# Cauchy Slice Holography:
A New AdS/CFT Dictionary

Goncalo<EMAIL_ADDRESS>, Rifath
<EMAIL_ADDRESS>, Aron C<EMAIL_ADDRESS>DAMTP,
University of Cambridge

###### Abstract

We investigate a new approach to holography in asymptotically AdS spacetimes,
in which time rather than space is the emergent dimension. By making a
sufficiently large $T^{2}$-deformation of a Euclidean CFT, we define a
holographic theory that lives on Cauchy slices of the Lorentzian bulk. (More
generally, for an arbitrary Hamiltonian constraint equation that closes, we
show how to obtain it by an irrelevant deformation from a CFT with suitable
anomalies.) The partition function of this theory defines a natural map
between the bulk canonical quantum gravity theory Hilbert space, and the
Hilbert space of the usual (undeformed) boundary CFT. We argue for the
equivalence of the ADM and CFT Hamiltonians. We also explain how bulk
unitarity emerges naturally, even though the boundary theory is not
reflection-positive. This allows us to reformulate the holographic principle
in the language of Wheeler-DeWitt canonical quantum gravity.

Along the way, we outline a procedure for obtaining a bulk Hilbert space from
the gravitational path integral with Dirichlet boundary conditions. Following
previous conjectures, we postulate that this finite-cutoff gravitational path
integral agrees with the $T^{2}$-deformed theory living on an arbitrary
boundary manifold—at least near the semiclassical regime. However, the
$T^{2}$-deformed theory may be easier to UV complete, in which case it would
be natural to take it as the definition of nonperturbative quantum gravity.

###### Contents

  1. 1 Introduction
    1. 1.1 The Unity of Physics
    2. 1.2 Holography at Finite Cutoff
    3. 1.3 Cauchy Slice Holography
    4. 1.4 WDW States
    5. 1.5 Relation to Tensor Network Models
    6. 1.6 Plan of Paper
  2. 2 The $T^{2}$ Deformation
    1. 2.1 Preliminaries
    2. 2.2 Asymptotics of the Flow
    3. 2.3 The General Deformation
    4. 2.4 Example: Scalar Field Coupled to Gravity
    5. 2.5 Normal Ordering Prescription
    6. 2.6 Exotic Properties of the Deformed Theory
  3. 3 Towards a Generalized Holographic Principle
    1. 3.1 Form of the Duality
    2. 3.2 The Need for a Contour Prescription
    3. 3.3 Challenges to the Validity of the GHP
    4. 3.4 $T^{2}$ as the Definition of the Bulk Theory?
  4. 4 QG Inner Product from the Path Integral
    1. 4.1 Lorentzian Transition Amplitude
    2. 4.2 Why we use both Signs for the Lapse
    3. 4.3 Example: Maximally Symmetric Slices
    4. 4.4 Properties of the Amplitude
  5. 5 Construction of the Bulk Hilbert Space
    1. 5.1 Quick and dirty version
    2. 5.2 Technical details for those who want more rigor
  6. 6 An Explicit AdS/CFT Dictionary
    1. 6.1 The Bulk $\to$ Boundary Map
    2. 6.2 Invertability of the Map
    3. 6.3 The Boundary $\to$ Bulk Map
  7. 7 Asymptotic Dynamics and Unitarity
    1. 7.1 Consistency of Time Evolution
    2. 7.2 ADM Hamiltonian $=$ CFT Hamiltonian
    3. 7.3 Emergence of Bulk Unitarity
  8. 8 Discussion
    1. 8.1 A New AdS/CFT Dictionary
    2. 8.2 Summary of Additional Results
    3. 8.3 Towards a UV completion
    4. 8.4 Holographic Cosmology
  9. A ADM Hamiltonian on Cauchy slices

##  1 Introduction

###  1.1 The Unity of Physics

When you start learning about physics, there seem to be a large number of
subfields that all seem quite different from each other. Later on, you begin
to notice the same mathematics popping up in different places. In particular,
there is a very deep structural analogy in physics, which relates mechanics,
thermodynamics, and field theory [1, 2]. One possible presentation of this
analogy is displayed in the table below:

Mechanics | Thermodynamics | Field Theory
---|---|---
($\vec{x}$, $\vec{p}$) | e.g. ($V$, $P$) | ($J$, $\mathcal{O}$)
positions, momenta | extensive, intensive | sources, operators
on-shell action $I$ | free energy $F$ | effective action $S$
$dI=p_{i}\,dx^{i}$ | $dF=P\,dV+\ldots$ | $dS=\int{\cal O}\,dJ$
QM: state $\Psi(x)$ | Stat. Mech: $Z(V\ldots)$ | QFT: $Z[J]$
space at time $t$ | space, in equilibrium | spacetime

In this analogy, the top row represents a $2N$ dimensional phase space on
which there is a naturally defined symplectic structure $\omega$. Such a phase
space naturally arises as the cotangent space of some configuration space of
variables, whenever you have a quantity which is extremized (second row). The
variation of this quantity gives a canonical 1-form (from which $\omega$ may
be derived by taking the exterior derivative). After “quantization” (or its
analogue “thermalization”), one ends up with a function on an $N$-dimensional
configuration space on which $\omega$ vanishes, which includes just _one half_
of the phase space (although there are rules for transforming $\Psi$ or $Z$ if
you wish to change your basis to a different $N$-dimensional configuration
space).444In the case of statistical mechanics, the entity playing the role of
the amplitude is the probability, which is constrained to be positive, unlike
the other two columns where the amplitude may be complex. This is the one
truly distinctive feature of Statistical Field Theory, as compared to QFT,
which is instead usually constrained to be unitary (or reflection-positive in
Euclidean signature). However, the $T^{2}$ deformed field theories considered
in this paper are not unitary!

The bottom row indicates the geometrical domain on which the symplectic
structure is defined. Strangely, this means that the relation between sources
and operators in a $d$ dimensional _spacetime_ field theory, are analogous to
concepts that involve _space alone_ in Thermodynamics or Mechanics.555It is of
course also possible to treat a relativistic field theory as a mechanical
system, by moving to a phase space over field variables, but this is not the
same as the third column. Hence, if we take this analogy seriously as a full-
fledged duality, it seems that there is a possibility of an emergent time
dimension!

This emergent time dimension is somewhat uninteresting in the case of
Thermodynamics, because in that case we normally assume the system is in
thermal equilibrium, so all expectation values remain the same as time passes.
On the other hand, if we reinterpret the ($J$, $\mathcal{O}$)’s in a field
theory as the $(\vec{x},\vec{p})$’s of a mechanical phase space, then it seems
that there is the possibility of a duality which implies an emergent _time
dimension_ with nontrivial dynamics.

In the quantum form of this duality, the _partition function_ of a field
theory in $d$ spatial dimensions would be dual to the _wavefunction_ of a QM
system which is defined in $d+1$ dimensions. Such a theory would therefore
obey a generalized form of the holographic principle:

$\displaystyle\Psi[J]=Z[J],$ (1)

where on the left side of the equation $J$ plays the role of a configuration-
space variable $x$.

The existence of an emergent time dimension is very reminiscent of the AdS/CFT
correspondence, in which a $d$-dimensional CFT is dual to a $d+1$-dimension
asymptotically AdS quantum gravity theory [3, 4, 5]. In this paper, we will
attempt to make this connection more precise.

As we shall see, in a class of field theories known as $T^{2}$-deformed
theories, this emergent dynamics is naturally described by a _Hamiltonian
constraint_ ${\cal H}(x)\Psi=0$, satisfied at each point $x$. Thus, the $d+1$
dimensional theory is a gravitational theory, even though $Z[J]$ was defined
on a fixed background (the metric $g_{ab}$ being one of the field theory
sources). This local constraint equation is inherited from the conformal
symmetry of the original, undeformed CFT.666More generally, in any field
theory, the existence of a scalar constraint equation acting on $Z$ may be
expected if there is a nontrivial action of a local renormalization group flow
on the field theory sources, so long as a CFT lies anywhere in the RG space of
the theory. Assuming the number of local constraint equations cannot change
discontinuously, this would imply that _any_ QFT which can be continuously
deformed to a CFT is dual to a quantum gravitational theory, although not
necessarily one with a good semiclassical regime. Hence, we are able to
reformulate the holographic principle in the language of Wheeler-DeWitt
canonical quantum gravity.

As usual in AdS/CFT, in order to have a good semiclassical regime, it is
necessary to assume that the field theory has a large number $N$ of color
degrees of freedom. Furthermore, the theory must be in an ‘t Hooft-like regime
[6], in which there exists a subset of “single-particle” operators which are
weakly coupled (i.e. approximately generalized free fields). The analogy to a
bulk phase space works best if we restrict our phase space to linear
combinations of single-particle sources $J_{\text{s.p.}}$ and single-particle
operators ${\cal O}_{\text{s.p.}}$.777We do not here use the more common term
“single-trace” since our $T^{2}$ deformation is double-trace (at leading
order) and as a result the single-particle states of the $T^{2}$ theory
correspond to multi-trace operators in the original CFT. Higher-particle
operators like $J_{\text{s.p.}}^{n}{\cal O}_{\text{s.p.}}^{m}$ are then
analogous to nonlinear product operators of the form $x^{n}p^{m}$. Such
operators obviously have to exist in the theory if the phase space is to have
an approximately classical limit. For reasons described in [7, 8, 9], writing
the beta functions in terms of single-trace operators is equivalent to
allowing quantum fluctuations over the bulk fields.888If one includes all
operators on equal footing, there is still an analogy between renormalization
and symplectic mechanics [10], but the Hamiltonian is linear in the momenta so
there are no fluctuation terms.

The most important source in the field theory is the background metric
$g_{ab}$, and its dual operator is the stress-tensor $T^{ab}$, which is a
single-particle operator in the weak-coupling regime. In the dual bulk
interpretation, these correspond to the _spatial_ metric and its canonically
conjugate ADM momentum $\Pi^{ab}$ respectively.

###  1.2 Holography at Finite Cutoff

In the context of AdS3/CFT2, it was observed that by deforming the CFT by an
irrelevant operator called $T\bar{T}$ which is quadratic in stress tensor, we
get a dual gravitational theory satisfying Dirichlet boundary conditions for
the metric, on a boundary $\partial\mathcal{M}$ which is now at finite
distance from the center of AdS [11]. In the case relevant to the AdS bulk
being pure gravity, Zamolodchikov showed that the $T\bar{T}$ operator has a
factorization property, which makes the spectrum of the theory on a cylinder
$S_{1}\times\mathbb{R}$ exactly solvable [12, 13].999This exact solubility is
presumably related to the fact that $D=3$ pure gravity has no local degrees of
freedom, since adding matter, e.g. a scalar field, to the $D=3$ bulk requires
additional double trace operators
($\mathcal{O}_{\text{s.t.}}\mathcal{O}_{\text{s.t.}}$) in the deformation,
which do not have this factorization property. In this case we must still
resort to large $N$. Also the factorization property fails to hold for a
curved background metric.

Later this deformation was generalized to $d>2$ dimensions [14, 15, 16, 17],
in which case it is called a $T^{2}$ deformation. In this case, the theory is
exactly solvable at leading order in a large $N$ expansion.

$\mu$$\partial\mathcal{B}$$\partial\mathcal{B}^{(\mu)}$ Figure 1: Illustration
of the usual finite-cutoff AdS/CFT duality. The $T^{2}$ deformed theory lives
on a finite radius brick wall, labelled by $\partial\mathcal{B}^{(\mu)}$.

In the usual way of thinking about the $T^{2}$ theory, one starts with a
Lorentzian boundary CFT. Turning on a finite value of the deformation
parameter $\mu$ makes it so the field theory lives on a cylinder at finite
sized bulk radius. On this cylinder, the metric satisfies Dirichlet boundary
conditions, so it is a “brick wall” on which quantum fluctuations are
suppressed, enclosing a finite volume interior (see Figure 1).

AdS/CFT is usually thought of as involving an emergent _space_
dimension.101010In the best understood versions of the AdS/CFT duality, there
are also some number of emergent compact (but large) Kaluza-Klein dimensions
that appear in the bulk theory, in order to bring the total number of bulk
dimensions to the $D=10$ of string theory or $D=11$ of M-theory. In this
article we ignore such extra dimensions, and treat the bulk theory as a KK
reduced $d+1$ dimensional theory. This means that, if we consider the case of
a pure gravitational theory in AdSd+1 (or a theory with a single scalar field
as in section 2.4), we are considering a model that does not, strictly
speaking, have a known CFT dual, although many properties of such field
theories are known. Although such CFTs probably do exist, most likely they
require fine-tuning to get a small cosmological constant, especially in the
absence of supersymmetry. We expect that with matter fields included, the
tower of KK fields appearing in realistic models can be consistently included
using our general approach to the deformation given in section 2.3. The usual
approach here is to consider “sideways” evolution in the bulk and treat
evolution with respect to $z$ as an analogue of the time direction [18, 19].
Despite the fact that this corresponds to a nonstandard Cauchy problem, one
can still make sense of the radial Wheeler-DeWitt evolution [20, 15, 21]. But
how might this relate to an emergent _real_ time dimension?

###  1.3 Cauchy Slice Holography

In this paper we will make a different choice and adopt a point of view where
the signature of the $d$-dimensional field theory remains Euclidean
throughout. Our perspective is similar to that of Caputa, Kruthoff, and
Parrikar [22]; however these authors end their deformation on the $t=0$
symmetric slice,111111Although these authors did suggest extending the flow
further, into Lorentzian signature, in their discussion section. See also [23]
where a spatial tensor network is introduced in a somewhat different way to
describe the $T^{2}$ deformation itself. while we wish to continue it all the
way to generic Lorentzian Cauchy slices.

2$\textbf{2}^{\textbf{*}}$$\mu$010 Figure 2: The evolution of the embedding of
the slice $\Sigma$, with a given fixed metric, under the deformation
parametrised by $\mu$, in a saddle-point approximation. The illustration shows
a (complexified) $\text{AdS}_{2}$ cross-section of the $\text{AdS}_{d+1}$
vacuum geometry. The Euclidean CFT boundary is labelled by 0, in a conformal
frame where its geometry is hyperbolic. Three different stages (0, 1 and
2/$\textbf{2}^{\textbf{*}}$) of the deformation are shown. Stage 0 corresponds
to no deformation of the CFT. As we perform the $T^{2}$ deformation the slice
moves inwards along the imaginary time direction, while still being embedded
in a Euclidean bulk space (stage 1). For a sufficiently large deformation
there is a phase transition to a Lorentzian saddle-point bulk geometry. By
time-reversal symmetry, the slice $\Sigma$ can be embedded in two different
ways in the same bulk geometry (with the same intrinsic metric). This
corresponds to stages 2 and $\textbf{2}^{\textbf{*}}$. From the perspective of
the field theory, this phase spontaneously breaks time-reversal (and therefore
$CPT$).

Starting with a _Euclidean_ holographic CFT (living at the infinite Euclidean
boundary), and then doing a $T^{2}$ deformation, we can move the slice
$\Sigma$ inward along the imaginary time direction.121212Since the $T^{2}$
deformation is irrelevant, the result depends on the choice of conformal frame
for the original CFT metric. After a large enough deformation, $\Sigma$ may
end up embedded in a Lorentzian signature spacetime (depending on the choice
of metric $g_{ab}$ on $\Sigma$). So in the end the dual holographic theory
lives not at spatial infinity, but rather it becomes a field theory living on
a _Cauchy slice_ of a Lorentzian AdS spacetime.131313In asymptotically AdS
spacetimes, $\Sigma$ is a Cauchy slice iff every complete timelike curve _that
does not go to the timelike boundary_ passes through $\Sigma$ exactly once.
(Alternatively, we can hold the deformation parameter $\mu$ fixed, in which
case the transition to Lorentzian signature occurs when the spatial metric
$g_{ab}$ is chosen suitably.) See Figure 2.141414A different way to try to
extend to Lorentzian signature metrics, involving constant mean curvature
slices, was sketched by [21]. However if our proposal is correct this
transition already happens in the $g_{ab}$ basis.

As we shall see, the $T^{2}$ deformed partition function can be used to define
a quantum gravity state on $\Sigma$. Since the Cauchy slice $\Sigma$ has a
boundary at infinity, its partition function $Z[\Sigma]$ also defines a state
in the usual holographic Lorentzian CFT (the one that lives at timelike
spatial infinity). This gives us an (in principle, complete) map between the
geometrodynamical degrees of freedom in the bulk, and the CFT degrees of
freedom on the boundary.

