aainstitutetext: Institute for Theoretical Physics, University of Cologne,
Zülpicher Straße 77, 50937 Köln, Germanybbinstitutetext: Department of Physics
and Haifa Research Center for Theoretical Physics and Astrophysics, University
of Haifa, Haifa 31905, Israel

# Reconstructing bulk equation of motion using CFT modular Hamiltonians

Nele Callebaut b and Gilad Lifschytz<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In the framework of bulk reconstruction, we elucidate the relationship between
the action of CFT modular Hamiltonians on bulk operators, the possible
equation of motion for the bulk operators, and the charge distribution at
infinity corresponding to such bulk fields. In particular for scalar fields
interacting with gravity or with gauge fields, we show how CFT considerations
of the action of the modular Hamiltonian constrain the possible bulk equation
of motion to be consistent with bulk gauge invariance and diffeomorphism
invariance. In fact we show that requiring that the action of the modular
Hamiltonian on a dressed bulk scalar operator be compatible with some unknown
simple equation of motion, fixes, under reasonable assumptions, both the
equation of motion and the action of the modular Hamiltonian once the first
order $\frac{1}{N}$ terms are known.

##  1 Introduction

In $AdS/CFT$ Maldacena:1997re , reconstruction of bulk operators in terms of
CFT data or ‘bulk reconstruction’ can proceed through various routes. One can
start from a bulk point of view, and solve the bulk equation of motion
perturbatively in the radial direction, with initial conditions on the
boundary of AdS Kabat:2011rz ; Heemskerk:2012mn . This is more a proof of
concept since one does not want to use that much bulk information, and the
resulting expressions need to be interpreted in terms of CFT operators. Once
bulk operators are constructed from boundary data in terms of CFT operators,
one can compute CFT correlation functions and find that they are well-defined
and obey appropriate bulk locality conditions.

The second path is to start with a CFT point of view. One constructs zeroth
order expressions Hamilton:2005ju ; Hamilton:2006az  from symmetry
considerations Nakayama:2015mva ; Goto:2017olq  or more generally from
intersecting modular Hamiltonian equations Kabat:2017mun 111One can also try
to use more sophisticated modular flow tools to reconstruct the bulk operator
Faulkner:2017vdd .. The zeroth order expression reproduces correctly the bulk
two-point function through a CFT computation. One then corrects this
expression by demanding that inserting the CFT representation of the bulk
operator into higher-point CFT correlation functions gives well-defined
results Kabat:2018pbj ; Kabat:2020nvj . This results in some expression for
the bulk field, from which one can then compute as an output the equation of
motion the bulk field obeys Kabat:2015swa ; Kabat:2016zzr ; Kabat:2020nvj .

Recently, another path was suggested for bulk reconstruction of scalars in
$AdS_{3}$ interacting with gauge or gravity fields Kaplan1708 ; Chen:2019hdv
(see also Guica:2015zpf  and Lewkowycz:2016ukf ; Nakayama:2016xvw  for
reconstruction proposals in this context). The employed condition on the bulk
operators comes from demanding a particular boundary distribution of charges,
i.e. a particular form of the commutator between the boundary conserved
current and the bulk operator.

Why do all these seemingly different approaches give answers that obey the
other conditions as a consequence? In this paper, we will see an answer to
this question starting to emerge. Central to our discussion will be the action
of the modular Hamiltonian.

In the CFT approach, the action of the modular Hamiltonian on bulk fields is a
very useful tool for the reconstruction program. First, the zeroth order term
can be derived from intersecting modular Hamiltonian equations. Secondly, for
scalar interaction alone, the action of the modular Hamiltonian constrains the
correction terms to be particular smearings of primary scalar operators.

What about interactions that come from local symmetries, can the action of the
modular Hamiltonian be useful for the reconstruction problem and moreover
elucidate the relationship to the approach of Kaplan1708 ; Chen:2019hdv ?
Further, bulk gauge transformations or coordinate transformations that fall
off fast enough near the boundary are supposed to be invisible to the boundary
CFT. However, these gauge redundancies constrain the possible equation of
motion that a bulk field can obey. The equations of motion (in a fixed gauge)
should be obeyed by bulk operators reconstructed from the CFT correlation
function, and one would like to understand if there is a simple way to
understand why (or how) such a structure arises from the CFT point of view,
without constructing explicitly well-defined bulk operators and then computing
their equation of motion.

We show in this paper that in the case of a scalar field interacting with
gravity or gauge fields, three sets of information about the bulk field are
connected by compatibility constraints. These are 1) the equation of motion
for the bulk operator, 2) the modular Hamiltonian action on the bulk scalar
operator, and 3) the charge distribution at infinity of the scalar operator.
The compatibility conditions they must obey tightly constrain any two of the
sets given the third. This explains the relationship between the different
bulk reconstruction approaches. For instance, we will show that knowledge of
how the CFT modular Hamiltonian $H_{mod}$ acts on bulk scalar operators
interacting with gauge fields or gravity constrains the possible equations of
motion the bulk operators can obey, in a way that is consistent with the
original bulk local symmetries. In fact we will demonstrate that one can build
up consistent pairs of the $H_{mod}$ action on scalar bulk operators and the
equation of motion they obey, order by order in $\frac{1}{N}$, resulting in a
good bulk operator. In addition, a simple charge distribution at infinity can
be related to the requirement of well-defined CFT correlation functions, as in
Kabat:2018pbj .

In order to explain the various points of views we find useful, we start with
simple examples and work our way up. In section 2 we start with a CFT
computation to leading order in $\frac{1}{N}$ of the action of $H_{mod}$ on a
scalar bulk operator interacting in $AdS_{3}$ with one abelian Chern-Simons
gauge field, dual to a CFT current $j_{-}$. The result is of the general form

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi+\Lambda\,\phi$ (1.1)

which we will encounter for each field $\phi$ that we consider, i.e. dressed
by interactions with gauge fields or gravity. The first term is the one
expected for a local bulk operator JLMS , while the second term carries
information about the non-locality of the dressed field. From the found
$H_{mod}$ action on $\phi$, we show there are two possible actions of the
Casimir operator on the bulk operator. Comparing both, one gets a correct bulk
equation of motion for the scalar bulk operator. This is then generalized in
section 2.1 to all orders in $\frac{1}{N}$. In section 2.2 we use a Jacobi
identity to generalize the construction of section 2.1 to the case where the
scalar operator is coupled to two Chern-Simons gauge fields such that the dual
CFT has both a $j_{-}$ and a $j_{+}$ conserved current. We then explain that
the procedure we used is equivalent to constraining the equation of motion
directly by the action of the modular Hamiltonian and reproduce the same
results from this new perspective in section 2.3. In section 3 we explain the
relationships between different approaches to construct the bulk operator. In
section 4 we use the point of view of section 2.3 to constrain the equation of
motion for a scalar operator coupled to a bulk gauge field in higher
dimensions. This is done using the $H_{mod}$ action on the bulk operator and
on the bulk gauge field, and this to all orders in $\frac{1}{N}$ for both the
scalar and the gauge field. We then argue in section 5 that our success of
getting any consistent bulk equation of motion relies on having a compatible
pair of $H_{mod}$ action and bulk equation of motion, and that one can use the
existence of a simple bulk equation of motion as a guiding principle to build
up order by order in $\frac{1}{N}$ both the action of $H_{mod}$ on the bulk
operator as well as the bulk equation of motion. This is demonstrated in
section 6 with the example of a scalar field interacting with gravity in
$AdS_{3}$. We end with some comments about this approach. Some computations
are presented in Appendix A-F.

##  2 Scalar field interacting with a gauge field in $AdS_{3}$

We start with the reconstruction program for a complex scalar field of charge
$\frac{q}{N}$, coupled to a gauge field in $AdS_{3}$, with metric

$\displaystyle ds^{2}=\frac{1}{Z^{2}}(dZ^{2}+dx^{+}dx^{-}),$ (2.1)

where the gauge field is described by a single Chern-Simons term in the bulk.
This situation is reconstructed in the CFT by a conserved left (or right)
moving current $j_{-}$ and a complex scalar operator ${\cal O}$ of dimension
$\Delta=2h$ which has a non trivial 3-point function

$<{\cal O}(x_{1})\bar{{\cal
O}}(x_{2})j_{-}(x_{3})>=-\frac{iq}{N}\frac{1}{(x_{12}^{-}x_{12}^{+})^{\Delta}}\frac{x_{12}^{-}}{x_{13}^{-}x_{23}^{-}}\,,$
(2.2)

where $x^{\pm}=x\pm t$ and $x_{ij}=x_{i}-x_{j}$. Our $i\epsilon$ prescription
is $t_{j}\rightarrow t_{j}-i\epsilon_{j}$ and the order of operators in the
correlation function from left to right is in order of decreasing
$\epsilon_{j}$. Throughout the paper, unless stated otherwise, we normalize
operators such that their two-point function is $O(1)$ and the 3-point
function is $O(\frac{1}{N})$ where $N^{2}\sim$ central charge of the CFT.

Promoting ${\cal O}$ to a bulk field $\phi$ was done in Kabat:2020nvj  up to
$1/N$ corrections, with the result that the bulk operator up to that order has
the form

$\phi=\phi^{(0)}+\frac{1}{N}\phi^{(1)}+\cdots\,.$ (2.3)

The first order correction $\phi^{(1)}=\sum_{l}a_{l}{\cal A}_{(0,l)}$ is
represented as a sum of smeared operators of the type

${\cal A}_{(0,l)}=\int K_{\Delta+2+2l}\ \partial^{l+1}_{+}J_{(0,l)}$ (2.4)

where $J_{(0,l)}$ is a primary operator of dimensions
$(l+1+\frac{\Delta}{2},\frac{\Delta}{2})$ of the form

$J_{(0,l)}=\sum_{k=0}^{l}d_{lk}\partial_{-}^{k}j_{-}\partial_{-}^{l-k}{\cal
O}$ (2.5)

with

$d_{lk}=\frac{(-1)^{k}}{\Gamma(k+1)\Gamma(l-k+1)\Gamma(k+2)\Gamma(l-k+\Delta)},$
(2.6)

and $K_{\Delta_{l}}$ is the scalar smearing function for a scalar operator of
conformal dimension $\Delta_{l}$ obeying
$\nabla^{2}K_{\Delta_{l}}=\Delta_{l}(\Delta_{l}-2)$. Here,
$\nabla^{2}=\nabla_{M}\nabla^{M}$ is the box operator in the AdS metric.

Since the bulk field is coupled to a gauge field one would expect that the
physical operator corresponding to it will not commute, due to Gauss law, with
the boundary current $j_{0}$, and will thus not transform under AdS isometries
as a local bulk scalar. This is manifested in the CFT construction by the fact
that the CFT operators from which the bulk operator is constructed are not
smeared primary scalars. In fact it is not possible to construct a primary
scalar from a conserved current $j$, a primary scalar $\mathcal{O}$ and their
derivatives for use in the construction at all Kabat:2012av .

It is useful to know how the bulk scalar operator will transform, in
particular it is useful to know how it will transform under the action of the
modular Hamiltonian. We take the modular Hamiltonian for a segment of length
$2R$ centered around the origin222Here and anywhere else in the paper the
modular Hamiltonian is the total modular Hamiltonian for the separation of
space into the region $A$ and its complement $\bar{A}$. Then
$H_{mod}=H_{mod,A}-H_{mod,\bar{A}}$., in the vacuum state of the CFT. It is
defined in terms of the CFT generators ($L_{1}$, $L_{-1}$ and $L_{0}$, or the
translations $P$ and special conformal transformations $Q$) as Casini:2011kv ;
deBoer:2016pqk

$\frac{1}{2\pi
i}H_{mod}=\frac{1}{2R}(Q_{0}-R^{2}P_{0})=\frac{1}{2R}\left((\bar{L}_{1}-L_{1})-R^{2}(\bar{L}_{-1}-L_{-1})\right).$
(2.7)

The bulk operator was constructed from CFT considerations in Kabat:2020nvj .
Given the form of the bulk operator as computed from the CFT, we can compute
the action of the CFT modular Hamiltonian on it (see Appendix A for details):

$\frac{1}{2\pi
i}[H_{mod},\phi^{(0)}+\frac{1}{N}\phi^{(1)}]=\xi^{M}\partial_{M}(\phi^{(0)}+\frac{1}{N}\phi^{(1)})+\frac{\beta
Z^{2}}{2RN}j_{-}\phi^{(0)},$ (2.8)

with

$\displaystyle\beta=-i\frac{q}{k}$ (2.9)

and $\xi$ the bulk Killing vector defined in (A.20) that vanishes on the Ryu-
Takayanagi surface Ryu:2006bv  of the considered region. From this expression
(or computing straightforwardly) one can read off the action of the conformal
generators on the bulk operator to be

$[L_{-1},(\phi^{(0)}+\frac{1}{N}\phi^{(1)})]=\partial_{-}(\phi^{(0)}+\frac{1}{N}\phi^{(1)}),\
\
[\bar{L}_{-1},(\phi^{(0)}+\frac{1}{N}\phi^{(1)})]=\partial_{+}(\phi^{(0)}+\frac{1}{N}\phi^{(1)}),$
$[L_{1},(\phi^{(0)}+\frac{1}{N}\phi^{(1)})]=(Zx^{-}\partial_{Z}+(x^{-}){{}^{2}}\partial_{-}-Z^{2}\partial_{+})(\phi^{(0)}+\frac{1}{N}\phi^{(1)}),$
$[\bar{L}_{1},(\phi^{(0)}+\frac{1}{N}\phi^{(1)})]=(Zx^{+}\partial_{Z}+(x^{+}){{}^{2}}\partial_{+}-Z^{2}\partial_{-})(\phi^{(0)}+\frac{1}{N}\phi^{(1)})+\beta
Z^{2}j_{-}\phi^{(0)}.$

