YITP-21-25

IPMU21-0022

Holographic Path-Integral Optimization

Jan Borucha, Pawel Caputaa, Dongsheng Gea and Tadashi Takayanagib,c,d

aFaculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw,
Poland.
bYukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
cInamori Research Institute for Science,
620 Suiginya-cho, Shimogyo-ku, Kyoto 600-8411 Japan
dKavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU),
University of Tokyo, Kashiwa, Chiba 277-8582, Japan

In this work we elaborate on holographic description of the path-integral
optimization in conformal field theories (CFT) using Hartle-Hawking wave
functions in Anti-de Sitter spacetimes. We argue that the maximization of the
Hartle-Hawking wave function is equivalent to the path-integral optimization
procedure in CFT. In particular, we show that metrics that maximize gravity
wave functions computed in particular holographic geometries, precisely match
those derived in the path-integral optimization procedure for their dual CFT
states. The present work is a detailed version of [1] and contains many new
results such as analysis of excited states in various dimensions including JT
gravity, and a new way of estimating holographic path-integral complexity from
Hartle-Hawking wave functions. Finally, we generalize the analysis to
Lorentzian Anti-de Sitter and de Sitter geometries and use it to shed light on
path-integral optimization in Lorentzian CFTs.

###### Contents

  1. 1 Introduction
  2. 2 Review of the Path-Integral Optimization
    1. 2.1 Two-Dimensional CFTs
    2. 2.2 Higher-dimensional CFTs
  3. 3 Optimization from Hartle-Hawking Wave Functions
    1. 3.1 Definition of Hartle-Hawking Wave Functions
    2. 3.2 Evaluation of $\Psi^{(T)}_{HH}[\phi]$
    3. 3.3 Maximization
    4. 3.4 Holographic Path-Integral Complexity
    5. 3.5 Maximization and Gravity Hamiltonian Constraint
    6. 3.6 Boundary UV Limit
  4. 4 More Euclidean Examples
    1. 4.1 Global $AdS_{d+1}$
    2. 4.2 Excited states in 3d
      1. 4.2.1 Conical Singularities
      2. 4.2.2 Thermofield double
    3. 4.3 Planar Black Holes in Higher Dimensions
  5. 5 JT Gravity
  6. 6 Lorentzian Spacetimes
    1. 6.1 Semi-classical wave functions in $AdS_{d+1}$
      1. 6.1.1 Timelike $Q$
      2. 6.1.2 Spacelike $Q$
    2. 6.2 Semi-classical wave-functions in $dS_{d+1}$
      1. 6.2.1 Timelike $Q$
      2. 6.2.2 Spacelike $Q$
    3. 6.3 Towards Lorentzian Path-Integral Optimization
  7. 7 Tension as an Emergent Time
    1. 7.1 Euclidean $AdS$
    2. 7.2 Lorentzian $AdS$
    3. 7.3 Lorentzian $dS$
  8. 8 Conclusions
  9. A Lorentzian determinant
  10. B Inhomogeneous slices in $AdS_{3}$
    1. B.1 Action and solutions
    2. B.2 Uniformized metric
    3. B.3 Inhomogeneous equations of motion
  11. C Properties of the gravitational action with modified Hayward term
  12. D Maps in $AdS_{3}$
    1. D.1 Spinning BTZ

##  1 Introduction

Holography, or the so-called Anti-de Sitter space/Conformal Field Theory
(AdS/CFT) correspondence [2], is one of the most promising approaches to
quantum gravity. The main reason is that it relates quantum theories of
gravity on Anti-de Sitter spacetimes to a class of CFTs, where there is no
gravity present. Therefore, in AdS/CFT, gravitational spacetime is emergent
and, as it is manifest in the holographic entanglement entropy [3, 4, 5], it
is encoded in the geometry of quantum entanglement in CFT. Thus we expect that
the dynamics of quantum entanglement in CFTs is directly related to that of
quantum gravity.

Surprisingly, some tensor networks provide useful playgrounds where
holographic correspondence is realized in discretized lattice models. This
direction of research started from the pioneering work [6], which conjectured
the relation between AdS gravity and a particular tensor network for CFT
states called MERA (multi-scale entanglement renormalization ansatz) [7, 8,
9]. Among others, this tensor network setup beautifully explains the geometric
calculation of entanglement entropy in AdS/CFT [3, 4]. Refer to e.g. [10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] for further progress and
improvements in this direction.

Even though the connection between tensor networks and AdS geometry leads to
an interesting progress in qualitative understanding of the mechanism behind
AdS/CFT, it still remains among toy models for holography (however, refer to a
few recent attempts to improve that [25, 26, 27, 28, 29]). The main obstacle
of tensor network approaches to AdS/CFT is the manifest presence of lattice
regularizations, which prohibits us from understanding analytical results in
the continuum limit. On the other hand current approaches to continuous tensor
networks such as e.g. cMERA (continuous MERA) are limited to free theories
(see however [30, 31, 32, 33, 34] for recent progress beyond free CFTs).
Therefore, in order to draw conclusions about a genuine holography, we need to
develop a tensor network-like formalism applicable to continuous and
interacting, holographic CFTs.

The path-integral optimization, which was proposed in [35] and is reviewed in
the next section, is a promising formalism for this purpose. The construction
starts with a wave function representation of a CFT state defined by Feynman’s
path-integral on a flat Euclidean space with prescribed boundary condition for
quantum fields. Then, utilizing the conformal symmetry, we perform a Weyl re-
scaling of the Euclidean metric while keeping the boundary condition fixed, so
that the final quantum state prepared by the path-integral remains unchanged.
Intuitively, we can interpret path-integrals on different Weyl-rescaled
geometries as different continuous (non-unitary) tensor networks preparing the
same quantum state (see e.g. [18] for relation between tensor networks and
geometry of path-integrals).

The “lattice structure” of the path-integral network is specified by the
metric such that there is a unit lattice site per unit area. This way, we can
simplify such continuous tensor networks by a coarse-graining procedure
implemented by choosing an appropriate metric (Weyl factor). The maximization
of this coarse-graining is called the “path-integral optimization”.
Quantitatively, the path-integral optimization corresponds to minimizing the
“path-integral complexity”, a functional of the background metric, given by
e.g. the Liouville action in two-dimensional CFTs.111A close connection
between the computational complexity and the AdS/CFT already started from
[36]. More generally, “complexity” gives us a notion of how hard it is to
prepare a given quantum state and in this context we can regard it as a size
of the path-integral tensor network.

After this procedure it was found that optimal metrics for CFT path-integrals
are indeed hyperbolic ($AdS$) [35]. This is a strong evidence that the path-
integral optimization provides a successful, continuum version of the
AdS/tensor network correspondence. Nevertheless, the emergence of $AdS$
geometries from this approach has been mysterious and precise connection (i.e.
the gravity dual) to the AdS/CFT has not been well understood. Moreover, up to
date, we have been able to fully analyze only two-dimensional CFT examples
while higher-dimensional generalization of this procedure relied only on a
conjectured path-integral complexity action.

In this article, we present a more direct relation between the path-integral
optimization and the AdS/CFT in the light of the Hartle-Hawking wave function.
We will argue that the path-integral optimization corresponds to the
maximization of the Hartle-Hawking wave function in gravity on a Euclidean AdS
background. Our construction will be based on generalized Hartle-Hawking wave
functions that require imposing a specific boundary condition on the
asymptotic boundary of AdS, dual to a specific CFT state. This maximization
naturally arises in the saddle point approximation when we calculate physical
observables using the Hartle-Hawking wave function. From this perspective, we
can also understand how to generalize the path-integral optimization to a
quantum regime beyond the semi-classical approximation.

The arguments below will not only explain the origin of the optimization
procedure from the viewpoint of the AdS/CFT but also shed new light on several
interesting directions in the path-integral optimization program. Firstly, the
Hartle-Hawking wave function method works well in higher dimensions and we use
it to extract information about the “correct” path-integral optimization in
any dimensions. Secondly, we extend the analysis to Lorentzian signature
including both Anti de-Sitter and de-Sitter spacetimes. This is a starting
point for exploring holography in general spacetimes using gravitational path-
integrals. Finally, in Euclidean as well as Lorentzian setups, we propose a
natural definition of holographic counterpart of the path-integral complexity
in terms of gravity action with the Hayward term. Below, we compute it in all
examples and discuss some of its interesting properties.

This paper is organized as follows. In section 2, we give a review of path-
integral optimization in two as well as in higher dimensions. In section 3, we
provide a detailed derivation of path-integral optimization geometry from
Hartle-Hawking wave function in $AdS_{d+1}$ using Poincare coordinates, define
the Euclidean holographic path-integral complexity and discuss a few important
points of our proposal. In section 4 we analyze more examples in Euclidean
$AdS$ geometries and in section 5 we test our proposal in the context of JT
gravity. In section 6 we study Lorentzian $AdS_{d+1}$ and $dS_{d+1}$
geometries where we derive Hartle-Hawking wave functions and use them as a
guidance for Lorentzian path-integral optimization in CFT. In section 7, we
discuss foliations of $AdS$ and $dS$ spacetimes by the slices that maximize
Hartle-Hawking wave functions and interpret their tension as an emergent time.
Finally, in section 8 we conclude and leave several technical details into
appendices.

##  2 Review of the Path-Integral Optimization

We start with a brief review of the path-integral optimization [35] that
provides a framework for constructing continuous tensor networks for CFT wave
functions. We will discuss only some of its important properties and new
insights that will be particularly relevant for the holographic interpretation
in later sections (see also [37, 38, 39, 25] for more details of the
formalism). We will separately discuss CFTs in two and in higher dimensions.

###  2.1 Two-Dimensional CFTs

The main idea of the path-integral optimization is as follows. We start with a
two-dimensional CFTs on the Euclidean plane $\mathbb{R}^{2}$ with coordinates
$(\tau,x)$ and denote all the fields in the CFT by $\Phi(\tau,x)$. The ground
state wave function $\Psi_{CFT}[\Phi(x)]$ of the CFT, that is formally a
functional of the boundary conditions imposed for the CFT fields at the time
slice $\tau=0$: $\Phi(x)\equiv\Phi(x,\tau=0)$, is defined by the Euclidean
path-integral as

$\displaystyle\Psi_{CFT}[\Phi(x)]=\int\prod_{-\infty<\tau\leq
0,x}[D\tilde{\Phi}(\tau,x)]e^{-S_{CFT}[\tilde{\Phi}]}\delta(\tilde{\Phi}(0,x)-\Phi(x)),$
(2.1)

where $S_{CFT}$ is the Euclidean action of the 2d CFT under consideration.
This exact definition of the wave functions is usually only a formal
expression and can be rarely used to derive the state beyond free CFTs. On the
other hand, this path-integral representation contains all the information
about the state, including $AdS$ dual geometry for holographic CFTs, structure
of entanglement, as well as data required to estimate state’s complexity. The
main problem is to extract this information directly from the path-integral.
The path-integral optimization is then an attempt to solve this problem and,
in particular, to extract information about holographic dual geometry and
complexity from the path-integral representation of CFT states.

The key step in the path-integral optimization is to deform the metric on the
two-dimensional Euclidean plane on which we perform the path-integral in the
following way

$\displaystyle ds^{2}=e^{2\phi(\tau,x)}(d\tau^{2}+dx^{2}).$ (2.2)

Then, the original path-integral computing $\Psi_{CFT}[\Phi(x)]$ is evaluated
on a flat $\mathbb{R}^{2}$ metric with $e^{2\phi(\tau,x)}=1/\epsilon^{2}$,
where $\epsilon$ is a UV regularization scale (i.e. lattice constant)
introduced once we discretize path-integrals of quantum fields into those on a
lattice. On the other hand, general Weyl factors $e^{2\phi(\tau,x)}$ can be
interpreted as a position-dependent choices of the lattice discretization. For
this interpretation we need to impose an extra rule, namely, that there is a
single lattice site per unit area. This way we can relate a given metric to a
(non-unitary) tensor network, which is a discretization of Euclidean path-
integral, and volume of the above geometry to the number of tensors.

Next, let us write the wave function obtained from the path-integral on the
curved space with metric (2.2) as $\Psi_{CFT}^{\phi}[\Phi(x)]$. If we impose
the boundary condition

$e^{2\phi(0,x)}=\frac{1}{\epsilon^{2}}\equiv e^{2\phi_{0}},$ (2.3)

then this wave function is proportional to the original one
$\Psi_{CFT}[\Phi(x)](=\Psi_{CFT}^{\phi_{0}}[\Phi(x)])$ computed by path-
integral on flat space $\mathbb{R}^{2}$. The reason for this is that CFTs are
invariant under Weyl rescalings up to the Weyl anomaly, so we have

$\displaystyle\Psi_{CFT}^{\phi}[\Phi(x)]=e^{S^{(e)}_{L}[\phi]-S^{(e)}_{L}[\phi_{0}]}\cdot\Psi_{CFT}^{\phi_{0}}[\Phi(x)],$
(2.4)

where the functional in the exponent $S^{(e)}_{L}[\phi]$ is the Liouville
action on a Euclidean flat space [40]

$\displaystyle
S^{(e)}_{L}[\phi]=\frac{c}{24\pi}\int^{\infty}_{-\infty}\\!dx\\!\int^{0}_{-\infty}\\!d\tau\\!\left[(\partial_{x}\phi)^{2}+(\partial_{\tau}\phi)^{2}+\mu
e^{2\phi}\right].~{}~{}$ (2.5)

Notice that Liouville action universally depends on the number of degrees of
freedom denoted by the central charge $c$ of the 2d CFT that appears as a pre-
factor. From the relation (2.4), we see that the quantum state still remains
the same CFT vacuum $|0\rangle$ for any choice of the metric (2.2) as long as
the boundary condition (2.3) at $\tau=0$ is satisfied. Moreover, the
assumption about the discretization, i.e. that one unit area of the metric
(2.2) corresponds to a single lattice site, fixes the value of the
“cosmological constant” $\mu$ to $\mu=1$ as in [35]. Nevertheless, we will
keep this constant parameter for later convenience and will also provide an
interpretation for $\mu\neq 1$ in the gravity dual setup.

One way to derive the Liouville action is to examine the expectation value of
the trace of energy stress tensor in CFT, which is given by (we write
$i=\tau,x$)

$\displaystyle\langle
T^{i}_{i}\rangle=-\frac{c}{12\pi}\partial_{i}\partial^{i}\phi+\mu_{0}e^{2\phi}.$
(2.6)

The first term is proportional to the Ricci scalar curvature and describes the
Weyl anomaly that is universal for 2d CFTs. The second term simply comes from
the UV divergence (see e.g. App. A). Since the expectation value of the trace
$T^{i}_{i}$ is the derivative of the partition function with respect to the
Weyl scaling factor $\phi$, we can reproduce (2.5) with setting
$\mu=12\pi\mu_{0}/c$. Note that here, we are interested in the bare action
(before renormalization) such as the Liouville function $S^{(e)}_{L}[\phi]$.
Indeed, the Liouville potential term gets quadratically divergent as
$e^{2\phi}\sim\epsilon^{-2}$. Nevertheless, we keep it and, as we will
momentarily see, it plays an important role in the path-integral optimization
story. Finally, since the Liouville potential term in (2.5) arises from the UV
divergence, it should dominate over the kinetic term when we take the UV limit
$\epsilon\to 0$. This condition is satisfied when

$\displaystyle(\partial_{i}\phi)^{2}\ll e^{2\phi}\ \ \ (i=x,\tau).$ (2.7)

We will come back to this limit in the gravitational construction in later
sections.

Next, we describe how the optimal metrics for Euclidean path-integrals are
chosen. The basic strategy of path-integral optimization is to coarse-grain
the discretization of path-integral as much as possible, such that e.g. the
cost of numerical computation of path-integral is minimal, while keeping the
final boundary condition that specifies the correct wave function fixed. For
this we need a measure of computational cost of a path-integration i.e. an
estimate of its computational complexity. In the original path-integral
optimization, the actual Liouville action $S^{(e)}_{L}[\phi]$ (with $\mu=1$)
was proposed as the complexity functional [35], which should be minimized in
the optimization procedure. This was motivated by the minimization of the
overall factor of the wave function, which is proportional to
$e^{S^{(e)}_{L}[\phi]}$ as in (2.4). Although, as usually, the overall factor
does not affect physical quantities in quantum mechanics, this quantity can be
interpreted as a measure of the number of repetitions of numerical integrals
when we discretise path-integrals into lattice calculations.222Remember that
our choice of lattice regularization is specified by the metric (2.2).

It was shown [35] that such optimization (minimization) procedure properly
selects the most efficient discretization of path-integral which produces the
correct vacuum state. More precisely, the minimization of $S^{(e)}_{L}[\phi]$
is achieved by solving the Liouville equation

$(\partial_{x}^{2}+\partial^{2}_{\tau})\phi=\mu e^{2\phi},$ (2.8)

which is equivalent to the condition for constant negative (hyperbolic) Ricci
scalar of the 2-dimensional metric (2.2)

$R=-2\mu.$ (2.9)

Recall that this conclusion is a consequence of the fact that we work with the
bare Liouville action with the potential term ($\mu\neq 0$).
The most general solution of this equation is given in terms of two functions:
$A(w)$ and $B(\bar{w})$, of $w=\tau+ix$ and its complex conjugate
$\bar{w}=\tau-ix$ as

$e^{2\phi}=\frac{4A^{\prime}(w)B^{\prime}(\bar{w})}{\mu(1-A(w)B(\bar{w}))^{2}}.$
(2.10)

For example, for the path-integral preparing the vacuum state of the CFT on a
line, with the boundary condition (2.3), the minimization leads to the
solution

$\displaystyle e^{2\phi(\tau,x)}=\frac{1}{(\sqrt{\mu}\tau-\epsilon)^{2}},\ \ \
(-\infty<\tau\leq 0).$ (2.11)

We regard the solution with $\mu=1$ as the genuine optimized metric, which
minimized the original Liouville action (remember we set $\mu=1$ for our
complexity functional [35]). In addition, in this work, we propose to
interpret solutions with $\mu<1$, for which the metric gets larger than the
original with $\mu=1$, as only partially optimized where the UV cut-off length
scale is taken to be larger. On the other hand, solutions with $\mu>1$, for
which the metric is smaller than the original with $\mu=1$, intuitively,
correspond to fine-graining the cut-off scale more than the current lattice
spacing (i.e. $\epsilon$). We would like to suggest that these solutions with
$\mu>1$ may not be allowed. One of the arguments is that, as $\mu$ gets
larger, the coarse-graining gets too fast (i.e. the curvature $R$ is “too
big”) to maintain the Weyl invariance which is only guaranteed in the
continuum limit and which leads to our basic equivalence relation (2.4).