In other words, this form of the duality can be thought of a specification of
the AdS/CFT dictionary involving arbitrary canonical gravity states in the
bulk, including degrees of freedom that lie arbitrarily deep in the bulk. This
includes behind horizons, inside bags of gold [24], etc. Obviously, this is
useful information for purposes of studying the black hole information puzzle
[25, 26, 27, 28, 29, 30], which we will discuss further in section 7.3. This
picture makes it quite manifest how all degrees of freedom in the bulk must
“flow” outwards in a spacelike direction, so as to be encoded in the boundary.

These time slices $\Sigma$ satisfy Dirichlet boundary conditions, not because
there is a physical end-of-time brane on them, but merely to represent the
quantum gravity wavefunction in a “position basis” in which the metric on
$\Sigma$ has a definite value $g_{1}$. By picking two such definite metrics,
we will show that one can calculate the gravitational inner product $\langle
g_{2}|g_{1}\rangle$ either by means of a bulk gravitational path integral, or
by a field theory calculation as a partition function.

###  1.4 WDW States

For some researchers, one of the attractions of defining quantum gravity by
means of a holographic duality, is that they don’t have to get their hands
dirty playing around with quantizing the metric. Even after much work,
geometrodynamics has had limited success resolving fundamental questions such
as the information paradox, or what happens at the Planck scale.

That being said, there is no way to express the _equivalence_ between
geometrodynamics and holography, without defining both sides of the equation.
Thus, in this article, we cannot avoid quantizing gravity directly. In this
paper we will mention both canonical and path integral techniques, pointing
out potential difficulties and pitfalls as they arise.

In a canonical theory of Quantum Gravity [31], the state is described by a
superposition of metrics, $\Psi[g]$, which satisfies the Hamiltonian and
momentum constraints. In the case of pure GR these are given by:

$\mathcal{H}(x)\Psi[g]:=\Bigg{\\{}\frac{16\pi
G_{N}}{\sqrt{g}}:\\!\Big{(}\Pi_{ab}\Pi^{ab}-\frac{1}{d-1}\Pi^{2}\Big{)}\\!:-\frac{\sqrt{g}}{16\pi
G_{N}}(R-2\Lambda)\Bigg{\\}}\Psi[g]=0,$ (2)
$\mathcal{D}^{a}(x)\Psi[g]:=-2(\nabla_{b}\Pi^{ab})\Psi[g]=0,$ (3)

where $g_{ab}(x)$ is the metric on a $d$-dimensional Cauchy slice $\Sigma$,
$\Pi^{ab}(x)$ is the canonical momentum conjugate to it:

$\Pi^{ab}(x)=-i\frac{\delta}{\delta g_{ab}(x)},$ (4)

and $:\>\,:$ is the normal ordering symbol, meaning that we need to subtract
off some divergences—we will discuss this in more detail in section 2.5. We
will henceforth refer to solutions of the above constraints as WDW-states.
From (4) we can deduce the commutator:

$\displaystyle[\Pi^{ab}(x),\,g_{cd}(y)]=-i\delta^{ab}_{cd}\,\delta(x-y),$ (5)

where
$\delta^{ab}_{cd}=\tfrac{1}{2}(\delta^{a}_{c}\delta^{b}_{d}+\delta^{a}_{d}\delta^{b}_{c})$.

Since both WDW-states $\Psi[g]$ and partition functions $Z[g]$ are functionals
of a metric, there is a natural analogy between the two concepts, as explained
in section 1.1. We will therefore set about to define a _field theory_ whose
partition function $Z$ naturally defines a WDW-state:

$\displaystyle Z[g]=\Psi[g].$ (6)

In this field theory, $\Pi^{ab}$ plays the role of the stress-tensor operator,
while the metric $g_{ab}$ plays the role of the conjugate source.

###  1.5 Relation to Tensor Network Models

Like [22], this project was conceptually inspired by the goal of coming up
with a tensor network model for AdS/CFT states. A number of holography
researchers have constructed various kinds of tensor network models, whose
geometry is thought of as being a discrete analogue of hyperbolic space; see
e.g. [32, 33, 34, 35, 36, 37, 38, 39, 40]. A significant motivator of this
line of research has been to construct a model which naturally explains the
Ryu-Takayagi formula [41] in the bulk:

$\displaystyle S_{\text{CFT}}[R]=\min_{\gamma}\text{Area}[\gamma]/4G_{N},$ (7)

where $\gamma$ is a codimension-2 bulk surface anchored to the boundary of the
CFT region $R$. Since this formula is only valid on Cauchy slices $\Sigma$
with a time-reversal symmetry (e.g. the $t=0$ slice of AdS), much of the work
on tensor networks is intended as a model only of such slices.

Unfortunately, building a tensor network on a time-reversal symmetric slice
$\Sigma_{0}$ is conceptually untenable as an accurate model of quantum
gravity, since it violates the Heisenberg uncertainty principle between
$g_{ab}$ and $\Pi^{ab}$. The characteristic feature of a moment-of-time
symmetric slice is that the extrinsic curvature $K_{ab}$ vanishes, and hence
$\Pi_{ab}=\sqrt{g}(K_{ab}-g_{ab}K)/8\pi G_{N}=0$. But if we wish for generic
surfaces $\gamma\in\Sigma_{0}$ to have well-defined areas, we also need to
have a well-defined metric $g_{ab}$ on $\Sigma_{0}$. By the commutator (5),
this is too much information!

It follows that, if we wish to take tensor networks seriously as a model of
quantum space, we necessarily have to be able to use them to describe a
generic dynamical Cauchy slice $\Sigma$. Even if you start with an initial
state which is tightly peaked around $K_{ab}=0$, performing a quantum
measurement can take you to another state with large $K_{ab}$. Even
classically, the $K_{ab}=0$ slice must still satisfy the Hamiltonian
constraint. So even to describe a moment-of-time symmetric slice, we need a
formalism in which dynamics is already implicitly present.151515This fact may
explain the paradoxical no-go result of [40] in which it was impossible to
construct a certain tensor network with crossing RT surfaces, using
approximate isometries.

In this article we will show how the $T^{2}$ deformation provides such a model
of holographic space, in which dynamics arises naturally. Unlike tensor
networks, this is a _continuum_ model.161616Another example of a continuum
approach to holography on a spatial slice is cMERA holography [42, 43, 44].
However, this approach is not rotationally-invariant, and is nonlocal at sub-
AdS scales. This is technically convenient as it allows us to define a local
stress-tensor, and to explore sub-AdS scales. However, for reasons we shall
discuss in sections 2.5 and 8.3, the $T^{2}$ deformed theory still requires a
UV completion in order to describe physics at finite $N$. (Otherwise we would
have just solved an even more famous problem: to provide a nonperturbative
definition of quantum gravity!) It is therefore still possible that space may
still be discrete at the Planck scale, in which case the fundamental model of
space might be literally a tensor network.171717There is a dynamical version
of the holographic entropy, given by [45]:
$S_{\text{HRT}}[R]=\min\mathrm{ext}_{\gamma}\text{Area}[\gamma]/4G_{N}$. In
the past it has proven difficult to accommodate this formula in a tensor
network model for a Cauchy slice $\Sigma^{\prime}$ that does not pass through
the HRT surface, as the minimal area of $\Sigma^{\prime}$ is always less than
the area of the HRT surface [46], which gives the boundary CFT entropy. The
resolution of this paradox in the $T^{2}$ deformed model will be described in
more detail in a forthcoming article [47], but an important clue will be
provided in section 2.6 where it is observed that the entropy of $T^{2}$ in
Lorentzian signature sometimes exceeds $\text{Area}/4G_{N}$ due to the
presence of extra states in the spectrum. In order to be compatible with the
HRT formula, a valid dynamical tensor network model must have bond dimensions
large enough to include these extra states. This happens whenever the
codimension-2 extrinsic curvature is timelike.

Since a field theory partition function gives a sum over states whenever it is
cut open in the middle, we expect that Cauchy slice holography will implement
some form of the surface-state correspondence in which information is
attributed to codimension 2 surfaces [36, 48]. But in this article we focus on
the use of the $T^{2}$ theory to construct a new form of the holographic
AdS/CFT dictionary, which maps bulk WDW states to boundary CFT states.

###  1.6 Plan of Paper

In Section 2, we study how the constraints of bulk gravity translate into
constraints on the holographic field theory meant to describe bulk states. We
provide a general prescription for obtaining the correct deformation away from
the boundary CFT181818We will generically refer to this deformation as the
$T^{2}$-deformation, as a reference to the usual deformation corresponding to
pure Einstein gravity in the bulk., which also provides us with a constraint
on the CFT anomaly, in the cases in which such an anomaly exists. We also work
through the explicit example of a bulk scalar field coupled to gravity in
$3+1$-dimensions with a negative cosmological constant.

In Section 3 we propose a generalized version of the holographic principle we
are used to from AdS/CFT, taking into account the presence of boundaries at
finite distances and times.

Section 4 is dedicated to a discussion of the bulk gravitational path integral
as defining a bulk inner product. To the extent that this bulk prescription is
ambiguous, we propose that the $T^{2}$ theory could in principle resolve such
ambiguities. In Section 5 we present a construction of the Hilbert space of
solutions to the constraint equations in quantum gravity, based on viewing the
path integral method as a linear map from kinematic states to WDW-states.

In Section 6, we use the $T^{2}$ partition function to construct linear maps
between the bulk and boundary Hilbert spaces. That is, given a bulk kinematic
Cauchy slice metric $g_{ab}$, we can define the dual CFT state. Conversely, we
can also define a map from the boundary states to solutions of the constraint
equations. (These two maps are not inverses; rather the composition of these
two maps—assuming the validity of the generalized holographic principle—is
equivalent to the bulk-to-bulk path integral map defined in Section 5.)

In Section 7 we show the consistency of boundary and bulk time evolution,
given the generalized holographic principle. We also show an equality between
the boundary and bulk Hamiltonians, to leading order in large N. Finally we
discuss the way in which bulk unitarity emerges naturally from the field
theory, despite the fact that the $T^{2}$ spectrum is nonunitary.

Finally, in the discussion 8 we review our results, and discuss the prospects
of UV completing the $T^{2}$ boundary theory into an ultraviolet finite
theory—a task which, if completed, might then serve as a satisfactory
definition of quantum gravity. We suggest that it does not actually matter
much _how_ the theory is UV completed, as different choices are likely to
provide dual descriptions of the same physics.191919For readers who wish to
get to the main AdS/CFT results as quickly as possible, we suggest the
following reading plan: start with 2.1–2.2 and 3.1 for an overview of $T^{2}$
holography, followed by 4.1, 5.1 to get the gist of our approach to quantum
gravity in the bulk. At this point the key AdS/CFT dictionary results
described in sections 6—8 should be accessible.

##  2 The $T^{2}$ Deformation

###  2.1 Preliminaries

Consider a holographic CFT which is dual to quantum gravity in an
asymptotically-AdS bulk ${\cal B}$. We assume the low energy bulk corresponds
to pure gravity with matter fields, collectively denoted by $\\{\Phi\\}$, or
in other words that the only low-dimension single-trace primary operators in
the CFT are the stress tensor $T_{ab}$ and the set $\\{\mathcal{O}\\}$ which
are in correspondence with the bulk matter fields. The leading order
Fefferman-Graham expansion of the metric near the boundary is:

$ds^{2}=g_{\mu\nu}\,dx^{\mu}dx^{\nu}=\frac{g^{\text{bdy}}_{ab}dx^{a}dx^{b}+dz^{2}}{z^{2}}+...,$
(8)

where $\mu,\nu$ are indices of the $D=d+1$ dimensional AdS bulk spacetime
$\cal B$. A Cauchy slice in this bulk is a noncompact manifold $\Sigma$ with
$\partial\Sigma\subset{\partial{\cal B}}$.

We will now evaluate the CFT partition function, not on the usual boundary at
infinity $\partial\cal B$, but rather on the Cauchy slice $\Sigma$. After we
deform this CFT partition function by a $T^{2}$ deformation, it will describe
quantum gravity in the bulk. We will choose a convention in which the coupling
constant $\mu$ (corresponding to the direction in theory space of this $T^{2}$
operator) is the only scale in the theory. This is particularly nice when
thinking in terms of a Wilsonian picture of RG. Also, any bulk gravity scales
will be written in terms of this $\mu$ and a set of dimensionless field theory
parameters, collectively denoted by $\\{\alpha_{i}\\}$, to fix the ratio
between bulk scales.

We pick a _complete commuting set of operators_ $\\{\chi\\}$ acting on the CFT
Hilbert space of $\partial\Sigma|_{\partial{\cal B}}$, i.e. on the boundary
side of $\partial\Sigma$. Importantly, this is _not_ necessarily the same as
evaluating the same operators $\\{\chi\\}$ at $\partial\Sigma|_{\Sigma}$,
because the sharp corner between the two geometries can cause certain
operators, such as the stress-tensor, to discontinuously jump across the
boundary (as will be discussed explicitly in section 7.2). In order to keep
our notation clean, without writing junction conditions everywhere, we choose
to always think of the Cauchy slice $\Sigma$ as making this transition to
$\partial{\cal B}$ _before_ we evaluate $\\{\chi\\}$—even in situations where
two Cauchy slices $\Sigma$ and $\Sigma^{{}^{\prime}}$ meet at the same
boundary time on $\partial{\cal B}$, with no lapse between them,202020In other
words we are implicitly identifying the Hilbert space of $\partial\Sigma$ with
that of $\partial{\cal B}|_{\partial\Sigma}$. Otherwise we would need to
explicitly include junction maps ${\cal H}_{\partial{\cal
B}|{\partial\Sigma}}\to{\cal H}_{\partial\Sigma}$ everywhere. as will happen
later.

This defines a basis of states labelled by the set of eigenvalues of
$\\{\chi\\}$212121We will be using the same symbol $\\{\chi\\}$ to denote both
the complete set of operators and the set of eigenvalues used as boundary
conditions for the partition function. The context should make this
distinction clear., while a general state on $\partial\Sigma$ will be a
superposition thereof: $\psi[\\{\chi\\}]$.222222These operators are _not_
necessarily local fundamental fields, since those are not usually good
operators on a Hilbert space in a strongly coupled theory. The set
$\\{\chi\\}$ can include eigenvalues of nonlocal operators e.g. the total
energy or angular momentum. The partition function
$Z_{\text{CFT}}[\\{J_{i}\\},\\{\chi\\}]$ is thus a functional of any field
theory sources $\\{J_{i}\\}$ and of the boundary conditions $\\{\chi\\}$. The
sources include the metric and any other sources which are mapped to bulk
matter fields in the low-energy limit. So $\\{J_{i}\\}=(g,\\{\phi\\})$, where
$\phi$ is the field theory source which is just a rescaled version of the bulk
field $\Phi$ (which is always to be understood as the value of the bulk field
induced on the Cauchy slice $\Sigma$). The correct rescaling depends on the
nature of the bulk field.