From these, the action of $L_{0}$ and $\bar{L}_{0}$ can be computed

$[L_{0},(\phi^{(0)}+\frac{1}{N}\phi^{(1)})]=(\frac{1}{2}Z\partial_{Z}+x^{-}\partial_{-})(\phi^{(0)}+\frac{1}{N}\phi^{(1)}),$
$[\bar{L}_{0},(\phi^{(0)}+\frac{1}{N}\phi^{(1)})]=(\frac{1}{2}Z\partial_{Z}+x^{+}\partial_{+})(\phi^{(0)}+\frac{1}{N}\phi^{(1)}).$

The previous expressions can be put into the expression for the quadratic
Casimir

$C_{2}=2(L_{0}^{2}+\bar{L}_{0}^{2}-\frac{1}{2}(L_{-1}L_{1}+L_{1}L_{-1}+\bar{L}_{-1}\bar{L}_{1}+\bar{L}_{1}\bar{L}_{-1}))=\sum_{i,j}(C^{ij}L_{i}L_{j}+\bar{C}^{ij}\bar{L}_{i}\bar{L}_{j}).$
(2.10)

We then define the operation

$C_{2}\circ\phi=\sum_{i,j}(C^{ij}[L_{i},[L_{j},\phi]]+\bar{C}^{ij}[\bar{L}_{i},[\bar{L}_{j},\phi]]).$
(2.11)

The Casimir operator then has the properties

$C_{2}\circ\phi^{(0)}=\nabla^{2}\phi^{(0)}$ (2.12)
$C_{2}\circ(\phi^{(0)}+\frac{1}{N}\phi^{(1)})=\nabla^{2}(\phi^{(0)}+\frac{1}{N}\phi^{(1)})-\frac{2\beta
Z^{2}}{N}j_{-}\partial_{+}\phi^{(0)}.$ (2.13)

We will now see that there is another expression for the action of $C_{2}$ on
the bulk operator. Every ${\cal A}_{(0,l)}$ obeys

$C_{2}\circ{\cal A}_{(0,l)}=(2(l+1+h)(l+h)+2(h(h-1)){\cal
A}_{l}=\frac{1}{2}(\nabla^{2}+m_{0}^{2}){\cal A}_{(0,l)}$ (2.14)

with $m_{0}^{2}=\Delta(\Delta-2)$ the mass of the free scalar $\phi^{(0)}$.
Since we can also write

$C_{2}\circ\phi^{(0)}=\frac{1}{2}(\nabla^{2}+m_{0}^{2})\phi^{(0)},$ (2.15)

we see that

$C_{2}\circ(\phi^{(0)}+\frac{1}{N}\phi^{(1)})=\frac{1}{2}(\nabla^{2}+m_{0}^{2})(\phi^{(0)}+\frac{1}{N}\phi^{(1)}).$
(2.16)

We can now compare the two expressions for the action of the Casimir. Since
they both are correct, it must be that

$(\nabla^{2}-m_{0}^{2})(\phi^{(0)}+\frac{1}{N}\phi^{(1)})-\frac{4\beta
Z^{2}}{N}j_{-}\partial_{+}\phi^{(0)}=0.$ (2.17)

This is indeed the correct equation of motion a bulk scalar coupled to a gauge
field obeys to this order.

###  2.1 All orders

Here we generalize the above construction to the situation where the bulk
scalar gets corrected to all orders in its interaction with the gauge field,
but the gauge field does not get back-reacted by the scalar field. This will
give a bulk scalar operator which is suitable to be inserted into correlation
functions with as many $j_{-}$ as we want, but only one additional $\bar{{\cal
O}}$, similar to what was done in Chen:2019hdv . The simplest possible
generalization of (2.8) is then333This might not have worked, see section 5
for a discussion on this point.

$\frac{1}{2\pi i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi+\frac{\beta
Z^{2}}{2NR}j_{-}\phi.$ (2.18)

where now

$\phi=\phi^{(0)}+\sum_{i=1}^{\infty}\frac{1}{N^{i}}\phi^{(i)}.$ (2.19)

Reading off from this the actions of the conformal generators, and inserting
those into the expression for the quadratic Casimir, one gets

$C_{2}\circ\phi=\nabla^{2}\phi-\frac{2\beta Z^{2}}{N}j_{-}\partial_{+}\phi.$
(2.20)

What CFT operators can we use to build up the $\phi^{(i)}$? If we have only
$j_{-}$ (and no $j_{+}$), then the only possible multi-trace primary operators
we can build from one ${\cal O}$, $(n+1)$ currents $j_{-}$, and $l$
derivatives are the operators $J_{(n,l)}$, which have conformal dimensions
$(h+n+l+1,h)$. The first descendant scalar operator is then
$\partial^{n+l+1}_{+}J_{(n,l)}$. Of course one might think that to construct
the bulk operators one can also use smearing of other scalar descendants such
as $(\partial_{-}\partial_{+})^{m}\partial^{n+l+1}_{+}J_{(n,l)}$, but this is
not compatible with the action of the modular Hamiltonian (2.18) on the bulk
operator. This is so since equation (2.18) indicates that $\phi$ transforms
under $L_{1}$ (but not under $\bar{L}_{1}$) as if it was a smeared primary
scalar. Thus the $\phi^{(n+1)}$ are built from an infinite sum over $l$ of

${\cal A}_{(n,l)}=\int K_{\Delta+2+2n+2l}\ \partial^{l+n+1}_{+}J_{(n,l)}.$
(2.21)

One can see that, as before,

$C_{2}\circ{\cal A}_{(n,l)}=(2(l+1+n+h)(l+h+n)+2h(h-1)){\cal
A}_{(n,l)}=\frac{1}{2}(\nabla^{2}+m_{0}^{2}){\cal A}_{(n,l)}.$ (2.22)

Since only smeared $\partial^{n+l+1}_{+}J_{(n,l)}$ operators can appear in
$\phi$, we then have

$C_{2}\circ\phi=\frac{1}{2}(\nabla^{2}+m_{0}^{2})\phi.$ (2.23)

Comparing again the two valid expressions for the action of $C_{2}$ on $\phi$
we find

$(\nabla^{2}-m_{0}^{2})\phi-\frac{4\beta Z^{2}}{N}j_{-}\partial_{+}\phi=0,$
(2.24)

which is the right bulk equation of motion when one solves to all orders in
$1/N$ for $\phi$ but keeps the bulk gauge field uncorrected beyond the free
limit (in $AdS_{3}$ one has $A_{-}(Z,x)\sim j_{-}(x^{-})$ at that order).

###  2.2 Including both $j_{-}$ and $j_{+}$

In this section we look at the situation where the CFT includes both a left
moving and a right moving conserved current $j_{-}$ and $j_{+}$. We will
consider the situation where the dual bulk theory has two Chern-Simons gauge
fields $A$ and $\bar{A}$ in a parity preserving combination

$\frac{k}{2}\
\int\epsilon^{abc}(A_{a}\partial_{b}A_{c}-\bar{A}_{a}\partial_{b}\bar{A}_{c}),$
(2.25)

and a scalar field coupled to them with the same charge. Note that this case,
from the scalar field point of view, is more similar to the higher-dimensional
case than the situation with just left or right CFT currents.

The simplest expected transformation of the bulk scalar field is then

$\frac{1}{2\pi i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi+\frac{\beta
Z^{2}}{2NR}j_{-}\phi-\frac{\beta Z^{2}}{2NR}j_{+}\phi.$ (2.26)

This transformation, to first order in $\frac{1}{N}$, follows from a CFT
computation and can also (to all orders in $\frac{1}{N}$) be derived from the
bulk point of view (as explained in Appendix C). Reading from this the action
of the conformal generators, and inserting into $C_{2}$, we get

$C_{2}\circ\phi=\nabla^{2}\phi-\frac{2\beta
Z^{2}}{N}j_{-}\partial_{+}\phi-\frac{2\beta Z^{2}}{N}j_{+}\partial_{-}\phi.$
(2.27)

In this case there are many types of primary operators one can build
(involving $j_{-}$, $j_{+}$, ${\cal O}$ and derivatives), and use in
constructing the $\phi^{(i)}$. There does not seem to be any constraint coming
from the action of the modular Hamiltonian that will give a nice equation like
(2.22) for each of the involved smeared operators individually. The way to
move forward is to notice that (2.23) cannot be of any form. It has to obey a
boundary condition and a Jacobi identity. The boundary condition is simple,
since as $Z\rightarrow 0$ then $\phi\rightarrow{\cal O}$ and we know that
$C_{2}\circ{\cal O}=\Delta(\Delta-2){\cal O}$. The Jacobi identity involves
$H_{mod}$, $C_{2}$ and $\phi$, and since in our case $[H_{mod},C_{2}]=0$, the
Jacobi identity becomes

$[C_{2},[H_{mod},\phi]]|0\rangle=[H_{mod},[C_{2},\phi]]|0\rangle$ (2.28)

where we have taken the expression to act on the vacuum state of the CFT. Note
that since $L_{i}|0>=\bar{L}_{i}|0>=0$,

$C_{2}\circ\phi|0>=[C_{2},\phi]|0>,$ (2.29)

and as always $H_{mod}|0>=0$. Indeed if we take the expressions (2.23) and
(2.18), then they both obey the Jacobi identity (2.28).

We will now use the Jacobi identity as our guiding principle. If we have both
$j_{+}$ and $j_{-}$ so that the action of the modular Hamiltonian is as in
(2.26), then the action of the Casimir as in (2.23) does not obey the Jacobi
identity. We know that when we have only $j_{-}$ or only $j_{+}$, things
reduce to equation (2.23). This suggests the simplest ansatz for the Casimir
action we can try is ($a$ and $\rho$ are some constants)

$C_{2}\circ\phi=(a\nabla^{2}+(1-a)m_{0}^{2})\phi-\rho Z^{2}j_{+}j_{-}\phi.$
(2.30)

Using

$\frac{1}{2\pi
i}[H_{mod},j_{+}]=\frac{1}{2R}(2x^{+}j_{+}+(x^{+}+R)(x^{+}-R)\partial_{+}j_{+}),$
(2.31)

we find that this expression for $C_{2}$ will obey the Jacobi identity (2.28),
only if $a=\frac{1}{2}$ and $\rho=-2\beta^{2}$.

Comparing now the expressions (2.27) and (2.30) gives an equation of motion

$(\nabla^{2}-m_{0}^{2})\phi=\frac{4\beta
Z^{2}}{N}(j_{-}\partial_{+}+j_{+}\partial_{-})\phi-\frac{4\beta^{2}Z^{2}}{N^{2}}j_{+}j_{-}\phi$
(2.32)

which is the correct equation of motion for a bulk scalar operator coupled
with two (zeroth order) Chern-Simons gauge fields in $AdS_{3}$, in holographic
gauge.

One can look for more general ansatze for $C_{2}\circ\phi$ obeying the Jacobi
identity. For instance one can add to the right hand side of (2.30) any
polynomial of the form

$\sum_{k}a_{k}\phi^{k+1}\bar{\phi}^{k}$ (2.33)

provided $\bar{\phi}$ obeys

$\frac{1}{2\pi
i}[H_{mod},\bar{\phi}]=\xi^{M}\partial_{M}\bar{\phi}-\frac{\beta
Z^{2}}{2NR}j_{-}\bar{\phi}+\frac{\beta}{2NR}j_{+}\bar{\phi}\,.$ (2.34)

This results in a potential term in the equation of motion for the scalar
field.

###  2.3 Constraints on the equation of motion

Given some known action of a known operator on a bulk field operator, this can
constrain the possible equations of motion that the bulk operator can obey.
For instance let us take the action of the modular Hamiltonian on the bulk
field to be just

$\frac{1}{2\pi i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi.$ (2.35)

Let us also assume that the bulk field obeys the equation

$\nabla^{2}\phi=m_{0}^{2}\phi.$ (2.36)

One then must have

$[H_{mod},\nabla^{2}\phi]=m_{0}^{2}[H_{mod},\phi].$ (2.37)

Since $H_{mod}$ does not depend on the coordinates $(Z,x,t)$ of the bulk
field, it can go through the differential operator and we get

$\nabla^{2}(\xi^{M}\partial_{M}\phi)=m_{0}^{2}\,\xi^{M}\partial_{M}\phi.$
(2.38)

That is, the equation of motion must still hold even under modular flow. This
will only be correct (without further restrictions on $\phi$) if

$[\nabla^{2},\xi^{M}\partial_{M}]=0$ (2.39)

which is correct in our case since $\xi^{M}$ is a Killing vector for the
metric that is used to write $\nabla^{2}$.

Now one can instead take the action of the modular Hamiltonian to be equation
(2.18)

$\frac{1}{2\pi i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi+\frac{\beta
Z^{2}}{2NR}j_{-}\phi.$ (2.40)

This time, the previous argument will not hold for the equation of motion

$\nabla^{2}\phi-m_{0}^{2}\phi=0.$ (2.41)

That is, commuting both sides of the equation of motion with $H_{mod}$ will
not give an equality, unless conditions on $\phi$ which are not compatible
with the equation of motion are met. So clearly the transformation under
$H_{mod}$ and the above equation of motion cannot both be correct. However, it
is not hard to find a different equation of motion that will be compatible
with the action of the modular Hamiltonian, and not surprisingly, the simplest
compatible extension of (2.41) is the one we found previously

$(\nabla^{2}-m_{0}^{2})\phi-\frac{4\beta Z^{2}}{N}j_{-}\partial_{+}\phi=0.$
(2.42)

The compatibility means that

$\frac{1}{2\pi i}[H_{mod},(\nabla^{2}-m_{0}^{2})\phi-\frac{4\beta
Z^{2}}{N}j_{-}\partial_{+}\phi]=(\xi^{M}\partial_{M}+\frac{\beta
Z^{2}}{2NR}j_{-})((\nabla^{2}-m_{0}^{2})\phi-\frac{4\beta
Z^{2}}{N}j_{-}\partial_{+}\phi).$ (2.43)

This line of reasoning is not fundamentally different than before (but
sometimes simpler computationally). The procedure of identifying an equation
of motion through two possible expressions of the action of the quadratic
Casimir on bulk operators (compatible with the Jacobi identity with
$H_{mod}$), results in an equation of motion that guarantees to obey the
constraint coming from the known action of $H_{mod}$, such as (2.43). This is
of course extended to the case with both $j_{-}$ and $j_{+}$, where the action
of the modular Hamiltonian is given by (2.26), which results in the equation
of motion (2.32).

