is created by acting a primary operator O (w; w) with the conformal dimension h = h at the center w = w = 0. Its behavior under the Weyl re-scaling is expressed as (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) e2 (w;w) = (1 4 jwj2)2 ; O(w; w) / e 2h : a = r 24h c : Since the angle of w coordinate is 2 periodic, this geometry has the de cit angle 2 (1 a). Now we compare this geometry with the time slice of the gravity dual predicted from AdS3=CFT2. It is given by the conical de cit angle geometry (3.15) with the identi cation Thus, the geometry from our optimization (3.13) agrees with the gravity dual (3.16) up to the rst order correction when h c, i.e. the case where the back-reaction due to the point particle is very small. It is intriguing to note that if we consider the quantum Liouville theory rather than the classical one, we nd the perfect matching. In the quantum Liouville theory, we introduce a parameter such that c = 1 + 3Q2 and Q 2 + . The chiral conformal dimension of 2 the primary operator e is given by the 2D CFT has a classical gravity dual, we nd (Q 2 ) . If the central charge is very large so that which indeed agrees with the gravity dual (3.16) even when h =c is nite. This agreement may suggest that the actual optimized wave functional is given by a HJEP1(207)9 `quantum' optimization de ned as follows: (3.17) (3.18) opt['~] = Z D (x; z)e SL[ ] ( gab= ab ['~]) 1 1 : If we take the semi-classical approximation when c is large, we reproduce our classical optimization. It is an important future problem to understand the exact for of the proposal at the quantum level. 3.3 Liouville equation and 3D AdS gravity In the above we have seen that the minimizations of Liouville action, which corresponds to the optimization of Euclidean path-integrals in CFTs, lead to hyperbolic metrics which t nicely with canonical time slices of bulk AdS in various setups of AdS3=CFT2. If this derivation of time slice metric in AdS3 really explains the mechanism of emergence of AdS in AdS/CFT, it should t nicely with the dynamics of AdS gravity for the whole 3D space-time. One natural coordinate system in 3D gravity for our argument is as follows ds2 = RAdS d 2 + cosh2 2 e2 dydy : (3.19) Indeed the Einstein equation R = e2 . + RA2dS 2 g It is also useful to remember that connections between Liouville theory and 3D AdS gravity were discussed in earlier papers [59{66] (refer to [67] for a review). Especially the direct connection between the equation of motion in the SL(2; R) Chern-Simons gauge theory description of AdS gravity [68] and that of Liouville theory was found in [61] (see also closely related arguments [62{65]). Indeed, we can nd a coordinate transformation which maps the metric (3.19) into the one from [61], where the map gets trivial only in the near boundary limit This shows that we can identify these two appearances of Liouville theory from 3D AdS gravity by a non-trivial bulk coordinate transformation. We presented the details of this ! 1. transformation in the appendix A. Notice also that we did not x the overall normalization of the optimized metric or equally the AdS radius RAdS because in our formulation it depends on the precise de nition of UV cut o . However, we can apply the argument of [14] for the symmetric = 0 is equivalent to the equation of motion in A+ AS2 A+ A R2 Optimize Optimize Identify A + A+ Ainto a sphere with a open cut depicted in the lower left picture. The upper right one is the one after the optimization and is equivalent to a geometry which is obtained by pasting two identical entanglement wedges along the geodesic (=the half circle) as shown in the lower right picture. orbifold CFTs and can heuristically argue that RAdS is proportional to the central charge c. This is deeply connected to the fact that we nd the sub AdS scale locality in gravity duals of holographic CFTs. 4 Reduced density matrices and EE Consider an optimization of path-integral representation of reduced density matrix A in a two dimensional CFT de ned on a plane R2. We simply choose the subsystem A to be an interval l x l at z(= ) = . A is de ned from the CFT vacuum by tracing out the complement of A (the upper left picture in gure 6). 4.1 Optimizing reduced density matrices The optimization procedure is performed by changing the background metric as in (2.5), where the boundary condition of is imposed around the upper and lower edge of the slit A. Refer to gure 6 for a sketch of this procedure. The plane R2 is conformally mapped into a sphere S2. Therefore the optimization is done by shrinking the sphere with an open cut down to a much smaller one so that the Liouville action is minimized. To make the analysis clear, let us divide the nal manifold into two halves by cutting along the horizontal line z = 0, denoted by + and . The boundary of consist of two parts: where manifold have e2 = 1= 2. = A [ A; (4.1) are identi ed so that the topology of the nal optimized + [ is a disk with the boundary A+ [ A . On the boundary A+ [ A we SLb = c Z 12 ds[K0 + where K0 is the (trace of) extrinsic curvature of the boundary @ in the at space. If we describe the boundary by x = f (z), then the extrinsic curvature in the at metric ds2 = dz2 + dx2, is given by K0 = boundary Liouville potential. Since we set B = 0 for our A optimization.10 + and (1+(f0)2)3=2 . On the other hand, the nal term is the are pasted along the boundary smoothly, Now, to satisfy the equation of motion at the boundary A, we impose the Neumann boundary condition11 of . This condition (when B = 0) is explicitly written as The optimization of each of is done by minimizing the Liouville action with boundary contributions. The boundary action in the Liouville theory [69] reads (4.2) (4.3) (4.4) where nx;z is the unit vector normal to the boundary in the at space. Actually this is simply expressed as K = 0, where K is the extrinsic curvature in the curved metric (2.5). This fact can be shown as follows. Consider a boundary x = f (z) in the two dimensional space de ned by the metric ds2 = e2 (z;x)(dz2 + dx2). The out-going normal unit vector N a is given by f 0(z)e p1 + f 0(z)2 (z;x) N z = e (z;x)nz = ; N x = e (z;x)nx = e (z;x) p1 + f 0(z)2 ; where na is the normal unit vector in the at space ds2 = dz2 +dx2. The extrinsic curvature (=its trace part) at the boundary is de ned by K = hab raNb, where all components are projected to the boundary whose induced metric is written as hab. Explicitly we can calculate K as follows: K = e (z;x) p1 + f 0(z)2 = e (4.5) In this way, the Neumann boundary condition requires that the curve A is geodesic. By taking the bulk solution given by the hyperbolic space = log z+const., the geodesic A is given by the half circle z2 + x 2 = l2. Thus, this geometry obtained from the optimization of A, coincides with (two copies of) the entanglement wedge [15, 16, 70{72]. Note that if we act a local operator inside the entanglement wedge in the original at space, then this excitation survives after the optimization procedure. However, if we act the operator outside, then the excitation is washed out under the optimization procedure and does not re ect the reduced density matrix A as long as we neglect its back-reaction. 