The stress-tensor $T^{ab}$ in any field theory is defined as a variation of
the metric:

$T^{ab}(x):=\frac{-2}{\sqrt{g}}\frac{\delta}{\delta
g_{ab}(x)}=\frac{-2i}{\sqrt{g}}\Pi^{ab}(x).$ (9)

If other sources, collectively denoted by $\\{\phi\\}$, are switched on in the
field theory the corresponding operators, collectively denoted by
$\\{\mathcal{O}\\}$, will be given by linear variation:

$\displaystyle\mathcal{O}(x)=\frac{1}{\sqrt{g}}\frac{\delta}{\delta\phi}=\frac{i}{\sqrt{g}}\Pi_{\phi}(x).$
(10)

Given any set of CFT sources $\\{J_{i}\\}$, the generator of diffeomorphism
symmetry is

$\displaystyle{\cal
D}_{a}:=\sum_{J}\frac{\delta}{\delta\xi^{a}}\\!\Big{[}{\cal L}_{\xi}(J){\cal
O}^{i}\Big{]}=\sum_{J}\Big{(}\partial_{a}J\cdot\Pi_{J}\>+\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\text{~{}~{}~{}~{}~{}~{}~{}upper
$J$
index}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{b}(J^{b}\cdot\Pi_{Ja})\>-\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\text{~{}~{}~{}~{}~{}~{}~{}lower
$J$
index}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{b}(J_{a}\cdot\Pi_{J}{{}^{b}})\Big{)},$
(11)

where ${\cal L}_{\xi}(J)$ is the Lie derivative of the tensor $J$, while the
generator of Weyl transformations is

$\displaystyle{\cal W}:=-\sum_{J}\Delta_{J}J\cdot\Pi_{J},$ (12)

where ${\cal O}^{i}$ is the operator conjugate to $J_{i}$ and $\Delta_{J_{i}}$
is the conformal weight of the source in the boundary CFT.232323In this
expression, we are assuming the sources transform as primaries under a Weyl
transformation. A generic CFT is Weyl invariant up to the usual Weyl anomaly
$\cal A$:

$\big{(}\mathcal{W}(x)-i\mathcal{A}(x)\big{)}\,Z^{(\epsilon)}_{CFT}[\\{J_{i}\\},\\{\chi\\}]=0,$
(13)

where $\epsilon$ is a regulator necessary to regulate the logarithmic
divergences (which cannot be eliminated by counterterms) arising from the
anomaly. However, this expression does not look like the ADM Hamiltonian
constraint, which we need in order to interpret $Z$ as a Wheeler-DeWitt state
on $\Sigma$. Relatedly, the closure relation of the conformal constraint
$\widetilde{\cal W}:=\mathcal{W}-i\mathcal{A}$ is trivial:

$\displaystyle\left[\widetilde{\mathcal{W}}(x),\widetilde{\mathcal{W}}(y)\right]=0,$
(14)

due to the Wess-Zumino consistency condition, whereas the commutator of the
Hamiltonian constraint on itself is proportional to the momentum constraint:

$\displaystyle\Big{[}\mathcal{H}(x),\mathcal{H}(y)\Big{]}=i\left(\mathcal{D}^{a}(x)\partial_{a}^{(x)}-\mathcal{D}^{a}(y)\partial_{a}^{(y)}\right)\delta(x-y).$
(15)

This encodes the principle of local Lorentz invariance in the bulk, since it
implies that when there is a nonzero $\cal H$ gradient (locally a boost) the
generator of time translations is shifted by the generator of space
translations.242424The action of the Lorentz boost on spatial translations is
fixed by the analogous commutator: $[\mathcal{H}(x),{\cal
D}_{a}(y)]=i\partial_{a}^{(x)}\\!({\cal H}(x)\delta(x-y))$. But this closure
relation is an automatic consequence of diffeomorphism symmetry, given that
$\mathcal{H}(x)$ is a scalar density on $\Sigma$.

Hence, to obtain the ADM constraint algebra, we need to deform the theory on
the Cauchy slice so that $\widetilde{\cal W}(x)$ becomes ${\cal H}(x)$ and
(14) deforms to (15). (The formula for ${\cal D}_{a}$ is unaffected so long as
the flow is covariant.) In other words, local Lorentz symmetry in the bulk
emerges from local conformal invariance in the boundary, as a deformed WZ
consistency condition [49, 9]:

$\displaystyle\text{Conformal
Invariance}\>\>\underset{\mu}{\Longrightarrow}\>\>\text{Local Lorentz
Invariance}.$ (16)

###  2.2 Asymptotics of the Flow

Freidel [20] showed that any radial WDW state of an asymptotically-AdS quantum
gravity theory tends to the following form in the superspace region
corresponding to an infinite uniform rescaling of the boundary metric
$\tilde{g}=g/\mu^{2/d}$:

$\displaystyle\lim_{\mu\to
0}\Psi\left[\tilde{g}\right]=A_{+}e^{i\text{S}[\tilde{g}]}Z_{+}[\tilde{g}]+A_{-}e^{-i\text{S}[\tilde{g}]}Z_{-}[\tilde{g}],$
(17)

where $g$ is the Lorentzian metric on the Dirichlet wall $\partial{\cal B}$,
$Z_{\pm}$ look like CFTs252525At least to the extent that they are conformally
invariant functionals of the metric, up to the anomaly term. Holography
asserts that, in a UV complete theory of quantum gravity, they should also
satisfy the axioms of local QFT. with opposite anomalies, and $\text{S}[g]$ is
an explicit real local action consisting of only relevant terms (_i.e._ terms
of dimension at most $d$) which we note to be the holographic counterterms.
Furthermore, this local action is the same for all WDW states. In the $\mu\to
0$ limit, the Dirichlet boundary $\partial{\cal B}$ goes to spatial infinity,
due to imposing Dirichlet boundary conditions with infinite volume.

The presence of two solutions is a consequence of the radial WDW equation
${\cal H}\Psi=0$ being of second order.262626Since ${\cal H}(x)=0$ is a second
order differential equation defined at each point $x$, one might think there
would be more than 2 solutions if we choose the sign differently for different
values of $x$. However, in the asymptotic limit, such wavefunctions would
always violate the diffeomorphism constraint ${\cal D}^{a}\Psi=0$ and so are
unphysical. (Here we are assuming that $\partial\cal B$ is connected,
otherwise we could pick a separate sign for each connected component.) Any of
the two is independently a solution and so is any linear combination.

If we Wick rotate $Z_{+}$ and $Z_{-}$ to Euclidean signature, one of these
partition functions (say $Z_{-}$) corresponds to a state $\Psi$ in which the
graviton fluctuations are unnormalizable. As we wish to define our Lorentzian
Cauchy slice theory by analytic continuation from the Euclidean regime, we are
therefore forced to restrict our attention to the other one ($Z_{+}$). We
believe this choice is closely related to the question of the quantum gravity
contour, which we will discuss in section 3.2.

The same behaviour also follows if we add matter fields or higher curvature
corrections, since sending the volume of $\partial{\cal B}$ to infinity
guarantees that we are in the IR limit of the bulk theory (at least in the
neighbourhood the boundary) and so the higher-order corrections to the WDW
equation become suppressed in an $M_{\text{Pl}}$ expansion. After picking the
branch with good Euclidean continuation, we have:

$\displaystyle\lim_{\mu\to
0}\Psi[\tilde{g},\tilde{\Phi}]=e^{i\text{S}[\tilde{g},\tilde{\Phi}]}Z_{+}[\tilde{g},\tilde{\Phi}],$
(18)

where $\tilde{\Phi}=\mu^{\Delta_{\phi}/d}\Phi$ is an appropriately rescaled
version of the matter field.

Also it is common knowledge in the literature [50] that taking the boundary to
infinity is equivalent to a renormalization group flow of the field theory
into the IR. So away from the fixed point, the radial WDW state must be given
by a deformation of the CFT by irrelevant (or at most marginal) operators, up
to relevant counterterms $\text{S}[g,\Phi]$. Hence we can reverse the
direction of argument and get a quantum gravity state from the CFT partition
function by the following flow:

$\displaystyle\Psi^{(\mu)}[g,\Phi]$ $\displaystyle=$ $\displaystyle
e^{\text{CT}(\mu)}\left(\text{P}\exp\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right)Z^{(\epsilon)}_{\text{CFT}}[g,\phi,\\{\chi\\}],$
(19) $\displaystyle\lim_{\mu\to\epsilon}\Psi^{(\mu)}[g,\Phi]$ $\displaystyle=$
$\displaystyle
e^{\text{CT}(\epsilon)}Z^{(\epsilon)}_{\text{CFT}}[g,\phi,\\{\chi\\}],$ (20)

where $O(\lambda)$ is a spatial integral of local irrelevant and marginal
operators, and $\text{CT}(\mu)$ is a counterterm, which is needed for the
Hamiltonian constraint equation to take the standard ADM form.272727The
solution corresponding to the other Freidel branch, at least formally, be
obtained by flipping the sign of the CFT trace anomaly, in which case the sign
of the the relevant part of the counterterms also flips, akin to Freidel, and
the deforming operator $O(\lambda)$ will itself be different. In section 2.3
we will calculate which $O(\lambda)$ we need so that the Hamiltonian
constraint equation is satisfied:

$\displaystyle{\cal H}(x)\Psi^{(1)}[g,\Phi]=0,$ (21)

with ${\cal H}(x)$ an arbitrary constraint that closes properly.

In a $T^{2}$ deformation, the irrelevant operators in $O(\lambda)$ always
includes a term quadratic in the stress tensor. The fact that the exponential
is _path ordered_ incorporates the usual rule that, at a given coupling
$\lambda$, the definition of the stress-tensor is used is that of the deformed
theory with coupling $\lambda$, rather than the stress-tensor of the original
CFT. In other words, if the path ordered exponential is expanded out as a
product, then the differential operator $\Pi^{ab}\sim\delta/\delta g_{ab}(x)$
acts on all structures appearing to its right. (This is equivalent to the more
standard $T^{2}$ notation, in which the deformation to the action is written
as a differential equation in $\lambda$.) A similar remark applies to any
other field theory operators ${\cal O}\sim\delta/\delta J$ appearing in
$O(\lambda)$.

The existence of marginal operators in $O(\lambda)$ produces log divergences,
which require regulation by the lower cutoff $\epsilon$. We take this to be
the same quantity as the UV cutoff $\epsilon$ used to regulate the log
divergences in the field theory partition function
$Z^{(\epsilon)}_{\text{CFT}}$, which are associated with the trace anomaly.
These log divergences cancel, so that the overall solution
$\Psi^{(\mu)}[g,\Phi]$ is ultimately independent of the regulator $\epsilon$.

As the simplest example (basically the original $T\overline{T}$ deformation),
if you want to obtain pure GR in AdS3 from a $d=2$ CFT with central charge
$c$, $O(\lambda)$ will look like this:

$O(\lambda)=\frac{1}{2}\int
d^{2}x\left[\frac{c}{24\pi}\sqrt{g}R-\frac{24\pi}{c}\frac{\lambda}{\sqrt{g}}\left(:\\!\Pi^{ab}\Pi_{ab}\\!:-:\\!\Pi^{2}\\!:\right)\right],$
(22)

and the counterterm is just a $d=2$ cosmological constant (not to be confused
with $\Lambda$, the bulk $2+1$ cosmological constant):

$\text{CT}(\mu)=-\frac{c}{12\pi}\mu^{-1}\int d^{2}x\sqrt{g},$ (23)

and at the end we obtain a theory satisfying the 2+1 dimensional constraint
equations (2) and (3) with

$\Lambda=-\frac{1}{\mu}=-\frac{1}{(L_{\text{AdS}})^{2}}.$ (24)

Furthermore, cancellation of log divergences requires that the central charge
of the original CFT satisfy the standard holographic relation:

$c=\frac{3L_{\text{AdS}}}{2G_{N}}.$ (25)

However, this deformation will not work in $d>2$ because $R$ looks like a
relevant term. So next we will, extending [15, 14], describe a general
procedure for defining the $T^{2}$ deformation in arbitrary dimension, with
general matter fields.

###  2.3 The General Deformation

Let us start with a Hamiltonian and momentum constraint system describing
gravity plus matter, in general bulk dimensions $D=d+1$. The form of the
momentum constraint ${\cal D}^{a}$ is determined by spatial covariance. We
allow for a general Hamiltonian constraint $\mathcal{H}$ which could be any
scalar density satisfying the standard ADM closure condition 15, which is
important enough to type again:282828It should be noted that the standard
closure conditions very strongly constrain the form of the theory. For
example, if the only phase space variables are $g_{ab}(x)$ and $\Pi_{ab}(x)$,
a series of rigidity theorems [51, 52, 53, 54, 55] suggest that (modulo field
redefinitions, and subject to the constraint that the kinetic term is
quadratic) general relativity may be the only allowed theory (but see [56] for
an attempt to deform the constraint algebra). These theorems require an exact
solution to the closure relation. Presumably higher curvature corrections,
such as Lovelock gravity, are allowed in the canonical formulation if one
treats them as perturbatively small, and only requires the closure relations
to be satisfied order-by-order.

$\displaystyle\left[\mathcal{H}(x),\mathcal{H}(y)\right]=i\left(\mathcal{D}^{a}(x)\partial_{a}^{(x)}-\mathcal{D}^{a}(y)\partial_{a}^{(y)}\right)\delta(x-y).$
(26)

It is reasonable not to try to modify the standard closure condition as
(modulo redefinitions of the fields and constraints) this must necessarily be
satisfied by _any_ locally Lorentz invariant bulk theory with a Hamiltonian
description.292929It is difficult to slightly deform special relativity while
preserving locality. One family of attempts, called “doubly special
relativity”, has been studied in detail (see [57] for a review); however it is
not known how to implement this proposal in a local field theory.

##### Irrelevant terms and anomalies:

Not every choice of $\mathcal{H}$ will be compatible with a given holographic
CFT, for as we shall see there are some anomaly matching conditions that need
to be satisfied, analogous to (25). In order to show that $\mathcal{H}$ is
compatible, we need to find a canonical transformation:303030We define any
similarity transformation of the operators to be a canonical transformation,
because in the classical limit ($\hbar\to 0$) this becomes a standard
canonical transformation, albeit with complex coefficients.

$\displaystyle\tilde{\mathcal{H}}(x):=e^{-\text{CT}}\mathcal{H}(x)e^{\text{CT}},$
(27)

such that $\tilde{\mathcal{H}}(x)$ has no relevant terms and it contains the
CFT trace equation as its marginal part (we will elaborate on how to achieve
this at the end of this subsection).313131This canonical transformation can
introduce delta function divergences from the reordering of the terms in
$\tilde{\mathcal{H}}$ (to put everything in a normal ordered form). These
divergences can be eliminated by point-splitting. That is,

$\displaystyle
i\mu^{1/d}\mathcal{\tilde{H}}^{(\mu)}(x)^{\text{marg}}=\widetilde{{\cal
W}}(x)={\cal W}(x)-i{\cal A}(x).$ (28)

The requirement of the above equation will allow us to relate the free
parameters in the bulk to the free parameters of the field theory (this will
be clear in an example shown in the following section).

Besides the above marginal couplings, in general $\tilde{\mathcal{H}}(x)$ will
also have dimensionful coupling constants in it. Following [15], we adopt a
convention in which the $T^{2}$ coupling $\mu$ is the only dimensionful
parameter in the flow,323232This is tantamount to adopting a very convenient
choice of RG scheme, in which the form of the beta functions is fixed in an
_entire neighborhood_ of the IR fixed point to exactly equal the leading-order
result. This convention is extremely useful for simplifying the algebra, as
without it a path ordered exponential will usually produce infinitely many
terms. which implies that the flow follows an RG trajectory. Hence we absorb
all other dimensionful couplings into $\mu$, so as to get a 1-parameter family
of Hamiltonian constraints: $\tilde{\mathcal{H}}^{(\mu)}(x)$, which differ
only by a rescaling of length. As the deformation parameter $\mu$ is varied,
the bulk scales will therefore flow, with the ratios between them being
determined by some dimensionless parameters in the $T^{2}$ theory.333333For an
explicit example see the powers of $\lambda$ appearing in (80)-(86) in section
2.4, which are fixed by dimensional analysis.