One can look at other constraints, for instance similar constraints can be
found by commuting both sides of the equation of motion with $j_{-}(x^{-})$.
Using

$[j_{-}(x),\phi(Z,y)]=-\frac{q}{N}\delta(x^{-}-y^{-})\phi(Z,y),\ \
[j_{-}(x),j_{-}(y)]=ik\partial_{x^{-}}\delta(x^{-}-y^{-}),$ (2.44)

we then find again that equation (2.41) is not compatible with this
relationship, but equation (2.42) is compatible with it for
$\beta=-i\frac{q}{k}$, as before. Here the compatibility means

$[j_{-}(x^{-}),(\nabla^{2}-m_{0}^{2})\phi-\frac{4\beta
Z^{2}}{N}j_{-}\partial_{+}\phi]=-\frac{q}{N}\delta(x^{-}-y^{-})((\nabla^{2}-m_{0}^{2})\phi-\frac{4\beta
Z^{2}}{N}j_{-}\partial_{+}\phi).$ (2.45)

Similarly, if one has both $j_{-}$ and $j_{+}$ with

$[j_{+}(x),\phi(Z,y)]=\frac{q}{N}\delta(x^{+}-y^{+})\phi(Z,y),\ \
[j_{+}(x),j_{+}(y)]=-ik\partial_{x^{+}}\delta(x^{+}-y^{+}),$ (2.46)

then a simple computation shows that this is compatible with equation (2.32).

##  3 Relationships between approaches

We would like to connect this line of thought to previous investigations of
the system under consideration. In Chen:2019hdv , the bulk operator was
constructed from a simple, physically motivated form of its asymptotic charge
distribution, which took the form (in our convention)

$[j_{-}(x),\phi(Z,y)]=-\frac{q}{N}\delta(x^{-}-y^{-})\phi(Z,y).$ (3.1)

As shown in Appendix E, the bulk operator computed in Kabat:2020nvj  agrees to
first order in $\frac{1}{N}$ with the one computed in Chen:2019hdv . Thus the
condition (3.1) leads to an operator obeying the bulk equation of motion
(2.17). The connection with the approach using the modular Hamiltonian comes
from looking for consistency of the action of the modular Hamiltonian on the
bulk field and the commutator of the boundary current with the bulk field. The
consistency condition is the Jacobi identity when applied to the operators
$H_{mod}$, $j_{-}$ and $\phi$.

Let us assume we know (as before)

$\frac{1}{2\pi i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi+\frac{\beta
Z^{2}}{2NR}j_{-}\phi.$ (3.2)

From the CFT we know the $H_{mod}$ action on the current

$\frac{1}{2\pi
i}[H_{mod},j_{-}]=-\frac{1}{2R}(2x^{-}j_{-}+(x^{-}+R)(x^{-}-R)\partial_{-}j_{-}),$
(3.3)

and the commutator of currents

$[j_{-}(x),j_{-}(y)]=ik\partial_{x^{-}}\delta(x^{-}-y^{-}).$ (3.4)

Equation (3.1) is consistent with equation (3.2) only if it obeys the Jacobi
identity

$[H_{mod},[j_{-},\phi]]+[[H_{mod},\phi],j_{-}]+[\phi,[H_{mod},j_{-}]]=0.$
(3.5)

A simple computation shows that it does, provided that $\beta=-i\frac{q}{k}$.
But it would not be consistent if we took a different action of $H_{mod}$, or
a different $[j_{-},\phi]$ with the same $H_{mod}$ action. There of course can
be other consistent pairs of $[H_{mod},\phi]$ and $[j_{-},\phi]$.

Thus the action of $H_{mod}$ on $\phi$ is tied to a particular commutator of
the conserved current with $\phi$ (i.e a particular charge distribution at
infinity) and as we saw in previous section, to a particular equation of
motion. This explains why the expression for the bulk operator one gets from
(3.1) will be the same as the one obeying the particular equation of motion we
got.

###  3.1 Connection to $i\epsilon$ issue

One view of bulk reconstruction is that one is looking for operators which
obey the intersecting modular equations at leading order in $\frac{1}{N}$ and
are corrected by operators such that they will produce well-defined
correlators. This procedure has many solutions that are connected by local
field redefinitions Kabat:2015swa ; Kabat:2020nvj . Indeed just taking
$\phi=\phi^{(0)}$ inside a 3-point function with $\bar{{\cal O}}$ and $j_{-}$
produces an ill-defined correlator in the sense that the CFT $i\epsilon$
prescription does not give a unique answer when $\phi^{(0)}$ is put in the
middle of the correlator Kabat:2018pbj . One can view this as a sign that the
commutator of $\phi^{(0)}$ with $j_{-}$ is an ill-defined operator. Indeed
equation (E.5) shows that the commutator $[j_{-},\phi^{(0)}]$, written in
terms of $\phi^{(0)}$, contains an infinite number of terms with increasing
number of derivatives on a delta function, each multiplied by integrals of
$\phi^{(0)}$. While this by itself does not mean that the operator is ill-
defined, it certainly gives this possibility, while any finite number of local
bulk terms will of course produce a well-defined operator. One can then view
the condition

$[j_{-}(x^{-}),(\phi^{(0)}+\frac{1}{N}\phi^{(1)})(Z,y^{-},y^{+})]=-\frac{q}{N}\delta(x^{-}-y^{-})\phi^{(0)}$
(3.6)

as a condition that may guarantee a well-behaved 3-point function

$\langle(\phi^{(0)}+\frac{1}{N}\phi^{(1)})\bar{\cal O}j_{-}\rangle,$ (3.7)

given that

$\langle\phi^{(0)}\bar{\cal O}\rangle$ (3.8)

is a well-defined correlator. As remarked above, the condition for a well-
defined correlator can be satisfied with many forms of $\phi^{(1)}$. For
instance another possibility (beyond that used in Appendix A) discussed in
Kabat:2020nvj  is,

$\phi^{(1)}\rightarrow\phi^{(1)}+\frac{i\alpha
q}{Nk}Z^{2}j_{-}\partial_{+}\phi^{(0)}$ (3.9)

with $\alpha$ any constant. This results in a different, but still simple
charge distribution at infinity

$[j_{-}(x^{-}),(\phi^{(0)}+\frac{1}{N}\phi^{(1)})(Z,y^{-},y^{+})]=-\frac{q}{N}\delta(x^{-}-y^{-})\phi^{(0)}-\frac{\alpha
q}{N}\partial_{-}\delta(x^{-}-y^{-})\partial_{+}\phi^{(0)}.$ (3.10)

This may explain why the approach demanding well-defined correlators
Kabat:2018pbj ; Kabat:2020nvj  seems to agree with the approach demanding
simple $[j_{-},\phi^{(0)}]$ Chen:2019hdv . Further, as seen in the previous
section, a specific $[j_{-},\phi^{(0)}]$ results in specific equations of
motion and a specific $H_{mod}$ action. This also explains why demanding a
well-defined $i\epsilon$ prescription results in a bulk operator solving some
simple bulk equation of motion.

We can generalize the charge distribution (3.6) to an expression valid for all
orders in $\frac{1}{N}$ corrections to $\phi$

$[j_{-}(x),\phi(Z,y)]=-\frac{q}{N}\delta(x^{-}-y^{-})\phi(Z,y).$ (3.11)

Then, because of the $\frac{1}{N}$ coefficient in this relationship, we can
argue for a well-defined correlator iteratively. If we start with $\phi$ to
some order in $\frac{1}{N^{k}}$, the commutator only depends on $\phi$ to
order $\frac{1}{N^{k-1}}$. So having a solution to (3.11) to order
$\frac{1}{N^{k}}$ indicates that the correlators

$\langle\phi\bar{\cal O}j_{-}(x^{-}_{1})\cdots j_{-}(x^{-}_{k})\rangle$ (3.12)

to order $\frac{1}{N^{k}}$ will have a well-defined $i\epsilon$ prescription
(and thus $\phi$ that solves (3.11) to $\frac{1}{N^{k}}$ is a well-defined
operator) given that

$\langle\phi\bar{\cal O}j_{-}(x^{-}_{1})\cdots j_{-}(x^{-}_{k-1})\rangle$
(3.13)

is a well-defined correlator with $\phi$ a solution of (3.11) to order
$\frac{1}{N^{k-1}}$.

##  4 Scalar field interacting with a gauge field in general dimensions

In this section we deal with the higher-dimensional case and all order
corrections in $\frac{1}{N}$ for both $\phi$ and $A_{\mu}$. We do this using
the ideas in section 2.3, that is, through looking for equations of motion
which are compatible with an assumed known action of the modular Hamiltonian
on the bulk operators.

The assumed action of the modular Hamiltonian
($\frac{1}{2R}(Q_{0}-R^{2}P_{0})$) on the bulk operators is

$\frac{1}{2\pi
i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi+\frac{iq}{2RN}\lambda_{0}\phi$ (4.1)

where

$\xi^{Z}=\frac{Zt}{R},\ \xi^{i}=\frac{x^{i}t}{R},\
\xi^{0}=\frac{1}{2R}(Z^{2}+t^{2}+x_{i}^{2}-R^{2}),$ (4.2)

and $\lambda_{0}$ is the compensating gauge transformation parameter KL1204
obeying $\partial_{Z}\lambda_{0}=2ZA_{0}$, which we can write as

$\lambda_{0}=2\int_{0}^{Z}dZ^{\prime}Z^{\prime}A_{0}(Z^{\prime},x_{i},t).$
(4.3)

We in addition need the action of the modular Hamiltonian on the bulk gauge
field operator. This is taken from a computation in Kabat:2018smf , which was
done there for the free gauge field case, but we will assume that it is valid
in general444See section 5 for a logical explanation.. It is given by

$\frac{1}{2\pi
i}[H_{mod},A_{\mu}]=\xi^{M}\partial_{M}A_{\mu}+A_{\nu}\partial_{\mu}\xi^{\nu}-\frac{1}{2R}\partial_{\mu}\lambda_{0}.$
(4.4)

The higher-dimensional analogue of equation (2.32) is

$(\nabla^{2}-m_{0}^{2})\phi=-2i\frac{q}{N}Z^{2}\eta^{\mu\nu}A_{\mu}\partial_{\nu}\phi+\frac{q^{2}}{N^{2}}Z^{2}\eta^{\mu\nu}A_{\mu}A_{\nu}\phi.$
(4.5)

We can now check compatibility with the modular Hamiltonian action. Let us
label

$EOM_{0}=(\nabla^{2}-m_{0}^{2})\phi+2i\frac{q}{N}Z^{2}\eta^{\mu\nu}A_{\mu}\partial_{\nu}\phi-\frac{q^{2}}{N^{2}}Z^{2}\eta^{\mu\nu}A_{\mu}A_{\nu}\phi$
(4.6)

for which we indeed find

$\frac{1}{2\pi
i}[H_{mod},EOM_{0}]=(\xi^{M}\partial_{M}+\frac{iq}{2RN}\lambda_{0})EOM_{0}\,,$
(4.7)

but only if

$\nabla^{2}\lambda_{0}=0.$ (4.8)

This condition is satisfied for $A_{\mu}$ which obey the free gauge field
equation of motion. This is the analogous case to the $AdS_{3}$ set-up where
the corresponding $\lambda_{0}=Z^{2}j_{0}$ also obeys (4.8). To go beyond the
free gauge field limit we would need a term that will cancel a term of the
form $\phi\nabla^{2}\lambda_{0}$, coming from $[H_{mod},\nabla^{2}\phi]$. It
is not hard to see that a possible term is of the form

$\phi Z^{2}\eta^{\mu\nu}\partial_{\mu}A_{\nu}$ (4.9)

which is zero in the free gauge field limit. Indeed if we now define

$EOM=(\nabla^{2}-m_{0}^{2})\phi+2i\frac{q}{N}Z^{2}\eta^{\mu\nu}A_{\mu}\partial_{\nu}\phi-\frac{q^{2}}{N^{2}}Z^{2}\eta^{\mu\nu}A_{\mu}A_{\nu}\phi+i\frac{q}{N}\phi
Z^{2}\eta^{\mu\nu}\partial_{\mu}A_{\nu}$ (4.10)

we find

$\frac{1}{2\pi
i}[H_{mod},EOM]=(\xi^{M}\partial_{M}+\frac{iq}{2RN}\lambda_{0})\ EOM.$ (4.11)

The equation of motion $EOM=0$ in (4.10) is of course the correct bulk gauge
invariant equation of motion in $A_{Z}=0$ gauge. Thus the imprint of bulk
gauge invariance from the CFT point of view can be thought of as a consequence
of the $H_{mod}$ action.

##  5 An alternative perspective

In this section we want to present an alternative perspective on the results
of the previous sections. We started with an expression for $[H_{mod},\phi]$
in equation (2.8), which was a result from a CFT computation to first order in
$\frac{1}{N}$. Using this action of the modular Hamiltonian on the bulk
operator, we arrived at an equation of motion (2.17) which is compatible with
the action of $H_{mod}$. Here there was no surprise.
$\phi^{(0)}+\frac{1}{N}\phi^{(1)}$ did solve a simple equation of motion. We
then made a reasonable guess in (2.18) as to what the action of the modular
Hamiltonian should be when considering higher order correction terms in
reconstructing $\phi$, while still keeping the gauge field reconstruction at
lowest order in $\frac{1}{N}$. We then used this to get an equation of motion
(2.24). This seemed reasonable but in reality it might have not worked, in the
sense that given the guess for the action of the modular Hamiltonian, there
was no guarantee that there would be a compatible equation of motion. Indeed
when starting with the action of the modular Hamiltonian (2.26), which is the
naive generalization of $[H_{mod},\phi]$ computed from the CFT point of view
for the case of having both $j_{-}$ and $j_{+}$, one does not end with the
naive generalization of the equation of motion. Instead one finds the equation
of motion (2.32) which has an extra term proportional to $j_{+}j_{-}\phi$.