The observation that (2.11) coincides with the time slice of a three-
dimensional AdS (AdS3) [35], offered the main implication that the path-
integral optimization, as a version of continuous tensor-network, can explain
the emergence of the AdS geometry purely from CFT. Indeed, the discretized
path-integral can be regarded as a (non-unitary) tensor network and its
relation to AdS provides a path-integral version of the conjectured
interpretation of AdS/CFT as a tensor network. Moreover, as a product of the
above procedure, the Liouville action $S^{(e)}_{L}[\phi]$ (the “path-integral
complexity” [35]) became the first workable definition of computational
complexity in continuous, interacting CFTs, that could be computed in
genuinely holographic models [36] (refer also to [41, 42, 43, 44, 45] for
further remarkable connections between path-integral and circuit
complexities). This method has been further generalized to various CFTs and
QFTs (see e.g. [38, 46, 39, 47, 48, 49, 50, 51, 52]) and also used to
successfully compute entanglement of purification in 2d CFTs [53], which was
recently verified numerically in [54].

Nevertheless, despite this emergence of hyperbolic geometries, a direct
relation between the path-integral optimization and AdS/CFT has remained an
open problem. Moreover, it has not been clear how to promote the classical
Liouville theory which appeared in the path-integral optimization to the
quantum regime. The need for that was found in [35] where, in order reproduce
the correct gravity metric dual to a primary state with $1/c$ corrections, one
would have to replace a classical Liouville result with a quantum one. The
main problem with “quantum” path-integral optimization is that integral
$\int[D\phi]e^{S^{(e)}_{L}[\phi]}$, which is expected to arise from (2.4) in
the quantized version, does not make sense as it is not bounded from below.
Indeed, the standard quantum Liouville is defined by the path-integral
$\int[D\phi]e^{-S^{(e)}_{L}[\phi]}$, but it has not been clear how such well-
defined expression arises from the above optimization. In other words, we
cannot get the minimization as a saddle point approximation of path-integrals
and its derivation within the AdS/CFT may appear too challenging. As we will
see, the new approach [1] further developed in this paper will naturally
resolve these two problems in the light of Hartle-Hawking wave functions.

Last but not least, we would like to point an important subtlety that, in the
solution (2.11), $(\partial_{i}\phi)^{2}$ and $e^{2\phi}$ are of the same
order, so the UV criterion (2.7) is not satisfied. This actually implies that
the path-integral optimization using the Liouville action is only qualitative.
Indeed, its derivation (2.6) involves the continuum limit and, generally, we
expect finite cut-off corrections to the Liouville action. As we will see
later, the gravity approach will naturally come with such corrections.

###  2.2 Higher-dimensional CFTs

In higher dimensions, the status of path-integral optimization is still not
entirely clear. At least in even dimensions, it is tempting to follow the
intuition from the Liouville action that, as discussed above, can be defined
as integrated anomaly. For example in four dimensions, this would lead us to
the so-called $Q$-curvature actions (see e.g. [55]) that naturally generalises
the Polyakov action from 2d. However, such actions, even though very
interesting, are genuinely higher order in derivatives (or the curvature) and
less intuitive from the tensor-network perspective. Instead, in [35], we
proposed a phenomenological generalisation to an “effective” path-integral
complexity action for higher-dimensional CFTs in the form

$I[\phi,\hat{g}]=N\int_{\Sigma}d^{d}x\sqrt{\hat{g}}\left[e^{(d-2)\phi}\hat{g}^{ab}\partial_{a}\phi\partial_{b}\phi+\frac{e^{(d-2)\phi}}{(d-1)(d-2)}R_{\hat{g}}+\mu
e^{d\phi}\right],$ (2.12)

where $R_{\hat{g}}$ is the Ricci scalar for the reference metric $\hat{g}$ and
the normalization

$N=\frac{d-1}{16\pi G_{N}},$ (2.13)

was fixed in order to reproduce entanglement entropy for spherical regions
[35].
We have argued that path-integral complexity computed by (2.12), as well as
improved Liouville complexity333with the volume term subtracted. are good
relative measures of complexity between two continuous tensor networks
described by metrics $g_{1}$ and $g_{2}$. Moreover, both of these complexity
actions satisfy the so-called “co-cycle conditions” in terms of three metrics

$I[g_{1},g_{2}]=-I[g_{2},g_{1}],\qquad
I[g_{1},g_{2}]+I[g_{2},g_{3}]=I[g_{1},g_{3}].$ (2.14)

Interestingly, from the above properties it is clear that $I[g_{1},g_{2}]$ can
be negative (unlike measures of complexity based on e.g. distance in the
Hilbert space).
One more comment we should make at this point is that in (2.12) we generalized
the action from [35] by including coupling $\mu$ in front of the potential. To
get a better intuition regarding this action and the meaning of $\mu$, let us
now rewrite it in a more transparent way. Namely, consider a general
$d$-dimensional metric of the form

$ds^{2}=e^{2\phi}\hat{g}_{ab}dx^{a}dx^{b},$ (2.15)

and its Ricci scalar curvature given by

$R^{(d)}=e^{-2\phi}(R_{\hat{g}}-2(d-1)\hat{\Box}\phi-(d-2)(d-1)\hat{g}^{ab}\partial_{a}\phi\partial_{b}\phi).$
(2.16)

We can check that, after integrating $\hat{\Box}\phi$ by parts, (2.12) can be
actually rewritten as

$I[\phi,\hat{g}]=\frac{1}{16\pi G^{(d)}_{N}(d-2)}\int
d^{d}x\sqrt{g}\left[R^{(d)}-2\Lambda^{(d)}\mu\right],$ (2.17)

where

$\Lambda^{(d)}=-\frac{(d-1)(d-2)}{2}.$ (2.18)

This is obviously the Einstein-Hilbert action with negative cosmological
constant $\Lambda^{(d)}\mu$ in $d$-dimensions, i.e. the same number of
dimensions as the CFT. Comparing with the standard gravity action, $\mu$ plays
the role of the $AdS$ radius $1/l^{2}$ that sits in the denominator of the
cosmological constant. Let us stress that this resemblance is only at the
formal level and our effective complexity action, by generalizing the
Liouville action, was aimed to describe the geometry of the continuous tensor
networks from path-integrals. Derivation of this action from the optimization
procedure in CFTs is still an open problem. Nevertheless, the holographic
approach [1], in principle, provides the full answer to the correct path-
integral complexity functional (including finite cut-off corrections) in
higher dimensions and will reproduce and justify (2.12) in the UV limit.

Setting aside the CFT derivation for the moment, this form of the action has
obvious consequences for optimal path-integral geometries. Clearly, by varying
the action with respect to $\phi$ we end up with the trace of Einstein’s
equations in $d$-dimensions

$\left(R^{(d)}_{ab}-\frac{1}{2}R^{(d)}g_{ab}+\Lambda^{(d)}\mu
g_{ab}\right)g^{ab}\delta\phi=0,$ (2.19)

which is of course equivalent to the constant negative Ricci curvature
condition

$R^{(d)}=-d(d-1)\mu=2\Lambda^{(d+1)}\mu.$ (2.20)

This provides a generalization of the optimization constraint for the path-
integral geometries (Liouville equation in 2d) to higher dimensions.
It is also interesting to see that the negative cosmological constant that
appears on the right hand side of this equation is $d+1$-dimensional. As we
will see later, this fact naturally comes from the Hamiltonian constraint in
the gravity proposal. Finally, the parameter $\mu$ that fixes the value of the
curvature will also find a holographic counterpart in terms of tension $T$ in
the Hartle-Hawking construction. After this review, we can proceed with the
gravity dual description.

##  3 Optimization from Hartle-Hawking Wave Functions

In this section we begin by explaining the construction reported in a short
article [1], then we solve the vacuum example in detail and discuss various
subtleties and properties of this new proposal. Our goal is to understand the
path-integral optimization procedure, reviewed above, in holographic CFTs that
are equivalently described by gravity in the one-dimension higher Anti-de
Sitter spacetime [2]. This correspondence, or the AdS/CFT for short, usually
involves super-conformal field theories and string theory or supergravity in
AdS, however, in this work, we will limit the discussion to large-$N$
conformal field theories and the pure gravity sector.

###  3.1 Definition of Hartle-Hawking Wave Functions

In what follows, we proceed with the analysis of the vacuum of a CFT on
$\mathbb{R}^{d}$ that, on the gravity side, corresponds to Euclidean Poincare
$AdS_{d+1}$ geometry444In Euclidean examples we will set the $AdS$ radius to
$l=1$.

$\displaystyle
ds^{2}=\frac{dz^{2}+d\tau^{2}+\sum_{i=1}^{d-1}dx_{i}^{2}}{z^{2}}.$ (3.1)

In this geometry, we focus on the shaded region $M$ in Fig.1 between surfaces
$\Sigma$ and $Q$ in the bulk of $AdS$. In this region, we will be interested
in a gravitational “Hartle-Hawking” wave function [56]555See e.g. [57, 58] for
previous important applications of gravity wave functions in the AdS/CFT
correspondence., denoted by $\Psi_{HH}[g_{ab}]$, which is a functional of the
induced metric $g_{ab}$ on $Q$. More precisely, we consider a path-integral of
Euclidean gravity from the asymptotic boundary $\Sigma$ of AdS, described by
the cut-off surface $z=\epsilon$ with $\tau<0$, up to the surface $Q$,
specified by $z=f(\tau)$ that extends from $z=\epsilon$ and $\tau=0$ (where it
meets $\Sigma$ at an angle $\theta_{0}$) towards the bulk as depicted in
Fig.1.

Figure 1: The on-shell action in the shaded region $M$ computes the Hartle-
Hawking wave function, related to the path-integral optimization for
holographic CFTs.

Note that, for simplicity, we focus on translationally-invariant surfaces $Q$.
Moreover, we assume the following form of the metric on $Q$

$\displaystyle ds^{2}=e^{2\phi(w)}(dw^{2}+\sum_{i=1}^{d-1}dx_{i}^{2}),$ (3.2)

where $-\infty<w<0$ (that is a function of $\tau$) renders the induced metric
on $Q$ conformally flat.666Note that the above construction assumes that a
gravity solution which calculates the Hartle-Hawking wave function, is given
by a sub-region in the Poincare AdS. The metric (3.2) obtained in this way
covers all possible metrics on $Q$ for $d=2$ with the condition (2.3) because
all solutions to the vacuum Einstein equation are locally equivalent to the
Poincare AdS3. However, for $d>2$, the above construction covers only a sub-
class of the metrics on the $d$-dimensional surface $Q$. Nevertheless, as we
will see below, this ansatz includes a large class of metrics of our direct
interest. More generally, we can extend target metrics to completely general
ones by directly solving the Einstein’s equations. In this setup, we now
define the Hartle-Hawking wave function $\Psi_{HH}[\phi]$ as follows

$\displaystyle\Psi_{HH}[\phi]=\int[Dg_{\mu\nu}]e^{-I_{G}[g]}\delta(g_{ab}|_{Q}-e^{2\phi}\delta_{ab}),$
(3.3)

where $I_{G}[g]$ is the $d+1$-dimensional Einstein-Hilbert action with
negative cosmological constant $\Lambda$ and the Gibbons-Hawking boundary term

$\displaystyle
I_{G}\\!=\\!-\frac{1}{2\kappa^{2}}\\!\int_{M}\\!\sqrt{g}(R\\!-\\!2\Lambda)\\!-\\!\frac{1}{\kappa^{2}}\\!\int_{Q\cup\Sigma}\\!\sqrt{h}K,~{}$
(3.4)

where we also denoted $\kappa^{2}=8\pi G_{N}$.777One of the defining features
of the Hartle-Hawking wave functions in canonical quantum gravity is the fact
that they solve the Wheeler-de Witt equation, i.e. the quantum Hamiltonian
constraint. In this work we will be mostly concerned with the semi-classical
regime and will not worry about subtleties of the quantum definition of (3.3)
such as e.g. sum over topologies etc. More generally, as we will see below,
one should also carefully add a Hayward term in regions with non-smooth
boundaries like our $M$ above.
Cautious reader will notice that we also (implicitly) imposed an initial
condition (i.e. fixed the metric) on $\Sigma$ that corresponds to the pure AdS
dual to the CFT vacuum. In principle, we can consider more general boundary
conditions on $\Sigma$ as well as $\Psi_{HH}[\phi(w,x)]$ corresponding to
excited states of a CFT (see following sections).

Given this definition, we are now ready to make the connection with the CFT
path-integral optimization. Namely, the key step of the proposal [1] is to
identify the metric (3.2) on $Q$ as the “holographic dual” of the metric (2.2)
in the path-integral optimization for holographic CFTs. Moreover, we propose
that the gravity dual of the optimization procedure corresponds to the
maximization of the Hartle-Hawking wave function $\Psi_{HH}[\phi]$ with
respect to $\phi$ (more generally, with respect to the metric on $Q$). This
maximization arises naturally when we evaluate e.g. generic gravity
correlation functions using the Hartle-Hawking wave functional

$\displaystyle\langle
O_{1}O_{2}\ldots\rangle=\int[D\phi]|\Psi_{HH}[\phi]|^{2}O_{1}O_{2}\ldots,$
(3.5)

by applying the saddle point approximation.888In fact, in AdS/CFT, this
computation could also naturally appear in the correlator of local bulk
operators where operators $O_{i}$ would be related to the CFT ones by the
standard HKLL prescription. The main advantage of this dual description is
that the maximization of Hartle-Hawking wave function works well even in the
presence of quantum fluctuations of $\phi$, as opposed to the minimization of
$e^{S^{(e)}_{L}[\phi]}$ as discussed before. Indeed, we will compute
explicitly in the next section that $\Psi_{HH}[\phi]$ is proportional to
$e^{-S^{(e)}_{L}[\phi]}$ up to finite cut-off corrections.

As we already mentioned, in this work, we will be able to discuss a
generalization of the path-integral optimization based on the Liouville action
(and its higher dimensional version (2.12)) with a non-trivial coefficient
$\mu$ of the exponential potential. For this new construction, on the gravity
side, it will be useful to introduce a tension term on the surface $Q$, which
also plays an important role in the AdS/BCFT [59] (we assume $T<0$ below)

$\displaystyle I_{T}[h]=\frac{T}{\kappa^{2}}\int_{Q}\sqrt{h}.$ (3.6)

This allows us to define a one-parameter family of “deformed” Hartle-Hawking
wave functions in the following way

$\displaystyle\Psi^{(T)}_{HH}[\phi]\\!=\\!\int\\![Dg_{\mu\nu}\\!]e^{-I_{G}[g]-I_{T}[e^{2\phi}]}\,\delta(g_{ab}|_{Q}\\!-\\!e^{2\phi}\delta_{ab}).~{}~{}$
(3.7)

The standard Hartle-Hawking wave functions are obtained by setting $T=0$. We
can regard $T$ as a chemical potential or the Legendre transformation for the
area of $Q$, acting on Hartle-Hawking wave functions. As we will see, the
tension term in gravity will be responsible for shifting the value of the
cosmological constant $\mu$ in the Liouville action.999Later, we will set
$T=0$ while evaluating the holographic path-integral complexity. Adding $T$
should then be thought of as just a trick for selecting partially optimized
configurations as solutions to the problem of maximization of the Hartle-
Hawking wave functions. More importantly, since the maximum of
$\Psi^{(T)}_{HH}[\phi]$ will correspond to a family of surfaces $Q$ in AdS
parameterized by the tension $T$, we will observe that $T$ plays the role of
an emergent holographic time.

###  3.2 Evaluation of $\Psi^{(T)}_{HH}[\phi]$

Computation of the wave function (3.7) in full quantum gravity is clearly
beyond the present technology but we can nevertheless make progress using
saddle point analysis. In the remaining discussions we will focus on
evaluating $\Psi^{(T)}_{HH}[\phi]$ using the semi-classical approximation i.e.
as an on-shell gravity action with tension term added

$\displaystyle\Psi^{(T)}_{HH}[\phi]\simeq e^{-I_{G}[g]-I_{T}[h]}|_{\mbox{on-
shell}},$ (3.8)

in region $M$. In this limit, the evaluation of the Hartle-Hawking wave
function boils down to a standard computation from the AdS/BCFT, of a
classical gravity action in a portion of $AdS_{d+1}$ up to an end-of-the-world
brane $Q$ with tension $T$. However, the important technical assumption in
this computation will be the fact that the (fictitious) “brane” that describes
surface $Q$ does not back-react on the bulk geometry. This allows us to
evaluate the wave function using the Poincare AdS metric (3.1) in a tractable
way. This assumption is sufficient to find solutions whose metrics take the
conformally flat form (3.2). For more general metrics we may need to solve the
back-reactions in the presence of the brane, which is also a well-defined
problem.