We define the following local operator $X^{(\mu)}$ (which will turn out to be
the operator needed to deform the CFT by an amount $\mu$):

$\displaystyle-idX^{(\mu)}(x):=i\mu^{1/d}\tilde{\mathcal{H}}^{(\mu)}(x)-{\cal
W}(x).$ (29)

##### Canonical transformation on the algebra:

As the canonical transformation preserves the commutators, we get the
following relation:

$\displaystyle
e^{-\text{CT}(\mu)}\left[i\mu^{1/d}\mathcal{H}^{(\mu)}(x),i\mu^{1/d}\mathcal{H}^{(\mu)}(y)\right]e^{\text{CT}(\mu)}$
$\displaystyle=\left[i\mu^{1/d}\tilde{\mathcal{H}}^{(\mu)}(x),i\mu^{1/d}\tilde{\mathcal{H}}^{(\mu)}(y)\right]$
(30) $\displaystyle=\left[{\cal W}(x)-idX^{(\mu)}(x),{\cal
W}(y)-idX^{(\mu)}(y)\right]$ (31) $\displaystyle=-id\left(\left[{\cal
W}(x),X^{(\mu)}(y)\right]-\left[{\cal W}(y),X^{(\mu)}(x)\right]\right)$
$\displaystyle+(id)^{2}\left[X^{(\mu)}(x),X^{(\mu)}(y)\right].$ (32)

We define the covariant operator:

$\displaystyle O(\mu):=\int d^{d}xX^{(\mu)}(x).$ (33)

##### Conformal classes:

We split $X^{(\mu)}(x)$ into conformal classes, with each of the terms in the
sum having a particular scaling $q_{n}$ with $\mu$ (determined by dimensional
analysis):

$\displaystyle X^{(\mu)}(x)=\sum_{n}\mu^{q_{n}}X_{n}(x),$ (34)

where $X_{n}(x)$ is a sum of terms with conformal weight $\omega_{n}$ (i.e.
$-2$ for every power of $g_{ab}$ and $\Delta_{\phi}$ for every power of
$\phi$, counting $g^{ab}$ or $\Pi^{ab}$ as $+2$ and $\Pi_{\phi}$ as
$-\Delta_{\phi}$) such that:

$\displaystyle\left[\int d^{d}y\,{\cal
W}(y),X_{n}(x)\right]=+i\omega_{n}X_{n}(x).$ (35)

Because $X_{n}(x)$ is a scalar density (as can be seen from (29)) its mass and
conformal dimensions differ by $d$: $\left[X_{n}\right]=\omega_{n}+d$, and so
we obtain the following universal relation by dimensional analysis:

$\displaystyle\omega_{n}=q_{n}\,d.$ (36)

Note that the operator $X^{(\mu)}(x)$ can be written in terms of its marginal
($\omega_{n}=0$) and irrelevant $(\omega_{n}>0)$ parts:

$\displaystyle X^{(\mu)}=\frac{\mathcal{A}}{d}+X^{(\mu)}_{\text{irrel}}.$ (37)

##### The power of demanding bulk algebra closure:

Now we integrate (32) with respect to $y$ to get:

$\displaystyle\int
d^{d}y\;e^{-\text{CT}(\mu)}\left[i\lambda^{1/d}\mathcal{H}^{(\lambda)}(x),i\lambda^{1/d}\mathcal{H}^{(\lambda)}(y)\right]e^{\text{CT}(\mu)}$
$\displaystyle=-id\left(\left[{\cal
W}(x),O(\lambda)\right]-\sum_{n}i\omega_{n}\lambda^{q_{n}}X_{n}(x)\right)$
(38) $\displaystyle\qquad+(id)^{2}\left[X^{(\lambda)}(x),O(\lambda)\right].$
(39)

Next we integrate the above equation with respect to $\log(\lambda)$. However,
this integral
$\left(\int_{0}^{\mu}\frac{d\lambda}{\lambda}\lambda^{q_{n}}\right)$ diverges
for $q_{n}=0$. So we need to regulate the lower bound of the integral by a
cutoff parameter $\epsilon$. By rearranging the terms, we get:

$\displaystyle\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\left[{\cal
W}(x),O(\lambda)\right]=\frac{1}{2}\left(\mu^{2/d}-\epsilon^{2/d}\right)e^{-\text{CT}(\mu)}\partial_{a}{\cal
D}^{a}(x)e^{\text{CT}(\mu)}+idX^{(\mu)}(x)-idX^{(\epsilon)}(x)$ (40)
$\displaystyle+id\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\left[X^{(\lambda)}(x),O(\lambda)\right],$
(41)

where we used $\int
d^{d}y\left[\mathcal{H}^{(\lambda)}(x),\mathcal{H}^{(\lambda)}(y)\right]=i\partial_{a}{\cal
D}^{a}(x)$, which comes from integrating (26).

We now further define a covariant operator from $O(\lambda)$ which is path-
ordered in the scale $\lambda$:

$\left(\text{P}\exp\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}O(\lambda)\right):=\lim_{\delta\to
0}\prod_{n=\epsilon}^{n=\mu/\delta}\left(1+\frac{1}{n}O(n\delta)\right),$ (42)

We will eventually be interested in the following “sandwiched” local operator
for the purposes of obtaining WDW states:

$\displaystyle\left(\text{P}\exp\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right){\cal
W}(x)\left({\text{P}\exp}-\\!\\!\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right)$
$\displaystyle={\cal
W}(x)-\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\left[{\cal
W}(x),O(\lambda)\right]$ (43)
$\displaystyle\\!\\!\\!\\!\\!\\!+\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\int_{\epsilon}^{\lambda}\frac{d\lambda^{\prime}}{\lambda^{\prime}}\left[\left[{\cal
W}(x),O(\lambda^{\prime})\right],O(\lambda)\right]+\ldots,$ (44)

where the RHS follows from the Baker-Campbell-Hausdorff formula.

We observe that relation (41) generates a neat cascade of cancellations in the
infinite series emerging from the closure relation of the Hamiltonian
constraint:

$\displaystyle{\cal W}(x)\,-$
$\displaystyle\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\left[{\cal
W}(x),O(\lambda)\right]+\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\int_{\epsilon}^{\lambda}\frac{d\lambda^{\prime}}{\lambda^{\prime}}\left[\left[{\cal
W}(x),O(\lambda^{\prime})\right],O(\lambda)\right]+\dots$ (45)
$\displaystyle={\cal
W}(x)-idX^{(\mu)}(x)-\frac{1}{2}\mu^{2/d}e^{-\text{CT}(\mu)}\partial_{a}{\cal
D}^{a}(x)e^{\text{CT}(\mu)}$ (46)
$\displaystyle+\left(\text{P}\exp\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right)idX^{(\epsilon)}(x)\left({\text{P}\exp}-\\!\\!\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right),$
(47)

where the commutator $\partial_{a}[{\cal D}^{a}(x),O(\lambda)]=0$ by
covariance of $O(\lambda)$. Therefore we get:

$\displaystyle\left(\text{P}\exp\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right)\left\\{{\cal
W}(x)-idX^{(\epsilon)}(x)\right\\}$ (48)
$\displaystyle=\Bigg{\\{}\mathcal{W}(x)-idX^{(\mu)}(x)-\frac{1}{2}\left(\mu^{2/d}-\epsilon^{2/d}\right)$
$\displaystyle e^{-\text{CT}(\mu)}\partial_{a}{\cal
D}^{a}(x)e^{\text{CT}(\mu)}\Bigg{\\}}\left(\text{P}\exp\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right),$
(49) $\displaystyle=\left\\{{\cal
W}(x)-idX^{(\mu)}(x)\right\\}\left(\text{P}\exp\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right),$
(50)

where we have used the momentum constraints in throwing out the
$\partial_{a}{\cal D}^{a}(x)$ term.343434The closure constraints form an
important consistency property on the allowed deformations $O(\lambda)$. For
example, if we replace the standard ADM kinetic term inside $X$ with
$\Pi_{ab}\Pi^{ab}-\alpha\Pi^{2}$ (51) with $\alpha\neq\frac{1}{d-1}$, and plug
that into (52), then these cancellations will no longer occur. The resulting
constraint ${\cal H}^{(\mu)}(x)$ at finite $\mu$ will have an infinite series
of total derivative terms in it, and in fact the coefficients of these terms
fall off slowly enough with the number of derivatives, that it resums to a
spatially nonlocal answer! (The nonlocality of the resulting expression is a
consequence of the fact that integrating with respect to $\lambda$ is
equivalent to dynamically evolving a finite conformal distance into the bulk.)
Such nonlocal terms are excluded on the hypothesis that the $T^{2}$ deformed
theory has a UV completion which is local down to very short length scales. A
similar argument shows that, if we do not throw away the ${\cal D}^{a}$ terms,
we would get a nonlocal integral over those as well. This is a necessary
consequence of the fact that the algebra of conformal transformations (Weyl
plus spatial diffeos) is not locally isomorphic to the ADM constraint algebra.

##### The resulting solution:

So now we can obtain a solution to an arbitrary Hamiltonian and momentum
constraints of gravity arbitrarily coupled with arbitrary matter fields, in
arbitrary dimensions $d+1$ by deforming an arbitrary $d$-dimensional CFT:

$Z^{(\mu)}[g,\phi,\\{\chi\\}]:=e^{\text{CT}(\mu)}\left(\text{P}\exp\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right)Z_{\text{CFT}}^{(\epsilon)}[g,\phi,\\{\chi\\}],$
(52)

where $O(\lambda)$ is defined in (33) and is determined by the choice of
starting Hamiltonian constraint $\mathcal{H}^{(\mu)}(x)$ and the choice of
canonical transformation $\text{CT}(\mu)$ in (27). This is then easily shown
to be a WDW state:

$\displaystyle\mathcal{H}^{(\mu)}(x)$ $\displaystyle
Z^{(\mu)}[g,\phi,\\{\chi\\}]=\mathcal{H}^{(\mu)}(x)e^{\text{CT}(\mu)}\left(\text{P}\exp\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right)Z_{\text{CFT}}^{(\epsilon)}[g,\phi,\\{\chi\\}]$
(53) $\displaystyle=-i\mu^{-1/d}e^{\text{CT}(\mu)}\left({\cal
W}(x)-idX^{(\mu)}(x)\right)\left(\text{P}\exp\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right)Z_{\text{CFT}}^{(\epsilon)}[g,\phi,\\{\chi\\}]$
(54)
$\displaystyle=-i\mu^{-1/d}e^{\text{CT}(\mu)}\left(\text{P}\exp\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right)\left({\cal
W}(x)-i\mathcal{A}(x)-idX^{(\epsilon)}_{\text{irrel}}(x)\right)Z_{\text{CFT}}^{(\epsilon)}[g,\phi,\\{\chi\\}].$
(55)

Now taking the limit $\epsilon\to 0$ we have that
$X^{(\epsilon)}_{\text{irrel}}(x)\to 0$ as a power law and so we learn that
the Hamiltonian constraint is satisfied:

$\displaystyle\mathcal{H}^{(\mu)}(x)Z^{(\mu)}[g,\phi,\\{\chi\\}]$
$\displaystyle=-i\mu^{-1/d}e^{\text{CT}(\mu)}\left(\text{P}\exp\int_{\epsilon}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right)\left({\cal
W}(x)-i\mathcal{A}(x)\right)Z_{\text{CFT}}^{(\epsilon)}[g,\phi,\\{\chi\\}],$
(56) $\displaystyle=0,$ (57)

where at the end, we used the trace anomaly equation of the CFT. This can be
viewed as a derivation of what CFT anomaly ${\cal A}(x)$ is required if a
holographic correspondence is to hold.

##### Fixing the counterterms:

We now give a systematic procedure to fix the counterterms $\text{CT}(\mu)$.
First we pick a canonical transformation to exactly generate $\mathcal{W}$,
since otherwise we will not obtain the trace anomaly equation at the end. Next
we choose a relevant term, say $Y(x)$, that we want to get rid of and perform
a further canonical transformation of the form $\sim\int d^{d}xY(x)$. This
works because:

$\displaystyle e^{-\int d^{d}yY(y)}\mathcal{W}(x)e^{\int d^{d}yY(y)}$
$\displaystyle=$ $\displaystyle\mathcal{W}(x)+\left[\mathcal{W}(x),\int
d^{d}yY(y)\right]$ (58) $\displaystyle=$ $\displaystyle\mathcal{W}(x)+\sim
Y(x)+\sim\text{total derivatives}.$ (59)

However, if such a counterterm has already been used to generate the correct
form of $\mathcal{W}$, then it might not be possible to eliminate such a
relevant term without also modifying $\mathcal{W}$. This leaves us with only
one possibility to eliminate such relevant terms and that is by fixing certain
relations between bulk and field theory parameters. This is crucial, since if
we do not do this then our starting CFT will not satisfy a well-defined trace
anomaly equation, as it would include left-over relevant terms. Therefore,
these parameter relations are a direct consequence of the trace anomaly
equation of the CFT. Having done that, we proceed with the next relevant terms
in an iterative process until we only have marginal and irrelevant terms left,
at which point we are in a position to define $X^{(\mu)}(x)$ as in equation
(29).353535It is worth noting that there exists another formalism in which one
allows relevant terms in $X$, e.g. in pure GR $X$ might for example look
identical to the ADM Hamiltonian $\cal H$, even in $d>2$. However, the price
of doing this is that one must also include a second set of counterterms
$e^{CT_{2}(\epsilon)}$ on the right hand side of the path ordered exponential
in (52). (Or equivalently, one must agree to throw out all power law
divergences in $\epsilon$ when evaluating the $\int_{\epsilon}^{\mu}$
integral.) This formalism has the charm of making the formula for $X$ less
dependent on specific dimensions, but it obscures the nature of the RG flow
near the CFT. Since it does not allow for any distinct theories beyond those
mentioned in the main text, we do not need to consider it further.

As long as we restrict attention to CFT sources and fields in the range
strictly within the interval $0<\Delta<d$ (corresponding to $m^{2}<0$) there
should be only a finite number of possible relevant terms, equal to the number
of possible counterterms, and the process is guaranteed to terminate. This is
what happens in the case of pure Einstein gravity in any dimension.

However, outside this range it could also happen that the newly generated
terms are either more relevant or of equal relevancy to the term we were
trying to eliminate. This is in fact what will happen for the case of a bulk
scalar with $m^{2}>0$. In such cases we would need an infinite series of
counterterms to get rid of all the relevant terms. However, this happens in
the regime where we turn on nonrenormalizable sources in the original
undeformed holographic QFT.363636This should not be confused with the question
of whether the $T^{2}$ deformation is itself renormalizable. Since the $T^{2}$
deformation consists of irrelevant terms, it is always at least superficially
nonrenormalizable. and hence the original QFT (which is no longer conformal)
has to be regarded as an effective field theory valid only at low coupling.
Hence, the deformed theory will also only be defined perturbatively in the
massive bulk matter fields. Defining the bulk theory nonperturbatively would
presumably require UV completion of the $T^{2}$ theory.

###  2.4 Example: Scalar Field Coupled to Gravity

Consider a scalar field $\Phi$ in $d+1$ dimensions coupled to gravity with the
Hamiltonian constraint:

$\displaystyle\mathcal{H}(x)\Psi[g,\Phi]$ $\displaystyle:=$
$\displaystyle\Bigg{\\{}\frac{16\pi
G_{N}}{\sqrt{g}}:\\!\Big{(}\Pi_{ab}\Pi^{ab}-\frac{1}{d-1}\Pi^{2}\Big{)}\\!:-\frac{\sqrt{g}}{16\pi
G_{N}}(R-2\Lambda)$ (60) $\displaystyle+$
$\displaystyle\frac{1}{2}\left(\frac{1}{\sqrt{g}}\\!:\\!\Pi_{\Phi}^{2}\\!:+\sqrt{g}\left(g^{ab}\nabla_{a}\Phi\nabla_{b}\Phi+m^{2}\Phi^{2}+2V(\Phi,R)\right)\right)\Bigg{\\}}\Psi[g,\Phi]=0,$
(61)

where $V(\Phi,R)$ is an even potential with arbitrary non-minimal coupling
(excluding terms already explicitly in the action).

By dimensional analysis, the bulk matter field is related to the field theory
source via (by dimensional analysis):

$\displaystyle\Phi=\mu^{(1/d)(\Delta_{\phi}-(d-1)/2)}\phi,$ (62)

where $\Delta_{\phi}$ is the conformal dimension of the field theory source
$\phi$. The Weyl and Diff generators in the CFT are then determined by (11)
and (12) to be:

$\displaystyle{\cal D}_{a}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!-2\nabla_{b}\Pi^{b}_{a}+\partial_{a}\phi\,\Pi_{\phi},$
(63) $\displaystyle{\cal W}\\!\\!$ $\displaystyle=$
$\displaystyle\phantom{-}2\Pi\>\>-\>\,\Delta_{\phi}\phi\Pi_{\phi}.$ (64)

##### Explicit counterterms:

The correct choice of canonical transformation to remove all relevant terms
depends on the dimensions $d$ and $\Delta_{\phi}$. In order to keep things
simple we will consider the following canonical transformation—which is valid
for a certain range of parameters to be discussed below:

$\displaystyle\text{CT}:=\int d^{d}y\sqrt{g}\left(a+b\ R+c\ \Phi^{2}\right).$
(65)

Under the above canonical transformation, we have:

$\displaystyle\Pi^{ab}\to e^{-\text{CT}}\Pi^{ab}e^{\text{CT}}$
$\displaystyle=$ $\displaystyle\Pi^{ab}-\frac{i}{2}\sqrt{g}\left(a+c\
\Phi^{2}\right)g^{ab}+i\sqrt{g}\ b\ G^{ab},$ (66) $\displaystyle\Pi_{\Phi}\to
e^{-\text{CT}}\Pi_{\Phi}e^{\text{CT}}$ $\displaystyle=$
$\displaystyle\Pi_{\Phi}-2i\sqrt{g}\ c\ \Phi.$ (67)

By performing the canonical transformation of $\mathcal{H}(x)$, we obtain a
marginal term of the form:

$\displaystyle 16\pi G_{N}\frac{ia}{d-1}\Pi(x)-2ic\Phi\Pi_{\Phi}(x).$ (68)

By demanding this to be equal to $-i\mu^{-1/d}\mathcal{W}(x)$, we fix $a$ and
$c$:

$\displaystyle a$ $\displaystyle=$ $\displaystyle-2\frac{d-1}{16\pi
G_{N}}\mu^{-1/d},$ (69) $\displaystyle c$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{\phi}}{2}\mu^{-1/d}.$ (70)

Now because the cosmological constant term is relevant and we have already
used the counterterm $\int d^{d}x\sqrt{g}$, we need to impose the following
relation between the cosmological scale and the deformation coupling:

$\displaystyle\Lambda=-\frac{d(d-1)}{2}\mu^{-2/d},$ (71)

which identifies the scale $\mu^{1/d}$ with $L_{\text{AdS}}$. Similarly, given
that for the range of $\Delta_{\phi}$ in which the counterterm (65) is valid
(as explained below), the mass term is relevant we need to impose the usual
holographic mass relation for consistency:

$\displaystyle m^{2}=\Delta_{\phi}(\Delta_{\phi}-d)\mu^{-2/d}.$ (72)

In order to eliminate the only left-over relevant term, $\sqrt{g}R$, in
$\mathcal{H}$ we need to fix $b$:

$\displaystyle b$ $\displaystyle=$ $\displaystyle-\frac{1}{(d-2)16\pi
G_{N}}\mu^{1/d}.$ (73)

The choice of complex sign for $\mu$ determines whether our deformed partition
function satisfies the Wheeler-DeWitt equation with positive or negative
cosmological constant. In particular,

$\displaystyle\Lambda<0$ $\displaystyle\longrightarrow$
$\displaystyle\mu^{1/d}\in\mathbb{R}_{+},$ (74) $\displaystyle\Lambda>0$
$\displaystyle\longrightarrow$ $\displaystyle\mu^{1/d}\in i\mathbb{R}_{+}.$
(75)

Also notice that $m^{2}$ goes to minus itself in the dS case. In this paper,
we will proceed with the case of $\Lambda<0$, corresponding to AdS/CFT.
However it appears that one might also use the same approach to deform the
dS/CFT duality [58, 59] to finite time Cauchy slices; although in the dS case,
the Euclidean CFT cannot be reflection-positive even prior to the $T^{2}$
deformation. (We plan a followup paper to discuss this case [60].)