Conceivably, there might not have been an equation of motion that is
compatible with (2.26) at all. Similar statements hold of course for the pair
(4.1) and (4.10). So when we start with an action of the modular Hamiltonian
which is appropriate to first order in $\frac{1}{N}$ and a compatible equation
of motion to that order, what is the best way to proceed to higher order
statements?

Here we want to suggest that instead of looking at the equation of motion as
an output from a compatibility condition with the action of $H_{mod}$, our
guiding principle can be to look for a pair of compatible actions. That is, we
start with the action of $H_{mod}$ and the equation of motion compatible with
it to leading order in $\frac{1}{N}$ and we then try to generalize both of
them to higher order in $\frac{1}{N}$, such that they are compatible. We will
see an example of this in the next section555This may or may not be a unique
procedure..

##  6 Scalar field interacting with gravity in $AdS_{3}$

In this section we consider a 2-dimensional CFT with a scalar operator
$\mathcal{O}$ that couples to a conserved CFT stress tensor $T_{\mu\nu}$,
corresponding in the dual theory to a gravitationally dressed scalar bulk
operator $\phi$ in an asymptotically $AdS_{3}$ background. That is, the CFT
stress tensor is dual to metric perturbations $h_{{\mu\nu}}$ of the
background, that the dual scalar $\phi$ interacts with.

We set out to determine the $H_{mod}$ action on the CFT representation of the
gravitationally dressed field $\phi$. The action of $H_{mod}$ captures the
non-locality of the physical, i.e. diffeomorphism invariant $\phi$, and we
will show that demanding compatibility of the $H_{mod}$ action with the
existence of a bulk equation of motion is enough to determine both the
$H_{mod}$ action on $\phi$ and the bulk equation of motion for $\phi$ order by
order in $1/N$, once the first term linear it $T_{\mu\nu}$ is known.

In the case of a conserved, chiral current of general spin $s$ being present
in the CFT, the form of $\phi^{(1)}$ was derived from CFT principles in
Kabat:2020nvj . Making use of this known expression, the $H_{mod}$ action on
$\phi=\phi^{(0)}+\frac{1}{N}\phi^{(1)}$ was determined in Appendix B to be
(B.11). In the presence of the spin 2 conserved current $T_{\mu\nu}$ in the
CFT, the result of (B.11) applies to linear order, with a contribution from
each chirality of the stress tensor as in equation (2.26). That is,

$\displaystyle\frac{1}{2\pi
i}[H_{mod},(\phi^{(0)}+\frac{1}{N}\phi^{(1)})]=\xi^{M}\partial_{M}(\phi^{(0)}+\frac{1}{N}\phi^{(1)})+\frac{Z^{4}}{2R}\frac{\gamma}{N}\tilde{T}_{--}\partial_{+}\phi^{(0)}-\frac{Z^{4}}{2R}\frac{\gamma}{N}\tilde{T}_{++}\partial_{-}\phi^{(0)},$
(6.1)

with $\xi$ the Killing vector (A.20) of $AdS_{3}$. It is important to note
that the normalization leading to this equation is such that all 2-point
functions are $O(1)$, without any $N\sim\sqrt{c}$ dependence, where $c$ is the
CFT central charge. However, the two-point function of the energy momentum
tensor $\sim c$ and this is why in the above equation there appears
$\tilde{T}_{\pm\pm}=\frac{1}{N}T_{\pm\pm}$. The constant $\gamma\sim O(1)$ in
(B.11) was left undetermined in Appendix B. We will choose normalizations that
fix the constant $\frac{\gamma}{N^{2}}=\frac{6}{c}$ at the end of this
section.

This $H_{mod}$ action is of the general form (1.1), with the first term the
contribution as if $\phi$ is just a local bulk field, and the second and third
term expressing the non-locality of the gravitationally interacting $\phi$ in
the form of stress tensor dependent derivative operators $\Lambda(T)$. Because
of the non-linearity of gravity, the $H_{mod}$ action can be expected to
receive contributions at each order in $1/N$,

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\phi]=\sum_{i=0}^{\infty}\left(\frac{1}{N}\right)^{i}\mathcal{L}_{H}^{(i)}\phi=\mathcal{L}_{H}\phi,$
(6.2)

having introduced the notation $\mathcal{L}_{H}^{(i)}$ for the $i$-th order in
$\tilde{T}_{\pm\pm}$ derivative operator acting on the all-order field $\phi$,
and $\mathcal{L}_{H}$ for the corresponding total derivative operator. The
total field $\phi$ is expanded as
$\sum_{i=0}^{\infty}(\frac{1}{N})^{i}\phi^{(i)}$. To first order, equation
(6.2) is just a rewriting of (6.1) to

$\displaystyle\mathcal{L}_{H}^{(0)}=\xi^{M}\partial_{M},\qquad\mathcal{L}_{H}^{(1)}=\frac{\gamma}{2R}\left(Z^{4}\tilde{T}_{--}\partial_{+}-Z^{4}\tilde{T}_{++}\partial_{-}\right)$
(6.3)

or

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\phi^{(0)}]=\mathcal{L}_{H}^{(0)}\phi^{(0)},\qquad\frac{1}{2\pi
i}[H_{mod},\phi^{(1)}]=\mathcal{L}_{H}^{(0)}\phi^{(1)}+\mathcal{L}_{H}^{(1)}\phi^{(0)}.$
(6.4)

The modular Hamiltonian action $[H_{mod},\phi]$ in the CFT knows about - or at
least strongly constrains - the bulk equation of motion for $\phi$. Namely, as
in equation (2.43), we impose that the bulk equation of motion should satisfy
the compatibility condition

$\displaystyle\frac{1}{2\pi i}[H_{mod},EOM]=\mathcal{L}_{H}\,EOM.$ (6.5)

Now it would have been enough if on the right hand side of (6.5) there would
appear any differential operator, but since we expect that there will be a
linear term with arbitrary constant coefficient in $\phi$ in the equation of
motion, this linear operator must be $\mathcal{L}_{H}$. With the knowledge of
the $H_{mod}$ action to first order in $1/N$, given in (6.1) or (6.3)-(6.4),
this condition indeed allows us to determine the bulk equation of motion to
first order in $1/N$. This we will now first demonstrate.

We take as an ansatz for the unknown bulk equation of motion for the
gravitationally interacting field $\phi$, the form

$\displaystyle\nabla^{2}\phi=RHS,\qquad
RHS=\sum_{i=0}^{\infty}\left(\frac{1}{N}\right)^{i}RHS^{(i)}$ (6.6)

with right hand side terms $RHS$, which we expand in a priori unknown $i$-th
order in $\frac{1}{N}$ terms. At zeroth order, the condition (6.5) is
consistent with

$\displaystyle RHS^{(0)}=b\,\phi^{(0)}$ (6.7)

for any constant $b$, if $\nabla^{2}$ in (6.6) is the box operator in
$AdS_{3}$ such that it commutes with $\mathcal{L}_{H}^{(0)}$.

With this ansatz, the condition (6.5) becomes

$\displaystyle\nabla^{2}(\mathcal{L}_{H}\phi)-\frac{1}{2\pi
i}[H_{mod},RHS]=\mathcal{L}_{H}\nabla^{2}\phi-\mathcal{L}_{H}\,RHS.$ (6.8)

###  6.1 First order

At first order in $1/N$, the condition (6.8) reads

$\displaystyle\mathcal{L}_{H}^{(0)}RHS^{(1)}-\frac{1}{2\pi
i}[H_{mod},RHS^{(1)}]=\mathcal{L}_{H}^{(1)}\nabla^{2}\phi-\nabla^{2}\mathcal{L}_{H}^{(1)}\phi-\mathcal{L}_{H}^{(1)}RHS^{(0)}.$
(6.9)

The right hand side of this condition is known and given by (with primes for
derivatives)

$\displaystyle\begin{split}&\frac{2Z^{4}\gamma}{R}\left(Z^{2}\tilde{T}_{++}^{\prime}\partial_{-}^{2}-Z^{2}\tilde{T}_{--}^{\prime}\partial_{+}^{2}+2\tilde{T}_{++}\partial_{-}-2\tilde{T}_{--}\partial_{+}+2Z\tilde{T}_{++}\partial_{Z}\partial_{-}-2Z\tilde{T}_{--}\partial_{Z}\partial_{+}\phantom{\frac{b}{4}}\right.\\\
&\qquad\left.+\frac{b}{4}(\tilde{T}_{++}\partial_{-}-\tilde{T}_{--}\partial_{+})\right)\phi^{(0)}.\end{split}$
(6.10)

We can build an ansatz for $RHS^{(1)}$ (and later on for all $RHS^{(n)},n>0$)
based on the following principles: Lorentz invariance in the $AdS_{3}$
conformal boundary indices ${\mu\nu}$, which take the values $+$ and $-$ (i.e.
the terms in the ansatz should contain equal numbers of $+$ and $-$ indices),
symmetry in exchanging $+$ and $-$ indices, and dimensional analysis (to fix
the accompanying orders of $Z$). It turns out to be sufficient for finding a
solution to include at most second order derivatives on $\phi$ and
$\tilde{T}_{\pm\pm}$ in the ansatze. We have not investigated including higher
order derivatives. The ansatz for $RHS^{(1)}$ is then given by
$aZ^{4}(\tilde{T}_{++}\partial_{-}^{2}\phi^{(0)}+\tilde{T}_{--}\partial_{+}^{2}\phi^{(0)})+b\,\phi^{(1)}$,
with the coefficient $a$ an undetermined constant. The ansatz also includes
the expected first order correction $b\,\phi^{(1)}$ to $RHS^{(0)}$ given in
(6.7), with $b$ a mass parameter that can take on any value.

With this ansatz, the left hand side of (6.9) can be worked out using the
action of $H_{mod}$ on $RHS^{(1)}$ given in terms of the action on $\phi$ in
(6.1) and that on the stress tensors as

$\displaystyle\begin{split}[H_{mod},RHS^{(1)}]&=\partial_{\tilde{T}_{++}}RHS^{(1)}\,[H_{mod},\tilde{T}_{++}]+\partial_{\tilde{T}_{--}}RHS^{(1)}\,[H_{mod},\tilde{T}_{--}]+b\,[H_{mod},\phi^{(1)}]\\\
&\,+\partial_{\partial_{-}^{2}\phi^{(0)}}RHS^{(1)}\,\partial_{-}^{2}[H_{mod},\phi^{(0)}]+\partial_{\partial_{+}^{2}\phi^{(0)}}RHS^{(1)}\,\partial_{+}^{2}[H_{mod},\phi^{(0)}].\end{split}$
(6.11)

The action of $H_{mod}$ on the CFT stress tensors
$[H_{mod},\tilde{T}_{\pm\pm}]$ is immediate from the $H_{mod}$ definition
(2.7) and the action of the conformal generators on the stress tensor, given
explicitly in (F.22)-(F.23). The left hand side of (6.9) then takes a form
that matches the right hand side (6.10) if and only if $a=4\gamma$ or

$\displaystyle
RHS^{(1)}=4Z^{4}\gamma\,(\tilde{T}_{++}\partial_{-}^{2}\phi^{(0)}+\tilde{T}_{--}\partial_{+}^{2}\phi^{(0)})+b\,\phi^{(1)}.$
(6.12)

###  6.2 Second order

We only have access at this stage to the CFT obtained modular Hamiltonian
action (6.1), valid to first order in $1/N$. That is, in the notation of
equation (6.2), the derivative operators $\mathcal{L}_{H}^{(i)}$ are unknown
for $i\geq 2$. We can proceed by _assuming_ that the modular Hamiltonian
action (6.1) is valid also to the next order, i.e. that
$\mathcal{L}_{H}^{(2)}=0$666This was indeed the case, in the gauge field
case.. (This assumption will turn out to be false.) If we do so, the
compatibility condition with the existence of a bulk equation of motion (6.5)
at second order in $1/N$ takes the form

$\displaystyle\begin{split}\mathcal{L}_{H}^{(0)}RHS^{(2)}-\frac{1}{2\pi
i}[H_{mod},RHS^{(2)}]&=\mathcal{L}_{H}^{(2)}\nabla^{2}\phi^{(0)}-\nabla^{2}\mathcal{L}_{H}^{(2)}\phi^{(0)}+\mathcal{L}_{H}^{(1)}\nabla^{2}\phi^{(1)}-\nabla^{2}\mathcal{L}_{H}^{(1)}\phi^{(1)}\\\
&\quad-\mathcal{L}_{H}^{(1)}RHS^{(1)}-\mathcal{L}_{H}^{(2)}RHS^{(0)}.\end{split}$
(6.13)

The right hand side is known in terms of $RHS^{(1)}$ in (6.12), $RHS^{(0)}$ in
(6.7), $\mathcal{L}_{H}^{(1)}$ in (6.3), $\nabla^{2}$ of $AdS_{3}$, and the
assumed $\mathcal{L}_{H}^{(2)}=0$.

Proceeding in an identical matter as at lower orders, we impose an ansatz for
$RHS^{(2)}$. It will contain terms: $(\tilde{T}_{++}\tilde{T}_{--})$
multiplying
($\phi^{(0)},\partial_{Z}\phi^{(0)},\partial_{Z}^{2}\phi^{(0)},\partial_{+}\partial_{-}\phi^{(0)})$,
and $\tilde{T}_{++}\tilde{T}^{\prime}_{--}\partial_{+}\phi^{(0)}$ and
$\tilde{T}_{++}\tilde{T}^{\prime\prime}_{--}\partial_{+}^{2}\phi^{(0)}$ (all
symmetrized over $+\leftrightarrow-$ exchange). Said otherwise, the ansatz for
$RHS^{(2)}$ will contain 8 undetermined coefficients. Additionally, it will
contain terms
$b\,\phi^{(2)}+4Z^{4}\gamma(\tilde{T}_{++}\partial_{-}^{2}\phi^{(1)}+\tilde{T}_{--}\partial_{+}^{2}\phi^{(1)})$
such that the previous orders in $RHS$ are consistently extended to include
higher orders in $\phi$.