Let us now compute the gravity action step by step. First, since we are using
the on-shell solution of Einstein’s equations (3.1), we have $R-2\Lambda=-2d$
and the bulk action contributes

$\displaystyle-\frac{1}{2\kappa^{2}}\int_{M}d^{d+1}x\sqrt{g}\left(R-2\Lambda\right)$
$\displaystyle=$ $\displaystyle\frac{d}{\kappa^{2}}\int d^{d-1}x\int
d\tau\int^{f(\tau)}_{\epsilon}\frac{dz}{z^{d+1}},$ (3.9) $\displaystyle=$
$\displaystyle\frac{V_{x}}{\kappa^{2}}\left(\frac{L_{\tau}}{\epsilon^{d}}-\int\frac{d\tau}{f(\tau)^{d}}\right),$

where we denoted the divergent volumes and the length of $\tau$ direction as

$V_{x}\equiv\int d^{d-1}x,\qquad L_{\tau}\equiv\int^{0}_{-\infty}d\tau.$
(3.10)

Clearly, the bulk action not only gives rise to a divergent part from the UV
limit but also contributes a part with $f(\tau)$ to the effective action that
will describe metric on $Q$. Moreover, notice that the bulk action vanishes
when we take $Q\to\Sigma$ i.e. $f(\tau)=\epsilon$.
Next, from the UV boundary surface $\Sigma:z=\epsilon$ with normal
$n^{\mu}=-z\delta^{\mu z}$, we have $K|_{\Sigma}=d,$ and the contribution from
the Gibbons-Hawking term

$-\frac{1}{\kappa^{2}}\int_{\Sigma}d^{d}x\sqrt{h}K|_{\Sigma}=-\frac{d}{\kappa^{2}}\frac{V_{x}L_{\tau}}{\epsilon^{d}}.$
(3.11)

Finally, we carefully evaluate the contribution from $Q$, specified by
function $z=f(\tau)$. Firstly, the induced metric on $Q$ is given by

$\displaystyle ds^{2}=\frac{(1+f^{\prime 2})d\tau^{2}+dx^{2}_{i}}{f^{2}}\equiv
e^{2\phi(w)}\left(dw^{2}+dx^{2}_{i}\right),$ (3.12)

where in the second step we defined

$e^{2\phi}\equiv\frac{1}{f^{2}},$ (3.13)

and, to make the metric diagonal, we introduced “conformal” coordinate $w$
with the range $-\infty<w<0$, satisfying

$\frac{dw}{d\tau}=\sqrt{1+f^{\prime 2}}=\left(1-\dot{f}^{2}\right)^{-1/2}.$
(3.14)

We also introduced a short-hand notation
$f^{\prime}=f^{\prime}(\tau)=\partial_{\tau}f(\tau)$,
$\dot{f}=\dot{f}(w)=\partial_{w}f(w)$ and the relation
$f^{\prime}(\tau)=w^{\prime}(\tau)\partial_{w}f(w)=w^{\prime}(\tau)\dot{f}$
was used to solve for the second relation. In order to rewrite formulas in
terms of $w$ coordinate, we will later use iteratively
$f^{\prime\prime}(\tau)=w^{\prime}(\tau)\partial_{w}f^{\prime}(\tau)$ and from
the definition

$\dot{f}=-e^{-\phi(w)}\dot{\phi}(w),\qquad\ddot{f}=-e^{-\phi(w)}\left(\ddot{\phi}(w)-\dot{\phi}(w)^{2}\right).$
(3.15)

Secondly, we can compute extrinsic curvature and its trace on $Q$. With
coordinates on $Q$: $y^{i}=\\{\tau,x^{a}\\}$, and projectors

$e^{\alpha}_{i}\equiv\frac{\partial x^{\alpha}}{\partial y^{i}},\qquad
e^{\alpha}_{\tau}=\\{1,\vec{0},f^{\prime}(\tau)\\},\qquad
e^{\alpha}_{x^{a}}=\delta^{\alpha}_{a},$ (3.16)

as well as the outward-pointing normal vector

$n^{\mu}=\frac{z}{\sqrt{1+f^{\prime}(\tau)^{2}}}\left\\{-f^{\prime}(\tau),\vec{0},1\right\\},$
(3.17)

we can compute the components of the extrinsic curvature

$K_{ij}=e^{\alpha}_{i}e^{\beta}_{j}\nabla_{\alpha}n_{\beta}.$ (3.18)

Expressed in terms of $f(\tau)$ they read

$\displaystyle K_{\tau\tau}$ $\displaystyle=$
$\displaystyle-\frac{1+f^{\prime}(\tau)^{2}+f(\tau)f^{\prime\prime}(\tau)}{f(\tau)^{2}\sqrt{1+f^{\prime}(\tau)^{2}}},\qquad
K_{\tau a}=0,$ $\displaystyle K_{ab}$ $\displaystyle=$
$\displaystyle-\frac{1}{f(\tau)^{2}\sqrt{1+f^{\prime}(\tau)^{2}}}\delta_{ab},$
(3.19)

and after taking the trace we obtain

$K|_{Q}=-\frac{f(\tau)f^{\prime\prime}(\tau)+d(1+f^{\prime}(\tau)^{2})}{(1+f^{\prime}(\tau)^{2})^{3/2}}.$
(3.20)

Note here that when $Q\to\Sigma$ ($f(\tau)\to\epsilon$), $K|_{Q}$ has the
opposite sign to $K|_{\Sigma}$ hence, by construction, the sum of the two
Gibbons-Hawking terms also vanishes when $Q\to\Sigma$. We will use this fact
in discussion on holographic path-integral complexity.
Then, using relations between $\tau$ and $w$, we can write (3.20) in terms of
$\phi(w)$ as

$K|_{Q}=\frac{e^{-2\phi}\left(\ddot{\phi}+(d-1)\dot{\phi}^{2}\right)-d}{\sqrt{1-e^{-2\phi}\dot{\phi}^{2}}}.$
(3.21)

Putting all these ingredients together, we have the on-shell gravity action
with the tension term expressed in terms of $f(\tau)$

$\displaystyle
I_{G}+I_{T}=-\frac{d-1}{\kappa^{2}}\frac{V_{x}L_{\tau}}{\epsilon^{d}}-\frac{V_{x}}{\kappa^{2}}\int\frac{d\tau}{f^{d}}w^{\prime}(\tau)\left[K|_{Q}+\frac{1}{w^{\prime}(\tau)}-T\right],$
(3.22)

with $w^{\prime}(\tau)$ given by the first relation in (3.14) and $K|_{Q}$
given by (3.20).
We can also rewrite this action in terms of $w$ and $\phi(w)$ field as

$\displaystyle
I_{G}+I_{T}=-\frac{d-1}{\kappa^{2}}\frac{V_{x}L_{\tau}}{\epsilon^{d}}+\frac{V_{x}}{\kappa^{2}}\int
dw\,e^{d\phi}\left[\frac{d-1-e^{-2\phi}\left(\ddot{\phi}+(d-2)\dot{\phi}^{2}\right)}{\sqrt{1-e^{-2\phi}\dot{\phi}^{2}}}+T\right].$
(3.23)

Furthermore, we can integrate by parts and rewrite this action as first-order
in derivatives

$\displaystyle I_{G}+I_{T}$ $\displaystyle=$
$\displaystyle-\frac{(d-1)V_{x}L_{\tau}}{\kappa^{2}\epsilon^{d}}+\frac{(d-1)V_{x}}{\kappa^{2}}\int
dwe^{d\phi}G(\dot{\phi})+\frac{V_{x}}{\kappa^{2}}\int dwe^{d\phi}T$ (3.24)
$\displaystyle-\frac{V_{x}}{\kappa^{2}}\left[e^{(d-1)\phi}\arcsin(\dot{\phi}e^{-\phi})\right]^{0}_{-\infty},$

where $G$ is the following function bounded from below

$\displaystyle
G(\dot{\phi})\\!=\\!\sqrt{1-e^{-2\phi}{\dot{\phi}}^{2}}\\!+\\!\dot{\phi}e^{-\phi}\arcsin(\dot{\phi}e^{-\phi}).$
(3.25)

This is our main result in this subsection and a general algorithm for
computing the Hartle-Hawking wave functions semi-classically. In order to
derive it, it was crucial that we treated $Q$ with tension $T$ as a probe in
Poincare $AdS_{d+1}$. With this action, we are now ready to maximize the semi-
classical wave function (3.8).

###  3.3 Maximization

The next step in our procedure is to maximize the Hartle-Hawking wave function
(3.8) by choosing the appropriate metric on $Q$. This step should be
understood as the gravity counterpart of the path-integral optimization on the
CFT side and we will justify this claim below. The maximization condition is
obtained by taking a variation of the on-shell action (3.24) with respect to
$\phi$, leading to

$\displaystyle\frac{e^{-2\phi}(\ddot{\phi}+(d-1)\dot{\phi}^{2})-d}{\sqrt{1-e^{-2\phi}{\dot{\phi}}^{2}}}=\frac{d}{d-1}T.$
(3.26)

Comparing with (3.21), we find that the left-hand side is given by $K|_{Q}$,
therefore (3.26) becomes

$K|_{Q}=\frac{d}{d-1}T.$ (3.27)

In other words, the maximization implies that $Q$ should be a constant mean-
curvature (CMC) slice of the $AdS_{d+1}$ metric (3.1) dual to the CFT vacuum.
More generally, the maximization procedure requires variation of the on-shell
gravity action with tension term with respect to the induced metric $h_{ij}$
on $Q$. On general grounds this gives

$(K_{ij}-Kh_{ij}+Th_{ij})\delta h^{ij}=0,$ (3.28)

i.e. the Neumann boundary condition on $Q$. In the “conformal” gauge and with
$\phi(w)$ being the only component of the metric, this variation is
proportional to the metric itself and gives the trace of the Neumann condition
(3.27). We will see explicitly below that our metrics will solve (3.27) as
well as all the components of (3.28) so the maximization procedure is
equivalent to imposing the full Neumann boundary condition on $Q$

$\displaystyle K_{ij}-Kh_{ij}=-Th_{ij},$ (3.29)

which is also usually imposed in the standard AdS/BCFT construction [59]. See
more in App. B.
Now, by definition, $f(w=0)=\epsilon$ (see Fig. 1), and we search for
solutions of (3.26) with the boundary condition

$\displaystyle e^{2\phi}|_{w=0}=\frac{1}{f^{2}(w=0)}=\frac{1}{\epsilon^{2}}.$
(3.30)

This condition also matches the two metrics, the one on $Q$ and the one on
$\Sigma$, at the corner $\tau=w=0$ where the two surfaces meet. The relevant
solutions can be found when $T<0$ and we derive

$\displaystyle
e^{2\phi}=\frac{1}{f^{2}(w)}=\frac{1}{\left(-\sqrt{1-\frac{T^{2}}{(d-1)^{2}}}w+\epsilon\right)^{2}}.$
(3.31)

Note that for $T=0$, metric (3.31), in the “conformal coordinate” $w$,
precisely matches the CFT metric from the path-integral optimization (2.11)
written in $\tau$ with $\mu=1$. Identifying the two solutions for all values
of the parameters suggests the dictionary between the two parameters

$\mu=1-\frac{T^{2}}{(d-1)^{2}},$ (3.32)

and we will elaborate on it below and check it in all the Euclidean examples.
By definition, our solution $\phi$ determines function $f(\tau)$. Indeed using
(3.14) we can rewrite (3.31) in terms of the original coordinate $\tau$ that
from integrating (3.14) is given by

$\tau=\frac{|T|}{d-1}w,$ (3.33)

so that surface $Q$ is parametrized by

$\displaystyle
z=f(\tau)=\sqrt{1-\frac{T^{2}}{(d-1)^{2}}}\frac{(d-1)}{T}\tau+\epsilon.$
(3.34)

Figure 2: Solutions (3.34) are half planes interpolating between the boundary
at $T=-(d-1)/l$ and the $\tau=0$ slice for $T=0$ (in the text we used $l=1$).
The inner angle $\theta_{0}$ is related to $\Theta$ and $T$ via (3.35).

Surfaces $Q$ that maximize Hartle-Hawking wave functions, shown on Fig. 2, are
then half-planes that interpolate between $z=\epsilon$ at $T=-(d-1)$ and the
constant time-slice of $AdS_{d+1}$ for $T=0$ given by $\tau(z)=0$.
The tension parameter $T$ is naturally related to the angle between $\Sigma$
and $Q$. More precisely, we have

$\cos\Theta=n_{\Sigma}\cdot
n_{Q}|_{\tau=0}=\frac{T}{d-1},\qquad\sin\theta_{0}=\sqrt{1-\frac{T^{2}}{(d-1)^{2}}},$
(3.35)

where angle $\Theta$ is simply related to the inner angle $\theta_{0}$ between
$\Sigma$ and $Q$ by $\Theta=\pi-\theta_{0}$ (see Fig 2). The allowed values of
$\theta_{0}\in[0,\pi/2]$ are reflected in the range of $T\in[-(d-1),0]$.
If we recall our interpretation of parameter $0<\mu\leq 1$ in the path-
integral complexity action and take the relation (3.32), these gravity
solutions provide an elegant geometrization of sub-optimal continuous tensor
networks from the path-integral optimization. Indeed for $T\to-(d-1)$, surface
$Q\to\Sigma$ represents the original, flat (most un-optimal) tensor network.
On the other hand, for $T\to 0$ the $\tau=0$ slice of the bulk corresponds to
the fully optimized path-integral geometry with $\mu=1$. All the CMC surfaces
in between with $-(d-1)<T\leq 0$ naturally correspond to partially optimized
path-integral tensor networks from the path-integral complexity action with
$0<\mu<1$.

###  3.4 Holographic Path-Integral Complexity

The important part of the CFT analysis was focused on the path-integral
complexity actions as relative measures of computational complexity. Now,
after the optimization we obtained a family of slices $Q$ labeled by tension
$T$ and we argued that they interpolate between the boundary $\Sigma$ and the
optimal slice for $T=0$. In this section we make this statement more
quantitative and propose how to estimate complexity of these sub-optimal
networks in the gravitational construction.
After what we have already discussed, it is natural to expect that the gravity
action in region M is a good candidate for the relative measure of complexity
between $\Sigma$ and $Q$. However, the full story is slightly more involved.
In order to prepare for our final quantity, let us then first evaluate the
classical action (3.22) or (3.24) that computes the Hartle-Hawking wave
function $\Psi^{(T)}_{HH}[\phi]$ on the solution (3.31). We can verify that
the boundary term, part of the action with $G(\dot{\phi})$ and the tension
term precisely cancel and we are only left with the divergent contribution

$\displaystyle
I_{G}+I_{T}=-\frac{(d-1)V_{x}L_{\tau}}{\kappa^{2}\epsilon^{d}},$ (3.36)

so the on-shell action with the tension term added is independent of the
tension parameter $T$. This way, in order to measure computational complexity
from the Hartle-Hawking wave function construction, we propose to consider the
value of the action without the tension term added. This is natural since the
role of the tension term is only to fix the intermediate slices and, by
definition, it also doesn’t contribute to the complexity of the $\tau=0$
slice.
The on-shell action without the tension term is then evaluated as follows

$\displaystyle
I_{G}=-\frac{(d-1)V_{x}L_{\tau}}{\kappa^{2}\epsilon^{d}}+\frac{V_{x}}{\kappa^{2}}\frac{|T|}{(d-1)\sqrt{1-\frac{T^{2}}{(d-1)^{2}}}}\frac{1}{\epsilon^{d-1}}.$
(3.37)

Still, this is not the end of the story. Recall that for the correct
variational principle, gravitational action in regions with non-smooth
boundaries should be supplemented by the so-called Hayward term [60] defined
as

$I_{H}=-\frac{1}{\kappa^{2}}\int\sqrt{\gamma}(\Theta+\varphi_{0}),$ (3.38)

where as shown on Fig. 2, $\Theta$ is defined in terms of the normal vectors
to the surfaces that meet at the corner (see (3.35)) and $\gamma_{ij}$ is the
induced metric on the corner. By adding $\varphi_{0}$ in (3.38) we wanted to
stress that, from the point of view of the variational problem, any constant
value of $\varphi_{0}$ works sensibly in this definition. In order to estimate
the complexity of partially optimized slices, we will argue that the
appropriate choice of $\varphi_{0}$ is given by $\varphi_{0}=-\pi$, which
leads to the following expression for the corner contribution

$I^{\prime}_{H}\equiv\frac{1}{\kappa^{2}}\int\sqrt{\gamma}\,\theta_{0},$
(3.39)

that is determined entirely by the data in region $M$. We will refer to this
term as “modified Hayward term” expressed by the inner angle $\theta_{0}$
between $\Sigma$ and $Q$.
A clear motivation for this way of fixing the ambiguity in the Hayward term
comes in fact from the path-integral complexity. Namely, as we pointed in the
careful derivation of the Hartle-Hawking wave functions, the gravity action
(bulk as well as the two Gibbons-Hawking contributions) vanish in the limit of
$Q\to\Sigma$. This is clearly a desired property of a relative complexity
between metrics on $\Sigma$ and $Q$. If we added the standard Hayward term to
the gravitational action on M, since it vanishes only for smooth boundaries
(in our setup $\theta_{0}=\pi$) it would spoil this property. This way, in
order to keep a well-defined variational principle in the cusped region M as
well as have a sensible holographic definition of a relative complexity
functional, we propose to fix the constant ambiguity in the Hayward term to
$\varphi_{0}=-\pi$ and use (3.39) in addition to the gravity action (3.37).

Finally, we propose to estimate the holographic path-integral complexity of
the partially optimized network $Q$ (relative to $\Sigma$) corresponding to
the Euclidean Hartle-Hawking wave function with fixed value of the tension $T$
as the sum of the contribution of the on-shell gravity action in region $M$
(without the tension term) and the modified Hayward term101010In fact a
similar definition was used before in [25] with the overall sign difference
since it was evaluated in the bulk region up to the surface $Q$ that
represented a cut-off in that work.

$\mathcal{C}^{(e)}_{T}\equiv I_{G}+I^{\prime}_{H}.$ (3.40)

In the following sections we will study this quantity in various explicit
examples of Euclidean geometries. Interestingly, before going to any specific
examples, we can verify that $\mathcal{C}^{(e)}_{T}$ indeed satisfies the same
co-cycle properties (2.14) as the CFT path-integral complexity (see App. C).
Evaluating $\mathcal{C}^{(e)}_{T}$ in the vacuum example gives

$\displaystyle\mathcal{C}^{(e)}_{T}=-\frac{(d-1)V_{x}L_{\tau}}{\kappa^{2}\epsilon^{d}}+\frac{V_{x}}{\kappa^{2}\epsilon^{d-1}}(\theta_{0}+\cot\theta_{0}),$
(3.41)

where $\theta_{0}$ and its relation to $T$ is computed in (3.35). This shows
that $\mathcal{C}^{(e)}_{T}$ is indeed a monotonically decreasing function of
$\theta_{0}$ or equally of $T$. Thus, it is minimized when $T=0$ ($\tau=0$
slice), as expected from the optimization procedure. The minimal value is
given by

$\mathcal{C}^{(e)}_{T=0}=-\frac{(d-1)V_{x}L_{\tau}}{\kappa^{2}\epsilon^{d}}+\frac{V_{x}}{\kappa^{2}\epsilon^{d-1}}\frac{\pi}{2}.$
(3.42)

Notice that the first negatively divergent term in (3.41) can be regarded as
the subtraction of the complexity for the reference state before the
optimization (i.e. $e^{\phi(w)}=1/\epsilon$ at $\Sigma$).
Let us also point a subtlety that, by construction, (3.41) should vanish when
$\theta_{0}\to 0$ or equivalently $T\to-(d-1)$. This was clear from the
definition of $\mathcal{C}^{(e)}_{T}$ but after evaluating it on-shell (and
doing the $\tau$-integral) in order to reproduce this fact we should take a
careful (formal) limit of

$\theta_{0}\to\cot^{-1}\left(\frac{(d-1)L_{\tau}}{\epsilon}\right),\qquad
T\to\frac{-(d-1)}{\sqrt{1+\frac{\epsilon^{2}}{(d-1)^{2}L^{2}_{\tau}}}},$
(3.43)

with $L_{\tau}/\epsilon\to\infty$.
This concludes our analysis of the holographic path-integral complexity from
the Hartle-Hawking wave function and in section 4 we will provide further
evidence and support for our definition in various Euclidean examples.