One significant difference between the two cases is that for dS deformations,
one can probe Lorentzian spacetime even when the parameter $\mu$ is small
(measured relative to other length scales set by the choice of metric
$g_{ab}$). But for the AdS case, it is necessary for $\mu$ to be sufficiently
large in order to cross a phase transition into Lorentzian signature (see
section 2.6).

##### Bulk scales:

As explained earlier, all the bulk scales of the gravity theory can be
expressed in terms of the only dimensionful coupling in the $T^{2}$ theory,
$\mu$, alongside a set of dimensionless parameters. We already saw how two of
these bulk scales (the AdS scale and the mass of the scalar field) are fixed
in this way, in (71) and (72), where $\Delta_{\phi}$ is the appropriate
dimensionless parameter. Similarly we can define another dimensionless
parameter, $\alpha$, that fixes the Planck scale relative to the AdS scale:

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{L_{\text{AdS}}^{d-1}}{16\pi G_{N}},$ (76)
$\displaystyle\implies 16\pi G_{N}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha}\mu^{\frac{d-1}{d}}.$ (77)

From the field theory perspective $\alpha$ will be related to things like the
anomaly coefficient, the stress tensor two-point function coefficient or the
coefficient of the divergence in the entanglement entropy. If we are in even
dimensions and our theory has a nonzero central charge, then we can fix this
ratio of scales at the current level of analysis, by comparing the anomaly
term $\mathcal{A}$ directly with the CFT result.

In odd dimensions, we need to work harder. This ratio can also be calculated
from the holographic CFT from e.g. relating the stress-tensor 2-point function
to gravitons in the bulk. So if our Cauchy slice theory is to describe a
consistent theory of gravity with a holographic dual, we must fix the
parameter $\alpha$ to get a consistent local bulk theory. If we pick the wrong
value of $\alpha$, all of the above equations will still hold (unless we fail
to satisfy an anomaly matching condition associated with another field theory
source), but we expect that the higher n-point functions of the stress-tensor
will not agree with a normal semiclassical gravitational bulk with Dirichlet
boundary conditions.373737This question is related to the “fake bulk” of [21,
61, 62].

##### Regime of validity:

We now discuss the regime of parameters in which the counterterms above are
valid. This requires that $X^{(\mu)}$ contain only irrelevant terms, and that
the marginal terms match with anomalies.

The remaining terms in
$i\mu^{1/d}\tilde{\mathcal{H}}^{(\mu)}-\mathcal{W}=-idX^{(\mu)}$ group into
conformal classes which are listed in Table 1 along with their conformal
weights $\omega$. For a term to be irrelevant, it must have positive conformal
weight.

Terms in $X^{(\mu)}$ | Conformal Weight $\omega$ | Condition for $\omega>0$
---|---|---
$\frac{1}{\sqrt{g}}\Big{(}\Pi_{ab}\Pi^{ab}-\frac{1}{d-1}\Pi^{2}\Big{)}$ | $d$ | $d>0$
$G_{ab}\Pi^{ab}-\frac{1}{d-1}G\Pi$ | $2$ | $2>0$
$\sqrt{g}\left(G^{ab}G_{ab}-\frac{1}{d-1}G^{2}\right)$ | $4-d$ | $d<4$
$\frac{1}{\sqrt{g}}\Pi_{\Phi}^{2}$ | $d-2\Delta_{\phi}$ | $\Delta_{\phi}<\frac{d}{2}$
$\sqrt{g}g^{ab}\nabla_{a}\Phi\nabla_{b}\Phi$ | $2-d+2\Delta_{\phi}$ | $\Delta_{\phi}>\frac{d-2}{2}$
$\sqrt{g}\Phi^{2}$ | $-d+2\Delta_{\phi}$ | $\Delta_{\phi}>\frac{d}{2}$
$\Phi^{2}\Pi$ | $2\Delta_{\phi}$ | $\Delta_{\phi}>0$
$\sqrt{g}\Phi^{2}G$ | $2-d+2\Delta_{\phi}$ | $\Delta_{\phi}>\frac{d-2}{2}$
$\sqrt{g}\Phi^{4}$ | $4\Delta_{\phi}-d$ | $\Delta_{\phi}>\frac{d}{4}$
Table 1:  “Relevancy” of the terms in $X^{(\mu)}$ for the counterterms chosen
in (65).

Notice that the conditions for $\frac{1}{\sqrt{g}}\Pi_{\Phi}^{2}$ and
$\sqrt{g}\Phi^{2}$ to be irrelevant are incompatible. That means that
depending on the value of $\Delta_{\phi}$ we will need a different counterterm
to cancel whichever one of the two is the relevant term.

The term $\sqrt{g}\left(G^{ab}G_{ab}-\frac{1}{d-1}G^{2}\right)$ will be
relevant for $d>4$, in which case we must include additional counterterms to
get rid of it. But it is irrelevant for $d=3$, so we make this choice for the
sake of illustration. Now we can read from the table the allowed range for
$\Delta_{\phi}$ such that all the terms in $X^{(\mu)}$ are irrelevant:

$\displaystyle\Delta_{\phi}\in\left(\frac{3}{4},\frac{3}{2}\right).$ (78)

Notice that for this range, the mass of the bulk scalar field will be negative
but this is allowed in AdS by boundary unitarity:

$\displaystyle
m^{2}\in\left(-\frac{9}{4},-\frac{27}{16}\right)L_{\text{AdS}}^{-2}.$ (79)

##### Explicit deformation:

As a result we have the deformation operator for $d=3$ and the above range of
$\Delta_{\phi}$ to be:383838The need to include the $\Pi_{\Phi}^{2}$ term was
emphasized in [63].

$\displaystyle X^{(\lambda)}(x)$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{3\alpha}\frac{1}{\sqrt{g}}:\left(\Pi^{ab}\Pi_{ab}-\frac{1}{2}\Pi^{2}\right):$
(80)
$\displaystyle+i\frac{2}{3}\lambda^{2/3}\left(G_{ab}\Pi^{ab}-\frac{1}{2}G\Pi\right)$
(81)
$\displaystyle+\frac{\alpha}{3}\lambda^{1/3}\sqrt{g}\left(G^{ab}G_{ab}-\frac{1}{2}G^{2}\right)$
(82)
$\displaystyle-\frac{1}{3}\lambda^{1-2\Delta_{\phi}/3}\frac{1}{2\sqrt{g}}\Pi_{\phi}^{2}$
(83)
$\displaystyle-\frac{1}{3}\lambda^{(2\Delta_{\phi}-1)/3}\frac{\sqrt{g}}{2}\left(g^{ab}\nabla_{a}\phi\nabla_{b}\phi+\frac{\Delta_{\phi}}{4}R\phi^{2}\right)$
(84)
$\displaystyle-\frac{1}{3}\lambda^{1/3}\sqrt{g}V\left(\lambda^{(\Delta_{\phi}-1)/3}\phi,R\right)$
(85)
$\displaystyle+i\frac{\Delta_{\phi}}{12\alpha}\lambda^{2\Delta_{\phi}/3}\phi^{2}\Pi-\frac{\Delta_{\phi}^{2}}{32\alpha}\lambda^{4\Delta_{\phi}/3-1}\sqrt{g}\phi^{4}.$
(86)

As shown in section 2.3, the deformed field theory partition function:

$\displaystyle
Z^{(\mu)}[g,\phi,\\{\chi\\}]:=e^{\text{CT}(\mu)}\left(\text{P}\exp\int_{0}^{\mu}\frac{d\lambda}{\lambda}\,O(\lambda)\right)Z_{\text{CFT}}[g,\phi,\\{\chi\\}],$
(87)

with the $\text{CT}(\mu)$ defined in (65) and $O(\lambda)$ defined in (33)
will satisfy the Hamiltonian constraint (61):

$\displaystyle\mathcal{H}^{(\mu)}(x)\ Z^{(\mu)}[g,\phi,\\{\chi\\}]=0.$ (88)

Note that here we did not need the $\epsilon$ regulator simply because in the
range of $\Delta_{\phi}$ for which this deformation is valid there are no
anomalies.

##### Addressing other ranges:

If we wish to consider a $\Delta_{\phi}$ outside the range
$(\frac{3}{4},\frac{3}{2})$, or $d\neq 3$, we need to adjust our deformation
accordingly. For generic values of $0<\Delta_{\phi}<\frac{d+2}{2}$
(corresponding to relevant sources in a unitary CFT), it should suffice to
simply choose a different set of relevant counterterms. This will change the
set of terms needed in $X^{(\mu)}$.393939We should also be careful about terms
in the potential $V$ becoming relevant as we change $\Delta_{\phi}$. We deal
with such terms as described for any relevant term. For example, for an
“alternative” boundary condition scalar with
$\frac{d}{2}<\Delta_{\phi}<\frac{d+2}{2}$, $\Pi_{\Phi}^{2}$ is relevant and
needs a counterterm, while $\Phi^{2}$ is irrelevant and can appear in
$X^{(\mu)}$. Under a canonical transformation of the form $\sim\int
d^{d}x\frac{1}{\sqrt{g}}\Pi_{\Phi}^{2}$ the source field $\Phi$ gets shifted
by:

$\displaystyle\Phi(x)\to e^{-\int
d^{d}y\frac{1}{\sqrt{g}}\Pi_{\Phi}^{2}}\Phi(x)e^{\int
d^{d}y\frac{1}{\sqrt{g}}\Pi_{\Phi}^{2}}$ $\displaystyle=$
$\displaystyle\Phi(x)+2i\frac{1}{\sqrt{g}}\Pi_{\Phi}(x).$ (89)

Notice that in this case, when we take the limit $\mu\to\epsilon$, our field
theory flows to a different IR fixed-point from
$Z_{\text{CFT}}^{(\epsilon)}[g,\phi]$ due to the presence of this operator-
dependent relevant counterterm.

On the other hand, for a measure zero subset of values of $\Delta_{\phi}$
(e.g. when any of the inequalities in Table 1 are saturated, or any of the
similar conditions that arise after doing additional canonical
transformations) there will be additional marginal terms in $X^{(\mu)}$, which
result in additional anomaly matching conditions to $\cal A$ in the CFT. For
example, when $\Delta_{\phi}=d/2$ exactly, there might be an anomaly in the
CFT proportional to $\phi^{2}$. Or when $d=4$, it is necessary for
$\sqrt{g}\left(G^{ab}G_{ab}-\frac{1}{2}G^{2}\right)$ to match with the
anomalies in the CFT—which implies that the 4d CFT has equal “central” charges
$a=c=\frac{L_{\text{AdS}}^{2}}{8G_{N}}$, as expected in AdS5/CFT4 duality
[64].404040These charges are not to be confused with the counterterm
coefficients of the rest of the section, which are labelled by the same
letters.

As $\Delta_{\phi}\to 0$, an increasingly large number of counterterms are
necessary, but they all take the form $f(\phi)$ times a finite number of
relevant expressions. On the other hand, for $\Delta_{\phi}<0$, there will be
an uncontrolled proliferation of counterterms needed, since every expression
becomes relevant when multiplied by a high enough power of $\phi$. (This is
related to the fact that the CFT becomes nonrenormalizable when perturbed by
the source $\phi$). It should still be possible to work out the deformation
order-by-order in perturbation theory in $\phi$, but we leave the details to
future work.

###  2.5 Normal Ordering Prescription

In order to explain why we need to consider a large $N$ theory, first let us
give a more complete description of the normal ordering symbol $:\>\>:$ in
e.g. (2) or (60)–(61). For us this symbol actually performs a double duty:

  1. 1.

It moves all factors of $\Pi^{ab}$ (with both indices raised) to the right of
arbitrary functions of the metric $g_{cd}$, thus removing unpleasant
divergences of the form
$[\Pi^{ab}(\mathbf{x}),g_{cd}(\mathbf{x})]=-i\delta^{ab}_{cd}\delta(\mathbf{0})$
which arise due to the commutator.414141Note that
$\Pi_{ab}=g_{ac}g_{bd}\Pi^{cd}$ has hidden factors of the metric, as does
$\Pi=g_{ab}\Pi^{ab}$. (This is the same result one would obtain if the
divergence is regulated by point-splitting.)

  2. 2.

It also subtracts off field theory divergences associated with the coincident
limit of the two $\Pi^{ab}$ operators as they approach each other in the field
theory. The way this works depends on whether we are at infinite or finite
$N$.

#### Infinite $N$:

In the $N\to\infty$ limit of an ’t Hooftian field theory, correlation
functions of stress-tensors $T_{ab}$ are dominated by Wick contractions. It
follows that the OPE of two stress-tensors is, up to terms subleading in
$1/N$, given by

$\Pi^{ab}(\mathbf{x})\Pi^{cd}(\mathbf{y})=C^{abcd}(\mathbf{x},\mathbf{y})\phantom{i}+:\\!\Pi^{ab}\Pi^{cd}\\!:(\mathbf{x})+\ldots,$
(90)

where $C^{abcd}(\mathbf{x},\mathbf{y})$ is a c-number which can depend on the
background metric and which diverges as $\mathbf{y}\to\mathbf{x}$, while the
second term is an operator of dimension $2d$ which is independent of the
separation. Hence, we can define the normal-ordering symbol as:

$:\\!\Pi^{ab}\Pi^{cd}\\!:(\mathbf{x}):=\lim_{\mathbf{y}\to\mathbf{x}}\left[\Pi^{ab}(\mathbf{x})\Pi^{cd}(\mathbf{y})-C^{abcd}(\mathbf{x},\mathbf{y})\right].$
(91)

This defines $:\\!\Pi^{ab}\Pi^{cd}\\!:$ (up to a possible curvature ambiguity
of weight $2d$).

This property holds even after doing a $T^{2}$ deformation with finite $\mu$,
since in the semiclassical approximation to gravity, fluctuations of the
metric are Gaussian. (Even though the $T^{2}$-deformed stress-tensor would be
multi-trace if written in terms of the original CFT fields.) Although the
theory is no longer conformal, (90) still picks out a unique definition of the
$:\\!\Pi^{ab}\Pi^{cd}\\!:$ operator, up to a c-number correction which can be
absorbed into other terms in $\cal H$.

A similar analysis would apply to the kinetic terms associated with matter
fields, e.g. $:\\!\Pi_{\Phi}^{2}\\!:$, but for simplicity we consider only the
metric in the remainder of this section.