10Non-zero 11On the cuts A B leads to a jump of the extrinsic curvature which will be used later. we imposed the Dirichlet boundary condition. The reason why we imposed the Neumann one on A is simply because the manifold is smoothly connected to the other side on A. A+ A= Deficit angle deformation z A+ Deficit angle A- deformation x A+ A= A+ Aassume the analytical continuation such that n is very close to 1 such that (1 n) this describes an in nitesimally small (negative) de cit angle deformation. After the optimization, we obtain the conical geometry in the lower right picture with = (1 n). Entanglement entropy ' nd the relation K ' Next we evaluate the entanglement entropy by the replica method. Consider an optimization of the matrix product nA. We assume an analytical continuation of n with jn 1 j The standard replica method leads to a conical de cit angle 2 (1 n) 2 at the two end points of the interval A. Thus, after the optimization, we get a geometry with the corner angle =2 + (n 1) instead of =2 (the lower right picture in gure 7). This modi cation of the boundary A is equivalent to shifting the extrinsic curvature from K = 0 to K = (n 1). Indeed, if we consider the boundary given by x2 + (z z0)2 = l2, we get K = z0=l. When z0 is in nitesimally small, we get x ' l + (z0=l) z + O(z2) near the boundary point (z; x) = (0; l). Therefore the corner angle is shifted to be =2 with z0=l (for the de nition of , refer also to lower pictures in gure 7). Therefore we . If we set the boundary Liouville term in (4.2) non-zero B 6= 0, the boundary condition is modi ed from (4.3) i.e. K = 0 into K + desired angle shift (or negative de cit angle) is realized by setting B = 0. Thus the B = (1 n). In the presence of in nitesimally small B we can evaluate the Liouville action by a probe approximation neglecting all back reactions. By taking a derivative with respect to n, we obtain the entanglement entropy12 SA: SA = ds e n=1 = 3 l log ; (4.6) reproducing the well-known result [73]. The lower left expression (4.6) 6c R@ + agrees with the holographic entanglement entropy formula [15, 16] as e precisely A has to be the geodesic due to the boundary condition. 12The abuse of notation for the entanglement entropy and the Liouville aciton should be clear form the context. HJEP1(207)9 Finally we would like to evaluate the value of Liouville action SL[ ] in the reduced subregion. It is natural to argue that this provides a de nition of complexity for the reduced density matrix A. For various earlier proposals for holographic subregion complexity refer to [27, 30]. As in the previous section we take A to be the interval l x l. By computing the action for two copies of the half disk x2 + z2 l2 with the solution = log z, we nd SL = = = c Z c Z l 12 6 6 c 2l dz + 2 l p 2 z2 z2 log + l 2 6 c Z =2 : c Z 6 =2 dsK0 (4.7) It will be interesting to compute and explore it further for more general states and we leave it as an open future problem. 5 Energy momentum tensor in 2D CFTs One of the most fundamental objects in two dimensional CFTs is the energy momentum tensor and in this section we show how to extract it from our optimization. Since we already know how to compute entanglement entropy, our derivation will be based on the rst-law of entanglement that relates changes in entanglement entropy of an interval to the energy momentum tensor. More precisely, as shown in [74], under small perturbations of a quantum state, the change of entanglement entropy of a small interval A = [ l=2; l=2] is proportional to Ttt On the other hand, in our approach, the change in entanglement entropy under a small variation of a quantum state is captured by the variation in the Liouville eld 0(z) + (z). Moreover, for small perturbations we can write such that the change in entanglement entropy in perturbed state becomes SA ' 6 c Z c 2 Z l=2 0 z dz p1 4z2=l2 = cl2 Comparing with the rst law, we can now match the energy momentum tensor SA ' l 2 3 Ttt: (z) = z 2 2 z 2 (z) + O(z4); c 8 Ttt = : 0(z) = written as and we obtain the well known result Let us now compare this with our explicit examples. The vacuum solution is given by log (z). Then, after a simple shift, the thermo eld double solution (3.5) can be (z) = log sin 2 2 z Similarly, writing our conical singularity solution (3.15) in coordinates w = exp(z + ix) (z) = log sinh (az) ' 0(z) a2z2 6 + O(z4); and the known energy momentum tensor Ttt = c : Ttt = a2c 24 ; (5.5) (5.6) (5.7) (5.8) (5.9) (5.10) that for a = 1 reproduces the Casimir energy. Let us also point the interesting consistency of the above result with the Liouville energy momentum tensor. Namely, it is well known that by varying the action with respect to the background \reference" metric one can derive the Liouville energy momentum tensor. The corresponding holomorphic and anti-holomorphic classical energy momentum tensors are 2 ; 2 : One can check that, for our solutions, these energy momentum tensors match the ones computed form the rst law. In general we can use the rst law for entanglement entropy in states conformally mapped to the vacuum (see e.g. [75, 76]) and show that the increase in the entropy is proportional to the (constant) Liouville energy momentum tensor. 