###  3.5 Maximization and Gravity Hamiltonian Constraint

In this subsection we give further comments on the maximization procedure. In
particular, we elaborate on the fact that path-integral geometries in CFT are
found from the constant Ricci scalar constraint (e.g. Liouville equation)
whereas surfaces $Q$ are obtained from the Neumann boundary condition. We will
see that it is in fact the gravity Hamiltonian constraint that elegantly
connects the two procedures.
As we argued on general grounds, as in AdS/BCFT [59], the maximization of the
Hartle-Hawking wave function is equivalent to imposing Neumann boundary
condition on $Q$. We can check this explicitly using components of the
extrinsic curvature tensor derived in (3.19). After writing (3.29) in
components we get

$\frac{1}{\sqrt{1+f^{\prime}(\tau)^{2}}}=-\frac{T}{d-1},\quad-\frac{f(\tau)f^{\prime\prime}(\tau)+(d-1)(1+f^{\prime}(\tau)^{2})}{(1+f^{\prime}(\tau)^{2})^{3/2}}\delta_{ab}=T\delta_{ab},$
(3.44)

or in terms of $w$ and $\phi(w)$

$\sqrt{1-\dot{\phi}^{2}e^{-2\phi}}=-\frac{T}{d-1},\qquad\frac{e^{-2\phi}\left(\ddot{\phi}+(d-2)\dot{\phi}^{2}\right)-(d-1)}{\sqrt{1-\dot{\phi}^{2}e^{-2\phi}}}\delta_{ab}=T\delta_{ab}.$
(3.45)

Plugging (3.34) to the first and (3.31) to the second pair of constraints, we
can verify that full Neumann boundary conditions are satisfied for $T<0$
straightforwardly.111111Note that to find $Q$ (i.e. $f(\tau)$) that maximizes
the Hartle-Hawking wave function, we could also vary the action (3.22) with
respect to $f(\tau)$ and get the constraint
$\frac{2ff^{\prime\prime}+d(1+f^{\prime 2})}{(1+f^{\prime
2})^{2}}=-\frac{T}{d-1}\frac{ff^{\prime\prime}+d(1+f^{\prime 2})}{(1+f^{\prime
2})^{3/2}}.$ (3.46) We can easily check that (3.34) is a solution of this
equation with $T<0$. See also App. B for more discussion on solving the
Neumann boundary condition.
Given the induced metric on (3.12) with maximizing solution (3.31), we can
compute the Ricci scalar curvature (intrinsic) of our bulk slices

$R^{(d)}=e^{-2\phi}(-2(d-1)\partial^{2}_{w}\phi-(d-2)(d-1)(\partial_{w}\phi)^{2})=2\Lambda^{(d+1)}\left(1-\frac{T^{2}}{(d-1)^{2}}\right),$
(3.47)

where $\Lambda^{(d+1)}=-d(d-1)/2$. Clearly, slices $Q$ have constant negative
curvature parametrized by tension $T$. We will come back to this result
momentarily and discuss the precise relation with the path-integral
optimization geometries.
Before doing that, we can verify that this negative curvature gives yet
another interpretation of the maximization procedure. In fact, we can check
that, by definition, when we impose the Neumann boundary condition on $Q$, the
extrinsic curvature tensor is proportional to the induced metric

$K_{ij}=(K-T)h_{ij}=\frac{T}{d-1}h_{ij},$ (3.48)

where in the second equality we used (3.27). Consequently, on-shell, the
following combination of the extrinsic curvature and the square of its trace
becomes

$K^{2}-K^{ij}K_{ij}=\frac{d}{d-1}T^{2}.$ (3.49)

This is nothing but the left-hand side of the “Hamiltonian” constraint in pure
gravity on $AdS_{d+1}$

$K^{2}-K^{ij}K_{ij}=R^{(d)}-2\Lambda^{(d+1)},$ (3.50)

where $\Lambda^{(d+1)}$ is the negative cosmological constant of $AdS_{d+1}$
and $R^{(d)}$ is the Ricci scalar curvature of the induced metric on the
gravity slice. Plugging (3.49), we can check that this constraint reproduces
the Ricci scalar $\eqref{Riccid}$. This local Hamiltonian constraint is the
contraction of the bulk Einstein’s equation121212This can be seen by using
Gauss-Codazzi equation, see e.g. [61]. that is satisfied from the beginning in
our construction

$n^{\mu}n^{\nu}\left(G_{\mu\nu}=-\Lambda^{(d+1)}g_{\mu\nu}\right),$ (3.51)

where $G_{\mu\nu}$ is the Einstein’s tensor and $n^{\mu}$ are the normal
vectors to an arbitrary slice of the bulk. Here, since the maximization of the
Hartle-Hawking wave functions leads to the Neumann boundary condition this
constraint automatically implies that the Ricci scalar curvature of our slices
is constant.

Recall that “Hamiltonian” constraint (3.50) in AdS/CFT usually refers to the
“radial Hamiltonian” where $K_{ij}$ and $K$ are computed locally on some cut-
off surface with Ricci scalar $R^{(d)}$. This constraint plays an important
role in the recent developments on the relations between
$T\bar{T}$-deformations of holographic CFTs and finite cut-off gravity with
Dirichlet boundary condition on the cut-off surface [62, 63, 64]. Imposing
Dirichlet boundary condition for the gravity action with holographic counter-
terms allows one to compute the holographic stress-tensor expressed by
$K_{ij}$, its trace $K$ and the counter-term contribution. Then, rewriting
$K_{ij}$ and $K$ in (3.50) in terms of $T_{ij}$ and $T^{i}_{i}$ reproduces
trace anomaly equation in a CFT deformed by the $T\bar{T}$-operator with the
coupling related to the finite-cut-off radius. This construction was recently
generalized to $T\bar{T}$ deformations with time-dependent couplings [29] (see
also [65] for discussion on the flow of quantum states under $T\bar{T}$
deformations that is relevant in our context) and studied from the perspective
of holographic Tensor Networks as constant $K$ slices of $AdS$. Interestingly,
the explicit time-dependent “folding $T\bar{T}$ deformation” in CFTs,
precisely corresponds to the bulk $AdS$ geometry with time-dependent cut-off
surfaces identical to the $Q$ in (3.34). We hope that this intriguing
connection will shed more light on the operational and circuit-type
interpretation of the path-integral optimization and path-integral complexity
and we leave it as an interesting future direction.

###  3.6 Boundary UV Limit

Let us finally finish with more evidence for the holographic relation with
field theory path-integral optimization. Firstly, one striking feature of the
proposal is that both metrics, Weyl flat metric from path-integral
optimization (2.2) (in arbitrary dimensions) and induced metric on $Q$ that
maximizes Hartle-Hawking wave function (3.2), are identical after we relate
$w$ and $\tau$. Indeed, they have both constant negative Ricci scalars equal
to

$R^{(d)}|_{PI}=2\Lambda^{(d+1)}\mu,\qquad
R^{(d)}|_{HH}=2\Lambda^{(d+1)}\left(1-\frac{T^{2}}{(d-1)^{2}}\right).$ (3.52)

We will analyze various, non-trivial, examples below that confirm this
observation. This again leads to the identification (3.32) between parameters
in our proposals. Remember that changing $\mu$ from $\mu=0$ to $\mu=1$ means
that we gradually increase the amount of optimization.131313Let us also point
that, from the perspective of complexity and gate counting [41], one could
think about $\mu$ as a relative “penalty factor” between unitaries and
isometries and we leave this alternative interpretation for future studies
(see also [66]). In the gravity dual, this corresponds to changing the tension
from $T=-(d-1)$ to $T=0$ which tilts the surface $Q$ from the asymptotic
boundary $\Sigma$ to the time slice $\tau=0$.

Secondly, the gravity construction with Hartle-Hawking wave function involves
extremizing action (3.24), that can be thought of as the boundary (UV) action
with infinite sum of finite cut-off corrections. Indeed we can write
$G(\dot{\phi})$ as a series in $\dot{\phi}e^{-\phi}$

$G(\dot{\phi})=1+\sum^{\infty}_{k=1}\frac{\Gamma\left(\frac{2k+1}{2}\right)}{\sqrt{\pi}(2k-1)^{2}\Gamma(k+1)}\,\dot{\phi}^{2k}e^{-2k\phi}.$
(3.53)

In the UV limit, taking only the leading $\dot{\phi}e^{-\phi}$ term, we end up

$e^{d\phi}G(\dot{\phi})=\frac{1}{2}e^{(d-2)\phi(w)}\dot{\phi}^{2}+e^{d\phi}+O(\dot{\phi}^{4}),$
(3.54)

and the gravity action (without the tension term) becomes

$\displaystyle I_{G}|_{UV}$ $\displaystyle\simeq$
$\displaystyle\frac{(d-1)V_{x}}{2\kappa^{2}}\int
dw\left[e^{(d-2)\phi(w)}\dot{\phi}^{2}+2e^{d\phi}-2\epsilon^{-d}+O(\dot{\phi}^{4})\right]$
(3.55) $\displaystyle-$
$\displaystyle\frac{V_{x}}{\kappa^{2}}\frac{\theta_{0}}{\epsilon^{d-1}},$

where $\theta_{0}$ is the value of $\arcsin(\dot{\phi}e^{-\phi})$ at $w=0$.
Interestingly, we can cancel $\theta_{0}$ dependence by adding the “modified
Hayward” term (3.39), localized on $\Sigma\cap Q$. This in fact will be true
in all the Euclidean examples below. This is another argument for its
important role in the holographic path-integral complexity. More importantly,
(3.55) is precisely the (homogeneous) form of our conjectured action (2.12),
therefore the construction with Hartle-Hawking wave function gives a strong
justification of the higher dimensional conjecture.141414Giving up the
homogeneity assumption $z=f(t)$ and taking $z=f(t,x_{i})$ should reproduce the
full action in the UV limit.151515Note that if we add the tension term
$I_{T}$, the total action $I_{G}+I_{T}$ in the UV limit leads to a “UV
relation”: $\mu=2+\frac{2T}{d-1}$. We can regard the minimizing $I_{G}+I_{T}$
as a partial optimization and this corresponds the changing the cosmological
constant $\mu$. However notice that the genuine complexity functional, whose
minimization gives the full optimization is obtained by removing the tension
term.
Nevertheless, note that the gravity action (3.24) gives the correct
holographic finite cut-off corrections to the UV Liouville action (as in
(3.41)) and thus should be thought of as the full answer to the path-integral
optimization program. Having said that, there is still a possibility that both
complexity actions, the Liouville and the holographic path-integral
complexity, even though apparently different and coming with their own
regularization schemes, are actually equivalent. The fact that they lead to
the same optimal geometries with constant Ricci scalars may suggest this
however, we do not have any definite way to prove/disprove this possibility at
the moment and leave it for future investigation.

##  4 More Euclidean Examples

In the following sections we study more examples that will confirm our
procedure in various Euclidean setups across dimensions. The algorithm for
computing semi-classical Hartle-Hawking wave functions using on-shell gravity
action with tension (3.8) is straightforward and we will spare the details of
the steps while only showing main results of analogous computations as we did
for the vacuum case.

###  4.1 Global $AdS_{d+1}$

We start by analyzing the gravity dual of the vacuum of a CFT on a circle
given by global $AdS$ spacetime. We can perform similar analysis to what we
did in Poincare coordinates starting now from the global $AdS_{d+1}$ metric

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left(1+r^{2}\right)d\tau^{2}+\frac{dr^{2}}{1+r^{2}}+r^{2}\,d\Omega^{2}_{d-1},$
(4.1)

where $\Omega_{d-1}$ stands for the volume element of a $(d-1)$-sphere.
The region $M$ in global coordinates is bounded by the cut-off surface
$\Sigma$ at $r=r_{0}=\frac{1}{\epsilon}\to\infty$ and a homogeneous surface
$Q$ defined by $r=1/f(\tau)$, so its induced metric can be written as

$ds^{2}=\frac{1}{f(\tau)^{2}}\left[\left(1+f(\tau)^{2}+\frac{f^{\prime}(\tau)^{2}}{1+f(\tau)^{2}}\right)d\tau^{2}+d\Omega^{2}_{d-1}\right]=e^{2\phi(w)}\left(dw^{2}+d\Omega^{2}_{d-1}\right).$
(4.2)

Again, in the second equality, we introduced coordinate $-\infty<w<0$ and
field $\phi(w)$ that render the induced metric on $Q$ conformally flat

$w^{\prime}(\tau)=\sqrt{1+f^{2}+\frac{f^{\prime
2}}{1+f^{2}}}=\frac{1+f^{2}}{\sqrt{1+f^{2}-\dot{f}^{2}}},\qquad
e^{2\phi}=\frac{1}{f^{2}}.$ (4.3)

The notation for $f^{\prime}$ and $\dot{f}$ is the same as in Poincare
coordinates (see below (3.14)).
Next, we evaluate the bulk Hartle-Hawking wave function semi-classically as
the on-shell bulk action with the tension term on $Q$ (3.8). To compute the
classical action, we will need the trace of the extrinsic curvature on
$\Sigma$

$K|_{\Sigma}=\frac{d+(d-1)\epsilon^{2}}{\sqrt{1+\epsilon^{2}}},$ (4.4)

as well as for surface $Q$ written in terms of $f(\tau)$

$K|_{Q}=-\frac{1}{w^{\prime 3}}\left[ff^{\prime\prime}+\frac{f^{\prime
2}((d-3)f^{2}+d)+(1+f^{2})^{2}(f^{2}(d-1)+d)}{(1+f^{2})}\right],$ (4.5)

or in terms of $w$ of $\phi(w)$ as

$K|_{Q}=\frac{e^{-2\phi}\left(\ddot{\phi}+(d-1)(\dot{\phi}^{2}-1)\right)-d}{\sqrt{1-e^{-2\phi}(\dot{\phi}^{2}-1)}}.$
(4.6)

With these ingredients we can evaluate the gravity action with tension term
and the answer becomes

$I_{G}+I_{T}=-\frac{(d-1)V_{\Omega}L_{\tau}}{\kappa^{2}\epsilon^{d}}\left(1+\epsilon^{2}\right)-\frac{V_{\Omega}}{\kappa^{2}}\int\frac{d\tau}{f^{d}}w^{\prime}(\tau)\left[K|_{Q}-T+\frac{1}{w^{\prime}(\tau)}\right],$
(4.7)

where $w^{\prime}(\tau)$ is given by the first equation in (4.3), $K|_{Q}$ by
(4.5) and $V_{\Omega}$ is the volume of the $(d-1)$ sphere.
Similarly, in terms of $w$ and $\phi$, after integrating by parts, we derive
the main result for the global $AdS_{d+1}$

$\displaystyle I_{G}+I_{T}$ $\displaystyle=$
$\displaystyle-\frac{(d-1)V_{\Omega}L_{\tau}}{\kappa^{2}\epsilon^{d}}\left(1+\epsilon^{2}\right)+\frac{(d-1)V_{\Omega}}{\kappa^{2}}\int
dwe^{d\phi}G(\dot{\phi})+\frac{V_{\Omega}}{\kappa^{2}}\int dwe^{d\phi}T$ (4.8)
$\displaystyle-$
$\displaystyle\frac{V_{\Omega}}{\kappa^{2}}\left[e^{(d-1)\phi}\arcsin\left(\frac{\dot{\phi}e^{-\phi}}{\sqrt{1+e^{-2\phi}}}\right)\right]^{0}_{-\infty},$

where now, function $G(\dot{\phi})$, with finite size corrections, becomes

$G(\dot{\phi})=\sqrt{1-e^{-2\phi}(\dot{\phi}^{2}-1)}+\dot{\phi}e^{-\phi}\arcsin\left(\frac{\dot{\phi}e^{-\phi}}{\sqrt{1+e^{-2\phi}}}\right).$
(4.9)

Note that, as we showed in the previous example in Poincare coordinates, the
gravity action (without the tension term) that computes the semi-classical
Hartle-Hawking wave functions (3.8) should be regarded as the CFT path-
integral complexity (Liouville action in 2d) together with finite cut-off
corrections from the bulk. These holographic finite cut-off corrections will
naturally depend on the details of the gravity geometry such as e.g. the
finite size in (4.8) or mass of excitations (black hole) that we will see in
the following examples.

Maximization of the wave function with respect to $\phi$ again leads to the
CMC slices

$K|_{Q}=\frac{d}{d-1}T,$ (4.10)

and we can solve it for $-(d-1)<T\leq 0$ by

$e^{2\phi(w)}=\frac{1}{f^{2}(w)}=\frac{1}{\left(1-\frac{T^{2}}{(d-1)^{2}}\right)\sinh^{2}\left(w-c_{1}\right)}.$
(4.11)

Constant $c_{1}$ is fixed such that the boundary condition (3.30),
$f(0)=\epsilon$, is satisfied

$c_{1}\simeq\frac{\epsilon}{\sqrt{1-\frac{T^{2}}{(d-1)^{2}}}},$ (4.12)

and in the computations of the on-shell action we will be taking the
$\epsilon<<1$ limit first.
Again, this result in conformal coordinate $w$ matches the solution that we
get from path-integral optimization with the Liouville action (or its
generalization to higher dimensions (2.12)) on the cylinder.161616Recall that
the solution of Liouville equation on the cylinder of circumference $L$ with
coordinates $\tau\in(-\epsilon,-\infty)$ and $\sigma=\sigma+L$ written in
coordinates $w=\tau+i\sigma$ is given by
$e^{2\phi(w,\bar{w})}=\frac{4\pi^{2}}{L^{2}\sinh^{2}\left(\frac{\pi(w+\bar{w})}{L}\right)}=\frac{4\pi^{2}}{L^{2}\sinh^{2}\left(\frac{2\pi\tau}{L}\right)},$
(4.13) and we have the boundary condition
$e^{2\phi(\tau=-\epsilon)}=\frac{1}{\epsilon^{2}}$ and
$e^{2\phi(\tau=-\infty)}\to 0$.
Moreover, we can integrate (4.3) and write the embedding function $f(\tau)$ of
the surface $Q$ in original global $AdS_{d+1}$ coordinates

$f(\tau)=\sqrt{1-\frac{T^{2}}{(d-1)^{2}}}\frac{\tanh(c_{2}-\tau)}{\sqrt{\frac{T^{2}}{(d-1)^{2}}-\tanh^{2}(\tau-
c_{2})}},$ (4.14)

where, from the boundary condition $f(0)=\epsilon$ we can fix $c_{2}$ to

$c_{2}\simeq\frac{\epsilon|T|}{(d-1)\sqrt{1-\frac{T^{2}}{(d-1)^{2}}}}+O(\epsilon^{3}).$
(4.15)

To analyze this profile more carefully, it is useful to parametrize $Q$ as
$\tau(r)$ and we derive

$\tau(r)=\sinh^{-1}\left(\frac{T}{(d-1)\sqrt{(1+r^{2})\left(1-\frac{T^{2}}{(d-1)^{2}}\right)}}\right)+c_{2}.$
(4.16)

We can verify that $Q$ has a minimum at $r=0$ where

$\tau_{min}=\tanh^{-1}\left(\frac{T}{d-1}\right)+O(\epsilon).$ (4.17)

These surfaces are shown on Fig. 3. They interpolate between the $\tau=0$
slice for $T=0$ and the boundary surface $\Sigma$ for $T=-(d-1)$.

Figure 3: Example slices $Q$ that maximize the Hartle-Hawking wave functions
in global $AdS_{d+1}$ for a fixed value of $T$. They interpolate between the
maximal, $\tau=0$ slice for $T=0$ (green) and one reaching to $\tau=-\infty$
for $T=-(d-1)$.