#### Finite $N$, $\mu=0$:

The above definition works only at $N=\infty$. To see why, first consider the
case of $N=$ finite, $\mu=0$. In such a CFT, there will be additional
interaction terms in the stress-tensor OPE, but more importantly, in a generic
$d>2$ CFT, there will be no operator with dimension exactly $2d$. Instead the
dimension of this operator is shifted by some anomalous dimension $\eta(N)$,
which would require us to take a limit of the form

$\displaystyle:\\!\Pi^{ab}\Pi^{cd}\\!:\\!(\mathbf{x})$ $\displaystyle:=$
$\displaystyle\lim_{\mathbf{y}\to\mathbf{x}}\left[\frac{1}{|\mathbf{x}-\mathbf{y}|^{\eta}}\Pi^{ab}(\mathbf{x})\Pi^{cd}(\mathbf{y})-\hat{C}^{abcd}(\mathbf{x},\mathbf{y})\right],$
(92)

where now $\hat{C}^{abcd}(\mathbf{x},\mathbf{y})$ can depend, not just on
c-numbers, but also on operators of dimension up to $2d+\eta$.

The anomalous power $\eta$ means that the operator can no longer be specified
without introducing an extra length scale into the problem, and we cannot use
$\mu$ for this purpose as it vanishes. Because of this, it will be more
convenient to proceed directly to the case of finite $\mu$. This is acceptable
since our deformation is defined in (52) as an integral over finite values of
$\mu$.

#### Finite $N$, finite $\mu$:

Unfortunately, we cannot directly apply the CFT definition of normal ordering
at finite values of $\mu$, because at finite $\mu$ the theory is not a CFT in
the ultraviolet, and so we do not have access to a well-defined OPE if we take
the limit $\mathbf{y}\to\mathbf{x}$ in a careless manner.

We would like to still be able to have a well-defined kinetic operator
$G_{abcd}\\!:\\!\Pi^{ab}\Pi^{cd}\\!:\\!(\mathbf{x})$ operator at finite $N$
and finite $\mu$; otherwise our theory would not be probing bulk quantum
gravity. We will attempt to do this directly for finite value of the
deformation parameter $\mu$, using a point-splitting prescription. However, we
cannot take the point separation close to the Planck length, as this regime is
not understood. The Planck length scales as $l_{\text{Pl}}\sim\mu^{1/d}/N^{p}$
for some power $p$ (which depends on the particular AdS/CFT duality). So we
define a dimensionless parameter $\varepsilon$, such that the proper distance
$|\mathbf{x}-\mathbf{y}|=\mu^{1/d}\varepsilon$ and
$\frac{1}{N^{p}}\ll\varepsilon\ll 1$. In this way, the separation is much
smaller than the AdS scale but much larger than the Planck scale. Of course
this is only possible when $N\gg 1$.

We now consider an OPE expansion of
$\Pi^{ab}(\mathbf{x})\Pi^{cd}(\mathbf{y})$, in order to rewrite it in terms of
local operators in the deformed theory. This OPE will in general have
relevant, marginal and irrelevant terms. Since $\mu$ is the only scale, the
relevancy is entirely determined by the power of $\mu$. Any relevant terms
must be cancelled by the addition of counterterms. The remainder can then be
carried through into the definition of ${\cal H}(\mu)$ or (after substituting
$\mu\to\lambda$) into the definition of $O(\lambda)$, which will then be a
well-defined operator at each value of $\lambda>0$.424242This definition of
$:\\!\Pi^{ab}\Pi^{cd}\\!:$ becomes singular at $\lambda=0$, but the
elimination of relevant terms ensures that the integral over $\lambda$ values
is convergent (up to a possible log divergence, which would cancel with the
CFT anomaly as usual). We can then perform the path-ordered exponential (52)
in order to define $Z^{(\mu)}$.434343In the above procedure it is not manifest
that the closure relation (26) for the Hamiltonian constraint (and hence
deformation flow) continues to hold. Most likely it is necessary to add
additional counterterms to eliminate all failures of closure order-by-order in
$\varepsilon$. Presumably this is always possible if the bulk theory has no
gravitational anomaly.

In order to ensure that the momentum constraint ${\cal D}_{a}=0$ is satisfied,
it is necessary to ensure that the point-splitting prescription is done in a
fully covariant manner. One possible way to do this, if $g_{ab}$ is
sufficiently smooth, is to average over all possible geodesic segments of
length $\mu^{1/d}\epsilon$ emanating from $\mathbf{x}$:

$G_{abcd}(\mathbf{x})\Pi^{ab}(\mathbf{x})\int
d\Omega\;\tilde{\Pi}^{cd}(\mathbf{y}(\Omega)),$ (93)

where $d\Omega$ is the rotationally-invariant measure, $\mathbf{y}(\Omega)$ is
the point which is a unit proper distance along a geodesic of length
$\mu^{1/d}\epsilon$ at angle $\Omega$, and $\tilde{\Pi}^{cd}$ is obtained by
parallel transporting $\Pi^{cd}$ along the geodesic.444444Recall that our
normal ordering prescription takes all metric dependence in the covariant
derivative to the left of $\Pi^{ab}(\mathbf{x})$. This means that the geometry
used to define the covariant point-splitting procedures is done using the
background metric, prior to the variations associated with $\Pi^{ab}$. An
alternative, smoother regulator would be to use the heat kernel, in which case
the point-split operator would be

$\displaystyle G_{abcd}(\mathbf{x})\Pi^{ab}(\mathbf{x})\int
d^{d}{\mathbf{y}}\sqrt{g}\;K^{cd}_{ef}(s,{\mathbf{x}},{\mathbf{y}})\Pi^{ef}({\mathbf{y}}),\qquad\sqrt{s}=\mu^{1/d}\varepsilon,$
(94)

where $K^{ab}_{cd}(s,{\mathbf{x}},{\mathbf{y}})$ is the heat kernel
coefficient at Schwinger time $s$ associated with some wave equation which
propagates symmetric rank-2 tensors.

There can also be UV divergences in the theory associated with the limit as
two or more _distinct_ deformation operators (like $:\\!\Pi^{ab}\Pi^{cd}\\!:$
or $G^{ab}\Pi^{cd}$) approach one another. These divergences can appear at
various powers of $\mu$ and with various curvature couplings. We assume that
any such divergences are also regulated using a cutoff of length
$\mu^{1/d}\epsilon$, in order to ensure that 1) the deformation retains the
correct dimensional scaling with $\mu$, and 2) the theory is allowed to
exhibit locality at sub-AdS scales.

###  2.6 Exotic Properties of the Deformed Theory

The $T^{2}$ theory exhibits some exotic phenomena which violate the usual
axioms of QFT: nonunitarity and (relatedly) spontaneous CPT violation. These
phenomena should be welcomed as they are crucial for describing the regime in
which the Cauchy slice $\Sigma$ is embedded in Lorentzian
signature.454545Another exotic property of the Lorentzian signature
$T^{2}$-deformed field theory is that it allows superluminal signalling.
However, we do not analyze this property here since we are primarily
interested in the Euclidean signature $T^{2}$-deformed theory, where this
issue does not arise.

#### Nonunitarity

The Euclidean Wick rotation of a unitary QFT is reflection positive [65].
However, the $T^{2}$ theory is not reflection positive.

One way to see the violation of unitarity is to calculate the energy levels of
the $T^{2}$-deformed theory on a stationary spacetime e.g. a cylinder
$S_{d-1}\times\mathbb{R}$, where the sphere has radius $R$ and volume
$\Omega_{d-1}R^{d-1}$. Hartman, Kruthoff, Shaghoulian and Tajdini [15] did
this calculation for states of zero angular momentum, obtaining (in our
notation):

$E(\mu;R)=-2(d-1)\alpha\Omega_{d-1}\frac{R^{d-1}}{\mu}\;\sqrt{1-\frac{\mathcal{E}\mu}{(d-1)\alpha\Omega_{d-1}R^{d}}+\frac{(d-2)\mu^{2/d}}{R^{2}}},$
(95)

where the $\mu=0$ boundary condition is $E(\mu=0;R)=\mathcal{E}/R$, with
$\mathcal{E}$ being the energy quantum number of the undeformed CFT. This
result is only valid in the large-N limit of the theory. The expression above
is the result after shifting by the counterterm $\text{CT}(\mu)$,
corresponding to the case of pure gravity bulk.

One can see that there is a branch cut when the energy $E=0$, at some critical
value of $\cal E$. At this critical value, the microcanonical density of
states $\rho$ is given by

$\displaystyle\ln\rho(E=0)=S_{\text{BH}}=\frac{A}{4G_{N}},$ (96)

the black hole entropy. Higher $\mathcal{E}$ states of the CFT are mapped to
imaginary eigenvalues of $E$, for which $S>S_{\text{BH}}$. [11] argued that
these imaginary eigenstates should simply be truncated from the spectrum.
However, this approach is incompatible with the continuation to Lorentzian
signature, since it would forbid a timelike sign for the extrinsic curvature
$K_{ab}$. Hence, in order to describe holography on Cauchy slices, it is
important to keep these states in the spectrum [47]. It follows that the
theory that lives on the Cauchy slices is inherently nonunitary.464646This
type of nonunitarity, associated with a complex Hamiltonian, should not be
confused with the totally distinct idea that pure states might evolve to mixed
states! Hence, our holographic model of Cauchy slices cannot be used to argue
that information is lost inside black holes, as Hawking originally argued
[26]. In fact our model implies the exact opposite: all the information on any
Cauchy slice $\Sigma$ “flows” outward to the boundary at infinity. See section
7.3.

In our conventions, the field theory always lives on a Euclidean slice
$\Sigma$, and the branch cut is associated with changing the signature of the
bulk spacetime that $\Sigma$ is embedded in. (Imaginary $E$ gives a Lorentzian
bulk.)474747There is an alternative picture, associated with the radial
$T^{2}$ deformation, in which the bulk spacetime is always Lorentzian. In this
picture, the imaginary values of $E$ correspond to going inside the horizon of
a bulk black hole [11]. In this picture it is the signature of $\Sigma$ that
changes since radial slices inside the black hole are Euclidean. See Figure 2
for an example of this transition (but for a more general $\Sigma$ geometry).

This non-unitarity may seem surprising, but it was argued in [66] that the
theory of gravitational subregions is necessarily nonunitary, due to the
spontaneous breaking of $CPT$ symmetry that is associated with Cauchy slices
in Lorentzian spacetime. Furthermore, dS/CFT dualities, which also have an
emergent time direction, are known to be nonunitary [58, 59]. So we have to
learn to live with this.

Fortunately, it turns out that the nonunitarity of the boundary $T^{2}$ field
theory is compatible with unitarity of the bulk theory 7.3. Actually, bulk
unitarity emerges in a natural way from the unitarity of the starting CFT,
together with the fact that CPT symmetry is not explicitly broken.

#### Spontaneous CPT breaking

$CPT$ is however, spontaneously broken in the boundary theory! Another way of
looking at the branch cut in (95), is that it corresponds to a transition
between two phases with different symmetry properties. When $\Sigma$ is
embedded in a Euclidean bulk, this corresponds to a time-reversal symmetric
phase of the gravitational path integral, while a Lorentzian signature bulk
corresponds to a time-reversal violating phase of the gravitational path
integral [66].

In fact even the combined $CPT$ symmetry is broken. In pure gravity, $C$ acts
trivially, while $P$ is unbroken by the above construction. Hence the
$T$-violating phase also breaks $CPT$.484848Note that on a given Cauchy slice
$\Sigma$, $CPT$ means the same thing for the bulk and boundary theories, even
if we add matter. For $P$ this is obvious. In Euclidean signature, $T$ acts as
complex conjugation, which reverses the sign of $\Pi_{ab}$ due to the
imaginary sign in 4; so this looks just like time reversal in the bulk. And
$C$ acts trivially if we write the fields in a real basis. If we include both
possible signs for the imaginary energy eigenvalues (as advocated in [47]),
this can be regarded as a form of spontaneous symmetry breaking. This is of
course not possible in a unitary, rotationally-symmetric theory, by the $CPT$
theorem. But since the energy eigenvalues are imaginary, it is clear the
$T^{2}$ theory is _not_ unitary, so this is no contradiction.

The boundary between the two phases is given by a moment-of-time symmetric
slice (sometimes called a “t = 0” slice), which lies on both the Euclidean and
Lorentzian contours.494949However, because of the branch cut, the boundary
between the two phases is _not_ a standard second order phase transition
corresponding to a CFT. (In fact it is even less smooth than a normal 1st
order phase transition.) The theory on the boundary is highly non-Gaussian,
which is related to the fact that, in the gravitational dual, the
semiclassical approximation breaks down there, due to the absence of an
elliptic boundary condition for the gravitons [67]. In a minisuperspace
approximation, the region near the $t=0$ slice looks like an Airy function
$\text{Ai}(x)$, and semiclassical approximation is just the WKB approximation.
This approximation diverges at the transition between the exponential and
oscillatory phases, although $\text{Ai}(x)$ itself is perfectly well-defined
there. This may have significant implications for tensor network models of
AdS/CFT constructed on the $t=0$ slice (cf. section 1.5.

Similar phase transitions ought to occur for non-stationary backgrounds as
well, in order to match the dual gravitational path integral, although in such
cases it is no longer expedient to analyze them using energy eigenvalues. Let
us be very explicit about the following point:

_In order for our proposed AdS/CFT dictionary in this paper to make sense, it
is necessary to assume that the $T^{2}$-deformed theory continues to make
sense on the Lorentzian side of this phase transition!_

##  3 Towards a Generalized Holographic Principle

From this point onwards we will be dealing with a pure gravity bulk for
simplicity of notation. Matter fields can be added everywhere with minimal
modifications.

###  3.1 Form of the Duality

Motivated by the AdS/CFT correspondence [3, 4, 5], holography at finite cutoff
[11] and the realization that deformed field theories define WDW-states [15,
21, 22], including on Cauchy slices, we now postulate a generalized form of
the holographic duality for partition functions with arbitrary boundaries:

The Generalized Holographic Principle (GHP): This hypothesis states that the
$T^{2}$-deformed partition function of this boundary theory is equal to the
gravitational path integral over space-filling manifolds
$(\mathcal{M},\mathbf{g})$ with Dirichlet boundary conditions on
$\partial\mathcal{M}$, as shown in Fig. 3. (A more precise definition of the
right-hand side will be provided in section 4.)

$\boxed{Z_{T^{2}}^{(\partial\mathcal{M})}[g]=\sum_{\mathcal{M}}\int_{\mathbf{g}|_{\partial\mathcal{M}}=g}\frac{D\textbf{g}}{\text{Diff}(\mathcal{M})}\,e^{iI_{\text{grav}}[\textbf{g}]}}$
(97)

Here, on the LHS, we have a $T^{2}$-deformed field theory living on
$\partial\mathcal{M}$ with metric $g$. This could potentially be a complex
metric (e.g. it could be Lorentzian in some parts of $\partial\mathcal{M}$ and
Euclidean in some other parts of $\partial\mathcal{M}$).

On the RHS, we sum over possible manifolds $\mathcal{M}$ and bulk metrics
$\mathbf{g}$ that are compatible with our boundary conditions—albeit with the
usual caveats and controversies associated with the gravitational path
integral, which are discussed in the remainder of this section.

=Bulk metricgBoundary metric $g$Partition function of the field
theoryGravitational path integral with Dirichlet boundary conditions
$\mathbf{g}|_{\partial\mathcal{M}}=g$$\partial\mathcal{M}$$\mathcal{M}$ Figure
3: The figure on the left represents the partition function of the
$T^{2}$-deformed field theory living on $\partial\mathcal{M}$ (the closed
curve) with a background metric $g$. The shaded figure on the right represents
the gravitational path integral over the space filling manifold $\mathcal{M}$.
The bulk metric $\mathbf{g}$ satisfies the Dirichlet boundary conditions (i.e
$\mathbf{g}|_{\partial\mathcal{M}}=g)$.

An initial reason to believe this hypothesis is that we have already shown
that the $T^{2}$ theory obeys the constraint equations of general relativity.
So morally speaking, we are simply requiring that this duality also hold at
the level of gravitational path integrals with a single boundary. However, we
do not in this paper claim to show the equivalence between the canonical and
path-integral approaches to quantum gravity—particularly since the path-
integral does more than just project onto states satisfying the constraint
equations $\mathcal{H}\Psi=\mathcal{D}_{a}\Psi=0$; it also provides a
dynamical definition of the inner product between two states, which will be
constructed more explicitly in section 4.505050A careful analysis of the
relationship between the $T^{2}$ deformation and the path integral was done in
[21], but in their analysis the manifold ${\cal M}$ lies in between
$\partial{\cal M}$ and the usual CFT boundary, so it is the “inverse” of the
usual path integral. This could be taken as an argument for the GHP, starting
with the original form of the AdS/CFT equivalence.