6 Evaluation of SL in 2D CFTs Here we rst explain the properties of Liouville action SL in general setups with boundaries. We will nd that it depends on the reference metric and it does not seem to be possible to de ne its absolute value, which is due to the conformal anomaly in 2D CFTs. Rather we are lead to introduce an functional de ned by a di erence of Liouville action denoted by IL[g2; g1], where g1 is the reference metric and g2 is the nal metric after the optimization. IL[g2; g1] is expected to measure of the complexity between the two path-integrals in g1 and g2. Having them in mind, we proceed to explicit evaluations of IL[g1; g2] in various cases. constant time evaluations. In [30, 38] the authors investigated the constant time behavior of the holographic complexity. More speci cally they studied the divergence structure, considering both the CV and CA-conjectures. Also a possible prescription to remove an ambiguity due to di erent parametrization of the null boundary surfaces in the WDW patch was found in [35]. This prescription was used to evaluate the holographic complexity in [38]. In appendix B, we summarize these results of holographic complexity by focusing on the vacuum states. Comparisons with our results. We are nally ready to compare the evaluation of holographic complexity with our proposal against the same computed with the existing proposals in the literature, presented in appendix B. First if we follow the \Complexity = Volume" conjecture (8.47), the complexity has the structure CV . This behavior agrees with our results of complexity C 0 presented in (8.46), (8.44) and (8.45), though the relative coe cients do not coincide in general. Next we turn to the \Complexity = Action" conjecture (8.48). The analysis in [30] evaluates it to be divergent, in fact a logarithmically enhanced divergence of the form log (d 1) for the CA-conjecture as opposed to the 1= d 1 divergence for the CV-conjecture. On the other side, the [38] proposal, which introduces an additional boundary contribution, produces a surprising result for the d = 2 case i.e. bulk AdS3: for both Poincare and global AdS3, the leading divergence vanishes, leading to a constant holographic complexity. In higher dimensions d = 3; 4, the holographic complexity has a leading divergence of the form 1= d 1 for both Poincare and global AdSd+1. Therefore the divergence structure in [38] for d > 2 is the same as ours, whereas, they di er in the numerical coe cients in general. Nevertheless, in the next subsection, we will point out an interesting relation between our complexity functional Idtot and the gravity action IWDW in the WDW patch. Since there is no precise de nition of computational complexity in quantum eld theories known at present, we cannot decide which of these prescriptions is true by consulting with rigorous results in eld theory. However, notice that our proposal of computational complexity C 0 , de ned in (2.4), is based on not any holography but a purely eld theoretic argument as is clear in two dimensional CFT case, where it is related to the normalization 1 of wave functional. 8.7 Relation to \complexity = action" proposal d(d 1) below. We have discussed in the previous subsection that, in [32, 33], it has been conjectured that the holographic complexity is measured by the bulk action being integrated over the WDW patch de ned above including suitable boundary terms. Here we would like to compare this quantity with our complexity functional. For simplicity, we set RAdS = 1 and thus Consider the following class of d + 1 dimensional space-time: ds2 = dt2 + cos2 t e2 (x)hij dxidxj ; (8.50) where t takes the range t =2 and i; j = 1; 2; ; d. The pure AdSd+1 which is a solution to the Einstein equation from IWDW, is obtained when the metric e2 (x)hij dxidxj coincides with a hyperbolic space Hd. For example, when d = 2, the Einstein equation = e 2 i.e. the Liouville equation. Note that in this pure AdSd+1 solution, the coordinate covered by (8.50) indeed represents the WDW patch.15 Motivated by this we identify this space (8.50) with MWDW. However, note that for generic choices of and hij (8.50) does not represent the WDW patch in the original de nition in [32, 33]. They coincide only on-shell. Now we would like to evaluate the gravity action (8.49) within the WDW patch, integrating out the time t coordinate. Here we can ignore the contribution from the boundary as the Gibbons-Hawking term of this boundary turns out to be vanishing. We nally nd that the nal action is proportional to our complexity functional Idtot[ ; g] (8.2) with the normalization (8.3) up to surface terms at the AdS boundary z = 0 due to partial integrations: where the numerical constant nd is de ned by IdWDW = (d 2) nd Idtot[ ; g] + (IR Surface Term); nd = Z =2 =2 dt(cos t)d 2 = p d 1 2 d 2 : In the above computation, by introducing the Gibbons-Hawking term for the d dimensional boundary time like surface given by z = , the surface terms on this surface which are produced by the partial integrations of bulk action are all cancelled with the GibbonsHawking term. Therefore in the surface terms in (8.51) is localized at the IR boundary, which is at z = 1 and gives the vanishing contribution for the Poincare AdS coordinate. For example, when d = 3 with hij = ij (setting x3 = z), so that it ts with the Poincare AdS4, we nd I3WDW = 1 Z d x 2 Z =2 =2 dt 6e3 (cos3 t cos t cos 2t) which reproduces (8.51) after we integrate t and perform a partial integration with the boundary term at z = cancelled by the Gibbons-Hawking term at z = . When d = 2 we nd I2WDW = 1 8GN Z which indeed leads to vanishing action up to partial integrations, where again the boundary term at z = is cancelled by the Gibbons-Hawking term at z = . Therefore we simply 15In Euclidean signature obtained by t ! i , this leads to the hyperbolic slice of Hd+1 which is precisely given by (3.19). (8.51) (8.52) (8.53) (8.54) nd I2WDW = 8GN have log z and log sinh z for Poincare and global AdS3, we get I2WDW = 0 4GN for CFT2 vacuum on R1 dual to the Poincare AdS3; for CFT2 vacuum on S1 dual to the Poincare AdS3: Interestingly, this agrees with the evaluation of holographic complexity with the prescription in [38]. The above relation shows that there is no di erence with respect to the equation of motion for between the \Complexity = Action" approach and our proposal. However in the d = 2 limit they di ers signi cantly due to (d 2) factor in (8.51). In our proposal, the complexity functional for 2D CFTs is obtained as limd!2(Idtot[ ; g] Itot[0; g]) = d limd!2 (IdWDW IWDW j =0)=(d On the other hand, there are no bulk contributions in I2WDW as clear from (8.54). This is essential reason why the former have the UV divergence O( 1), while the latter does not. 2) , which coincides with the Liouville action IL[ ; g]. Higher derivative terms and anomalies In our optimization of two dimensional CFTs, we minimized the overall factor of wave functional, which is the same as the partition function Z for the region < z < 1. The Liouville action which we minimize is given by the log of this partition function SL = log Z. Therefore even for higher dimensional CFTs one may naively suspect that the complexity functional Id may also be written as Id = log Zd for d-dimensional CFTs. This indeed works for d = 3 as the UV divergent terms produces the two terms in the action (8.2). The situation is di erent for d = 4 due to the presence of conformal anomaly [86] and we need to have forth derivative terms in addition to the action (8.2). As we have explained in appendix C, here we just mention the nal form of I4 that correctly reproduces the anomalies in a four dimensional CFT, I4 = Z d4xpg 2 HJEP1(207)9 ) ; 3 where the terms with corresponding coe cients 4 5 denote the fourth derivative terms, responsible for producing correct anomalies. It should be mentioned that here we only consider metrics which are of the Weyl scaling type (2.5), g = e2 h at; with h at corresponding to Euclidean at space. In appendix C, we rstly explain in detail how this action in (8.55) produces the correct anomalies for four dimensional CFT. Next we also explain how the equations of motion following from this action allows time slice of AdS5, i.e. hyperbolic space H4, as a solution. Here we should admit that the action (8.55) is not bounded from below and hence cannot be minimized, therefore we can only extremize it. In this aspect, the action (8.2) without higher derivatives as we assumed in section 2.3 has an advantage over the modi ed action we are discussing here. (8.55) (8.56) So far we have studied stationary quantum states in CFTs. For further understandings of the dynamics of CFTs, we would like to turn to time dependent states in this section, focusing on 2D CFTs for simplicity. In particular we consider a simple but non-trivial class of time-dependent states, given by the time evolution of thermo eld double states (TFD states) in 2D CFTs: jT F D(t)i = 1 pZ (t) n X e 4 (H1+H2)e it(H1+H2)jni1jni2; (9.1) where the total Hilbert space is doubled Htot = H1 H2 (H1 is the original CFT Hilbert space and H2 is its identical copy). Its density matrix16 is given by (t) = jT F D(t)ihT F D(t)j and if we trace out H2, then the reduced density matrix 1 is time-independent, given by the standard canonical distribution 1 / e H1 . However the TFD state jT F D(t)i shows very nontrivial time evolution and is closely related to quantum quenches as pointed out in [89]. Motivated by this, let us study the path-integral expression of jT F D(t)i. First we can create the initial TFD state jT F D(0)i by the Euclidean path-integral for the range of Euclidean time : 4 4 : (9.2) After this path-integration, we can perform the Lorentzian path-integral by it on each CFT. This integration contour is depicted as the left picture in gure 8. However, as we will see later, there is an equivalent but more useful contour given by the right picture in gure 8. This is because we can exchange the Euclidean time evolution e (H1+H2)=4 with the real time one e it(H1+H2). Now we consider an optimization of this path-integral. For the Euclidean part we can apply the same argument as before and minimize the Liouville action. Next we need to consider an optimization of the real time evolution. However, we would like to argue that this Lorentzian path-integral cannot be optimized. A heuristic reason for this is that if the nal state even after a long time evolution, is still sensitive to the initial state as opposed to the Euclidean path-integral. On the other hand, if we perform an Euclidean time evolution for a period , then the nal state is insensitive to the high momentum mode k 1= of the initial state. Once we assume this argument, we can understand the reason why we place the Lorentzian time evolution in the middle sandwiched by the Euclidean evolution as this obviously reduces the value of SL. It is an intriguing future problem to verify these intuitive arguments using the tensor network framework. 16However note that (t) can not be obtained from the analytic continuation = it of Euclidean TFD density matrix ( ) = jT F D( )ihT F D( )j de ned by the Euclidean path-integral for the Euclidean time region =4 =4 + . Instead it is obtained from 0( ) = jT F D( )ihT F D( ) . j 4 4 it it Im[t] 4 it 4 4 it it 0 it it 4 it 0 it 4 it 2t Optimization in the rst CFT H1. The left and right choices are equivalent. is optimized by minimizing the Liouville action. We assumed that the Lorentzian one cannot be optimized. Assuming that the above prescription of optimization is correct at least semiquantitatively, we can nd the following solution (remember we set z = ): e 2 = ( 4 22 cos 2 2 Re[z] ; 4 22 ; ( it < z < it): ( 4 it < z < it; it < z < it + 4 ) (9.3) This is depicted in gure 9. It is also intriguing to estimate the complexity. For the Euclidean part, we proposed that it is given by the Liouville action as we explained before. For Lorentzian part, there is no obvious candidate. However since we assumed that is constant during the real time evolution, we can make a natural identi cation: the Liouville potential term gives the complexity. This is clear from the fact that the complexity should be proportional to the number of tensors. Thinking this way, we nd SL(t) = SL(0) + 2 This linear t growth is consistent with the basic idea in [26]. Since the energy in our 2D CFT at nite temperature T = 1= is given by (9.4) (9.5) (9.6) (9.7) (9.8) Interestingly this growth is equal to a half of the gravity action IWDW on the WDW patch for holographic complexity found in [32, 33], where the holographic complexity CA is conjectured to be CA = IWDW (8.48). Note that we are shifting both the time in the rst and second CFT at the time. This relation dIWDW = 2 ddStL may be natural because the partition dt function of CFTs Z eA is the square of that of wave functional in CFTs j j 2 e2SL . It is also intriguing to consider a pure state which looks thermal when we coarse-grain its total system. One typical such example in CFTs is obtained by regularizing a boundary j Bi = NBe H=4jBi; where NB is the normalization such that h Bj Bi = 1. This can also be regarded as an approximation of global quenches [87, 88]. This quantum state is dual to a single-sided black hole [89] shows the linear growth of holographic entanglement entropy which matches with the 2D CFT result in [87]. This state after our path-integral optimization is clearly given by a half of TFD (6.28) for 0 < z < =4 . The boundary at z = 0 corresponds to that of the boundary state jBi which matches with the AdS/BCFT formulation [90, 91]. Thus the growth of the complexity functional is simply given by a half of the TFD case (9.6). 