#### 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.
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