###  3.2 The Need for a Contour Prescription

What we have said does not quite suffice to fully specify the Generalized
Holographic Principle. For one thing, there are controversies about the
correct choice of contour of the integral over metrics in quantum gravity. As
indicated above, we sometimes want to use complex contours, for example those
associated with Euclidean signature saddles. But in the literature, there are
(at least!) two divergent philosophies for dealing with these Euclidean
saddles:

  1. 1.

Hartle and Hawking [68] assume that, in cases where a Euclidean saddle is
necessary, we always rotate the contour in the direction of the standard Wick
rotation $\tau=it$, even in cases where the resulting saddle is exponentially
enhanced. This has the advantage that the horizon entropy is always $S=A/4G$,
but sometimes the connection to the Lorentzian picture is unclear.

Fluctuations around such Euclidean saddles are not bounded below, because of
the ‘conformal mode problem’, where conformal modes have negative action. So
to calculate the quantum determinant $\Delta_{\text{1-loop}}$ one must also
deal with this, perhaps by rotating the contour of the conformal mode to
imaginary values [69].

  2. 2.

Other authors, including Vilenkin [70, 71] and Feldbrugge, Lehners, & Turok
[72], have argued that one should instead take the Lorentzian definition of
the path integral as fundamental. In this approach, the contour may be taken
to be complex only when, using Picard–Lefschetz theory, one can show that the
resulting saddle is a resummation of oscillatory Lorentzian contributions.

Invariably, the saddles that resum an oscillatory path integral are
exponentially suppressed, rather than exponentially large. In some cases, this
forces you to take the opposite sign of imaginary time, corresponding to
$\tau=-it$. In such anti-Euclidean saddles, while the conformal modes have
positive action, the transverse-traceless modes (corresponding to
gravitational waves) and ordinary matter fluctuations would have negative
action, leading to inverse Gaussian, unsuppressed fluctuations in the
resulting state [73].515151Cf. [74] for a proposed solution to this problem.
On the other hand, [75] argued that, in the context of dS cosmology, there is
_no_ choice of contour that leads to physically good results! But even if this
no-go result is valid, it would not necessarily rule out a holographic
cosmology theory defined on Cauchy slices, since the holographic theory only
needs to agree with a semiclassical calculation in the neighborhood of some
particular saddle—whichever one _it_ agrees with—and could be quite different
from any semiclassical calculation in the far off-shell regime. Furthermore,
this choice leads to an unphysical value of the Gibbons-Hawking entropy in AdS
or dS (corresponding to $-A/4G$ instead of $+A/4G$).

In our opinion, the Hartle-Hawking approach gives results that seem more
physically reasonable for defining a ground state. Since the conformal mode
corresponds to a non-propagating degree of freedom, it can always be gauge
fixed by a suitable choice of the time variable. On the other hand, having
unnormalizable fluctuations for the _physical_ modes around a supposedly
dominant saddle, in vacuum dS or AdS, would be disastrous, and of course most
of us will want the horizon entropy to have the correct sign.

Of course, the $\mu\to 0$ limit of the $T^{2}$ field theory corresponds to
going back to the original CFT. And it is generally accepted that in AdS/CFT,
a Euclidean CFT is dual to the Hartle-Hawking saddle, since this is the saddle
that gives rise to the correct boundary entropy and free energy. So by
continuity, it appears that the $T^{2}$ field theory must force us to pick the
Hartle-Hawking saddle.

###  3.3 Challenges to the Validity of the GHP

That being said, there is also a few potential reasons to doubt the GHP as
we’ve stated it. Here are some of them:

##### 1\. Nonrenormalizability (and other technical problems):

Obviously, the gravitational path integral is subject to all the usual
ambiguities of quantum gravity, including nonrenormalizability of the UV
divergences, among other technical issues. These issues should be resolvable
by taking the bulk theory to be a UV complete model, such as (presumably)
string theory.

Other potential technical problems include the oscillatory nature of the path
integral in Lorentzian signature, the fact that the Euclidean action is
unbounded below; questions concerning singularities and topology change, and
possible IR issues due to the unrestricted range of time integration. To
define a precise duality at the nonperturbative level, we would obviously need
to assume that such issues can be resolved.

##### 2\. Factorization Problem:

If $\partial{\cal M}$ contains 2 disconnected boundary components (i.e.
${\partial\cal M}=B_{1}\sqcup B_{2}$), then any local field theory partition
function must factorize:

$Z[\partial{\cal M}]:=Z[B_{1}\sqcup B_{2}]=Z_{1}[B_{1}]Z_{2}[B_{2}].$ (98)

In the usual way of doing holography one runs into the problem that it is not
at all manifest that the gravitational path integral has this factorization
property, since it would appear to include sums over geometries in which the
bulk $\cal M$ contains a single connected component, because one or more “baby
universes” connect the two sides. For the analogous problem of disconnected
CFT boundaries, in some cases one can use curvature conditions to argue [76]
that there are no on-shell saddles violating factorizability (98), but even
then one might worry about the contributions from off-shell metrics.525252See
[77, 78] for further discussion of Euclidean AdS wormholes solutions.

However, it has also been argued that the sole effect of such baby universes
is to shift the value of the allowed counterterms in the local gravitational
action [79, 80]. In a bulk theory such as string theory which has no free
parameters, this means that baby universes processes cannot really affect
anything at all! Indeed, if one accepts the GHP stated in (97) in complete
generality, a baby universe—or more generally any closed universe—would
correspond to a closed partition field theory function $Z$, which can always
be evaluated to get a complex number. This would imply that baby universes
cannot in fact transmit any information; in other words any connected geometry
in the path integral should be gauge-equivalent to some process in which the
two sides are disconnected. (It might be, however, that the equivalent
disconnected geometry requires using some type of end-of-the-world brane, or
other structures, which only appear in string theory.)535353Recently there has
been interest in relating the gravitational path integral to ensembles of
theories with varying coupling constants, because that allows disconnected
partition functions not to factorize. This is of no use with classic forms of
AdS/CFT where the CFT coupling constants take definite values. But if e.g. the
pure gravity path integral does not factorize, this could be because it
represents a nontrivial ensemble over holographic theories [81, 82, 83, 84].
This is probably why baby universes can play a nontrivial role in the replica
wormhole calculations of the Page curve [27, 85, 86].

##### 3\. Information Paradox:

Every field theory partition function generates a state of the boundary. We
will define this bulk-to-boundary map more explicitly in section 6.

Because of this map, the $T^{2}$-deformed theory makes it manifest that any
information which falls across the horizon of a one-sided black hole (because
it lies on a Cauchy slice which is connected to the boundary) is still
holographically encoded in the boundary theory.

In this respect it is opposed to the usual geometrodynamics perspective, which
in an evaporating black hole regime makes this information escape seem
unlikely [87], but see [88, 89, 90, 91]. One way that the information can
nevertheless escape, is if the $T^{2}$ partition function implements
additional constraint equations on the spacetime (associated with the finite
dimensionality of its Hilbert space at fixed energy) [47] which the purely
Lorentzian geometrodynamics theory knows nothing about.545454Although it has
been argued that Euclidean path integrals implement some nontrivial additional
constraints [92]. If this is true, and if we take the GHP to be valid, then
this imposes restrictions on the regime of validity for a semiclassical
approximation of the gravitational path integral. For example, this tells us
that for an evaporating black hole after the Page time, the semiclassical
approximation becomes invalid, and we should instead use the $T^{2}$ theory to
study how the information escapes to the boundary.

###  3.4 $T^{2}$ as the Definition of the Bulk Theory?

In order to deal with all of these problems, we shall adopt the following
point of view: We assume that as long as we evaluate (97) semiclassically,
using metrics that are close to dominant classical saddles, that both sides of
the duality are well-defined. In the field theory, this corresponds to a large
N expansion. In that semiclassical regime, this duality could potentially be
falsified, or require modifications, if the two sides were found to disagree.

As an even more ambitious proposal, we suggest that the GHP is likely to be
valid at all orders in a 1/N expansion. This corresponds to an $\hbar G$
expansion in the bulk.

On the other hand, whenever we go _beyond_ the regime in which
geometrodynamics is likely to be valid, we can take the $T^{2}$ boundary
theory as a _definition_ of the gravitational amplitude.555555A similar
attitude has sometimes been proposed towards the standard AdS/CFT duality,
that the CFT should simply be regarded as the _definition_ of quantum gravity.
This is unsatisfying because it gives no way, even in principle, to answer any
of the most perplexing questions about quantum gravity, like what happens
behind horizons or near singularities. On the other hand, taking the
$T^{2}$-deformed theory as the definition of quantum gravity seems potentially
more fruitful, since the $T^{2}$-deformed theory is _already_ embedded inside
the AdS bulk—so it doesn’t require you to give up on the idea of having a
physical picture which is valid deep inside event horizons! Of course, this
attitude raises the question of whether the $T^{2}$-deformed theory _itself_
makes sense non-perturbatively. The fact that the $T^{2}$ couplings are
irrelevant strongly suggests that it may not be, except in those special cases
($d=2$ or $N=\infty$) where it is exactly solvable.565656Another potential
problem is that the $\Pi_{ab}^{2}$ terms look like inverse diffusion, which is
not always well-defined after a finite amount of flow. It might be that
analytically continuing around such obstructions is the origin of the branch
cuts in the $T^{2}$ spectrum. It would be helpful if someone could do an
explicit calculation to confirm or deny this hypothesis.

If it is not fully well-defined, then the theory would need to be UV completed
somehow, just like its associated bulk quantum gravity theory. However, it is
probably much easier to UV complete a field theory defined on a fixed
Euclidean background spacetime, than it is to UV complete a background-
independent model of quantum gravity in Lorentizan signature! We will say more
about this issue in the Discussion.

##  4 QG Inner Product from the Path Integral

In section 4.1, we start by defining the Lorentzian path integral from a
purely gravitational perspective, without making use of the GHP proposed in
(97).

Section 4.2 describes the importance of integrating over both signs of the
Lorentzian lapse $N$ in order to preserve gauge-invariance. At the end we make
a few comments about the possibility of non-Lorentzian contours. In section
4.3 we discuss the example of maximally symmetric metrics $g$ to illustrate
how both Euclidean and Lorentzian contours can arise.

Finally, in section 4.4, we discuss some additional important properties of
the amplitude. It is quite easy to show that the amplitude is linear and
hermitian. We also present an argument that the path integral is positive, at
least in a saddle-point approximation, when $\Lambda<0$. We also discuss the
issue of when the amplitude is finite. (These properties will all be important
in section 5, when we construct the Hilbert space.)

###  4.1 Lorentzian Transition Amplitude

We will start by defining the Lorentzian transition amplitude (but we will
later discuss how things generalize if the contour becomes complex).

We can formally define the Lorentzian transition amplitude as:

$\langle
g_{2}\,|\,g_{1}\rangle_{I}=\sum_{\mathcal{M}}\int^{\mathbf{g}|_{\Sigma_{2}}=g_{2}}_{\mathbf{g}|_{\Sigma_{1}}=g_{1}}\frac{D\textbf{g}}{\text{Diff}(\mathcal{M})}\,e^{\pm
iI_{\text{grav}}[\textbf{g}]},$ (99)

where $\mathbf{g}$ is any $D=d+1$ dimensional Lorentzian metric, on a
globally-hyperbolic submanifold ${\cal M}\subset{\cal B}$ of a possible
asymptotically-AdS bulk geometry on ${\cal B}$. Note that the metric
$g_{ab}^{\text{bdy}}$ on $\partial{\cal B}$ is fixed, so the asymptotic
behavior of $\mathcal{B}$ is given by a Fefferman-Graham expansion, whose
leading order behavior is given by (8).

$\mathcal{M}$ is the manifold connecting $\Sigma_{1}$ to $\Sigma_{2}$. We
require that $\Sigma_{1}$ and $\Sigma_{2}$ are anchored to the same boundary
Cauchy slice $\Sigma^{\text{bdy}}\subseteq\partial\cal B$ (i.e. a Cauchy slice
of the usual boundary CFT). $\text{Diff}(\mathcal{M})$ is the group of
spacetime diffeomorphisms acting in the interior of ${\cal M}$.575757For
purposes of doing explicit calculations it is often convenient to gauge-fix in
order to remove the (noncompact) volume integral over the diff group
$\text{Diff}(\mathcal{M})$ in the denominator, which necessitates the
introduction of Faddeev-Popov ghosts. Since our present concern is not to
manipulate the amplitude into a convenient form for calculation, we will
retain the current form, sometimes called “unitary gauge”. 585858If the
topologies of $\Sigma_{1}$ and $\Sigma_{2}$ are different in Eq. (99), then
this must be interpreted as transition amplitude between different topologies
of bulk Cauchy slices, as well as different metrics.

The gravitational action is:

$\displaystyle I_{\text{grav}}=\frac{1}{16\pi
G_{N}}\int_{\mathcal{M}}\\!d^{D}x\sqrt{-\mathbf{g}}\,(\mathbf{R}-2\Lambda)\>-\>\frac{1}{8\pi
G_{N}}\int_{\partial\mathcal{M}}\\!d^{d}x\sqrt{g}\,K$
$\displaystyle\>+\>\int_{\mathcal{M}}\mathcal{L}_{\text{RG}}+\int_{\partial\mathcal{M}}\mathcal{L}^{\text{bdy}}_{\text{RG}}+\int_{\Sigma^{\text{bdy}}}\mathcal{L}_{\text{lapse}},$
(100)

where $\partial\mathcal{M}=\Sigma_{1}\cup\Sigma_{2}$,
$\mathcal{L}_{\text{RG}}$ and $\mathcal{L}^{\text{bdy}}_{\text{RG}}$ refer to
RG counterterms needed to cancel UV divergences of bulk fluctuations.

$\mathcal{L}_{\text{lapse}}$ refers to counterterms needed to deal with IR
divergences of the action associated with the corner terms between the Cauchy
slices $\Sigma_{1}$ and $\Sigma_{2}$ as one approaches the asymptotic AdS
boundary $\partial{\cal B}$. In Fefferman-Graham coordinates (8), we may
define a conformally-rescaled relative lapse as $\tilde{N}=zN$, where $N$ is
the proper time difference between $\Sigma_{1}$ and $\Sigma_{2}$. There is
then a divergence in the spacetime volume (and hence the action) if
$\tilde{N}\sim z^{d}$, or any lower power of $z$. If we consider the class of
Cauchy slices which are conformally smooth at $z=0$, dimensional analysis
indicates that only a finite number of IR counterterms will be
required.595959We could also restrict attention only to a subset of Cauchy
slices whose relative lapse falls off quickly enough that there are no
divergences. An alternative, somewhat more convenient approach (which we will
use in section 7.2) is to take $\partial\Sigma$ to be a small distance
$z=\epsilon$ away from the conformal boundary $\partial{\cal B}$.

$D\mathbf{g}$ is a local covariant measure on the space of metrics, which can
be defined formally as the volume induced by the following metric on the space
of metrics:606060There is one other possible derivative-free covariant
integrand,
$\sqrt{-\mathbf{g}}\,\delta\mathbf{g}_{\mu\nu}\,\delta\mathbf{g}_{\sigma\tau}\,\mathbf{g}^{\mu\nu}\mathbf{g}^{\sigma\tau}$,
which would change the measure of the conformal mode. But since all we care
about is the overall volume, this parameter can be absorbed into a rescaling
of $C$.

$d\mathbf{s}^{2}=C\int_{\cal
M}d^{D}x\,\sqrt{-\mathbf{g}}\,\delta\mathbf{g}_{\mu\nu}\,\delta\mathbf{g}_{\sigma\tau}\,\mathbf{g}^{\mu\sigma}\mathbf{g}^{\nu\tau},$
(101)

where $C$ is a dimensionful parameter needed to define the path
integral.616161$C$ is not actually an independent variable, since rescaling it
to a new value $C^{\prime}$ is equivalent to adding an _imaginary_ counterterm
to the action $I$ which is proportional to $i\ln(C^{\prime}/C)$ times a
(regulator-dependent) local integral. Such imaginary terms in the action are
not arbitrary, but must be fixed to whatever value is required for unitarity
to hold. (This is one reason why it is simpler to regulate in Euclidean
signature, where such measure-dependent counterterms are instead real. More
generally, as discussed in section 7.3, imposing $CPT$ symmetry fixes the
signs of terms in the action, in a way that guarantees that bulk unitarity
cannot be spoiled by local counterterms.) This measure $D\mathbf{g}$ is well-
defined when considering a finite-dimensional space of metric variations, so
it is reasonable to think that the measure can be defined in effective field
theory after imposing a UV regulator which cuts off short distance modes.