Next, we write out the left hand side and right hand side of (6.13)777 For
this, we make use of the action of the modular Hamiltonian on a general,
second order derivative ansatz for the $n$-th order in $1/N$ contribution to
the $RHS$ of the bulk equation of motion:  $\displaystyle[H_{mod},RHS^{(n)}]$
$\displaystyle=(\partial_{\tilde{T}_{++}}RHS^{(n)})[H_{mod},\tilde{T}_{++}]+(\partial_{\tilde{T}_{--}}RHS^{(n)})[H_{mod},\tilde{T}_{--}]$
$\displaystyle+(\partial_{\tilde{T}_{++}^{\prime}}RHS^{(n)})\partial_{+}[H_{mod},\tilde{T}_{++}]+(\partial_{\tilde{T}^{\prime}_{--}}RHS^{(n)})\partial_{-}[H_{mod},\tilde{T}_{--}]$
$\displaystyle+(\partial_{\partial_{-}\phi}RHS^{(n)})\partial_{-}[H_{mod},\phi]+(\partial_{\partial_{+}\phi}RHS^{(n)})\partial_{+}[H_{mod},\phi]+(\partial_{\partial_{Z}\phi}RHS^{(n)})\partial_{Z}[H_{mod},\phi]$
$\displaystyle+(\partial_{\partial_{+}^{2}\phi}RHS^{(n)})\partial_{+}^{2}[H_{mod},\phi]+(\partial_{\partial_{-}^{2}\phi}RHS^{(n)})\partial_{-}^{2}[H_{mod},\phi]$
$\displaystyle+(\partial_{\phi}RHS^{(n)})\,[H_{mod},\phi]+(\partial_{\partial_{+}\partial_{-}\phi}RHS^{(n)})\partial_{+}\partial_{-}[H_{mod},\phi].$
(6.14) . Equating the prefactors of the different derivative terms of $\phi$
gives rise to a set of ten equations, which does not have a solution for the
undetermined coefficients in the ansatz for $RHS^{(2)}$. We conclude that the
assumption $\mathcal{L}_{H}^{(2)}=0$ was incorrect.

We can instead allow for the most general $\mathcal{L}_{H}^{(2)}$ in
$[H_{mod},\phi]$ we can think of. Besides it being quadratic in the stress
tensor, we will impose that it contains a component in $\partial_{+}$ and one
in $\partial_{-}$, just as $\mathcal{L}_{H}^{(1)}$ in (6.3) does. A
$\partial_{Z}$ term is in principle not disallowed, but since it turns out not
to be necessary for finding a solution, we do not include it in the ansatz.
The ansatz for $\mathcal{L}_{H}^{(2)}$ will then contain 6 unknown
coefficients, for the possible terms obtained from multiplying
$Z^{6}(\tilde{T}_{++}\tilde{T}_{--},\tilde{T}_{++}^{2},\tilde{T}_{--}^{2})$
with $(\partial_{+},\partial_{-})$. Condition (6.13) can now be reordered into
an unknown left hand side equal to a known right hand side:

$\displaystyle\begin{split}\mathcal{L}_{H}^{(0)}RHS^{(2)}-\frac{1}{2\pi
i}[H_{mod},RHS^{(2)}]-&(\mathcal{L}_{H}^{(2)}\nabla^{2}\phi^{(0)}-\nabla^{2}\mathcal{L}_{H}^{(2)}\phi^{(0)})+\mathcal{L}_{H}^{(2)}RHS^{(0)}\\\
=&(\mathcal{L}_{H}^{(1)}\nabla^{2}\phi^{(1)}-\nabla^{2}\mathcal{L}_{H}^{(1)}\phi^{(1)})-\mathcal{L}_{H}^{(1)}RHS^{(1)}.\end{split}$
(6.15)

We can write out the left hand side using the ansatze for $RHS^{(2)}$ and
$\mathcal{L}_{H}^{(2)}$, and the right hand side using $RHS^{(1)}$ in (6.12),
$\mathcal{L}_{H}^{(1)}$ in (6.3) and $\nabla^{2}$ of $AdS_{3}$. Subsequently
equating the prefactors of the different derivative terms of $\phi$ results in
a set of equations that can uniquely be solved for the coefficients in the
ansatze. The resulting $RHS^{(2)}$ and $\mathcal{L}_{H}^{(2)}$ are given by

$\displaystyle RHS^{(2)}$
$\displaystyle=\gamma^{2}\left(\tilde{T}_{--}\tilde{T}_{++}(4Z^{5}\partial_{Z}\phi^{(0)}-12Z^{6}\partial_{+}\partial_{-}\phi^{(0)})-4Z^{6}(\tilde{T}_{--}\tilde{T}^{\prime}_{++}\partial_{-}\phi^{(0)}+\tilde{T}_{++}\tilde{T}^{\prime}_{--}\partial_{+}\phi^{(0)})\right)$
$\displaystyle\quad+4Z^{4}\gamma\,(\tilde{T}_{++}\partial_{-}^{2}\phi^{(1)}+\tilde{T}_{--}\partial_{+}^{2}\phi^{(1)})+b\,\phi^{(2)}$
(6.16)

and

$\displaystyle\mathcal{L}_{H}^{(2)}=\frac{\gamma^{2}}{2R}Z^{6}\tilde{T}_{--}\tilde{T}_{++}(\partial_{+}-\partial_{-}).$
(6.17)

We have shown that the compatibility condition (6.5) fixes _both_ the action
of $H_{mod}$ on $\phi$ and the bulk equation of motion for $\phi$ at second
order in $1/N$.

###  6.3 Higher order

The process of fixing both the action of $H_{mod}$ on $\phi$ and the bulk
equation of motion for $\phi$ at second order in $1/N$ can be repeated at each
order. Indeed, the compatibility condition (6.8) takes the form of an
iterative equation for solving the $n$-th order in $\tilde{T}_{\pm\pm}$
contribution to the equation of motion (namely $RHS^{(n)}$) as well as the
$n$-th order in $\tilde{T}_{\pm\pm}$ contribution to $[H_{mod},\phi]$ (namely
$\mathcal{L}_{H}^{(n)}$), if the previous orders $RHS^{(n-1)}$, …, $RHS^{(0)}$
and $[H_{mod},\phi]$ to $(n-1)$-st order in $\tilde{T}_{\pm\pm}$ are known:

$\displaystyle\begin{split}&[\nabla^{2},\mathcal{L}_{H}^{(n)}]\phi^{(0)}+\mathcal{L}_{H}^{(n)}RHS^{(0)}+\mathcal{L}_{H}^{(0)}RHS^{(n)}-\frac{1}{2\pi
i}[H_{mod},RHS^{(n)}]\\\
&=\sum_{i=1}^{n-1}\left([\mathcal{L}_{H}^{(i)},\nabla^{2}]\phi^{(n-i)}-\mathcal{L}_{H}^{(i)}RHS^{(n-i)}\right).\end{split}$
(6.18)

The right hand side of this condition is known from the previous orders, and
the left hand side has unknowns $\mathcal{L}_{H}^{(n)}$ and $RHS^{(n)}$ for
which we use appropriate ansatze. The ansatze we use are the most general
ones, up to including at most second order derivatives on $\phi$ in
$RHS^{(n)}$ and not including a $\partial_{Z}$ term in
$\mathcal{L}_{H}^{(n)}$. This means that in principle there could be more
consistent pairs of $H_{mod}$ action and possible equations of motion.

Let us give one more example. For $n=3$, the ansatz for the third order
contribution $RHS^{(3)}$ to the equation of motion for $\phi$ will contain the
expression for $RHS^{(2)}$ with $\phi^{(i)}$ replaced by $\phi^{(i+1)}$. It
will also contain $\partial_{-}\phi$ times
$\tilde{T}_{++}^{2}\tilde{T}_{--}^{\prime}$ and $\partial_{-}^{2}\phi$ times
$\tilde{T}_{++}^{2}\tilde{T}_{--}$ (symmetrized over $+\leftrightarrow-$
exchange). This leaves 4 coefficients to be determined. The ansatz for
$\mathcal{L}_{H}^{(3)}$ contains 8 undetermined coefficients, for the possible
combinations of factors
$Z^{8}(\tilde{T}_{--}^{3},\tilde{T}_{--}^{2}\tilde{T}_{++},\tilde{T}_{--}\tilde{T}_{++}^{2},\tilde{T}_{++}^{3})$
and $(\partial_{+}\phi,\partial_{-}\phi)$. The condition (6.5), rewritten into
(6.18), uniquely determines the coefficients. The resulting addition to the
equation of motion of $\phi$ is

$\displaystyle
RHS^{(3)}=\gamma^{3}\left(4Z^{8}\tilde{T}_{++}^{2}\tilde{T}^{\prime}_{--}\partial_{-}\phi^{(0)}+8Z^{8}\tilde{T}_{++}^{2}\tilde{T}_{--}\partial_{-}^{2}\phi^{(0)}+\,\,\,(+\leftrightarrow-)\,\,\,\right)+\cdots$
(6.19)

where $\cdots$ are the terms in $\phi^{(1)}$, $\phi^{(2)}$ and $\phi^{(3)}$
that are immediate from the previous orders in the equation of motion. And the
resulting addition to the modular Hamiltonian action on $\phi$ is

$\displaystyle\mathcal{L}_{H}^{(3)}=\frac{\gamma^{3}}{2R}Z^{8}(\tilde{T}_{++}\tilde{T}_{--}^{2}\partial_{+}\phi-\tilde{T}_{--}\tilde{T}_{++}^{2}\partial_{-}\phi).$
(6.20)

We now fix our normalization to $\frac{\gamma}{N^{2}}=\frac{6}{c}$, such that
$\frac{\gamma}{N}\tilde{T}_{\pm\pm}=\frac{6}{c}T_{\pm\pm}$. The results we
find at each order are consistent with the _all-order_ bulk equation of
motion, as well as the _all-order_ modular Hamiltonian action (D.14) that is
obtained from a bulk perspective in Appendix D:

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi+\epsilon^{+}\partial_{+}\phi+\epsilon^{-}\partial_{-}\phi$
(6.21)

in terms of the stress tensor dependent functions

$\displaystyle\epsilon^{+}=\frac{1}{2R}\frac{\frac{6}{c}Z^{4}T_{--}+\frac{36}{c^{2}}Z^{6}T_{--}T_{++}}{1-\frac{36}{c^{2}}Z^{4}T_{--}T_{++}},\qquad\epsilon^{-}=-\frac{1}{2R}\frac{\frac{6}{c}Z^{4}T_{++}+\frac{36}{c^{2}}Z^{6}T_{--}T_{++}}{1-\frac{36}{c^{2}}Z^{4}T_{--}T_{++}},$
(6.22)

and

$\displaystyle\nabla^{2}\phi-m_{0}^{2}\,\phi$
$\displaystyle=\frac{4Z^{8}\frac{6^{3}}{c^{3}}T_{++}^{2}T_{--}^{\prime}-4Z^{6}\frac{36}{c^{2}}T_{--}T_{++}^{\prime}}{(1-Z^{4}\frac{36}{c^{2}}T_{--}T_{++})^{3}}\partial_{-}\phi+\frac{4Z^{4}\frac{6}{c}T_{++}}{(1-Z^{4}\frac{36}{c^{2}}T_{--}T_{++})^{2}}\partial_{-}^{2}\phi$
$\displaystyle+\frac{4Z^{8}\frac{6^{3}}{c^{3}}T_{--}^{2}T^{\prime}_{++}-4Z^{6}\frac{36}{c^{2}}T_{++}T^{\prime}_{--}}{(1-Z^{4}\frac{36}{c^{2}}T_{--}T_{++})^{3}}\partial_{+}\phi+\frac{4Z^{4}\frac{6}{c}T_{--}}{(1-Z^{4}\frac{36}{c^{2}}T_{--}T_{++})^{2}}\partial_{+}^{2}\phi$
$\displaystyle+\frac{4Z^{6}\frac{36}{c^{2}}T_{--}T_{++}(-3+Z^{4}\frac{36}{c^{2}}T_{--}T_{++})}{(1-Z^{4}\frac{36}{c^{2}}T_{--}T_{++})^{2}}\partial_{+}\partial_{-}\phi+\frac{4Z^{5}\frac{36}{c^{2}}T_{--}T_{++}}{1-Z^{4}\frac{36}{c^{2}}T_{--}T_{++}}\partial_{Z}\phi.$
(6.23)

Here we have $b=m_{0}^{2}$ in $RHS^{(0)}$ for a massive field, and the right
hand side contains the $(T_{\pm\pm}\phi)$-interaction terms that, from the CFT
point of view, are allowed by the procedure of demanding compatibility with
the modular Hamiltonian action on the dressed $\phi$.

The all-order bulk equation of motion above is a rewriting of the, perhaps not
unexpected, equation of motion of a field $\phi$ in a Banados geometry (D.9)

$\displaystyle\nabla^{2}_{Banados}\phi-m_{0}^{2}\phi$ $\displaystyle=0$ (6.24)

when the Brown-York stress tensor is replaced by the CFT stress tensor. We
note that the full vector field $\xi+\epsilon$ in the all-order $H_{mod}$
action (6.21) is an asymptotic, or ‘Brown-Henneaux’, Killing vector of the
Banados geometry in the sense that it preserves holographic gauge, but is not
an actual Killing vector of the Banados geometry. This is further detailed in
Appendix D on the bulk perspective.

The all-order expressions (6.21) and (6.23) for the modular Hamiltonian action
and bulk equation of motion of a gravitationally dressed scalar field $\phi$
are checked to satisfy the compatibility condition (6.5).