Finally, for obtaining complexity $\mathcal{C}^{(e)}_{T}$ (3.40), we first
evaluate the on-shell gravity action (without the tension term). It can be
computed in arbitrary dimensions in a closed form written in terms of special
functions but, since it is not particularly illuminating, we just show a
couple of results for specific dimensions.
On the other hand, the modified Hayward term in general dimensions given by

$I^{\prime}_{H}=\frac{1}{\kappa^{2}}\int_{\gamma}\sqrt{\gamma}\theta_{0}=\frac{V_{\Omega}}{\kappa^{2}\epsilon^{d-1}}\theta_{0},\qquad\sin\theta_{0}=\sqrt{1-\frac{T^{2}}{(d-1)^{2}(1+\epsilon^{2})}}.$
(4.18)

This way, e.g. in global $AdS_{3}$ we derive

$\mathcal{C}^{(e)}_{T}=-\frac{V_{x}L_{\tau}}{\kappa^{2}\epsilon^{2}}+\frac{V_{x}}{\kappa^{2}\epsilon}\left(\theta_{0}+\frac{|T|}{\sqrt{1-T^{2}}}\right)-\frac{V_{x}}{\kappa^{2}}\left(\frac{|T|}{1-T^{2}}+\tanh^{-1}(T)\right)+O(\epsilon).$
(4.19)

In odd $d$, we also get an additional logarithmic divergence from the gravity
action part. E.g. in global $AdS_{4}$ we derive

$\displaystyle\mathcal{C}^{(e)}_{T}$ $\displaystyle=$
$\displaystyle-\frac{2V_{\Omega}L_{\tau}}{\kappa^{2}\epsilon^{3}}\left(1+\epsilon^{2}\right)+\frac{V_{\Omega}}{\kappa^{2}}\left[\frac{|T|}{2\sqrt{1-\frac{T^{2}}{4}}}\frac{1}{\epsilon^{2}}-\text{arctan}\frac{|T|}{2\sqrt{1-\frac{T^{2}}{4}}}\right]$
(4.20) $\displaystyle-$
$\displaystyle\frac{V_{\Omega}}{\kappa^{2}}\frac{|T|^{3}}{8\left(1-\frac{T^{2}}{4}\right)^{3/2}}\log\left[\frac{2}{\epsilon}\sqrt{1-\frac{T^{2}}{4}}\right]+O(\epsilon).$

From the analytic formula in general $d$ we can verify that the complexity
action action is always minimized for $T=0$. The minimal value can be computed
in all dimensions and is given by

$\mathcal{C}^{(e)}_{T=0}=-\frac{(d-1)V_{\Omega}L_{\tau}}{\kappa^{2}\epsilon^{d}}\left(1+\epsilon^{2}\right)+\frac{V_{\Omega}}{\kappa^{2}\epsilon^{d-1}}\frac{\pi}{2}.$
(4.21)

The leading divergent term and the modified Hayward term have the same form as
in Poincare coordinates (with $V_{x}$ there and $V_{\Omega}$ here) and the new
finite-size signature enters via the sub-leading contribution (or constant
term in $d=2$).

###  4.2 Excited states in 3d

Next, we consider a general family of spacetimes that will allow us to
reproduce all the metrics found in the path-integral optimization for 2d CFTs
[35]. More precisely, we repeat the computation of the semi-classical Hartle-
Hawking wave functions (3.8) using the on-shell gravity solution given by
Euclidean BTZ-type metrics in three dimensions

$\displaystyle
ds^{2}=(r^{2}-r^{2}_{h})d\tau^{2}+\frac{dr^{2}}{r^{2}-r^{2}_{h}}+r^{2}dx^{2},$
(4.22)

where $r^{2}_{h}=M-1$ can be positive or negative depending on the mass $M$ of
the excitation. For $M=0$ or $r^{2}_{h}=-1$ we will formally recover the above
results in global $AdS_{3}$ (with $d=2$).
We will now compute $\Psi^{(T)}_{HH}$ for region $M$ specified by
$\frac{1}{f(\tau)}\leq r\leq\frac{1}{\epsilon}$, where $r=1/f(\tau)$ and
$r=1/\epsilon$ describe the surface $Q$ and the asymptotic boundary $\Sigma$,
respectively. The $x$ direction can be taken periodic or a real line and will
only appear inside the overall factor $V_{x}$ as before.
Following our procedure, for $\Sigma$ at $r=1/\epsilon$ we derive

$K_{\Sigma}=\frac{2-r^{2}_{h}\epsilon^{2}}{\sqrt{1-r^{2}_{h}\epsilon^{2}}}.$
(4.23)

On the other hand, the induced metric on $Q$ is given by

$ds^{2}=\frac{1}{f^{2}}\left[\left(1-r^{2}_{h}f(\tau)^{2}+\frac{f^{\prime
2}}{1-r^{2}_{h}f(\tau)^{2}}\right)d\tau^{2}+dx^{2}\right]=e^{2\phi}(dw^{2}+dx^{2}),$
(4.24)

where we introduced $-\infty<w<0$ such that

$w^{\prime}(\tau)=\sqrt{1-r^{2}_{h}f^{2}+\frac{f^{\prime
2}}{1-r^{2}_{h}f^{2}}}=\frac{1-r^{2}_{h}f^{2}}{\sqrt{1-r^{2}_{h}f^{2}-\dot{f}^{2}}}.$
(4.25)

The trace of extrinsic curvature in terms of $\tau$ and $f(\tau)$ is now

$K|_{Q}=-\frac{1}{w^{\prime
3}}\left[ff^{\prime\prime}+\frac{f^{\prime}(\tau)^{2}(2+r^{2}_{h}f^{2})+(1-r^{2}_{h}f^{2})^{2}(2-r^{2}_{h}f^{2})}{1-r^{2}_{h}f^{2}}\right],$
(4.26)

or in terms of $w$ and $\phi(w)\equiv-\log(f(w))$

$K|_{Q}=\frac{e^{-2\phi}(\ddot{\phi}+\dot{\phi}^{2})+r^{2}_{h}e^{-2\phi}-2}{\sqrt{1-e^{-2\phi}(\dot{\phi}^{2}+r^{2}_{h})}}.$
(4.27)

The action that computes the Hartle-Hawking wave function is now given in
terms of $f(\tau)$ as

$I_{G}+I_{T}=-\frac{V_{x}L_{\tau}}{\kappa^{2}\epsilon^{2}}\left(1-r^{2}_{h}\epsilon^{2}\right)-\frac{V_{x}}{\kappa^{2}}\int\frac{d\tau}{f(\tau)^{2}}w^{\prime}\left[K|_{Q}-T+\frac{1}{w^{\prime}}\right].$
(4.28)

As before, we can integrate by parts and derive the main result for excited
states in asymptotically $AdS_{3}$ geometry (4.22)

$\displaystyle I_{G}+I_{T}=$ $\displaystyle-$
$\displaystyle\frac{V_{x}L_{\tau}}{\kappa^{2}\epsilon^{2}}\left(1-r^{2}_{h}\epsilon^{2}\right)+\frac{V_{x}}{\kappa^{2}}\int
dw\,e^{2\phi}G(\dot{\phi})+\frac{V_{x}}{\kappa^{2}}\int dw\,e^{2\phi}T$
$\displaystyle-$
$\displaystyle\frac{V_{x}}{\kappa^{2}}\left[e^{\phi}\arcsin\left(\frac{\dot{\phi}e^{-\phi}}{\sqrt{1-r^{2}_{h}e^{-2\phi}}}\right)\right]^{0}_{-\infty},$

with

$\displaystyle G(\dot{\phi})$ $\displaystyle=$
$\displaystyle\sqrt{1-e^{-2\phi}\left(\dot{\phi}^{2}+r^{2}_{h}\right)}+\dot{\phi}e^{-\phi}\arcsin\left(\frac{\dot{\phi}e^{-\phi}}{\sqrt{1-r^{2}_{h}e^{-2\phi}}}\right).$
(4.30)

Variation with respect to $\phi$ yields the CMC condition $K|_{Q}=2T$ and is
again equivalent to the Neumann condition (3.29). For negative tension, we can
solve it by

$\displaystyle
e^{2\phi}=\frac{r^{2}_{h}}{\left(1-T^{2}\right)\sin^{2}\left(r_{h}(w-c_{1})\right)},$
(4.31)

where $c_{1}$ is tuned appropriately to reproduce the boundary condition
(3.30).
This family of solutions precisely matches those in the path-integral
optimization [35] via the identification (3.32). For
$r^{2}_{h}=-(1-M)=-\alpha^{2}$ we reproduce excited states from the
optimization for primary operators in 2d CFT i.e. conical singularity
geometries, including the finite size vacuum. For $r_{h}=2\pi/\beta$ and $w$
on the strip, we reproduce the optimal geometry for the thermofield double
(TFD) state that was found to describe the time slice of eternal black hole
(Einstein-Rosen bridge) [67]. We can verify that all these solutions have
constant negative curvature

$R^{(2)}=-2(1-T^{2}),$ (4.32)

which is just a special case of (3.47) with $d=2$ and follows from the the
Hamiltonian constraint supplemented by the Neumann boundary condition. We now
analyze two of these solutions in more detail.

####  4.2.1 Conical Singularities

Let as first take the solutions with $r^{2}_{h}=-\alpha^{2}$, often referred
to as conical singularities

$e^{2\phi}=\frac{\alpha^{2}}{\left(1-T^{2}\right)\sinh^{2}\left(\alpha(w-c_{1})\right)}.$
(4.33)

In fact this solution has a constant Ricci scalar as (4.32) with an extra
$\delta$-function proportional to $(1-\alpha)$ at $w\to-\infty$. This can be
seen by e.g. mapping the 2d surface into a disc.
The boundary condition for $\phi$ at $w=0$ implies

$c_{1}\simeq\frac{\epsilon}{\sqrt{1-T^{2}}}+O(\epsilon^{3}).$ (4.34)

In terms of $\tau$ the embedding function of $Q$ reads

$f(\tau)=\frac{\sqrt{1-T^{2}}\tanh(\alpha(c_{2}-\tau))}{\alpha\sqrt{T^{2}-\tanh^{2}(\alpha(\tau-
c_{2}))}},$ (4.35)

with

$c_{2}\simeq\frac{\epsilon|T|}{\sqrt{1-T^{2}}}+O(\epsilon^{3}).$ (4.36)

It is again convenient to parametrize $Q$ as

$\tau(r)=\frac{1}{\alpha}\tanh^{-1}\left(\frac{\alpha
T}{\sqrt{r^{2}(1-T^{2})+\alpha^{2}}}\right)+c_{2}.$ (4.37)

These rotationally symmetric solutions have minima at $r=0$

$\tau_{min}=\frac{1}{\alpha}\tanh^{-1}(T)+O(\epsilon),$ (4.38)

and interpolate between $\tau=0$ surface for $T=0$ and a surface with minimum
at $\tau=-\infty$ (and $r=0$) for $T\to-1$. For $\alpha=1$ we recover the
global $AdS_{3}$ solution.

For the Euclidean complexity $\mathcal{C}^{(e)}_{T}$, the on-shell action
without the tension term is equal to

$I_{G}=-\frac{V_{x}L_{\tau}}{\kappa^{2}\epsilon^{2}}+\frac{V_{x}}{\kappa^{2}}\left(\frac{|T|}{\sqrt{1-T^{2}}\epsilon}-\frac{\alpha|T|}{1-T^{2}}-\alpha\tanh^{-1}(T)\right)+O(\epsilon),$
(4.39)

while the modified Hayward term is given by

$I^{\prime}_{H}=\frac{1}{\kappa^{2}}\int\sqrt{\gamma}\theta_{0}=\frac{V_{x}}{\kappa^{2}\epsilon}\theta_{0},\qquad\theta_{0}=\sin^{-1}\left(\sqrt{1-\frac{T^{2}}{1+\alpha^{2}\epsilon^{2}}}\right).$
(4.40)

Adding these contributions together, we arrive at

$\mathcal{C}^{(e)}_{T}=-\frac{V_{x}L_{\tau}}{\kappa^{2}\epsilon^{2}}+\frac{V_{x}}{\kappa^{2}\epsilon}\left(\theta_{0}+\frac{|T|}{\sqrt{1-T^{2}}}\right)-\frac{V_{x}\alpha}{\kappa^{2}}\left(\frac{|T|}{1-T^{2}}+\tanh^{-1}(T)\right)+O(\epsilon).$
(4.41)

This action is monotonic and has a minimum at $T=0$ so the $\tau=0$ slices of
(4.22) have the lowest holographic path-integral complexity. The minimal value
is given by

$\mathcal{C}^{(e)}_{T=0}=-\frac{V_{x}L_{\tau}}{\kappa^{2}\epsilon^{2}}+\frac{V_{x}}{\kappa^{2}}\frac{\pi}{2\epsilon}+O(\epsilon).$
(4.42)

Again the $\epsilon^{-2}$ divergence originates from the choice of the
“reference metric” at $\Sigma$.

####  4.2.2 Thermofield double

Next, we can use the solution (4.31) to analyze the gravity dual of the path-
integral optimization for the TFD state

$|\Psi\rangle_{TFD}=\sum_{n}e^{-\frac{\beta}{2}E_{n}}|E_{n}\rangle|E_{n}\rangle,$
(4.43)

studied in [35]. We take $r_{h}=2\pi/\beta$ and Euclidean time between
$\tau\in(-\beta/4,\beta/4)$, while imposing the usual boundary conditions
(3.30) at both ends. If we then take

$w\in\left[-\frac{\beta}{4}+\frac{\epsilon}{\sqrt{1-T^{2}}},\frac{\beta}{4}-\frac{\epsilon}{\sqrt{1-T^{2}}}\right],$
(4.44)

we can write the solution as

$e^{2\phi}=\frac{4\pi^{2}}{\left(1-T^{2}\right)\beta^{2}\cos^{2}\left(\frac{2\pi
w}{\beta}\right)},$ (4.45)

and it clearly satisfies the boundary condition at each end. This solution is
equivalent to the optimized metric from the path-integrals preparing TFD state
in 2d CFTs [35]171717Note that analogous analysis could be done for the pure
state $|\Psi\rangle=e^{-\beta/4H}|B\rangle$ with $\tau\in[0,\beta/4)$ and
$|B\rangle$ representing a boundary state in CFT.
Using (4.25), we can derive $f(\tau)$ and surfaces $Q$, plotted on Fig. 4, are
given by the embedding function

$f(\tau)=\frac{\sqrt{1-T^{2}}}{r_{h}\sqrt{1+T^{2}\tan^{2}\left(r_{h}\tau\right)}},$
(4.46)

where the range of $\tau$ is now

$\tau\in\left[-\frac{\beta}{4}-c_{2},\frac{\beta}{4}+c_{2}\right],$ (4.47)

with cut-off

$c_{2}\simeq\frac{\epsilon T}{\sqrt{1-T^{2}}}+O(\epsilon).$ (4.48)

Again in the analysis we take $\epsilon\to 0$ first.
In order to estimate the complexity $\mathcal{C}^{(e)}_{T}$, we first compute
the on-shell action without tension term

$I_{G}=-\frac{V_{x}\beta}{2\kappa^{2}\epsilon^{2}}+\frac{V_{x}}{\kappa^{2}}\frac{2|T|}{\sqrt{1-T^{2}}\epsilon}+O(\epsilon).$
(4.49)

Figure 4: Solutions (4.46) shown on the cigar geometry (left) and its
projection (right) for a few sample values of $T$ interpolating between $T=0$
(red) and $T=-1$ (green). The semi-circles have the range
$\tau\in[-\beta/4,\beta/4]$.

Similarly, the modified Hayward terms from both corners are given by

$I^{\prime}_{H}=\frac{1}{\kappa^{2}}\int_{\tau=-\beta/4}\sqrt{\gamma}\theta_{0}+\frac{1}{\kappa^{2}}\int_{\tau=\beta/4}\sqrt{\gamma}\theta_{0}=\frac{2V_{x}}{\kappa^{2}\epsilon}\theta_{0},$
(4.50)

where the inner angles are

$\theta_{0}=\sin^{-1}\left(\sqrt{1-\frac{T^{2}}{1-\frac{4\pi^{2}\epsilon^{2}}{\beta^{2}}}}\right).$
(4.51)

Adding the two contributions gives the complexity for the holographic TFD
setup

$\mathcal{C}^{(e)}_{T}=-\frac{V_{x}\beta}{2\kappa^{2}\epsilon^{2}}+\frac{V_{x}}{\kappa^{2}\epsilon}\left(\frac{2|T|}{\sqrt{1-T^{2}}}+2\theta_{0}\right)+O(\epsilon).$
(4.52)

We can check that this expression is minimal for $T=0$ and again the
$\epsilon^{-2}$ divergence comes from the “reference metric” at $\Sigma$’s.
The minimal value is given by

$\mathcal{C}^{(e)}_{T=0}=-\frac{V_{x}\beta}{2\kappa^{2}\epsilon^{2}}+\frac{V_{x}}{\kappa^{2}}\frac{\pi}{\epsilon}.$
(4.53)

Note that this result has a similar structure to the vacuum example in
Poincare $AdS_{3}$ and the only dependence on the temperature enters via the
range of $\tau$: $L_{\tau}=\beta/2$. This is a similar behaviour to
holographic computations of complexity in three dimensions [36, 68] and
reflects the fact that in three dimensions all of the above metrics are
locally $AdS_{3}$ (see also App. D for more details on three-dimensional
solutions). We will see that in higher dimensions, black holes lead to a more
involved result for holographic path-integral complexity with respect to the
vacuum state.