9.2 Comparison with eternal BTZ black hole The time evolution of TFD state provides an important class of time-dependent states and here we would like to discuss possible connections between our optimization procedure and its gravity dual given by the eternal BTZ black hole. In this section we set = 2 for simplicity. First let us try to assume that the dual geometry for this time-dependent quantum states has a property that each time slice is given by a space-like geometry which is a solution to Liouville equation. Any solution to the Liouville equation is always a hyperbolic space with a constant curvature. Such a hyperbolic space at each time t is obtained by taking a union of all geodesic which connects two points at the time t with the same space coordinate in the two di erent boundaries, given explicitly by ECFT = dSL(t) dt 2 3 cT 2; = 2ECFT: ds2 = e2 (z)(dz2 + dx2); e2 (z) = 1 sinh2 dt2 + d 2 + cosh2 dx2; (9.11) However if we evaluate its action (as in the computation of (6.31)) we nd (we recovered dependence) 1 h cosh 1 z sin cosh SL = 8 d 2 + dx2 + (z tanh (d ) dz)2i ; tanh t = tanh z cos cosh : the metric is rewritten as cosh 1 can be rewritten into the metric ds2 = via the coordinate transformation z sin cosh cosh c d(Vol(t)) 24 dt c 12 ; (9.9) (9.10) (9.12) (9.13) (9.14) (9.15) (9.16) (9.17) Thus there is no linear t growth. In this way, this surface does not seem to have an expected property which supports the linearly growing complexity argued in many papers [25, 26, 32, 33, 35]. Now we would like to turn to another candidate: maximal time slice, whose volume was conjectured to be one candidates of holographic complexity [25, 26]. Note that this maximal time slice is not a solution to Liouville equation as opposed to the previous hyperbolic space (9.8), which is constructed from geodesics. The BTZ metric behind the horizon can be obtained by the analytic continuation = i , t~ = t + 2i equation . Maximal volume surface with boundary time t is determined by the s2(t) = cosh2 sinh4 cos2 sin4 _ 2 + sin2 : s(t)2 increases monotonically as t ( 0) increases, with boundary value s(0) = 0 and s(1) = 1=2. The induced metric on the maximal volume time-slice is ds2 = cosh2 sinh2 s(t)2 + sinh2 cosh2 d 2 + dx2 : The curvature of the maximal volume time slice is not constant, therefore the time slice is not hyperbolic. Then, we nd that the volume term increases linearly in time. Finally we obtain x z HJEP1(207)9 timization of path-integral for the TFD states. At low temperature the two CFTs are connected through a microscopic bridge with entanglement entropy O(1) in the tensor network. At high temperature the bridge gets macroscopic and has entanglement entropy O(c). at late time (here we used the same normalization as our proposal for the Liouville action). This behavior is in contrast to the previous hyperbolic time slice, where the action approaches monotonically to some constant value. In summary, the above arguments imply that for a generic time dependent background, the assumption that a preferred time slice in a gravity dual is described by Liouville equation, is not compatible with the requirement that the Liouville action gives a measure of complexity. Thus an extension of our proposal in this paper to time-dependent backgrounds looks highly non-trivial and deserves future careful studies. Comment on phase transition It is also intriguing to discuss how we can understand the con nement/decon nement phase transition in our approach. For this, we focus on the initial state jT F D(t = 0)i. Since our approach is based on pure states we need to consider the wave functional of TFD state (at temperature T ) and see how the corresponding tensor network changes as a function of T . It is obvious that at high temperature, the connected network which looks like macroscopic wormhole is realized and this should be described by the optimized pathintegral on the Einstein-Rosen bridge (3.5). As we make the temperature lower, the neck of bridge gets squeezed and eventually disconnected in a macroscopic sense. Here we mean by the macroscopic the quantum entanglement of order O(c) = O(1=GN ). Refer to gure 10. Since the TFD state has non-zero (but sub-leading order O(1)) entanglement entropy between the two identical CFTs even at low temperature, there should be a microscopic bridge or wormhole (following ER=EPR conjecture [92]) which connects the two sides in the tensor network description. In this low temperature, the bridge is due to the singlet sector of the gauge theory and is in its con ned phase. In large c holographic CFTs, there should be a phase transition of the macroscopic form of the tensor network at the value = 1=T = 2 predicted by AdS3=CFT2. Naturally, we expect that the favored phase of a given quantum state is the one which has smaller complexity C . However, in the current form of our arguments based on the path-integral optimization, it is not straightforward to compare the value of the complexity (i.e. Liouville action) for the con nement/decon nement phase transition. This is because we can only de ne the di erence of complexity which depends on the reference metric. In this phase transition, the topology of the reference space changes and it is di cult to know how to compere them precisely. Nevertheless, it might be useful to try to roughly estimate the complexity. For this we assume that the complexity for the decon ned phase (denoted by Cdec) is estimated by the bridge solution (6.30) and that for the con ned phase (denoted by Ccon) is by the twice of the vacuum result (6.16), which leads to c 3 c 3 : Qualitatively, this has an expected behavior that Cdec < Ccon for 2 and vise versa, though the phase transition temperature reads the gravity result 2 . Another interesting interpretation of the phase transition can be found from a property in the Liouville CFT. It is known that the (chiral) conformal dimension h of any local operators in Liouville theory has an upper bound (so called Seiberg bound [57]): h c 24 1 ; which implies the non-normalizability of the corresponding state. The operator which violates the bound should be regarded as a (normalizable) quantum state. In the large c limit, this bound (9.19) agrees with the condition that the conical de cit angle parameter a given by (3.16), takes a real value, for which the metric is that for con ned phase. When it is violated, a becomes imaginary and the metric changes into that for the decon ned phase (Einstein-Rosen bridge). This behavior seems to t very nicely with the gravity dual prediction and to proceed this further is an important future problem. 