###  4.2 Why we use both Signs for the Lapse

The reason for the $\pm$ sign in front of the gravitational action, is that we
are including both signs of the lapse in our gravitational path integral.

A famous paper by Claudio Bunster (formerly Teitelboim) [93] argues that one
ought to restrict the quantum gravitational path integral only to histories
that have positive values of the lapse, which is equivalent to saying that
$\Sigma_{2}$ lies to the future of $\Sigma_{1}$. Bunster frames this question
as a choice between “causality” and “gauge-invariance”, however we
respectfully disagree with his interpretation.

Bunster correctly observes that if we restrict the path integral only to
positive values of the lapse, that the resulting final state will no longer
obey the Hamiltonian constraint equation ${\cal H}(x)\Psi=0$, violating gauge-
invariance. Normally, the Hamiltonian constraint is guaranteed because it is
the Euler-Lagrange equation associated with the variation of the lapse $N(x)$.
But if we integrate over the domain $N(x)\geq 0$, then at the boundary of this
domain (where $N(x)=0$ for some $x$, i.e. $\Sigma_{1}$ and $\Sigma_{2}$
touch), we no longer have the ability to freely vary the lapse, hence ${\cal
H}(x)=0$ no longer follows. (Additionally, the resulting transition amplitude
would no longer be Hermitian.) In our opinion, this is an excellent reason to
include both signs of $N(x)$ in the variation. Such a contour over the whole
real line (with a small deformation away from the essential singularity at
$N=0$) has in fact already been proposed in [94].

Bunster resists this conclusion by arguing that if we do include both signs of
the lapse in the path integral, the theory will no longer be causal. We do not
share this opinion. Causality has to do with the propagation of physical
signals _within_ the manifold $\cal M$ (which is why we demand that $\cal M$
is globally hyperbolic). It does not have anything to do with the statement
that, for two abstractly defined Cauchy slices, $\Sigma_{2}$ lies to the
future of $\Sigma_{1}$ (a statement that is false in general). We are merely
trying to define a physical quantum gravity inner product $\langle
g_{2}\,|\,g_{1}\rangle_{I}$ with nice properties, and nothing about this task
requires the bra state $\langle g_{2}\,|$ to necessarily be to the future of
the ket state $|\,g_{1}\rangle$.626262If a supernatural being created a whole
universe, with some particular initial condition at time $t_{0}$, and _if_
causality holds in the sense that this being’s action at $t_{0}$ only affects
times to the future $t\geq t_{0}$ of that Cauchy slice, then the use of
Bunster’s “causal” propagator might be appropriate. But this is not the sort
of scenario we normally consider in physics. Usually we assume that there was
a past era, prior to the time of whatever experiment is being performed. For
example, in most discussions of AdS/CFT, it is assumed that the AdS spacetime
has existed forever. Even in cosmology, it is usually assumed that the
beginning of time was an initial singularity, rather than a Cauchy slice. In
fact, if we restrict to positive lapse solutions, the resulting inner product
will no longer be hermitian.

Hence, to obtain a gauge-invariant inner product, we must allow all possible
time relationships between the two Cauchy slices. Since $N(x)$ is a function
of spatial position, there are actually three possible cases to
consider:636363This classification ignores degenerate cases where $\Sigma_{1}$
and $\Sigma_{2}$ entirely coincide in some open region.

  1. 1.

$\Sigma_{2}$ lies entirely to the future of $\Sigma_{1}$,

  2. 2.

$\Sigma_{2}$ lies entirely to the past of $\Sigma_{1}$,

  3. 3.

$\Sigma_{2}$ straddles $\Sigma_{1}$ so that it lies partly to the future (in
some open spatial region $R$ of $\Sigma_{1}$) and partly to the past (in the
complementary open region $\overline{R}$ of $\Sigma_{1}$). (See Fig. 4.)

$\Sigma_{2}$$\Sigma_{2}$$\partial\Sigma$$\quad\partial
R$$\Sigma_{1}$$\Sigma_{1}$ Figure 4: The third case, in which $\Sigma_{2}$
straddles $\Sigma_{1}$, is shown. We are using the convention that we fix the
time orientation of $\mathcal{M}$ to always be upward so that the time-
ordering of the two slices is opposite in the two regions.

In portions of $\cal M$ where $\Sigma_{2}$ is to the future of $\Sigma_{1}$,
one should use the $+i$ sign for the exponential of the action, while where
$\Sigma_{2}$ is to the past, one should use the opposite sign $-i$.646464It is
possible to verify that these are the correct sign rules by considering simple
toy models in which one takes a direct sum of $n$ non-interacting QM systems,
each with its own time variable $t_{n}$ generated by a Hamiltonian $H_{n}$,
and then imposing the Hamiltonian constraint $H_{n}\Psi(x)=0$ on each system
separately by defining  $\langle
x_{2}\,|\,x_{1}\rangle=\prod_{n}\int_{-\infty}^{+\infty}dt_{n}\,\langle
x_{2}\,|e^{\sum_{n}iH_{n}t_{n}}|\,x_{1}\rangle,$ (102) and then switching to
the Lagrangian formulation. Additionally, [95] checked that these are the
correct rules in a discrete (but not holographic) model of spacetime that
allows for causal propagation.

What happens to this argument if we consider quantum gravity along a complex
contour which takes us away from Lorentzian signature? In this case, the lapse
$N(x)$ becomes complex. If the Hartle-Hawking prescription is correct, then we
must allow spacetimes with $N(x)$ having sign $+i$ but not $-i$. However, so
long as the integration contour through the space of metrics $\bf g$ is chosen
to have no endpoints, the constraint equations will still be
satisfied.656565In the Hartle-Hawking no boundary proposal, the sign of the
Euclidean imaginary-time lapse $\Delta\tau$ is restricted to be always
positive. This is sort of like a contour endpoint. However, taking $\tau\to 0$
corresponds to space disappearing entirely (this is the “no boundary” part of
the no-boundary proposal) so we never violate the Hamiltonian constraint on
any given nondegenerate spatial metric $g$.

###  4.3 Example: Maximally Symmetric Slices

As a simple example, suppose that we choose the slices $\Sigma_{1}$ and
$\Sigma_{2}$ such that their metrics $g_{1}$ and $g_{2}$ are homogeneous and
isotropic, with (spatial) scalar curvature $R$. It is important to note that
we are _not_ considering a minisuperspace cosmology here, as the history $\cal
M$ between the two slices not constrained to be isotropic, and thus there will
be quantum fluctuations of the anisotropic modes.666666The reason we can
impose isotropy on $g_{1}$ and $g_{2}$, is simply that these metrics are
whatever we want them to be. We are merely evaluating the full wavefunction on
special values of $g$. We assume that this isotropy symmetry is not
spontaneously broken by the dominant saddle. Then the ADM momentum is a pure
trace: $\Pi_{ab}=g_{ab}\Pi/d$. Plugging this relation into the Hamiltonian
constraint $\eqref{ham}$ we obtain $(R-2\Lambda)\propto-\Pi^{2}$. The
imposition of this constraint on the slices $\Sigma_{1}$ or $\Sigma_{2}$ will
determine if the classical geometry that interpolates between them is
Euclidean or Lorentzian.

There are three possibilities:

  * •

$(R-2\Lambda)<0$. This allows for a real solution for $\Pi^{ab}$, and as such
the slice geometry can be embedded in a Lorentzian spacetime.

  * •

$(R-2\Lambda)>0$. In this case we get a purely imaginary solution for
$\Pi^{ab}$, which corresponds to an embedding in an Euclidean classical
spacetime.

  * •

$(R-2\Lambda)=0$. This is the transition point between Lorentzian and
Euclidean embeddings. In the case of AdS ($\Lambda<0$) this transition occurs
at negative spatial curvature, when the radius of $\Sigma$ equals that of the
bulk spacetime $\cal B$.

So we see that the particular state $\ket{g}$ determines whether the saddle-
point is Euclidean or Lorentzian in the neighbourhood of the slice. If we get
opposite answers for $\Sigma_{1}$ and $\Sigma_{2}$, then the signature of the
spacetime must change somewhere in between.676767Although there are many
equivalent complex contours that would give such a result, it is nicest to
chose a contour in which the metric is everywhere either Euclidean or
Lorentzian. Whenever there are Lorentzian saddles, we have a choice about
whether to take the lapse $N$ to be positive or negative, as argued in 4.2.
This choice spontaneously breaks $CPT$ symmetry, as discussed in section 2.6.
The two contributions are complex conjugates. Both of these saddles have to be
included together and so they contribute with a real but oscillatory
functional to the transition amplitude [66].

Now if both the slices are consistent with a Lorentzian embedding, then we
will have four saddles occurring in two complex conjugate pairs, as shown in
Fig 5, arising from all the possible time orientations relative to the time-
symmetric slice. The sum over all 4 saddles gives the amplitude:

$\displaystyle\langle
g_{2}|g_{1}\rangle_{I}=\sum_{\pm}\Delta_{\text{gibbous}}e^{\pm
i(I_{1}+I_{2})}+\Delta_{\text{crescent}}e^{\pm i(I_{1}-I_{2})},$ (103)

where $I_{n}$ is the action for a positive lapse history from the slice
$g_{n}$ to the $K=0$ slice, and $\Delta$ is the quantum determinant associated
with the two possible geometries for $\cal M$ shown in Fig 5, which could be
calculated by e.g. heat kernel methods.686868As stated in section 4.1, these
amplitudes are IR divergent and require regulation by moving $\partial\Sigma$
to a finite value of $z$. In evaluating $I_{1}$ and $I_{2}$, we have also
ignored in (103) the Gibbons-Hawking contribution coming from the Euclidean
$i\pi$-rotation at $\partial\Sigma$. This contributes a thermodynamically
suggestive (but IR divergent) factor of
$e^{\text{Area}[\partial\Sigma]/4G_{N}}$ to the inner product $\langle
g_{2}|g_{1}\rangle_{I}$. Since this factor is the same regardless of the
initial and final metrics, it may be ignored when constructing the Hilbert
space.

$\Sigma_{1},g_{1}$$\Sigma_{2},g_{2}$$\partial\Sigma$$K=0$

$\Sigma_{1},g_{1}$$\Sigma_{2},g_{2}$$\partial\Sigma$$K=0$

$\Sigma_{1},g_{1}$$\Sigma_{2},g_{2}$$\partial\Sigma$$K=0$

$\Sigma_{1},g_{1}$$\Sigma_{2},g_{2}$$\partial\Sigma$$K=0$

Figure 5: For the case of the slices consistent with Lorentzian embedding, we
get 4 saddles. The top two “gibbous” saddles are time reversals of each other
and so they together contribute as a complex conjugate pair to the transition
amplitude. Similarly for the bottom two “crescent” saddles. In this figure the
time orientation is directed upwards and the $K=0$ dashed line is the time
symmetric slice. It can be seen that when $g_{1}=g_{2}$, the crescent saddles
go to zero lapse, leading to divergences of the 1-loop quantum determinant
$\Delta_{\text{crescent}}$, which will be discussed in section 4.4.

However, if only one of the slices is consistent with Lorentzian embedding,
while the other is embedded in an Euclidean geometry, then there are only two
contributing saddles, arising from flipping the time orientation of the
Lorentzian piece:

$\displaystyle\langle
g_{2}|g_{1}\rangle_{I}=\sum_{\pm}\Delta_{\text{mixed}}e^{-I_{E}\pm iI_{L}},$
(104)

where $I_{E}$ is the Euclidean part of the action and $I_{L}$ is the
Lorentzian part. Lastly, if both slices are consistent with an Euclidean
embedding, then there is only one dominant saddle,696969The dominant Euclidean
saddle is the one where both slices are on opposite sides of the $K=0$
surface, similar to the top row of Fig. 5 but in Euclidean signature. In
addition to having a larger amplitude (not necessarily a decisive point since
not every saddle necessarily lies on the physically correct contour) once the
Gibbons-Hawking term is taken into account, this saddle is clearly the one
which limits to the usual AdS/CFT saddles in the limit that $\mu\to 0$.
contributing with a real and positive functional:

$\displaystyle\langle
g_{2}|g_{1}\rangle_{I}=\Delta_{\text{Euclidean}}e^{-(I_{1}+I_{2})},$ (105)

where the $I_{n}$ are now Euclidean signature actions from $g_{n}$ to the
$K=0$ slice.707070It can be seen that if we neglect the quantum determinants
by setting $\Delta=1$, the resulting norm is nonnegative for arbitrary
superpositions of maximally-symmetric metrics. In fact it is highly
degenerate, with the dynamical state space being 1-dimensional. A more careful
check of positivity would therefore require knowing something about the
$\Delta$ factors.

We leave further analysis of this symmetric case to future work.

###  4.4 Properties of the Amplitude

In order for our construction of the Hilbert space in section 5 to succeed, we
will need for our inner product (the previously defined transition amplitude)
to satisfy the following four conditions: (a) linearity, (b) conjugate
symmetry and (c) positive semi-definiteness, and (d) a subspace on which the
inner product is finite.717171See [95] for discussion of these four
requirements in a discrete model of quantum gravity.

Below we prove the first two properties, and make some arguments concerning
positivity and finiteness, leaving more convincing proofs for future work.

##### Linear.

Equation (99) defines the inner product on the basis elements of the dual
space of distributions on functionals of the metric. This space will be later
properly defined and called $K^{*}$. Its action on a general element in
$K^{*}$ is defined from this in a linear fashion. The map is thus linear by
construction.

##### Hermitian.

We are using a convention in which the time orientation of the space-filling
manifold $\cal M$ is fixed. For the inner product $\langle
g_{2}\,|\,g_{1}\rangle_{I}$ we have the straddle expansion described above,
each diagram coming with a specific sign for the exponential of the action.
Now consider $\langle g_{1}\,|\,g_{2}\rangle_{I}$. This is equivalent to
flipping the time orientation of $\cal M$. The effect is to flip the sign of
the exponent for each component of the path integral. So we see that $\langle
g_{2}\,|\,g_{1}\rangle_{I}=\langle g_{1}\,|\,g_{2}\rangle^{*}_{I}$.

##### Positive Semi-definite.

The inner product cannot be positive definite, because it projects onto states
obeying the constraint equations ${\cal H}(x)=0$ and ${\cal D}_{a}(x)=0$, so
there ought to be lots of null states. This is good because it implements the
gauge-invariance of the theory. So at best, the path integral can be positive
semi-definite, so that for any superposition $\Psi$ of spatial metrics $g$:

$\langle\Psi\,|\,\Psi\rangle_{I}\geq 0.$ (106)

In what follows we will be assuming that this positivity property holds, at
least in the context of complete Cauchy slices $\Sigma$ in asymptotically
locally AdS spacetimes.727272See [66] for an argument that positivity fails
for _partial_ Cauchy slices which have boundaries in the interior of black
holes.

This property is difficult to show in general, due to the fact that $I$
includes in its definition arbitrary dynamical spacetimes. In order to prove
it, we need to somehow factor out the role of time evolution.

However, an argument for (106) can be made that positivity is at least true
within the semiclassical regime. We adopt an argument based on the conformal
method of solving the Einstein constraint equations [96, 97, 98], in which one
switches to an alternative quantum gravity basis $(\gamma_{ab},K)$ where
$\gamma_{ab}=g^{-1/d}g_{ab}$ is the conformal part of the metric, and $K$ is
the trace of the extrinsic curvature (below we will choose $K=0$).

This argument for the equivalence of this basis is only valid for the case of
negative cosmological constant ($\Lambda<0$).737373Perhaps this is related to
the fact that the CFTs which appear in dS/CFT are not unitary? More precisely,
we need to assume that gravity is attractive in all classical saddles relevant
for the computation of (106), which would follow if the generic Strong Energy
Condition is satisfied:

$\textbf{T}_{\hat{t}\hat{t}}+\tfrac{1}{D-2}\textbf{T}\>>\>\frac{\Lambda}{8\pi
G_{N}},$ (107)

for all unit timelike vectors $\hat{t}$.747474It might be possible to violate
this equation if we turn on an inflating scalar field $\Phi$ deep in the bulk,
or by means of a quantum fluctuation. However, when $\Lambda<0$, it requires a
finite stress-tensor excursion to violate (107) and so this requires a very
large quantum fluctuation. We can neglect this possibility in the