##  7 Comments

One may think that the constraints put on the equation of motion of the bulk
operator in the CFT vacuum state are a result of the modular Hamiltonian being
a sum of conformal generators which are symmetries of the CFT. We believe this
is not the case, but rather that there is a more general story. Consider some
non-vacuum state in the CFT which has a holographic dual. The modular
Hamiltonian need not be (and is generally not) a sum of conformal generators.
Nevertheless, near the HRT Hubeny:2007xt  surface one expects the modular
Hamiltonian to acts as boosts JLMS . So very near the HRT surface it seems
reasonable to expect

$\frac{1}{2\pi i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi+\cdots$ (7.1)

where $(\partial_{N}\xi^{M})|_{HRT}=\Lambda_{N}^{M}$ is the appropriate boost
matrix in the normal directions to the HRT surface. As argued in Kabat:2018smf
this constrains the background metric expected to be dual to the CFT state. In
this paper, we saw that it also constrains the possible equation of motion for
a bulk scalar operator, since equation (7.1) has to be compatible with the
equation of motion. In particular the Laplacian operator would have to be the
appropriate one compatible with the metric (as we saw in equation (2.39) in
the simple case of the vacuum state of the CFT). Note that this only clearly
constrains the equation of motion for the state for which $H_{mod}$ was
computed, since only then its action on $\phi$ is expected to have the simple
geometric form (7.1) near the HRT surface. This state is of course invariant
under the action of $H_{mod}$ and so one can consider this a symmetry of the
particular state, but not of the whole theory. So in essence the invariance of
a state under modular flow constrains the possible equation of motion of bulk
operators in the dual holographic picture. What we showed in this paper is
that also both general covariance and gauge invariance (in a fixed gauge) are
implemented through the action of $H_{mod}$, through extra terms on the right
of (7.1), and our argument indicates that this should be true also in
backgrounds which are not empty AdS.

In the picture of Kabat:2018smf , the bulk spacetime manifold arises from
properties of the algebra of modular Hamiltonians and the background metric
arises from the compatibility of the metric $g_{MN}$ with the various boost
matrices near the HRT surface (as implemented on bulk operators), representing
the invariance of the state under modular flow. So the construct of a metric
seems rather “classical”. In that picture a question arises: why should we
consider the metric as having quantum fluctuations and why should we consider
the properly smeared energy momentum operator as this fluctuation (in some
semiclassical approximation)?

Here we see a possible answer. The action of the modular Hamiltonian on the
bulk scalar field operator interacting with gravity includes a “classical
term” which is $\xi^{M}(g_{MN})\partial_{M}\phi$ and a quantum term
$\epsilon^{M}(T_{\mu\nu})\partial_{M}\phi$. So this structure suggests
combining the action of the modular Hamiltonian as
$(\xi^{M}(g_{MN})+\epsilon^{M}(T_{\mu\nu}))\partial_{M}\phi$, which looks as
if smearing of $T_{\mu\nu}$ (through $\epsilon^{M}(T_{\mu\nu})$) can be
interpreted as a fluctuation of the classical metric. This however is not
quite the case, rather it comes about in a somewhat indirect way through the
equation of motion. As we saw in the $AdS_{3}$ case in equation (6.21), given
a state with some expectation value for $\langle T_{\mu\nu}\rangle$ one can
combine $\xi^{M}(g_{MN})+\epsilon^{M}(\langle T_{\mu\nu}\rangle)$ but this
will not be $\xi^{M}(g^{(new)}_{MN})$. This is reasonable since the $H_{mod}$
we are dealing with is that of the vacuum state with no expectation value.
What actually happens is that the action of the modular Hamiltonian (classical
plus quantum part) constrains the equation of motion, and only in the equation
of motion, putting expectation values for $T_{\mu\nu}$ translates
straightforwardly into the correct equation of motion for a scalar field with
a deformed metric.

Acknowledgements

GL would like to thank D. Kabat for discussions. GL is supported in part by
the Israel Science Foundation under grant 447/17. NC was supported in part by
the Israel Science Foundation under grant 447/17.

##  Appendix A CFT calculation of the $H_{mod}$ action on $\phi^{(1)}$ in
conserved current case

A free bulk scalar has a CFT representation

$\displaystyle\phi^{(0)}(Z,x^{+},x^{-})$
$\displaystyle=\frac{1}{2\pi}\int_{t^{\prime 2}+y^{\prime
2}<Z^{2}}dt^{\prime}dy^{\prime}\,\left(\frac{Z^{2}-t^{\prime 2}-y^{\prime
2}}{Z}\right)^{\Delta-2}\mathcal{O}(x^{+}+t^{\prime}+iy^{\prime},x^{-}-t^{\prime}+iy^{\prime})$
(A.1)

with normalization such that $\phi^{(0)}\stackrel{{\scriptstyle Z\rightarrow
0}}{{\sim}}\frac{Z^{\Delta}}{2(\Delta-1)}\mathcal{O}(x)$, and $x^{\pm}=x\pm
t$.

In Kabat:2020nvj , the first order in $1/N$ correction to the CFT
representation of a scalar field in $AdS_{3}$ interacting with a Chern-Simons
gauge field, was determined to be given by a linear combination of operators
${\cal A}_{(0,l)}$

$\displaystyle\phi^{(1)}$ $\displaystyle=\sum_{l=0}^{\infty}a_{l}\,{\cal
A}_{(0,l)}.$ (A.2)

The ${\cal A}_{(0,l)}$ operators are obtained from smearing the boundary
operator $\partial_{+}^{1+l}J_{(0,l)}$ as if it is a primary scalar (even
though it is not actually primary)

$\displaystyle{\cal A}_{(0,l)}(Z,x^{+},x^{-})$ $\displaystyle=\int_{t^{\prime
2}+y^{\prime
2}<Z^{2}}dt^{\prime}dy^{\prime}\,K_{\Delta+2+2l}\,\partial_{+}^{1+l}J_{(0,l)}(x^{+}+t^{\prime}+iy^{\prime},x^{-}-t^{\prime}+iy^{\prime})$
(A.3)

with smearing function

$\displaystyle
K_{\Delta+2+2l}(Z,x^{+},x^{-};t^{\prime},y^{\prime})=\frac{1}{2\pi}\left(\frac{Z^{2}-t^{\prime
2}-y^{\prime 2}}{Z}\right)^{\Delta+2l},$ (A.4)

and $J_{(0,l)}$ defined in (2.5) as a primary, spin $(l+1)$ operator with
conformal weights $(l+\frac{\Delta}{2}+1,\frac{\Delta}{2})$ made of one
conserved boundary current $j_{-}$, derivatives and the primary scalar
operator $\mathcal{O}$.

From the standard action of the conformal generators $L_{1}$ and $\bar{L}_{1}$
on the current $j_{-}$ (with conformal weights $(1,0)$) and $\mathcal{O}$
(with conformal weights $(\frac{\Delta}{2},\frac{\Delta}{2})$), it follows
that the scalar descendants $\partial_{+}^{1+l}J_{(0,l)}$ transform as

$\displaystyle\bar{L}_{1}\,\partial_{+}^{l+1}J_{(0,l)}=\text{(primary
scalar)}+(1+l)(\Delta+l)\partial_{+}^{l}J_{(0,l)}$ (A.5)

with (primary scalar) the transformation of $\partial_{+}^{l+1}J_{(0,l)}$ as
if it was a primary scalar, given by
$(x^{+})^{2}\partial_{+}(\partial_{+}^{l+1}J_{(0,l)})+2(l+\frac{\Delta}{2}+1)x^{+}(\partial_{+}^{1+l}J_{(0,l)})$.
Note that $L_{1}\,\partial_{+}^{l+1}J_{(0,l)}$ will only consist of such a
(primary scalar) contribution, and so will the action of the other global
conformal generators.

We set out to determine $[H_{mod},\phi^{(1)}]$ by calculating the correlator
$\langle[H_{mod},\phi^{(1)}]\,\bar{\mathcal{O}}\,j_{-}\rangle$. With $H_{mod}$
defined in terms of the conformal generators in (2.7) and knowledge of the
$\bar{L}_{1}$ action (A.5), we find

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\partial_{+}^{1+l}J_{(0,l)}]=\frac{1}{2\pi
i}[H_{mod},\partial_{+}^{1+l}J_{(0,l)}]_{\text{(primary
scalar)}}+\frac{1}{2R}(1+l)(\Delta+l)\partial_{+}^{l}J_{(0,l)}$ (A.6)

with $\frac{1}{2\pi
i}[H_{mod},\partial_{+}^{1+l}J_{(0,l)}(t,x)]_{\text{(primary
scalar)}}=\frac{1}{2R}((t^{2}+x^{2}-R^{2})\partial_{t}+2tx\partial_{x}+2t(2l+\Delta+2))\partial_{+}^{1+l}J_{(0,l)}$.
It is the second contribution that carries information on the non-locality of
the bulk scalar field, dressed by its gauge field interaction. We focus in the
following on that non-(primary scalar) contribution, and want to evaluate

$\displaystyle\frac{1}{2\pi
i}\left\langle[H_{mod},\mathcal{A}_{(0,l)}]_{\text{non-(prim
scalar)}}\,\bar{\mathcal{O}}\,j_{-}\right\rangle=\frac{(1+l)(\Delta+l)}{2R}\left\langle(\int
K_{\Delta+2+2l}\,\partial_{+}^{l}J_{(0,l)})\,\bar{\mathcal{O}}\,j_{-}\right\rangle$
(A.7)

at the suppressed locations $\partial_{+}^{l}J_{(0,l)}(x_{1}^{+},x_{1}^{-})$,
$\mathcal{O}(x_{2}^{+},x_{2}^{-})$ and $j_{-}(x_{3}^{-})$. For this, we start
from the CFT correlator

$\displaystyle\left\langle
J_{(0,l)}(x_{1}^{+},x_{1}^{-})\,\bar{\mathcal{O}}(x_{2}^{+},x_{2}^{-})\,j_{-}(x_{3}^{-})\right\rangle=\frac{\alpha_{l}\,(-1)^{\Delta}(x_{23}^{-})^{l}}{(x_{12}^{-})^{\Delta+l}(x_{12}^{+})^{\Delta}(x_{13}^{-})^{l+2}}$
(A.8)

with OPE coefficient $\alpha_{l}$, and smear it to the correlator on the right
hand side of (A.7). We then evaluate the smearing integral in (A.7) by a
completely analogous calculation as the one detailed in Appendix A.2 of
Kabat:2020nvj  for calculating
$\langle\mathcal{A}_{(0,l)}\,\mathcal{O}\,j_{-}\rangle$. The result is

$\displaystyle\frac{1}{2\pi
i}\left\langle[H_{mod},\mathcal{A}_{(0,l)}]_{\text{non-(prim
scalar)}}\,\bar{\mathcal{O}}\,j_{-}\right\rangle=\frac{(1+l)(\Delta+l)}{2R}\frac{(-1)^{\Delta}\alpha_{l}\,(x_{23}^{-})^{l}}{2(x_{12}^{-}x_{12}^{+})^{l+\Delta}(x_{13}^{-})^{l+2}}\frac{\Gamma(l+\Delta)Z^{2l+\Delta+2}}{\Gamma(\Delta)(2l+\Delta+1)}\,\mathcal{B},$
(A.9)
$\displaystyle\mathcal{B}=\left(1+\frac{Z^{2}}{x_{12}^{+}x_{13}^{-}}\right)^{-(l+\Delta)}{{}_{2}F_{1}}\left(l+\Delta,l+\Delta,2+2l+\Delta,\frac{\frac{-Z^{2}}{x_{12}^{+}x_{12}^{-}}+\frac{Z^{2}}{x_{12}^{+}x_{13}^{-}}}{1+\frac{Z^{2}}{x_{12}^{+}x_{13}^{-}}}\right).$
(A.10)

Now moving on to $\left\langle[H_{mod},\phi^{(1)}]_{\text{non-(prim
scalar)}}\,\bar{\mathcal{O}}\,j_{-}\right\rangle=\sum
a_{l}\,\left\langle[H_{mod},\mathcal{A}_{(0,l)}]_{\text{non-(prim
scalar)}}\,\bar{\mathcal{O}}\,j_{-}\right\rangle$, an expression is obtained
that depends on the combination $a_{l}\alpha_{l}$ of the coefficients $a_{l}$
in (A.2) and the OPE coefficients $\alpha_{l}$. The same combination appears
in the correlator $\langle\phi^{(1)}\,\bar{\mathcal{O}}\,j_{-}\rangle$, when
calculated similarly from
$\langle\mathcal{A}_{(0,l)}\,\bar{\mathcal{O}}\,j_{-}\rangle$, as referred to
in the previous paragraph. Requiring the absence of unwanted branch cuts in
$\langle\phi^{(1)}\,\bar{\mathcal{O}}\,j_{-}\rangle$ (that appear because of
the hypergeometric function in its expression), fixes the combination
$a_{l}\alpha_{l}$ up to an $l$-independent constant $c_{1}$. This constant is
determined by requiring the absence of an unphysical pole in
$\langle\phi^{(0)}\,\bar{\mathcal{O}}\,j_{-}\rangle$ (see Kabat:2020nvj )

$c_{1}=\frac{1}{2\pi}\frac{-iq}{2\Gamma(\Delta)(\Delta-1)}.$ (A.11)

Using the expression thus obtained for $a_{l}\alpha_{l}$ in Kabat:2020nvj ,

$\displaystyle
a_{l}\alpha_{l}=2\,c_{1}(1+2l+\Delta)\frac{\Gamma(\Delta)\Gamma(l+\Delta)}{\Gamma(1+2l+\Delta)},$
(A.12)

we arrive at

$\displaystyle\frac{1}{2\pi
i}\left\langle[H_{mod},\phi^{(1)}]_{\text{non-(prim
scalar)}}\,\bar{\mathcal{O}}\,j_{-}\right\rangle$
$\displaystyle=\frac{c_{1}}{2R}\left(\frac{x_{13}^{-}}{Z}\right)^{\Delta-2}\frac{(-1)^{\Delta}}{(x_{23}^{-})^{\Delta}}\sum_{l=0}^{\infty}\left\\{(l+1)\frac{\Gamma(l+\Delta+1)\Gamma(l+\Delta)}{\Gamma(1+2l+\Delta)}\right.$
$\displaystyle\quad\left.\phantom{\frac{1}{2}}\times(-\mathcal{Y})^{l+\Delta}\,{{}_{2}F_{1}}\left(l+\Delta,l+2,2+2l+\Delta,\mathcal{Y}\right)\right\\}$
(A.13)

for

$\displaystyle\mathcal{Y}=Z\,\frac{x_{23}^{-}}{x_{13}^{-}}\,\frac{Z}{Z^{2}+x_{12}^{-}x_{12}^{+}}.$
(A.14)