###  4.3 Planar Black Holes in Higher Dimensions

Next, we can study an interesting generalization of the above computations to
the Euclidean planar black-hole metric in $d+1$ dimensions

$ds^{2}=\frac{1}{z^{2}}\left((1-Mz^{d})d\tau^{2}+\frac{dz^{2}}{1-Mz^{d}}+dx^{2}_{i}\right),$
(4.54)

where $M^{1/d}=(4\pi)/(d\,\beta)$ and the Euclidean time is periodic
$\tau\in[0,\beta]$ such that when we consider the wave function for the TFD
state, we focus on $\tau\in[-\frac{\beta}{4},\frac{\beta}{4}]$.
In the following, we compute the semi-classical Hartle-Hawking wave function
(3.8) using the usual algorithm. For the asymptotic boundary surface $\Sigma$
at $z=\epsilon$ we have the trace of the extrinsic curvature in this geometry

$K|_{\Sigma}=\frac{d\left(2-M\epsilon^{d}\right)}{2\sqrt{1-M\epsilon^{d}}}.$
(4.55)

On the other hand, for surface $Q$ defined by the embedding function
$z=f(\tau)$ the induced metric is

$ds^{2}=\frac{1}{f(\tau)^{2}}\left[\left(1-Mf^{d}+\frac{f^{\prime}(\tau)^{2}}{1-Mf^{d}}\right)d\tau^{2}+dx^{2}_{i}\right]\equiv
e^{2\phi}\left(dw^{2}+d^{2}x_{i}\right),$ (4.56)

where, as before for the TFD in two dimensions, we introduced the conformal
coordinate $w$ satisfying

$w^{\prime}(\tau)=\sqrt{1-Mf^{d}+\frac{f^{\prime}(\tau)^{2}}{1-Mf^{d}}}=\frac{1-Mf^{d}}{\sqrt{1-Mf^{d}-\dot{f}(w)^{2}}},$
(4.57)

and its range will be specified below.
The trace of the extrinsic curvature on $Q$ is now expressed in terms of
$f(\tau)$ as

$K|_{Q}=-\frac{1}{w^{\prime
3}}\left[ff^{\prime\prime}+\frac{d(2+Mf^{d})}{2(1-Mf^{d})}f^{\prime
2}+\frac{d}{2}\left(1-Mf^{d}\right)(2-Mf^{d})\right],$ (4.58)

or in terms of $\phi(w)=-\log(f(w))$ we get a generalization of the previous
expressions

$K|_{Q}=\frac{e^{-2\phi}\left(\ddot{\phi}+(d-1)\dot{\phi}^{2}\right)-d\left(1-\frac{1}{2}Me^{-d\phi}\right)}{\sqrt{1-Me^{-d\phi}-\dot{\phi}^{2}e^{-2\phi}}}.$
(4.59)

Then the on-shell action that computes the Hartle-Hawking wave function (3.8)
takes the standard form

$\displaystyle
I_{G}+I_{T}=-\frac{(d-1)V_{x}L_{\tau}}{\kappa^{2}}\left(\frac{1}{\epsilon^{d}}-\frac{dM}{2(d-1)}\right)-\frac{V_{x}}{\kappa^{2}}\int\frac{d\tau}{f^{d}}w^{\prime}(\tau)\left[K|_{Q}+\frac{1}{w^{\prime}(\tau)}-T\right].$
(4.60)

Finally, rewriting it in terms of $w$ and $\phi(w)$ and integrating by parts
we arrive at the main result for higher-dimensional black holes (4.54)

$\displaystyle I_{G}+I_{T}$ $\displaystyle=$
$\displaystyle-\frac{(d-1)V_{x}L_{\tau}}{\kappa^{2}}\left(\frac{1}{\epsilon^{d}}-\frac{dM}{2(d-1)}\right)+\frac{(d-1)V_{x}}{\kappa^{2}}\int
dwe^{d\phi}G(\dot{\phi})+\frac{V_{x}}{\kappa^{2}}\int dwe^{d\phi}T$ (4.61)
$\displaystyle-$
$\displaystyle\frac{V_{x}}{\kappa^{2}}\left[e^{(d-1)\phi}\arcsin\left(\frac{\dot{\phi}e^{-\phi}}{\sqrt{1-Me^{-d\phi}}}\right)\right]_{bdr},$

with a closed formula

$G(\dot{\phi})=\frac{1-\frac{d}{2(d-1)}Me^{-d\phi}}{1-Me^{-d\phi}}\sqrt{1-Me^{-d\phi}-\dot{\phi}^{2}e^{-2\phi}}+\dot{\phi}e^{-\phi}\arcsin\left(\frac{\dot{\phi}e^{-\phi}}{\sqrt{1-Me^{-d\phi}}}\right).$
(4.62)

This generalizes the result from $d=2$ and we can see that only for three-
dimensional geometry the pre-factor of the square-root trivializes leading to
a simpler equation of motion.
Indeed, the variation with respect to $\phi$ that gives the maximization
constraint can still be written in a compact form as

$\frac{d}{d-1}T-K|_{Q}=\frac{(d-2)Me^{-d\phi}\left(2e^{-2\phi}(\dot{\phi}^{2}-\ddot{\phi})+dMe^{-d\phi}\right)}{4(d-1)\left(1-Me^{-d\phi}-\dot{\phi}^{2}e^{-2\phi}\right)^{3/2}},$
(4.63)

where $K|_{Q}$ is given by (4.59). For $d=2$ the right hand side vanishes so
that we reproduce the condition for the CMC slice. Naively, we can expect that
CMC slices will no longer maximize the Hartle-Hawking wave functions in this
higher-dimensional background. This conclusion is too quick as we can see from
analyzing the full Neumann boundary condition.
Indeed, we can again consider the Neumann boundary condition (3.29) (rewritten
in terms of $\phi(w)$) that in this setup gives two constraints. First, the
$(\tau,\tau)$ components

$h_{\tau\tau}\left(T+(d-1)\sqrt{1-Me^{-d\phi}-\dot{\phi}^{2}e^{-2\phi}}\right)=0,$
(4.64)

and the $(a,b)$ components yield

$h_{ab}\left[T-\sqrt{1-Me^{-d\phi}-\dot{\phi}^{2}e^{-2\phi}}-K|_{Q}\right]=0.$
(4.65)

If we solve the first one, then the second is equivalent to the trace CMC
condition $K|_{Q}=\frac{d}{d-1}T$. Solving the first constraint is equivalent
to the equation

$\dot{\phi}^{2}+Me^{-(d-2)\phi}=\left(1-\frac{T^{2}}{(d-1)^{2}}\right)e^{2\phi},$
(4.66)

and for $T<0$, $\dot{\phi}$ also solves the second equation.
To solve (4.66), for convenience, we first shift

$\displaystyle\phi\to\tilde{\phi}-\frac{1}{2}\log\left(1-\frac{T^{2}}{(d-1)^{2}}\right),\qquad\tilde{M}_{d}=M\left(1-\frac{T^{2}}{(d-1)^{2}}\right)^{\frac{d-2}{2}},$
(4.67)

to absorb the constant on the right-hand side and write the equation in terms
of $\tilde{\phi}=-\log(F(w))$ as

$F^{\prime 2}+\tilde{M}_{d}F^{d}-1=0.$ (4.68)

This equation can be solved by standard methods and requires inverting the
integral

$w+c=\int^{F}\frac{dF}{\sqrt{1-\tilde{M}_{d}F^{d}}}=F\,_{2}F_{1}\left(\frac{1}{2},\frac{1}{d},\frac{d+1}{d},\tilde{M}_{d}F^{d}\right),$
(4.69)

in order to find $F(w)$. This way, in $d=2$ we recover the previous result
with $\sin$ function (4.31).
Interestingly, for $d=4$, we can also obtain analytic result in terms of
Jacobi sine Elliptic function $sn(x,-1)$. Namely, for $-3<T<0$, we find the
solution of the Neumann boundary condition and $K|_{Q}=4T/3$ given by

$e^{2\phi(w)}=\frac{\sqrt{M}}{\sqrt{1-\frac{T^{2}}{9}}sn^{2}\left(M^{1/4}\left(1-\frac{T^{2}}{9}\right)^{1/4}w,-1\right)}.$
(4.70)

This metric is a new result that on top of constant $K|_{Q}$ has a constant
Ricci scalar (3.47) for $d=4$ hence, after the map (3.32), it is also a
solution of the four dimensional path-integral optimization problem (2.20) for
the TFD state.
Let us now fix the appropriate range of $w$. Since the real periodicity of
$sn(x,m)$ is $4K(m)$, we can consider the 4d counterpart of the previous TFD
solution by taking $w$ in the interval

$w\in\left[c_{1},\frac{2K(-1)}{M^{1/4}\left(1-\frac{T^{2}}{9}\right)^{1/4}}-c_{1}\right],$
(4.71)

where $K(-1)=K(m=-1)$ is a numerical constant and $c_{1}$ is expressed in
terms of the inverse Jacobi $sn^{-1}$ as

$c_{1}=\frac{1}{M^{1/4}\left(1-\frac{T^{2}}{9}\right)^{1/4}}sn^{-1}\left(\frac{M^{1/4}\epsilon}{\left(1-\frac{T^{2}}{9}\right)^{1/4}},-1\right)\simeq\frac{\epsilon}{\sqrt{1-\frac{T^{2}}{9}}}+O(\epsilon^{5}),$
(4.72)

so that at the end points we satisfy the boundary condition $f(w)=\epsilon$.
Finally, this analytic result allows us to evaluate the holographic path-
integral complexity from the Hartle-Hawking wave function in this five-
dimensional black-hole geometry. The on-shell action without the tension term
can now be computed as

$\displaystyle I_{G}$ $\displaystyle=$
$\displaystyle-\frac{3V_{x}L_{\tau}}{\kappa^{2}}\left(\frac{1}{\epsilon^{4}}-\frac{2M}{3}\right)-\frac{V_{x}}{\kappa^{2}}\int
dwe^{4\phi}\left[\frac{4}{3}T+\frac{1}{w^{\prime}(\tau)}\right],$ (4.73)
$\displaystyle=$
$\displaystyle-\frac{3V_{x}L_{\tau}}{\kappa^{2}}\left(\frac{1}{\epsilon^{4}}-\frac{2M}{3}\right)+\frac{2V_{x}}{3\kappa^{2}}\left[\frac{|T|}{\sqrt{1-\frac{T^{2}}{9}}\epsilon^{3}}+\frac{|T|\,M^{3/4}K(-1)}{\left(1-\frac{T^{2}}{9}\right)^{5/4}}\right]+O(\epsilon),$

where $L_{\tau}=\beta/2=\pi/(2M^{1/4})$.
Moreover, the modified Hayward term again consists of two equal contributions
from each inner-angle

$I^{\prime}_{H}=2\frac{V_{x}}{\kappa^{2}\epsilon^{3}}\theta_{0},\qquad\theta_{0}=\sin^{-1}\left(\sqrt{1-\frac{T^{2}}{9\left(1-M\epsilon^{4}\right)}}\right).$
(4.74)

Adding two contributions we can verify that the Euclidean complexity
$\mathcal{C}^{(e)}_{T}$ is minimized for $T=0$. Its minimal value is given by

$\mathcal{C}^{(e)}_{T=0}=\frac{V_{x}}{\kappa^{2}}\left(-\frac{\beta}{2\epsilon^{4}}+\frac{\pi}{\epsilon^{3}}+\frac{\pi^{4}}{3\beta^{3}}\right)+O(\epsilon).$
(4.75)

Interestingly, unlike in $d=2$, we obtain a constant ($\epsilon^{0}$),
$\beta$-dependent term that increases the value of complexity for large
temperatures. In fact, as also noted in [68], the constant term is
proportional to the entropy.181818We thank the anonymous referee for pointing
this to us.
For other dimensions, inverting the hypergeometric function would have to be
done numerically and we leave this analysis as an interesting future problem.
It is also useful to comment on the relation between the on-shell values of
our Euclidean complexity actions and the complexity equals volume [36]
computations. Since, on-shell, the extrinsic curvature is proportional to $T$,
our complexity action contains the contribution from the pure volume of $Q$ as
well as from the bulk and the Hayward term. However, once we further minimize
over $T$, the contributions become more difficult to disentangle.
Nevertheless, e.g. similarity with [68] may suggest that the final result
contains (parts of) the CV answer and we leave determining these details and
connections with holographic complexity conjectures as an interesting future
problem.

##  5 JT Gravity

In this section we will test our construction in the context of Euclidean JT
gravity dual to the SYK model [69, 70, 71, 72, 73, 74]. Most of the
computations presented above goes through in this two dimensional gravity
model as well. However, it turns out that in order to naturally generalize our
discussion, it is more advantageous to introduce the tension term on surface
$Q$ by coupling to the dilaton191919From the three dimensional origin of JT
action such term naturally arises by adding a brane with tension in 3d in
which $\sqrt{h}$ yields the dilaton field. field $\Phi$, such that the action
computing the Hartle-Hawking wave functions (3.8) in JT will be given by

$\displaystyle I_{JT}+I_{T_{\Phi}}$ $\displaystyle=$
$\displaystyle-\Phi_{0}\chi(M)-\int_{M}\sqrt{g}\Phi(R+2)-2\int_{\partial
M}\sqrt{h}\Phi K+2T_{\Phi}\int_{Q}\sqrt{h}\Phi,$ (5.1)

with $\chi(M)$ being the Euler characteristic of the region $M$.
On-shell solutions of this theory (again we work in the probe limit and have
in mind solutions of pure JT gravity without $T_{\Phi}$) have constant
negative curvature $R=-2$ and dilaton equation of motion is set by the
vanishing energy-momentum tensor. As an example solution, we can take the 2d
Schwarzschild metric with dilaton profile

$ds^{2}=(r^{2}-r^{2}_{h})d\tau^{2}+\frac{dr^{2}}{r^{2}-r^{2}_{h}},\qquad\Phi(r)=\Phi_{b}r.$
(5.2)

We will take $r_{h}=2\pi/\beta$ and be interested in
$\tau\in[-\beta/4,\beta/4]$. By analogy to higher dimensions, we consider
region $M$ bounded by $\Sigma$ at $r=\frac{1}{\epsilon}\to\infty$ and $Q$
specified by $r=1/f(\tau)$. The induced metric on $Q$ becomes

$ds^{2}=e^{2\phi}dw^{2},\qquad e^{\phi}=f^{-1},$ (5.3)

where $w^{\prime}$ is given by the same expression as in (4.25).
Following the same steps as before, we can derive the on-shell action that
computes the semi-classical Hartle-Hawking wave function

$\displaystyle
I_{JT}+I_{T_{\Phi}}=-\frac{2L_{\tau}\Phi_{b}}{\epsilon^{2}}-2\Phi_{0}\int
dwe^{\phi}\left[K|_{Q}+\frac{1}{w^{\prime}}\right]-2\Phi_{b}\int
dwe^{2\phi}\left[K|_{Q}-T_{\Phi}\right],$ (5.4)

with $w^{\prime}$ as in (4.25) and

$K|_{Q}=\frac{\ddot{\phi}e^{-2\phi}-1}{\sqrt{1-e^{-2\phi}(\dot{\phi}^{2}+r^{2}_{h})}}.$
(5.5)

Similarly, the action written in terms of $w$ and $\phi(w)$ in the first-
derivative form can be obtained as

$\displaystyle I_{JT}+I_{T_{\Phi}}$ $\displaystyle=$
$\displaystyle-\frac{2L_{\tau}\Phi_{b}}{\epsilon^{2}}+2\Phi_{b}\int_{Q}e^{2\phi}G(\dot{\phi})+2\Phi_{b}\int_{Q}e^{2\phi}T_{\Phi}$
(5.6) $\displaystyle-$ $\displaystyle
2\left[\left(\Phi_{0}+\Phi_{b}e^{\phi}\right)\arcsin\left(\frac{\dot{\phi}e^{-\phi}}{\sqrt{1-r^{2}_{h}e^{-2\phi}}}\right)\right]_{bdr},$

with monotonic a function of $\dot{\phi}$

$\displaystyle G(\dot{\phi})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{1-e^{-2\phi}(\dot{\phi}^{2}+r^{2}_{h})}}{1-r^{2}_{h}e^{-2\phi}}+\dot{\phi}e^{-\phi}\arcsin\left(\frac{\dot{\phi}e^{-\phi}}{\sqrt{1-r^{2}_{h}e^{-2\phi}}}\right).$
(5.7)

With the intuition from previous examples we can again consider the UV limit.
Interestingly, for small $\dot{\phi}$ as well as small $r_{h}$, we get the
following action in the leading order

$I_{JT}|_{UV}=\Phi_{b}\int\left(\dot{\phi}^{2}+2e^{2\phi}\right)+...$ (5.8)

With the usual replacement of the reparametrization function
$e^{\phi}\sim\tilde{w}^{\prime}(w)$, this action precisely describes the
Schwarzian sector of the SYK model (compare e.g. [72, 73]).

The saddle point equation from the maximization of the wave-function is again
equivalent to the Neumann boundary condition. More explicitly, varying the
action with respect to $\phi$ yields the equation

$K|_{Q}-T_{\Phi}=-\frac{\left(1-\dot{\phi}^{2}e^{-2\phi}\right)\left(r^{2}_{h}+\left(\dot{\phi}^{2}-\ddot{\phi}\right)\right)e^{-2\phi}}{2\left(1-e^{-2\phi}(r^{2}_{h}+\dot{\phi}^{2})\right)^{3/2}},$
(5.9)

where $K|_{Q}$ is given by (5.5).
We can verify that for $-1<T_{\Phi}<0$, this equation is solved by

$e^{2\phi}=\frac{r^{2}_{h}}{(1-T^{2}_{\Phi})\cos^{2}\left(r_{h}w\right)},$
(5.10)

with

$w\in\left[-\frac{\beta}{4}+\frac{\epsilon}{\sqrt{1-T^{2}}},\frac{\beta}{4}-\frac{\epsilon}{\sqrt{1-T^{2}}}\right].$
(5.11)

Similarly to the higher-dimensional planar black holes, the right hand side of
(5.9) vanishes on-shell and the surface is again described by the CMC slice
$K|_{Q}=T_{\Phi}$. This solution is a natural generalization of its higher
dimensional TFD cousins.

Finally, we can compute holographic path-integral complexity in the Euclidean
JT model. First, we evaluate the on-shell JT gravity action without tension
$T_{\Phi}$ term

$\displaystyle I_{JT}$ $\displaystyle=$
$\displaystyle-\frac{\beta\Phi_{b}}{\epsilon^{2}}-2\Phi_{0}\int
dwe^{\phi}\left[T_{\Phi}+\frac{1}{w^{\prime}}\right]-2\Phi_{b}T_{\Phi}\int
dwe^{2\phi},$ (5.12) $\displaystyle=$
$\displaystyle-\frac{\beta\Phi_{b}}{\epsilon^{2}}+\frac{4|T_{\Phi}|\Phi_{b}}{\sqrt{1-T^{2}_{\Phi}}\epsilon}-4\Phi_{0}\left(\pi-\arccos(T_{\Phi})\right).$

On the other hand, the internal angles $\theta_{0}$ that appear in the
modified Hayward terms are now related to $T_{\Phi}$ by

$\theta_{0}=\arcsin\left(\sqrt{1-T^{2}_{\Phi}}\right).$ (5.13)

From the form of the JT action (5.1), it is then natural to generalize the
modified Hayward term from section 3.4 to include the coupling to $\Phi_{0}$
as well as the dilaton $\Phi(r)$ at each corner202020See also e.g. [74] where
standard Hayward/corner terms in JT gravity were included in the same form and
[75] for related analysis.

$I^{\prime}_{H}=2\sum_{i}\theta^{i}_{0}\left(\Phi_{0}+\Phi(r)\right)|_{C_{i}}=2\left(\Phi_{0}+\frac{\Phi_{b}}{\epsilon}\right)2\theta_{0},$
(5.14)

where $C_{i}$ stands for the $i$-th corner in the wedge region $M$. In fact
this term plays the same role as its higher-dimensional counterparts and, when
added to the JT action, cancels the boundary term in (5.6).
This way, the holographic path-integral complexity action in JT model becomes

$\displaystyle\mathcal{C}^{(e)}_{T}$ $\displaystyle=$
$\displaystyle-\frac{\beta\Phi_{b}}{\epsilon^{2}}+\frac{2\Phi_{b}}{\epsilon}\left(\frac{2|T_{\Phi}|}{\sqrt{1-T^{2}_{\Phi}}}+2\theta_{0}\right)$
(5.15) $\displaystyle-$ $\displaystyle
4\Phi_{0}\left(\pi-\cos^{-1}(T_{\Phi})-\theta_{0}\right).$

We can then check that after inserting (5.13) the second line of this
expression, as well as dependence on $\Phi_{0}$, vanishes. Therefore,
analogously to the TFD example in the previous sections, the complexity is
mimimized at $T_{\Phi}=0$. The minimum value is given by

$\mathcal{C}^{(e)}_{T=0}=-\frac{\beta\Phi_{b}}{\epsilon^{2}}+\frac{2\pi\Phi_{b}}{\epsilon}.$
(5.16)

With this example we finish the analysis of the Euclidean Hartle-Hawking wave
functions and confirm our procedure, including the estimation of holographic
path-integral complexity for $T\neq 0$ slices, across dimensions. In the next
section, we will proceed to analyze gravitational path-integrals in Lorentzian
geometries.