10 Conclusions In this work, we proposed an optimization procedure of Euclidean path-integrals for quantum states in CFTs. The optimization is described by a change of the background metric on the space where the path integral is performed. The optimization is completed by minimizing the complexity functional I for a given state j i, which is argued to be given by the Liouville action for 2D CFTs. The Liouville eld corresponds to the Weyl scaling of the background metric. Since this complexity is de ned from Euclidean path-integrals, we propose to call this \Path-Integral Complexity". Through calculations in various examples in 2d CFTs, we observed that optimized metrics for static quantum states coincide with those of time slices of their gravity duals. Thus we argued that our path-integral optimization o ers a continuous version of the tensor network interpretation of AdS3/CFT2 correspondence. Moreover, we also nd a simple formula to calculate the energy density for each quantum state. (9.18) (9.19) At the same time, we provide a eld theory framework for evaluating the computational complexity of any quantum states in CFTs. Note however, that in 2D CFTs, due to the conformal anomaly, the complexity functional (i.e. Liouville action) depends on the reference metric. Therefore, we proposed to use the di erence of the action, which is expected to give a relative di erence of complexity between the optimized network and the initial un-optimized one. We evaluated this quantity in several examples. In order to calculate the entanglement entropy, we studied an optimization of reduced density matrices. After the optimization we nd that the geometry is given by two copies of entanglement wedge and this nicely ts into the gravity dual. The entanglement entropy is nally reduced to the length of the boundary of the entanglement wedge and precisely reproduces the holographic entanglement entropy. Even though in most parts of this paper our analysis is devoted to static quantum states, we also discussed how our optimization of path-integrals can be applied to time-dependent backgrounds in 2D CFTs. Especially, we considered the time evolution of thermo- eld states which describe nite temperature states as a basic example. Our heuristic arguments show that an wormhole throat region linearly grows under the time evolution, which is consistent with holographic predictions. Moreover, we discussed how to interpret the con nement/decon nement phase transition in terms of tensor networks and our path-integral approach, whose details will be an interesting future problem. However, a precise connection between Liouville action and time-dependent states in 2D CFTs is still not clear and this was left as an important future problem. In the latter half of this paper, we investigated the application of our optimization method to CFTs in other dimensions than two. In one dimension, we nd that 1D version of Liouville action naturally arises from the conformal symmetry breaking e ect in NAdS2=CFT1 and this explains the emergence of extra dimension as in the AdS3=CFT2 case. In higher dimensions, we expect that the optimization procedure gets very complicated as we need to change not only the scaling mode but also other components of the metric as opposed to the 2D case. We focused on the Weyl scaling mode and proposed a complexity functional which looks like a higher dimensional version of Liouville action. However, notice the crucial di erence from the 2D case that the higher dimensional action does not depend on the reference metric. We con rmed that this reproduces the correct time slice metric for the vacuum states and correct holographic entanglement entropy when the subsystem is a round sphere. We pointed out an interesting direct connection to earlier proposal of holographic complexity [32, 33], which may suggest we should optimize with respect to all components of the metric. We also analyzed the spherically symmetric excited states and found that the optimized metric agrees with the AdS Schwarzschild one up to the rst order contribution of the mass parameter. We observed that for CFTs in any dimensions (including 2D), in order to take into account higher order back-reactions, we need to treat the Liouville mode in a quantum way. It is also possible to include higher derivative corrections without losing the above properties as we discussed in appendix (C). One advantage of this is that we can realize the higher dimensional conformal anomaly. However there is also a disadvantage that the action is no longer positive de nite and cannot be minimized but extremized. These issues on higher dimensional CFTs should deserve further studies. Last but not least, our approach based on the optimization of path-integrals is a modest but important step towards understanding of the basic mechanism of the AdS/CFT correspondence. For the future, apart from the questions we already mentioned above, there are many new directions for investigations like e.g. computation of correlation functions, generalizations to non-conformal eld theories and understanding a precise connection to AdS/CFT including 1=c expansions etc. Acknowledgments We thank Bartek Czech, Glen Evenbly, Rajesh Gopakumar, Kanato Goto, Yasuaki Hikida, HJEP1(207)9 Veronika Hubeny, Satoshi Iso, Esperanza Lopez, Alex Maloney, Rob Myers, Yu Nakayama, Tatsuma Nishioka, Yasunori Nomura, Tokiro Numasawa, Hirosi Ooguri, Alvaro-Veliz Osorio, Fernando Pastawski, Mukund Rangamani, Shinsei Ryu, Brian Swingle, Joerg Teschner, Erik Tonni, Tomonori Ugajin, Herman Verlinde, Guifre Vidal, Spenta Wadia, Alexander Westphal, Kazuya Yonekura for useful discussions and especially Rob Myers, Beni Yoshida and Alvaro Veliz-Osorio for comments on the draft. MM and KW are supported by JSPS fellowships. PC and TT are supported by the Simons Foundation through the \It from Qubit" collaboration. NK and TT are supported by JSPS Grant-in-Aid for Scienti c Research (A) No.16H02182. TT is also supported by World Premier International Research Center Initiative (WPI Initiative) from the Japan Ministry of Education, Culture, Sports, Science and Technology (MEXT). PC, MM, TT and KW thank very much the long term workshop \Quantum Information in String Theory and Many-body Systems" held at YITP, Kyoto where this work was initiated. NK would like to acknowledge the hospitality of the theory group at TIFR, Mumbai during an academic visit and for useful discussions during a seminar where parts of this work was presented. TT is very much grateful to the conference \Recent Developments in Fields, Strings, and Gravity" in Quantum Mathematics and Physics (QMAP) at UC Davis, the international symposium \Frontiers in Mathematical Physics" in Rikkyo U., the conference \String Theory: Past and Present (Spenta Fest)" in ICTS, Bangalore, the workshop \Tensor Networks for Quantum Field Theories II" at Perimeter Institute, the workshop \Entangle This: Tensor Networks and Gravity" at IFT, Madrid, the \Universitat Hamburg - Kyoto University Symposium"at DESY in Hamburg University, where the contents of this paper were presented. A Comparison with earlier Liouville/3D gravity relation Here we would like to compare our Liouville theory obtained from an optimization of Euclidean path-integrals with the earlier relation [61] between 3D gravity and Liouville theory. For simplicity we set the AdS radius to unit R = 1 below. We employ the ChernSimons description of 3D gravity [68], the two SL(2; R) gauge elds A and A correspond to the triad e and spin connection ! via A = ! + e and A = ! e. If we choose the solution: A = dr 2r rdx+ T++(x+) dxr+ ! dr 2r T dr 2r (x ) dxr rdx dr 2r ! ; (A.1) A = we obtain a series of solutions which describe gravitational waves on a pure AdS space (called Banados geometry) [85]: ds2 = becomes a BTZ black hole. Review of earlier argument r2 + T++T r 2 dx+dx T++(dx+)2 T (dx )2: (A.2) This satis es the equation of motion i T++ and T are functions of x+ and x respec= 0. If we set T++ and T to be constants, the geometry In the paper [61], motivated by the asymptotic behavior of BTZ black hole solutions, the following gauge choices are imposed: A = (G1) 1dG1 and A = (G2) 1dG2 (note that there is no bulk degrees of freedom in Chern-Simons gauge theories), where G1 and G2 are G2 = g2(x+; x ) p r 0 ! 0 p1 r In the above expression g1 and g2 are SL(2; R) matrices and describe the boundary degrees of freedom. Note that we can show Ar = Ar ; A 1 2r 0 0 1 2r a(3) a(+)=r ! ra( ) ; A = a(3) a(+)=r ! ra( ) ; where we de ned a = (g1) 1@+g1 and a = (g2) 1@ g2. Next we impose the chiral gauge choices a = a+ = 0. In this case the gauge theory for A and A becomes equivalent to the chiral and anti-chiral SL(2; R) WZW model, respectively [61]. Thus, by combining g1 and g2 as g = g1 1g2 we obtain a SL(2; R) WZW model. If we describe the SL(2; R) group element by (A.3) (A.4) (A.5) (A.6) g = Z 1 X ! 0 1 0 e 0 ! 1 0 Y 1 ! ; then the WZW model is described by the action SW ZW = dx+dx We can nd the solutions to the equation of motion for SW ZW such that (x+) e2 ; @ X = (x ) e2 ; Now we set (x+) = (x ) = 1 via a coordinate transformation. Note that the nal equation in (A.6) coincides with the equation of motion of Liouville theory and this provides the connection between the 3D gravity and Liouville theory. Finally, the gauge eld A and A for this solution read A = 1 2r rdx+ 1 2r ; A = 1 2r )2 dx r Thus we nd that the serious of the above solutions correspond to the Banados geometry (A.2) with the energy stress tensor in the Liouville CFT: T rdx 2r ! Now let us compare the above earlier argument to our metric ansatz (3.19), which ts naturally with our path-integral optimization argument. We work in Euclidean signature and consider the Euclidean version of Banados metric (A.2) given by dz2 ds2 = z2 + z2 + T (w)T (w)z 2 dwdw + T (w)dw2 + T (w)dw2: This metric is mapped into the standard Poincare AdS3 metric ds2 = d 2+dx2+d 2 2 via + ix = A(w) ix = B(w) 2A0(w)2B00(w) 2B0(w)2A00(w) 4z2A0(w)B0(w) + A00(w)B00(w) 4z2A0(w)B0(w) + A00(w)B00(w) ; ; 4z(A0(w)B0(w))3=2 4z2A0(w)B0(w) + A00(w)B00(w) : (A.7) (A.8) (A.10) Here A and B are holomorphic and anti holomorphic functions, respectively and the energy stress tensors are expresses as 3A00(w)2 2A0(w)A000(w) 4A0(w)2 T (w) = 3B00(w)2 2B0(w)B000(w) 4B0(w)2 : (A.9) On the other hand, the metric (3.19) with the general solution to the Liouville equation 2 = 4A0(y)B0(y) (A(y) + B(y))2 ; is mapped into the same Poincare AdS3 via the map: sinh A(y) + B(y) = 2p 2 + 2 A(y) B(y) = 2ix: (A.11) Note that the energy stress tensor for the Liouville eld (A.10) agrees with (A.9) as it should be. Therefore, by combining (A.8) and (A.11) we obtain a coordinate transformation between the Banados metric (z; w; w) and our metric ( ; y; y). Notice that the map is trivial near the AdS boundary such that y = w + O(z2) and y = w + O(z2) when z is very small. B Holographic complexity in the literature As mentioned in section 8.5, in this appendix we will consider both CV and CA-conjectures for the computation of holographic complexity and will explicitly determine them for some speci c cases like Poincare and global AdS in order to compare them with our set-up. In what follows we will summarize the behavior of holographic complexity in di erent situations and with both the CV and CA conjectures.17 17In this appendix, for the sake of convenience, we are using a convention where we put the AdS radius RAdS = 1. 1. Poincare AdSd+1: from [38], where the complexity action IWDW is evaluated with the null boundary term found in [35], we see CV conjecture: CA conjecture: CV = CA = IWDW = Vx (d IWDW 4Vx 1)GN d 1 log(d 1) d 1 with Vx =Volume of the (d 1)-dim spatial extent of CF T(d 1). global AdSd+1, we note that the leading divergence in CA behaves as CA d 1 16 2GN log p d 1 + where is the UV cut-o and d 1 being the volume of unit sphere S d 1. The subleading contributions include terms starting from 1= d 1, but strikingly enough the leading term has an additional and stronger logarithmic divergence. As explained in [30] this comes from one of the joint contributions but su ers from the ambiguity of a parametrization of the null boundary of the WDW patch, and is denoted by the free parameter . In [38], a prescription to resolve this ambiguity was proposed and following their construction we see CV conjecture: CA conjecture: d 1 Z cut d CV = CA = IWDW = GN IWDW 1 cos 4 d 1 tand 1 Z cut 0 dt0 tand t0 + 4 d 1 ln(d 1) + 1 d tand 1 cut with cut = =2 . For some explicit cases, we see that d=2 d=3 d=4 CV = CV = CV = IWDW = IWDW = 1 2 GN GN 3 GN 2 3 3 3 1 ; 8GN 4 log 3 4 log 