The sum over $l$ in (A.13) evaluates to $\Gamma(\Delta)\mathcal{Y}^{\Delta}$,
thanks to the hypergeometric identity Hansen

$\displaystyle\sum_{k}\frac{(a)_{k}(b)_{k}(c)_{k}}{\Gamma(k+1)(c)_{2k}}\,x^{k}\,F_{2,1}(a+k,b+k,c+2k+1,-x)=1$
(A.15)

with Pochhammer symbols $(a)_{k}=\frac{\Gamma(a+k)}{\Gamma(a)}$. The resulting
correlator is given by

$\displaystyle\frac{1}{2\pi
i}\left\langle[H_{mod},\phi^{(1)}]_{\text{non-(prim
scalar)}}\,\bar{\mathcal{O}}\,j\right\rangle$
$\displaystyle=\frac{c_{1}\Gamma(\Delta)}{2R}\frac{Z^{2}}{(x_{13}^{-})^{2}}\left(\frac{Z}{x_{12}^{-}x_{12}^{+}+Z^{2}}\right)^{\Delta}.$
(A.16)

On the other hand, we know

$\displaystyle\left\langle
j_{-}(x_{1}^{-})\phi^{(0)}(Z,x_{1}^{+},x_{1}^{-})\bar{\mathcal{O}}(x_{2}^{+},x_{2}^{-})j_{-}(x_{3}^{-})\right\rangle=\frac{1}{2\pi}\frac{1}{2(\Delta-1)}\frac{k}{(x_{13}^{-})^{2}}\left(\frac{Z}{x_{12}^{-}x_{12}^{+}+Z^{2}}\right)^{\Delta}$
(A.17)

from contracting
$\langle\phi^{(0)}(Z,x_{1})\bar{\mathcal{O}}(x_{2})\rangle=\frac{1}{2(\Delta-1)}\left(\frac{Z}{x_{12}^{-}x_{12}^{+}+Z^{2}}\right)^{\Delta}$
and $\langle
j_{-}(x_{1}^{-})j_{-}(x_{3}^{-})\rangle=\frac{k}{2\pi(x_{13}^{-})^{2}}$. It
follows from comparison between (A.16) and (A.17) that

$\displaystyle\frac{1}{2\pi i}[H_{mod},\phi^{(1)}]_{\text{non-(prim
scalar)}}=-\frac{iq}{k}\frac{Z^{2}}{2R}\,j_{-}\phi^{(0)}.$ (A.18)

Now let us consider the (primary scalar) contribution to
$[H_{mod},\partial_{+}^{1+l}J_{l}]$ in (A.6). We know from Kabat:2018smf ,
that this will give rise to a corresponding (primary scalar) contribution to
$[H_{mod},\mathcal{A}_{(0,l)}]$ given by

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\mathcal{A}_{(0,l)}(Z,t,x)]_{\text{(primary scalar)}}$
$\displaystyle=\xi^{M}\partial_{M}\mathcal{A}_{(0,l)}(Z,t,x),$ (A.19)

with $\xi$ the bulk Killing vector

$\displaystyle\xi=\xi^{M}\partial_{M}=\frac{1}{2R}(Z^{2}+x^{2}-R^{2}+t^{2})\partial_{t}+\frac{t}{R}(Z\partial_{Z}+x\partial_{x})$
(A.20)

of empty $AdS_{3}$

$\displaystyle ds^{2}=\frac{1}{Z^{2}}(dZ^{2}-dt^{2}+dx^{2}),$ (A.21)

that vanishes on the Ryu-Takayanagi surface Ryu:2006bv . We conclude that the
action of $H_{mod}$ on $\phi=\phi^{(0)}+\frac{1}{N}\phi^{(1)}$ is given by

$\displaystyle\frac{1}{2\pi
i}[H_{mod},(\phi^{(0)}+\frac{1}{N}\phi^{(1)})]=\xi^{M}\partial_{M}(\phi^{(0)}+\frac{1}{N}\phi^{(1)})+\frac{1}{N}\frac{\beta
Z^{2}}{2R}j_{-}\phi^{(0)},\qquad\beta=-\frac{iq}{k}.$ (A.22)

This is equation (2.8).

##  Appendix B $H_{mod}$ action on $\phi^{(1)}$ for general spin

We generalize here the discussion in Appendix A to the case of a conserved,
chiral spin $s$ current $j_{(s,0)}$ in the CFT, with $s>1$. This more general
set-up was considered in Kabat:2020nvj , where the $s>1$ generalization of
$\phi^{(1)}$ in (A.2)-(A.4) is given explicitly, and equation (A.5)
generalizes to

$\displaystyle\bar{L}_{1}\,\partial_{+}^{s+l}J^{(s)}_{(0,l)}=\text{(primary
scalar)}+(s+l)(\Delta+s+l-1)\partial_{+}^{s+l-1}J^{(s)}_{(0,l)}$ (B.1)

with $J^{(s)}_{(0,l)}$ now a primary spin ($s+l$) field with conformal
dimensions $(l+\frac{\Delta}{2}+s,\frac{\Delta}{2})$.

The derivation of (A.13) can straightforwardly be repeated for this more
general case, with the result

$\displaystyle\begin{split}&\frac{1}{2\pi
i}\left\langle[H_{mod},\phi^{(1)}]_{\text{non-(prim
scalar)}}\mathcal{O}j\right\rangle=\frac{(-1)^{\Delta}}{4R\Gamma(\Delta)}\frac{(x_{12}^{-})^{s-1}(x_{13}^{-})^{\Delta-s-1}}{(x_{23}^{-})^{s+\Delta-1}Z^{\Delta-2}}(-\mathcal{Y})^{s+\Delta-1}\\\
&\quad\times\sum_{l=0}^{\infty}a_{l}\alpha_{l}^{(s)}\frac{(s+l)\Gamma(l+s+\Delta)}{2s+2l+\Delta-1}(-\mathcal{Y})^{l}\,\,{{}_{2}F_{1}}\left(l+s+\Delta-1,l+2s,2s+2l+\Delta,\mathcal{Y}\right),\end{split}$
(B.2)

with $\mathcal{Y}$ defined in (A.14). Here $\alpha_{l}^{(s)}$ is the now
$s$-dependent OPE coefficient in (A.8), and the combination
$a_{l}\alpha_{l}^{(s)}$ has to be determined from the CFT regularity condition
in equation (78) of Kabat:2020nvj :

$\displaystyle\sum_{l=0}^{\infty}a_{l}\alpha_{l}^{(s)}(-1)^{l}\frac{\Gamma(l+s+\Delta)\Gamma(2l+2s+\Delta-1)}{2\Gamma(\Delta)\Gamma(l+\Delta)\Gamma(l+s)}\,\,{{}_{2}F_{1}}(l+\Delta+s,1-l-s,1+s,z)=0.$
(B.3)

We numerically check that

$\displaystyle\begin{split}&{{}_{2}F_{1}}(-1+\Delta+s,2-s,1+s,z)=\frac{2\Gamma(\Delta)\Gamma(\Delta-1)(\Delta+s-2)(2s-1)\Gamma(s)}{\Gamma(\Delta+2s-2)}\\\
&\quad\times\sum_{l=0}^{\infty}\left\\{\frac{(-1)^{l}(2s+2l+\Delta-1)\Gamma(s+\Delta+l-1)\Gamma(2s+l)\Gamma(2s+\Delta+l-2)}{2\Gamma(\Delta)\Gamma(s+\Delta-1)\Gamma(2s)\Gamma(l+2)\Gamma(l+\Delta)\Gamma(l+s+1)}\right.\\\
&\qquad\qquad\left.\phantom{\frac{1}{2}}\times{{}_{2}F_{1}}\left(l+\Delta+s,1-l-s,1+s,z\right)\right\\}.\end{split}$
(B.4)

Writing this identity as $F_{lhs}=F_{rhs}$, working with the derivative
operator
$\mathcal{L}=-z(1-z)\partial_{z}^{2}+[(2+\Delta)z-(1+s)]\partial_{z}+[2(\Delta-1)+s(3-\Delta)-s^{2}]$
gives $\mathcal{L}\,F_{lhs}=0$ and
$\mathcal{L}F_{rhs}=(1+l)(-2+\Delta+l+2s)F_{rhs}$, leading to

$\displaystyle\begin{split}&\sum_{l=0}^{\infty}\left\\{(-1)^{l}(2s+2l+\Delta-1)\frac{\Gamma(s+\Delta+l-1)\Gamma(2s+l)\Gamma(2s+\Delta-1+l)}{2\Gamma(\Delta)\Gamma(s+\Delta-1)\Gamma(2s)\Gamma(l+1)\Gamma(l+\Delta)\Gamma(l+s+1)}\right.\\\
&\qquad\left.\phantom{\frac{1}{2}}\times{{}_{2}F_{1}}(l+\Delta+s,1-l-s,1+s,z)\right\\}=0\end{split}$
(B.5)

(where we dropped an $l$-independent constant). It follows from comparison
with (B.3) that

$\displaystyle
a_{l}\alpha_{l}^{(s)}=\frac{2s+2l+\Delta-1}{s+l}\frac{c_{s,\Delta}\,\,\Gamma(s+\Delta-1+l)\Gamma(2s+l)\Gamma(2s+\Delta-1+l)}{\Gamma(l+s+\Delta)\Gamma(s+\Delta-1)\Gamma(2s)\Gamma(l+1)\Gamma(2s+\Delta-1+2l)}$
(B.6)

where $c_{s,\Delta}$ is an $l$-independent (but in general $\Delta$\- and
$s$-dependent) constant. Plugging this into (B.2), the sum on the second line
of (B.2) evaluates to 1 by the use of (A.15). We have thus found

$\displaystyle\frac{1}{2\pi
i}\left\langle[H_{mod},\phi^{(1)}]\mathcal{O}j\right\rangle$
$\displaystyle=\frac{c_{s,\Delta}}{4R\Gamma(\Delta)}Z^{s+1}\frac{(x_{12}^{-})^{s-1}}{(x_{13}^{-})^{2s}}\left(\frac{Z}{x_{12}^{-}x_{12}^{+}+Z^{2}}\right)^{s+\Delta-1}.$
(B.7)

On the other hand,

$\displaystyle\left\langle
j_{(s,0)}(x_{1}^{-})\partial_{x_{1}^{+}}^{s-1}\phi^{(0)}(Z,x_{1})\mathcal{O}(x_{2})j_{(s,0)}(x_{3}^{-})\right\rangle\sim\frac{1}{(x_{13}^{-})^{2s}}\left(\frac{x_{12}^{-}}{Z}\right)^{s-1}\left(\frac{Z}{x_{12}^{-}x_{12}^{+}+Z^{2}}\right)^{\Delta+s-1}$
(B.8)

from contracting $\langle
j_{(s,0)}(x_{1}^{-})j_{(s,0)}(x_{2}^{-})\rangle\sim\frac{1}{(x_{12}^{-})^{2s}}$
and
$\langle\phi^{(0)}(Z,x_{1})\mathcal{O}(x_{2})\rangle\sim\left(\frac{Z}{x_{12}^{-}x_{12}^{+}+Z^{2}}\right)^{\Delta}$,
and using

$\displaystyle\partial_{x_{1}^{+}}^{s-1}\left(\frac{Z}{x_{12}^{-}x_{12}^{+}+Z^{2}}\right)^{\Delta}\sim\left(\frac{x_{12}^{-}}{Z}\right)^{s-1}\left(\frac{Z}{x_{12}^{-}x_{12}^{+}+Z^{2}}\right)^{\Delta+s-1}.$
(B.9)

From comparison between (B.7) and (B.8), we conclude

$\displaystyle\frac{1}{2\pi i}[H_{mod},\phi^{(1)}]_{\text{non-(prim
scalar)}}\sim\frac{Z^{2s}}{2R}\,j_{(s,0)}\partial_{+}^{s-1}\phi^{(0)}.$ (B.10)

Finally, in the presence of a conserved spin-$s$ current in the CFT, the
action of $H_{mod}$ on the first order in $1/N$ CFT representation
$\phi=\phi^{(0)}+\frac{1}{N}\phi^{(1)}$ of a charged bulk scalar field is
given by

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi+\frac{1}{N}\frac{\gamma\,Z^{2n}}{2R}j_{(s,0)}\partial_{+}^{s-1}\phi$
(B.11)

with a constant $\gamma$ that we leave undetermined.

##  Appendix C Bulk calculation of the $H_{mod}$ action on gauge dressed
$\phi$

In this Appendix, we use the bulk perspective to derive the $H_{mod}$ action
on a charged scalar field $\phi$ in $AdS_{3}$ spacetime (A.21), with
coordinates $x^{M}=(Z,x^{+},x^{-})$, that is coupled to a $U(1)$ Chern-Simons
gauge field. The theory under consideration has the action

$\displaystyle
S=\int\sqrt{|g|}\left((D_{M}\phi)(D^{M}\phi)^{*}+m^{2}|\phi|^{2}-\frac{k}{2}\epsilon^{MNP}A_{M}\partial_{N}A_{P}\right)$
(C.1)

with covariant derivative $D_{M}=\partial_{M}+i\frac{q}{N}A_{M}$ (the charge
being defined by equation (3.1)). The field $\phi$ satisfies the bulk equation
of motion

$\displaystyle(\nabla^{2}-m^{2})\phi=-i\frac{q}{N}\left[(\nabla_{M}A^{M})\phi+2A_{M}\partial^{M}\phi\right]-\left(\frac{iq}{N}\right)^{2}A_{M}A^{M}\phi$
(C.2)

with
$\nabla^{2}\phi=\frac{1}{\sqrt{-g}}\partial_{M}\left(\sqrt{-g}\partial^{M}\phi\right)$
and
$\nabla_{M}A^{M}=\frac{1}{\sqrt{-g}}\partial_{M}\left(\sqrt{-g}A^{M}\right)$.