##  6 Lorentzian Spacetimes

The main important motivation for generalizing the above construction to
Lorentzian signature has to do with understanding how general, time-dependent,
holographic geometries emerge from CFT. An interesting clue in this direction
was given in [25] that proposed to interpret general slices of holographic
spacetimes as different quantum circuits constructed from path-integrals on
these geometries (Euclidean or Lorentzian) that prepare quantum states in CFTs
(see also [18, 19]). Nevertheless, in order to test this proposal, the
optimization of Lorentzian path-integrals in CFTs requires a much better
understanding. In this section, with the gravitational construction at hand,
we will first perform the Lorentzian computations in $AdS$ and then use them
to shed some top-down light on how one could advance the optimization of path-
integrals in Lorentzian CFTs.

Moreover, the advantage of the Hartle-Hawking approach is that we can apply it
also to gravity on a de Sitter (dS) spacetime. Up to date, there have been
various attempts to formulate holographic duals of de Sitter gravity, the so
called dS/CFT correspondence [81, 82], in which the dual CFTs are usually
argued to be associated with the future spacelike boundary (refer also to
other approaches e.g. [15, 83]). However, comparing AdS/CFT, understanding of
dS/CFT holography has been highly limited and it is fair to say that the
validity of the holographic duality with $dS$ itself is under debate at
present. Therefore, working under the assumption that the dual theory is
associated with the spacelike boundary region $\Sigma$, we hope that the
Hartle-Hawking approach may provide a new way to probe how holography on de
Sitter spaces looks like. In this section, we will also analyze Hartle-Hawking
wave functions in de Sitter spacetimes and extract new lessons for possible
path-integral optimization in dS/CFT.

Conceptually, the standard Lorentzian path-integral in gravity is computing
transition amplitudes between two geometries: some reference metric $g^{(1)}$
and a different metric $g^{(2)}$ (see e.g. [84] for interesting applications
in quantum gravity). If we naturally extend our proposal to Lorentzian
signature, the gravitational path-integral will be evaluating transition
amplitudes between a geometry on a timelike boundary $\Sigma$ of $AdS_{d+1}$
(or spacelike boundary $\Sigma$ for $dS_{d+1}$) and a timelike or spacelike
surface $Q$ with tension $T$ up to which we integrate the bulk gravity action

$\Psi^{(T)}_{HH}[g^{(2)},g^{(1)}]=\langle
g^{(2)}|g^{(1)}\rangle^{(T)}=\int[Dg_{\mu\nu}]e^{i\,(I_{G}+I_{T})}\delta(g|_{Q}-g^{(2)})\delta(g|_{\Sigma}-g^{(1)}),$
(6.1)

where $I_{G}$ is the Lorentzian Einstein-Hilbert action with the Gibbons-
Hawking term and $I_{T}$ is the tension term on $Q$. With a slight abuse of
language, we will still refer to these transition amplitudes as Hartle-Hawking
wave functions.

To make progress, we will again only be able to compute the Lorentzian Hartle-
Hawking wave function semi-classically by the gravity on-shell action with the
tension term. Namely, after introducing the conformal coordinate $w$ that will
put the metric on $Q$ (specified by embedding homogeneous function $f(t)$) in
the Weyl Minkowski (for timelike $Q$) or Weyl flat (for spacelike $Q$) form,
we will evaluate the Hartle-Hawking wave functional of the Weyl factor
$\phi(w)=-\log(f(w))$ as

$\Psi^{T}_{HH}[\phi]\sim e^{i\left(I_{G}+I_{T}\right)}|_{on-shell}.$ (6.2)

Then, the “maximization” will be performed by finding the extremum of the
classical action with respect to $\phi$ with analogous boundary condition to
(3.30).
Last but not least, we propose a generalization of the holographic path-
integral complexity from Hartle-Hawking wave functions to Lorentzian signature
given by

$\mathcal{C}^{(l)}_{T}=-\left(I_{G}+I^{\prime}_{H}\right),$ (6.3)

where $I_{G}$ is the Lorentzian gravity action without the tension term and
$I^{\prime}_{H}$ stands for an appropriate modified Lorentzian Hayward term
that, as in the Euclidean case (3.40), will be careful computed for spacelike
and timelike surfaces $Q$ below. The overall minus sign in (6.3) is chosen
such that it is consistent with the holographic analysis in [25].

Before we proceed with explicit examples, let us also review the argument with
the Hamiltonian constraint in gravity and the curvature of the slices that
maximize the Hartle-Hawking wave function. In Lorentzian signature, we will be
interested in slices of solutions to Einstein’s equations in $d+1$ dimensions
with negative ($AdS_{d+1}$) or positive $(dS_{d+1})$ cosmological constants

$G_{\mu\nu}=-\Lambda^{(d+1)}_{AdS/dS}g_{\mu\nu}.$ (6.4)

We will then consider arbitrary non-null slices $Q$ with normal vector
$n^{\mu}$ such that $n_{\mu}n^{\mu}=\varepsilon$, where $\varepsilon=+1$
corresponds to timelike and $\varepsilon=-1$ to spacelike slices respectively.
We can again project Einstein’s equations to the normal components and derive
a local “Hamiltonian” constraint (see e.g. [61])

$\varepsilon\left(K^{2}-K^{ij}K_{ij}\right)=R^{(d)}-2\Lambda^{(d+1)}_{AdS/dS}.$
(6.5)

As will see below, the maximization of the Hartle-Hawking wave functions will
again be equivalent to imposing Neumann boundary condition on $Q$, that
together with its trace being constant, implies (3.49). This way, from the
Hamiltonian constraint, we can derive that Ricci scalar curvature of slices
$Q$ is constant and given by

$R^{(d)}=2\Lambda^{(d+1)}_{AdS/dS}+\varepsilon\frac{d}{d-1}T^{2}.$ (6.6)

We will verify this formula in the examples below and indeed see that timelike
CMC slices of both $AdS_{d+1}$ and $dS_{d+1}$ are positively curved whereas
spacelike CMC slices in these geometries have constant negative curvatures. We
will then speculate on possible extensions of the path-integral optimization
to Lorentzian CFTs that could give rise to continuous tensor networks with
both of these constant curvatures.

###  6.1 Semi-classical wave functions in $AdS_{d+1}$

In this section we evaluate the semi-classical Hartle-Hawking wave function
(6.2) in $AdS_{d+1}$ using the on-shell Lorentzian action

$I_{G}+I_{T}=\frac{1}{2\kappa^{2}}\int_{M}\sqrt{-g}\left(R-2\Lambda\right)+\frac{1}{\kappa^{2}}\int_{Q}\sqrt{|h|}(K-T)+\frac{1}{\kappa^{2}}\int_{\Sigma}\sqrt{|h|}K.$
(6.7)

For simplicity, we analyze the $d+1$-dimensional $AdS$ in Poincare coordinates
dual to the vacuum of Lorentzian $CFT_{d}$

$ds^{2}=l^{2}\frac{dz^{2}-dt^{2}+dx^{2}_{i}}{z^{2}},$ (6.8)

where, in order to compare the results with those in de-Sitter spacetime in
the next section, we keep the radius $l$ of $AdS_{d+1}$ explicit.

As in the Euclidean computation, we focus on the asymptotic region $M$ of
$AdS_{d+1}$ between $\Sigma$: at $z=\epsilon$, that is timelike, and a
spacelike or timelike surface $Q$ in the bulk defined by radial coordinate as
a function of the boundary time $t$: $z=f(t)$. In M, we compute the bulk
action together with boundary terms and tension (6.7) that will give the
Lorentzian semi-classical Hartle-Hawking wave function $\Psi^{(T)}_{HH}[\phi]$
in (6.2).212121Again we keep the boundary condition on $\Sigma$ implicit.

To evaluate the result, we will need the trace of the extrinsic curvature at
$\Sigma$ given by

$K|_{\Sigma}=\frac{d}{l}.$ (6.9)

Then, we will have to pay a special attention to the signature of the induced
metric on $Q$. More precisely, the induced metric on $Q$: $z=f(t)$ is given by

$ds^{2}=l^{2}\frac{-(1-f^{\prime}(t)^{2})dt^{2}+dx^{2}_{i}}{f^{2}(t)},$ (6.10)

and in what follows, we will distinguish two cases: timelike when
$|f^{\prime}(t)|<1$ and spacelike when $|f^{\prime}(t)|>1$, and analyze them
separately below.

####  6.1.1 Timelike $Q$

In the timelike case with $|f^{\prime}(t)|<1$, we write the metric (6.10) on Q
as

$ds^{2}=\frac{l^{2}}{f(w)^{2}}\left(-dw^{2}+dx^{2}_{i}\right)\equiv
e^{2\phi(w)}\left(-dw^{2}+dx^{2}_{i}\right),$ (6.11)

where we defined coordinate $w$ with the range $-\infty<w\leq 0$, related to
$t$ by

$w^{\prime}(t)=\sqrt{1-f^{\prime}(t)^{2}}=\frac{1}{\sqrt{1+\dot{f}(w)^{2}}}.$
(6.12)

The trace of the extrinsic curvature on timelike $Q$ is then given by

$K|_{Q}=\frac{f(t)f^{\prime\prime}(t)-d(1-f^{\prime}(t)^{2})}{l(1-f^{\prime}(t)^{2})^{3/2}},$
(6.13)

or in terms of $w$ and $\phi(w)$

$K|_{Q}=-\frac{le^{-2\phi}(\ddot{\phi}+(d-1)\dot{\phi}^{2})+\frac{d}{l}}{\sqrt{1+l^{2}\dot{\phi}^{2}e^{-2\phi}}}.$
(6.14)

With these ingredients, the Lorentzian on-shell action that computes the
Hartle-Hawking wave function (“transition amplitude”)(6.2) between timelike
$\Sigma$ and timelike $Q$ can be written as

$I_{G}+I_{T}=\frac{(d-1)l^{d-1}}{\kappa^{2}}\frac{V_{x}L_{t}}{\epsilon^{d}}+\frac{V_{x}l^{d}}{\kappa^{2}}\int\frac{dt}{f(t)^{d}}w^{\prime}(t)(K|_{Q}+\frac{1}{lw^{\prime}(t)}-T),$
(6.15)

with all the expressions spelled in terms of $t$ and $f(t)$ above and $V_{x}$
and $L_{t}$ defined as in the Euclidean examples. We can again rewrite this
action in terms of $w$ and $\phi(w)$ as

$\displaystyle
I_{G}+I_{T}=\frac{(d-1)l^{d-1}}{\kappa^{2}}\frac{V_{x}L_{t}}{\epsilon^{d}}+\frac{V_{x}}{\kappa^{2}}\int
dwe^{d\phi(w)}\left[-\frac{le^{-2\phi}(\ddot{\phi}+(d-2)\dot{\phi}^{2})+\frac{d-1}{l}}{\sqrt{1+l^{2}\dot{\phi}^{2}e^{-2\phi}}}-T\right].$
(6.16)

Integrating by parts, allows to write the action in the first derivative form
and we arrive at

$\displaystyle I_{G}+I_{T}$ $\displaystyle=$
$\displaystyle\frac{(d-1)l^{d-1}}{\kappa^{2}}\frac{V_{x}L_{t}}{\epsilon^{d}}+\frac{(d-1)V_{x}}{l\kappa^{2}}\int
dwe^{d\phi(w)}G(\dot{\phi})-\frac{V_{x}}{\kappa^{2}}\int dwe^{d\phi(w)}T$
(6.17) $\displaystyle-$
$\displaystyle\frac{V_{x}}{\kappa^{2}}\left[e^{(d-1)\phi}\sinh^{-1}\left(l\dot{\phi}e^{-\phi}\right)\right]^{0}_{-\infty},$

with

$G(\dot{\phi})=l\dot{\phi}e^{-\phi}\sinh^{-1}\left(l\dot{\phi}e^{-\phi}\right)-\sqrt{1+l^{2}\dot{\phi}^{2}e^{-2\phi}}.$
(6.18)

This is the main result for timelike surfaces $Q$ in Lorentzian $AdS_{d+1}$.

To gain some intuition on what to expect from the Lorentzian path-integral
optimization, we can again use this result to extract the CFT action in the UV
limit. Expanding the answer for small $\dot{\phi}$ we derive

$e^{d\phi}G(\dot{\phi})\simeq\frac{1}{2}l^{2}e^{(d-2)\phi}\left(\dot{\phi}^{2}-\frac{2}{l^{2}}e^{2\phi}+O(\dot{\phi}^{4})\right).$
(6.19)

In two-dimensions, this limit reproduces the standard Lorentzian Liouville
action $S^{(l)}_{L}$, which takes the following form222222If we include the
tension term also, then $I_{G}+I_{T}$ gives the cosmological constant
$\mu=2+\frac{2T}{d-1}$, where we set $l=1$. Note that for $T<-(d-1)$ which is
assumed in the present case, we find $\mu<0$. This is the reason why we can
have the dSd solutions with positive Ricci scalar. However for the evaluation
of complexity (6.3) we always need to remove the tension term contribution as
we did in Euclidean case, which leads to $\mu=2$.

$\displaystyle S^{(l)}_{L}=\frac{c}{24\pi}\int
d^{2}x\left[(\partial_{t}\phi)^{2}-(\partial_{x}\phi)^{2}-\mu
e^{2\phi}\right].$ (6.20)

The Euclidean continuation of this action is the standard Liouville action
(2.5). We will come back to this action at the end of this section.
The equation of motion from the maximization of the Hartle-Hawking wave
function with (6.17) becomes

$-\frac{le^{-2\phi}(\ddot{\phi}+(d-1)\dot{\phi}^{2})+\frac{d}{l}}{\sqrt{1+l^{2}\dot{\phi}^{2}e^{-2\phi}}}=\frac{d}{d-1}T.$
(6.21)

Comparing with (6.14), this is again the constraint for the CMC slices in
Lorentzian geometry (6.8)

$K|_{Q}=\frac{d}{d-1}T.$ (6.22)

The solution with correct boundary condition, $f(0)=\epsilon$, is obtained for
the negative range of the tension parameter $-\infty<T<-(d-1)/l$ and yields
the timelike metric on $Q$

$ds^{2}=\frac{l^{2}}{\left(-\sqrt{\frac{l^{2}T^{2}}{(d-1)^{2}}-1}w+\epsilon\right)^{2}}\left(-dw^{2}+dx^{2}_{i}\right).$
(6.23)

These are the Lorentzian d-dimensional de Sitter slices of $AdS_{d+1}$
geometry (6.8). Indeed, we can verify that the Ricci scalar of this metric is
constant positive

$R^{(d)}=\frac{d(d-1)}{l^{2}}\left(\frac{l^{2}T^{2}}{(d-1)^{2}}-1\right)=-2\Lambda^{(d+1)}_{AdS}\left(\frac{l^{2}T^{2}}{(d-1)^{2}}-1\right),$
(6.24)

which is consistent with the general result (6.6) from the gravity Hamiltonian
constraint.
After integrating (6.12), we can find the embedding function $f(t)$ of the
timelike $Q$ in original time coordinate

$z=f(t)=\sqrt{\frac{l^{2}T^{2}}{(d-1)^{2}}-1}\frac{d-1}{lT}t+\epsilon.$ (6.25)

These are half-planes that interpolate between the timelike boundary $\Sigma$
for $T\to-(d-1)/l$ and the null plane in the limit $T\to-\infty$, see blue
lines on Fig. 5. The inner angle (i.e. “rapidity”) between $\Sigma$ and $Q$ is
denoted by $\eta_{0}$ in blue. We can also verify that they satisfy the full
Neumann boundary condition.

Figure 5: Solutions of the maximization in Lorentzian $AdS_{d+1}$. Timelike
half-planes (6.25), shown in blue, interpolate between the boundary at
$T=-(d-1)/l$ and null plane, dashed red, in the limit of $T\to-\infty$. The
counterpart of the inner angle (“rapidity”) $\eta_{0}$ given by (6.28) is
shown in blue. Spacelike half planes (6.42), shown in green, interpolate
between the null sheet for $T\to-\infty$ and the $t=0$ slice at $T=0$. Their
inner angle $\tilde{\eta}_{0}$, given by (6.45), is shown in green.

Up to now, the above results are natural generalizations of the Euclidean
derivation so in the next step, in analogy with section 3.4, we would like to
define the Lorentzian holographic path-integral complexity
$\mathcal{C}^{(l)}_{T}$ and investigate which slices $Q$ are favoured by this
candidate for a relative measure of complexity.

Similarly to the Euclidean $AdS_{d+1}$, the dependence on tension $T$ in the
on-shell action with the tension term evaluated on (6.25) cancels. Therefore,
as before, in the definition of Lorentzian complexity we will only take
gravity action (without tension term) evaluated on region $M$. For the
solution (6.25) the answer becomes

$I_{G}=\frac{(d-1)l^{d-1}}{\kappa^{2}}\frac{V_{x}L_{t}}{\epsilon^{d}}+\frac{V_{x}l^{d-1}}{\kappa^{2}\epsilon^{d-1}}\frac{lT}{(d-1)\sqrt{\frac{l^{2}T^{2}}{(d-1)^{2}}-1}}.$
(6.26)

Again, for the correct variational principle we should add an appropriate
Hayward term in the definition of complexity. For consistency with the notion
of relative complexity, we will again require that the Hayward term does not
contribute to the complexity action when $Q\to\Sigma$. Moreover, mimicking the
Euclidean construction, we will require that the Lorentzian modified Hayward
term should cancel the boundary term in (6.17). These two conditions uniquely
single out the candidate in terms of the “inner” rapidity parameter $\eta_{0}$
defined by the two normal vectors $n_{\Sigma}$ and $n_{Q}$ (normal to $\Sigma$
and $Q$ respectively) at the AdS boundary follows

$I^{\prime}_{H}=\frac{1}{\kappa^{2}}\int_{\gamma}\sqrt{\gamma}\,\eta_{0},\qquad
n_{\Sigma}\cdot n_{Q}=-\cosh\eta_{0}.$ (6.27)

For the solution (6.25), this modified Hayward term is the following function
of the tension parameter

$I^{\prime}_{H}=\frac{V_{x}l^{d-1}}{\kappa^{2}\epsilon^{d-1}}\eta_{0},\qquad\sinh\eta_{0}=\sqrt{\frac{l^{2}T^{2}}{(d-1)^{2}}-1}.$
(6.28)

Finally, with these two ingredients we define the Lorentzian holographic path-
integral complexity as minus the sum of the two above contributions

$\mathcal{C}^{(l)}_{T}=-\left(I_{G}+I^{\prime}_{H}\right).$ (6.29)

The overall minus sign is fixed so that it is consistent with the previous
holographic analysis in [25]. We will test this proposal and its implications
in the Lorentzian examples below.