3 8 log 2 3 2 ; (B.1) (B.2) (B.3) (B.4) As was mentioned in section 8.8, in this appendix we would like to explore the possibility of working with complexity functional Id such that it correctly produces the anomalies for even dimensional CFTs and hence can be considered as the partition function Id = log Zd for d-dimensional CFTs. Motivated by this, we analyze the AdS5=CFT4 case assuming the relation I4 = log Z4. We will con rm that the equation of motion for the new action again produces the hyperbolic time slice H4 and moreover its rst order perturbation agrees with the AdS5 Schwarzschild black hole solution. The possibility of having extra higher derivative terms in the action functional can be related to the trace anomaly in CF T4 I4 = Z d4xpg cW 2 aE4 + br r R I4 = Z d4xpg where W 2 is the square of Weyl tensor and E4(= R2 4R2 + R2) is the topological Euler density in 4-dimensions and = z; x1; x2; x3. Also, note the last term can be taken care of through a local counter term, see [86]. As mentioned before, we restricts ourselves here only to the metrics which are of the Weyl scaling type (2.5), g = e2 h at: with h at corresponding to Euclidean at space. It can be shown that the action I4, which correctly reproduces (C.1), becomes such that 3 = 6a 3b; 4 = 3b; 5 = 4a + 6b and g is the determinant of the metric g in (C.2). Next we will extremize the action (C.3) for the Poincare and global AdS5 respectively. C.1 Poincare AdS5 with higher derivatives For the time slice of Poincare AdS5 we consider the form of the metric as given in (8.10), and with that the action in (C.3) becomes (upto some total derivatives) I4 = Z dz 1e 2 4 where we de ned ~b = 3b + 2a and also assumed that is a function of z only. Extremizing the action in (C.4) we demand that the time slice of Poincare AdS5 is a solution to that. In other words, e = `=z extremizes the action if the following condition is satis ed 1`4 = 2` 2 6a: (C.1) (C.2) (C.3) (C.4) (C.5) For time slice of global AdS5 we again consider the metric as in (8.11) and the corresponding action functional (C.3), turns out to be I4 = Z 1e 4 + 2e 2 0 2 ~b 04 + 4~b 02 00 6b 002 3b 0 000 and we have also assumed that is a function of r only. It is straightforward to check that e = 2`=(1 r2) is a solution to the equation of obtained by extremizing (C.6), provided 1`4 = 2 6a; (C.6) (C.7) (C.8) (C.9) (D.1) (D.2) (D.3) which is same as (C.5) and hence we prove that the time slice of global AdS5 is indeed obtained by extremizing (C.6). C.3 Excitation in global AdS5 with higher derivatives Consider excited states in CFT4 dual to AdS5 Schwarzschild black holes (8.18). In Euclidean path integral analysis, we consider a spherically symmetry excitation and write its metric perturbation as e = 2 1 + M (r) : Working up to linear order in M , we substitute (C.8) in the equation of motion for that follows from the action in (C.6) and solve for (r). We use the restriction on the parameters as in (C.5) for the zeroth order solution. Also demanding that the solution be regular at r = 1 we check that (r) = (r) is indeed a allowed solution, where (r) is given in (8.24). Therefore we conclude that even in the presence of the higher derivative terms in (C.6), once the condition (C.5) is maintained the rst order perturbed metric of the AdS BH agrees with the extremization of I4. D Entanglement entropy and Liouville eld In our approach with the Liouville action and the metric ds2 = e2 (z) dz2 + dx2 ; we compute entanglement entropy as a line integral along the geodesic in the hyperbolic plane that is attached to the endpoints of the interval l and for a general geodesic parametrized by (z(t); x(t)), we have Sl = c Z 6 e (z)ds ds = px02 + z02dt: Moreover, it is important to note that all our \optimized" vacuum solutions not only satisfy the Liouville equation but also Notice also, that because we are interested in the regularized curve, we can just compute the entanglement entropy by (twice) the integral from the boundary to some distance in the bulk (turning point of the geodesic). That implies, using (D.4) Sl ' 3 c Z L~ c Z L~ e (z)dz = c ~ [ (z)]L This is clear for the vacuum solution log (z) and for L~ = l we obtain the usual result for the entropy. In general we can consider an arbitrary conformal transformation of the Liouville eld of the \vacuum" by chiral and anti chiral functions (w; w) ! (f (w); g(w)). Under such transformation, Liouville eld itself transforms as (f; g) = (w; w) log f 0(w)g0(w) : 1 2 This is still a solution of the Liouville equation with negative curvature (hyperbolic) and, in our approach, leads to a particular CFT state. Interestingly, we can then compute the entanglement entropy for such solution and after the line integral (D.2), we obtain Sl = log c f (w2))2 f 0(w1)f 0(w2) 2 c log g0(w1)g0(w2) 2 Curiously, from the general solution of the Liouville equation, we can now see that this result itself can also be written as a Liouville eld and satis es the Liouville equation but with positive curvature [76] and the space described by the end-points of the interval. It appears that these two Liouville elds can obtained form each other by simple analytic continuation (see also [56]) but the physical signi cance of this fact is far from obvious and remains to be elucidated. Nevertheless, given (D.8), we can still apply the rst law and compute the stress-tensor. Namely, if we set w2 = w1 + l and w2 = w1 + l, we can expand for small interval l Sl = c + c 6 log l l 2 6 c 12 ff (w1); w1g + 12 fg(w1); w1g c + O(l3) (D.9) where the expressions in the brackets are the Schwarzian derivatives ff (w); wg = f 000(w) f 0(w) 2 f 00(w) 2 f 0(w) : (D.4) (D.7) (D.10) On the other hand, we would like to extract this date from the original Liouville eld (hyperbolic) that enters in the optimization procedure. This can be done as follows: note that the entropy in the new geometry is computed by Sl = c Z 6 e (w;w)e 21 log(f0(w)g0(w))ds: If we then consider the exponent of the change in the Liouville eld, the stress tensor (Schwarzian derivative) can be read of from the simple equation 1 2 ff (w); wge 21 log(f0(w)g0(w)); and analogously for g. Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. (D.11) (D.12) Theor. 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Pawel Caputa, Nilay Kundu, Masamichi Miyaji, Tadashi Takayanagi, Kento Watanabe. Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT, Journal of High Energy Physics, 2017, 97, DOI: 10.1007/JHEP11(2017)097