In our first example, the Lorentzian holographic path-integral complexity
becomes

$\mathcal{C}^{(l)}_{T}=-\frac{(d-1)l^{d-1}}{\kappa^{2}}\frac{V_{x}L_{t}}{\epsilon^{d}}-\frac{V_{x}l^{d-1}}{\kappa^{2}\epsilon^{d-1}}\left(\eta_{0}-\coth\eta_{0}\right).$
(6.30)

Interestingly, this action is minimized in the limit $\eta_{0}\to\infty$ or
$T\to-\infty$. This is the light sheet surface and, in fact, it was argued to
be naturally related to geometry on which Lorentzian path-integral corresponds
to MERA-type quantum circuits [19, 25]. Notice also that this optimization
limit is equivalent to maximizing the absolute value of the phase of the
Hartle-Hawking wave function at zero tension $T=0$. This can be regarded as a
Lorentzian counterpart of the optimization in the Euclidean case, where we
maximize the amplitude of the Hartle-Hawking wave function.
Observe that again we chose the modified Hayward term so that in the limit of
$Q\to\Sigma$ action vanishes. This is now obtained in the limit

$\eta_{0}\to\coth^{-1}\left(\frac{(d-1)L_{t}}{\epsilon}\right),$ (6.31)

that generalizes the Euclidean limit (3.43).

####  6.1.2 Spacelike $Q$

Now we briefly revisit the above analysis for the spacelike surfaces $Q$. In
this case we have instead the spacelike induced metric on $Q$ given by

$ds^{2}=\frac{l^{2}}{f(w)^{2}}\left(dw^{2}+dx^{2}_{i}\right)\equiv
e^{2\phi(w)}\left(dw^{2}+dx^{2}_{i}\right),$ (6.32)

where we defined coordinate $w$ with range $-\infty<w<0$ that puts the induced
metric in a conformally flat form and is expressed by relations

$w^{\prime}(t)=\sqrt{f^{\prime}(t)^{2}-1}=\frac{1}{\sqrt{\dot{f}(w)^{2}-1}}.$
(6.33)

The only new ingredient that we need to evaluate the action is the trace of
the extrinsic curvature of $Q$. It is given in terms of the embedding function
as

$K|_{Q}=-\frac{f^{\prime\prime}(t)f(t)+d(f^{\prime}(t)^{2}-1)}{l\left(f^{\prime}(t)^{2}-1\right)^{3/2}},$
(6.34)

or equivalently in terms of $w$ and $\phi(w)=\log(l/f(w))$ as

$K|_{Q}=-\frac{le^{-2\phi}(\ddot{\phi}+(d-1)\dot{\phi}^{2})-\frac{d}{l}}{\sqrt{l^{2}\dot{\phi}^{2}e^{-2\phi}-1}}.$
(6.35)

The on-shell action that computes (6.2) is given in terms of w and $\phi(w)$
as

$\displaystyle
I_{G}+I_{T}=\frac{(d-1)l^{d-1}}{\kappa^{2}}\frac{V_{x}L_{t}}{\epsilon^{d}}-\frac{V_{x}}{\kappa^{2}}\int
dwe^{d\phi(w)}\left[\frac{le^{-2\phi}(\ddot{\phi}+(d-2)\dot{\phi}^{2})-\frac{d-1}{l}}{\sqrt{l^{2}\dot{\phi}^{2}e^{-2\phi}-1}}+T\right].$
(6.36)

Finally, after integrating by parts, we derive the main result for spacelike
$Q$

$\displaystyle I_{G}+I_{T}$ $\displaystyle=$
$\displaystyle\frac{(d-1)l^{d-1}}{\kappa^{2}}\frac{V_{x}L_{t}}{\epsilon^{d}}-\frac{(d-1)V_{x}}{l\kappa^{2}}\int
dwe^{d\phi(w)}G(\dot{\phi})-\frac{V_{x}}{\kappa^{2}}\int dwe^{d\phi(w)}T$
(6.37) $\displaystyle-$
$\displaystyle\frac{V_{x}}{\kappa^{2}}\left[e^{(d-1)\phi}\sinh^{-1}\left(i\,l\dot{\phi}e^{-\phi}\right)\right]^{0}_{-\infty},$

with function $G(\dot{\phi})$ now given by

$G(\dot{\phi})=\sqrt{l^{2}\dot{\phi}^{2}e^{-2\phi}-1}-l\dot{\phi}e^{-\phi}\sinh^{-1}\left(i\,l\dot{\phi}e^{-\phi}\right).$
(6.38)

Observe that since these surfaces cannot reach the boundary surface $\Sigma$,
we cannot perform the UV expansion $l\dot{\phi}e^{-\phi}<<1$ for this action.
The equation of motion from (6.36) is now given by

$-\frac{le^{-2\phi}(\ddot{\phi}+(d-1)\dot{\phi}^{2})-\frac{d}{l}}{\sqrt{l^{2}\dot{\phi}^{2}e^{-2\phi}-1}}=\frac{d}{d-1}T,$
(6.39)

which is again the trace of the Neumann boundary condition
$K|_{Q}=\frac{d}{d-1}T$ requiring spacelike slices $Q$ to be CMC. For the
range of the tension parameter $-\infty<T<0$ we can solve it for $\phi(w)$
that gives the metric on the spacelike $Q$

$ds^{2}=\frac{l^{2}}{\left(-\sqrt{1+\frac{l^{2}T^{2}}{(d-1)^{2}}}w+\epsilon\right)^{2}}\left(dw^{2}+dx^{2}_{i}\right).$
(6.40)

These in turn are the $d$-dimensional hyperbolic slices of the Lorentzian
$AdS_{d+1}$ (6.8) and their constant negative Ricci curvature is given by

$R^{(d)}=-\frac{d(d-1)}{l^{2}}\left(1+\frac{l^{2}T^{2}}{(d-1)^{2}}\right)=2\Lambda^{(d+1)}_{AdS}\left(1+\frac{l^{2}T^{2}}{(d-1)^{2}}\right),$
(6.41)

which can be computed explicitly or using (6.6).
Integrating (6.33), we can find the embedding function expressed in original
coordinates

$z=f(t)=\sqrt{1+\frac{l^{2}T^{2}}{(d-1)^{2}}}\frac{d-1}{lT}t+\epsilon.$ (6.42)

These are half-planes that interpolate between the null sheet in the limit of
$T\to-\infty$ and the $t=0$ slice for $T=0$, shown by green lines on Fig. 5.

Let us now discuss the holographic path-integral complexity (6.3) for
spacelike $Q$. Firstly, the action without tension term in this case is given
in terms of the negative tension $T$ as

$I_{G}=\frac{(d-1)l^{d-1}}{\kappa^{2}}\frac{V_{x}L_{t}}{\epsilon^{d}}+\frac{V_{x}}{\kappa^{2}\epsilon^{d-1}}\frac{T}{(d-1)\sqrt{1+\frac{l^{2}T^{2}}{(d-1)^{2}}}}.$
(6.43)

Secondly, we add the modified Hayward term expressed by the rapidity
$\tilde{\eta}_{0}$ defined in terms of normal vectors $n_{\Sigma}$ to timelike
$\Sigma$ and $n_{Q}$ to spacelike $Q$ (see Fig. 5) as

$I^{\prime}_{H}=\frac{1}{\kappa^{2}}\int_{\gamma}\sqrt{\gamma}\,\tilde{\eta}_{0},\qquad
n_{\Sigma}\cdot n_{Q}=-\sinh\tilde{\eta}_{0}.$ (6.44)

For the solution (6.42) this modified Hayward term for spacelike $Q$ becomes

$I^{\prime}_{H}=\frac{V_{x}l^{d-1}}{\kappa^{2}\epsilon^{d-1}}\tilde{\eta}_{0}\qquad\sinh\tilde{\eta}_{0}=\frac{l|T|}{d-1}.$
(6.45)

Let us point that for the spacelike $Q$ this modified Hayward term was chosen
slightly different than for timelike $Q$.232323Perhaps it would then be more
appropriate to denote it as $\tilde{I}^{\prime}_{H}$ but we avoid that for the
sake of simplicity of our formulas and hope that it will be clear from the
context which term we use.. Indeed the on-shell value of the boundary term in
(6.37) is given by

$-\frac{V_{x}}{\kappa^{2}}\left[e^{(d-1)\phi}\sinh^{-1}\left(i\,l\dot{\phi}e^{-\phi}\right)\right]^{0}_{-\infty}=-\frac{V_{x}l^{d-1}}{\kappa^{2}\epsilon^{d-1}}\left(\sinh^{-1}\left(\frac{l|T|}{d-1}\right)+\frac{i\pi}{2}\right).$
(6.46)

Therefore, if we insisted on canceling it, we would rather have to add the
modified Hayward term (6.44) with rapidity $\eta_{0}$ related to
$\tilde{\eta}_{0}$ by $\eta_{0}=\tilde{\eta}_{0}+\frac{\pi}{2}i$. From the
perspective of our holographic complexity this would lead to complex on-shell
values of $\mathcal{C}^{(l)}_{T}$ with imaginary part independent on the
tension $T$. This complexity would clearly give the same solution for optimal
$T$ but we would need to take the absolute value for the final answer to be
real. Instead, since we do not consider the UV limit for spacelike $Q$ where
the above cancelation was a consistency check, for spacelike slices we propose
to evaluate real-valued holographic path-integral complexity in terms of
rapidity $\tilde{\eta}_{0}$. We will study this proposal below as well as in
the de Sitter example and leave its test to more complicated time-dependent
geometries for future works.

Finally, the relative holographic path-integral complexity between timelike
$\Sigma$ and spacelike $Q$ for the solution (6.42) is given by

$\mathcal{C}^{(l)}_{T}=-(I_{G}+I^{\prime}_{H})=-\frac{(d-1)l^{d-1}}{\kappa^{2}}\frac{V_{x}L_{t}}{\epsilon^{d}}-\frac{V_{x}l^{d-1}}{\kappa^{2}\epsilon^{d-1}}\left(\tilde{\eta}_{0}-\tanh\tilde{\eta}_{0}\right).$
(6.47)

This action is minimized for $\tilde{\eta}_{0}\to\infty$ or $T\to-\infty$,
where the surface $Q$ becomes light-like so the optimal slice coincides with
the time-like case (6.30). Notice also that the absolute phase of the Hartle-
Hawking wave function is maximized in this limit.

Let us also point the value of $\mathcal{C}^{(l)}_{T}$ for
$\tilde{\eta}_{0}=0$ or $T=0$ that is given by

$\mathcal{C}^{(l)}_{0}=-\frac{(d-1)l^{d-1}}{\kappa^{2}}\frac{V_{x}L_{t}}{\epsilon^{d}}.$
(6.48)

This result is different that Euclidean counterpart (3.42). Even though
naively one could expect that both relative complexities are concerned with
slice $Q$ with the same induced metric, (6.48) hints on a very different
nature of our Euclidean and Lorentzian complexities. We leave better
understanding of this nature in the language of Euclidean and Lorentzian
continuous tensor networks as an interesting future problem.

###  6.2 Semi-classical wave-functions in $dS_{d+1}$

In this section we will evaluate semi-classical Hartle-Hawking wave functions
(6.2) for $d+1$ dimensional de Sitter spacetimes, using the gravity action

$I_{G}+I_{T}=\frac{1}{2\kappa^{2}}\int_{M}\sqrt{-g}\left(R-2\Lambda\right)+\frac{1}{\kappa^{2}}\int_{Q}\sqrt{|h|}(K-T)+\frac{1}{\kappa^{2}}\int_{\Sigma}\sqrt{|h|}K.$
(6.49)

This time, we consider a classical solution given by the de Sitter spacetime
in the Poincare coordinates

$ds^{2}=l^{2}\frac{-dt^{2}+dx^{2}+dy^{2}_{i}}{t^{2}},$ (6.50)

where we again keep the radius $l$ of $dS_{d+1}$ explicit. This metric solves
vacuum Einstein’s equations with

$R=\frac{d(d+1)}{l^{2}},\qquad\Lambda=\Lambda^{(d+1)}_{dS}=\frac{d(d-1)}{2l^{2}}.$
(6.51)

Generalization of our construction to de Sitter spacetimes will involve region
$M$ between the spacelike boundary cut-off surface $\Sigma$ given by
$t=-\epsilon$, that we assume is again associated with the dual theory on the
fixed flat metric that specifies the state dual to (6.50),242424This seems
like a natural assumption but it is possible that holography for dS works in a
completely different manner as the AdS counterpart. and a surface $Q$ defined
by

$t=h(x)<0.$ (6.52)

The Hwartle-Hawking wavefunction will again implicitly depend on the fixed
flat metric on $\Sigma$ and will be an explicit functional of $h(x)$ (or
$\phi$) with respect to which we will extremize it.
Our choice (convention) is such that $M$ will be in the negative $t$ and
negative $x$ region of the geometry (see Fig. 6).
The trace of extrinsic curvature of $\Sigma$ will always be given by
$K_{\Sigma}=-d/l$. For surfaces $Q$ we again assume the translational
invariance in $y_{i}$ directions and write their infinite volumes as $V_{y}$.
Similarly to $AdS_{d+1}$, we will now consider the two cases of timelike and
spacelike $Q$’s separately.

####  6.2.1 Timelike $Q$

For the timelike surface $Q$ we get the induced metric

$ds^{2}=l^{2}\frac{-(h^{\prime 2}-1)dx^{2}+dy^{2}_{i}}{h^{2}}\equiv
e^{2\phi}\left(-dw^{2}+dy^{2}_{i}\right),$ (6.53)

where we introduced conformal coordinate $w$ with range $-\infty<w<0$. In this
example, it is related to $x$ by derivatives

$w^{\prime}(x)=\sqrt{h^{\prime 2}-1}=\frac{1}{\sqrt{\dot{h}^{2}-1}},$ (6.54)

where as in previous examples we denote $h^{\prime}=\partial_{x}h(x)$ and
$\dot{h}=\partial_{w}h(w)$.

The relevant trace of extrinsic curvature on $Q$ can then be computed in terms
of $h(x)$ or $\phi(w)$ as

$K|_{Q}=\frac{h^{\prime\prime}h+d(h^{\prime 2}-1)}{l(h^{\prime
2}-1)^{3/2}}=\frac{le^{-2\phi}(\ddot{\phi}+(d-1)\dot{\phi}^{2})-\frac{d}{l}}{\sqrt{l^{2}e^{-2\phi}\dot{\phi}^{2}-1}}.$
(6.55)

Observe that in accordance with [82], this result can be formally obtained
from its $AdS$ counterpart (6.14) by continuation $l_{dS}=-il_{AdS}$.
With these ingredients, we have the on-shell action in the Lorentzian semi-
classical Hartle-Hawking wave function (6.2) in de Sitter geometry (6.50)

$\displaystyle I_{G}+I_{T}$ $\displaystyle=$
$\displaystyle-\frac{(d-1)V_{y}L_{x}l^{d-1}}{\kappa^{2}\epsilon^{d}}+\frac{V_{y}}{\kappa^{2}}\int
dx\frac{l^{d}}{h^{d}}w^{\prime}(x)\left(K|_{Q}-\frac{1}{lw^{\prime}(x)}-T\right).$
(6.56)

More explicitly, we can write the action in terms of $w$ and $\phi(w)$ as

$I_{G}+I_{T}=-\frac{(d-1)V_{y}L_{x}l^{d-1}}{\kappa^{2}\epsilon^{d}}+\frac{V_{y}}{\kappa^{2}}\int
dwe^{d\phi}\left[\frac{le^{-2\phi}(\ddot{\phi}+(d-2)\dot{\phi}^{2})-\frac{d-1}{l}}{\sqrt{l^{2}e^{-2\phi}\dot{\phi}^{2}-1}}-T\right],$
(6.57)

and comparing to the AdS counterpart, we have a formal relation
$l_{dS}=-il_{AdS}$ for the action of $Q$.
As for the spacelike $AdS_{d+1}$, we can rewrite it in the first derivative
form as follows

$\displaystyle I_{G}+I_{T}$ $\displaystyle=$
$\displaystyle-\frac{(d-1)V_{y}L_{x}l^{d-1}}{\kappa^{2}\epsilon^{d}}+\frac{(d-1)V_{y}}{l\kappa^{2}}\int
dwe^{d\phi(w)}G(\dot{\phi})-\frac{V_{y}}{\kappa^{2}}\int dwe^{d\phi(w)}T$
(6.58) $\displaystyle+$
$\displaystyle\frac{V_{y}}{\kappa^{2}}\left[e^{(d-1)\phi}\sinh^{-1}\left(i\,l\dot{\phi}e^{-\phi}\right)\right]^{0}_{-\infty},$

with the function

$G(\dot{\phi})=\sqrt{l^{2}\dot{\phi}^{2}e^{-2\phi}-1}-l\dot{\phi}e^{-\phi}\sinh^{-1}\left(i\,l\dot{\phi}e^{-\phi}\right).$
(6.59)

The form of $G(\dot{\phi})$ is analogous to the one for spacelike surfaces in
Lorentzian $AdS_{d+1}$ (6.38) and the reason that we cannot expand it for
$\dot{\phi}e^{-\phi}$ is a manifestation of the fact that timelike slices in
Lorentzian de Sitter geometry will not reach the spacelike boundary $\Sigma$
(see Fig. 6).

The maximization of the Hartle-Hawking wave function again leads to the CMC
constraint in de Sitter geometry

$\frac{le^{-2\phi}(\ddot{\phi}+(d-1)\dot{\phi}^{2})-\frac{d}{l}}{\sqrt{l^{2}e^{-2\phi}\dot{\phi}^{2}-1}}=\frac{d}{d-1}T,$
(6.60)

and in equivalent to imposing the Neumann boundary condition on $Q$.

Figure 6: Solutions in Lorentzian $dS_{d+1}$. Timelike half-planes (6.63),
shown in blue, interpolate between the the timelike plane $x=0$ at $T=0$ and
null plane, dashed red, in the limit of $T\to+\infty$. Their inner angle
$\xi_{0}$ given by (6.65) is shown in blue. Spacelike half planes (6.79),
shown in green, interpolate between the spacelike boundary $\Sigma$ at
$T=(d-1)/l$ and the null sheet for $T\to+\infty$. Their inner angle
$\tilde{\xi}_{0}$, given by (6.81), is shown in green.

This equation with boundary condition $h(0)=-\epsilon$ can be solved for the
range of tension parameter $0<T<+\infty$ by $\phi(w)$ that corresponds to the
induced metric on $Q$

$ds^{2}=\frac{l^{2}}{\left(\sqrt{1+\frac{l^{2}T^{2}}{(d-1)^{2}}}w-\epsilon\right)^{2}}\left(-dw^{2}+dy^{2}_{i}\right).$
(6.61)

These timelike geometries are the $d$-dimensional de Sitter slices of the
$(d+1)$-dimensional de Sitter geometry (6.50). Their constant positive Ricci
scalar curvature is given by

$R^{(d)}=\frac{d(d-1)}{l^{2}}\left(1+\frac{l^{2}T^{2}}{(d-1)^{2}}\right)=\Lambda^{(d+1)}_{dS}\left(1+\frac{l^{2}T^{2}}{(d-1)^{2}}\right),$
(6.62)

which is indeed the result (6.6) for timelike slices in $dS_{d+1}$ spacetime.
Integrating (6.54), and finding $w(x)$ allows us to write the embedding
function for this surface given by

$t=h(x)=\sqrt{1+\frac{l^{2}T^{2}}{(d-1)^{2}}}\frac{d-1}{lT}x-\epsilon.$ (6.63)