In the bulk, there is a gauge redundancy under $A_{M}\rightarrow
A_{M}+\partial_{M}\lambda$ and $\phi\rightarrow e^{-i\frac{q}{N}\lambda}\phi$.
A physical charged scalar field should therefore be ‘dressed’ into a gauge
invariant object. One simple choice of dressing is to attach a Wilson line
that extends to the boundary along the $Z$-direction Donnelly:2015hta

$\displaystyle\Phi=e^{i\frac{q}{N}\int_{0}^{Z}A_{Z}(Z^{\prime})dZ^{\prime}}\phi.$
(C.3)

The obtained physical field $\Phi$ equals the scalar $\phi$ in the
‘holographic gauge’ $A_{Z}=0$. We will choose to deal with the gauge
redundancy by fixing the gauge in this way. Once we have done that, the field
$\phi$ is physical and has a dual CFT representation.

The holographic gauge is not preserved under all $AdS_{3}$ isometries. The
generators of AdS isometries are given by

$\displaystyle\xi_{1}$
$\displaystyle=Zx^{-}\partial_{Z}+(x^{-})^{2}\partial_{-}-Z^{2}\partial_{+},\qquad\xi_{0}=\frac{1}{2}Z\partial_{Z}+x^{-}\partial_{-},\qquad\xi_{-1}=\partial_{-}$
(C.4) $\displaystyle\xi_{\bar{1}}$
$\displaystyle=Zx^{+}\partial_{Z}+(x^{+})^{2}\partial_{+}-Z^{2}\partial_{-},\qquad\xi_{\bar{0}}=\frac{1}{2}Z\partial_{Z}+x^{+}\partial_{+},\qquad\xi_{{-\bar{1}}}=\partial_{+}.$
(C.5)

That is, the Lie derivative of the $AdS_{3}$ metric (A.21) along these vectors
vanishes, $\mathcal{L}_{\xi_{0,\pm
1}}g_{MN}=\mathcal{L}_{\xi_{\bar{0},\pm\bar{1}}}g_{MN}=0$. This is dual to the
conformal invariance of the dual CFT vacuum state. The action of the conformal
generators on a CFT representation of a free bulk scalar field (A.1)
accordingly is given by (see e.g. Nakayama:2015mva ; Kabat:2018smf )

$\displaystyle[L_{1},\phi^{(0)}]$
$\displaystyle=\mathcal{L}_{\xi_{1}}\phi^{(0)},\quad[L_{0},\phi^{(0)}]=\mathcal{L}_{\xi_{0}}\phi^{(0)},\quad[L_{-1},\phi^{(0)}]=\mathcal{L}_{\xi_{-1}}\phi^{(0)}$
(C.6) $\displaystyle[\bar{L}_{1},\phi^{(0)}]$
$\displaystyle=\mathcal{L}_{\xi_{\bar{1}}}\phi^{(0)},\quad[\bar{L}_{0},\phi^{(0)}]=\mathcal{L}_{\xi_{\bar{0}}}\phi^{(0)},\quad[\bar{L}_{-1},\phi^{(0)}]=\mathcal{L}_{\xi_{-\bar{1}}}\phi^{(0)}.$
(C.7)

The Lie derivative of the gauge field is given, in general notation, by
$\mathcal{L}_{\chi}A_{M}=\chi^{N}\partial_{N}A_{M}+A_{N}\partial_{M}\chi^{N}$.
It follows that the special conformal transformations $L_{1}$ and
$\bar{L}_{1}$ (in dual conformal language) take you out of the holographic
gauge

$\displaystyle\left.\mathcal{L}_{\xi_{1}}A_{Z}\right|_{A_{Z}=0}=-2ZA_{+},\qquad\left.\mathcal{L}_{\xi_{\bar{1}}}A_{Z}\right|_{A_{Z}=0}=-2ZA_{-}.$
(C.8)

Since $A_{+}$ and $A_{-}$ are canonically conjugate to each other, only one of
them can fluctuate and the other is fixed as a boundary condition. We choose
to set $A_{+}=0$. It is then only the $\bar{L}_{1}$ transformation that
doesn’t preserve holographic gauge.

In $A_{Z}=0$ gauge, the other components of the gauge field transform under
AdS transformations as a combination of a Lie derivative and a compensating
gauge transformation $\lambda$, where $\lambda$ is determined by the condition
KL1204 ; Kabat:2018smf

$\displaystyle\left.\mathcal{L}_{\xi_{\bar{1}}}A_{Z}\right|_{A_{Z}=0}+\partial_{Z}\lambda=0.$
(C.9)

In the fixed gauge, the physical scalar field $\phi$ will then transform under
AdS transformations by a combination of a Lie derivative and the compensating
gauge transformation. For instance under the action of $\bar{L}_{1}$:

$\displaystyle[\bar{L}_{1},\phi]=\mathcal{L}_{\xi_{\bar{1}}}\phi-i\frac{q}{N}\bar{\lambda}\phi,$
(C.10)

where $\partial_{Z}\bar{\lambda}=2ZA_{-}$. In $AdS_{3}$, in particular, we
have that the conserved boundary current becomes the bulk gauge field
$A_{\mu}(Z,x^{+},x^{-})=\frac{1}{k}j_{\mu}(x^{+},x^{-})$, and thus

$\displaystyle\bar{\lambda}=\frac{1}{k}Z^{2}j_{-}.$ (C.11)

Next, let’s consider the action of $H_{mod}$, defined in (2.7) in terms of the
conformal generators. It will be given by the regular action
$\mathcal{L}_{\xi}\phi$, with the bulk Killing vector $\xi$ defined in (A.20),
supplemented with the above determined compensating gauge contribution

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\phi]=\mathcal{L}_{\xi}\phi+\frac{1}{2RN}(-iq\bar{\lambda})\phi$
(C.12)

or

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi-\frac{iq}{kN}\frac{Z^{2}}{2R}j_{-}\phi.$
(C.13)

This is equation (2.18).

We can include both currents $j_{-}$ and $j_{+}$ by considering a bulk action
with two Chern-Simons gauge fields $A_{M}$ and $\bar{A}_{M}$, with a relative
sign between them, see e.g. Jensen:2010em ; Keranen:2014ava . For the second
Chern-Simons gauge field one chooses $A_{-}=0$, resulting in a boundary
current $j_{+}$. The same reasoning as above applies and straightforwardly
gives rise to

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi-\frac{iq}{kN}\frac{Z^{2}}{2R}j_{-}\phi+\frac{iq}{kN}\frac{Z^{2}}{2R}j_{+}\phi\,,$
(C.14)

where the sign difference is due to that now it is the $L_{1}$ (and not
$\bar{L}_{1}$) transformation which gets an additional contribution. This is
just equation (2.26).

##  Appendix D Bulk calculation of the $H_{mod}$ action on gravitationally
dressed $\phi$

We use the bulk perspective to obtain the $H_{mod}$ action on a
gravitationally dressed scalar field $\phi$ in an asymptotically $AdS_{3}$
spacetime with coordinates $x^{M}=(Z,x^{+},x^{-})$. The field $\phi$ interacts
with gravity, corresponding to a scalar operator $\mathcal{O}$ coupling to a
conserved CFT stress tensor $T_{\mu\nu}$ in the dual theory. The CFT
perspective on this set-up is discussed in section 6.

If $\phi$ were non-interacting, the $H_{mod}$ action would be given by the Lie
derivative with respect to the bulk Killing vector $\xi$ defined in (A.20)
JLMS ; Kabat:2018smf . Turning on gravitational interactions, there will be
corrections to the $H_{mod}$ action. We derive in this Appendix the general,
all-order form of these corrections $\Lambda$ in

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi+\Lambda\,\phi.$ (D.1)

Diffeomorphism invariance in the bulk is invisible from the CFT perspective
and the gauge should be fixed. The gauge choice we make is that of
‘holographic gauge’ or Fefferman-Graham gauge, in which the asymptotically
$AdS_{3}$ bulk metric $ds^{2}=g_{MN}dx^{M}dx^{N}$ (with bulk indices
$M,N=Z,+,-$ and boundary indices $\mu,\nu=+,-$) takes the form

$\displaystyle
ds^{2}=\frac{\ell_{AdS}^{2}}{Z^{2}}(dZ^{2}+dx^{+}dx^{-})+h_{\mu\nu}dx^{\mu}dx^{\nu}.$
(D.2)

In this gauge, $h_{ZZ}=h_{Z\mu}=0$ and the remaining metric perturbations
$h_{\mu\nu}(Z,x^{+},x^{-})$ have a specific expansion in $Z$ deHaro:2000vlm ;
FG .

The holographic gauge is not preserved under the action of the modular
Hamiltonian, because it is not preserved under the action of special conformal
transformations $L_{1}$ and $\bar{L}_{1}$. That is, the corresponding
$AdS_{3}$ isometry generators $\xi_{1}$ and $\xi_{\bar{1}}$ (as defined in
(C.4) and (C.5)) take us out of holographic gauge:

$\displaystyle\mathcal{L}_{\xi_{1}}h_{Z+}=-2Zh_{++},\qquad\mathcal{L}_{\xi_{1}}h_{Z-}=-2Zh_{+-}$
(D.3)
$\displaystyle\mathcal{L}_{\xi_{\bar{1}}}h_{Z+}=-2Zh_{+-},\qquad\mathcal{L}_{\xi_{\bar{1}}}h_{Z-}=-2Zh_{--},$
(D.4)

with $\mathcal{L}_{\chi}h_{MN}=\nabla_{M}\chi_{N}+\nabla_{N}\chi_{M}$ the Lie
derivative along the vector $\chi$. The same will be true for the bulk Killing
vector $\xi$ associated with $H_{mod}$, defined in (A.20).

In holographic gauge, the isometry action

$\displaystyle\mathcal{L}_{\xi_{1}+\epsilon_{1}}h_{Z\mu}=0,\quad\mathcal{L}_{\xi_{\bar{1}}+\epsilon_{\bar{1}}}h_{Z\mu}=0$
(D.5)

therefore includes compensating gauge transformation contributions of the form
$\mathcal{L}_{\epsilon}$, with the vectors $\epsilon_{1}$ and
$\epsilon_{\bar{1}}$ determined by the condition (D.5) and that they vanish at
the boundary $Z\rightarrow 0$.

This determines the compensating gauge vector field $\epsilon_{1}$ to be of
the form

$\displaystyle\epsilon_{1}$
$\displaystyle=Zc_{1}\partial_{Z}+\int_{0}^{Z}dz\,\mathcal{F}\left(h_{\mu\nu}(z,x^{+},x^{-});c_{1}\right)\partial_{+}+\int_{0}^{Z}dz\,\tilde{\mathcal{F}}\left(h_{\mu\nu}(z,x^{+},x^{-});c_{1}\right)\partial_{-}$
(D.6)

with $c_{1}$ an arbitrary function $c_{1}(x^{+},x^{-})$, and $\mathcal{F}$ and
$\tilde{\mathcal{F}}$ given explicitly by

$\displaystyle\mathcal{F}$
$\displaystyle=\frac{-2z\left(h_{--}\left(4z^{4}h_{++}-2z^{2}\partial_{+}c_{1}\right)+\partial_{-}c_{1}\right)-4z^{3}h_{+-}\left(\partial_{-}c_{1}-1\right)+8z^{5}h_{+-}^{2}}{-4z^{4}h_{--}h_{++}+4z^{4}h_{+-}^{2}+4z^{2}h_{+-}+1}$
(D.7) $\displaystyle\tilde{\mathcal{F}}$
$\displaystyle=\frac{4z^{3}\left(\partial_{-}c_{1}+1\right)h_{++}-2z\partial_{+}c_{1}\left(2z^{2}h_{+-}+1\right)}{-4z^{4}h_{--}h_{++}+4z^{4}h_{+-}^{2}+4z^{2}h_{+-}+1}.$
(D.8)

In three bulk dimensions with a negative cosmological constant, the most
general solution of vacuum Einstein equation in the Fefferman-Graham
expansion, takes a truncated form, known as the Banados geometry
Banados:1998gg

$\displaystyle
ds^{2}=\frac{\ell_{AdS}^{2}}{Z^{2}}(dZ^{2}+dx^{+}dx^{-})+\frac{6}{c}\langle
T_{++}\rangle(dx^{+})^{2}+\frac{6}{c}\langle
T_{--}\rangle(dx^{-})^{2}+\frac{36}{c^{2}}\langle T_{++}\rangle\langle
T_{--}\rangle Z^{2}dx^{+}dx^{-}.$ (D.9)

Here we have already written the 3-dimensional Newton constant in terms of the
CFT central charge using $c=\frac{3\ell_{AdS}}{2G}$, and $\langle
T_{\pm\pm}\rangle$ are the Brown-York stress tensor components of
asymptotically $AdS_{3}$ gravity. The Banados metric gives the most general
solution of vacuum Einstein equations with negative cosmological constant. It
captures the full non-linear interactions of the metric perturbations
$h_{\mu\nu}$, but does not take into account backreaction of $\phi$ on the
metric. The field $\phi$ then gives rise to well-defined CFT correlators with
any number of CFT stress tensors $\langle\phi\,\mathcal{O}\,T_{--}\cdots
T_{++}\cdots\rangle$ (but no other operator insertions). Said otherwise, using
the Banados metric is the gravity equivalent of approximating the gauge field
$A_{\mu}$ by its non-backreacted (i.e. non-corrected with $1/N$ corrections)
form in section 2.1.

In the same way that the bulk gauge field $A_{\mu}$ of section 2.1 asymptotes
to the CFT current $j_{\mu}$, we have that the bulk metric perturbation
$h_{\mu\nu}$ asymptotes to the CFT stress tensor $T_{\mu\nu}$ at leading order
in $1/N$ KL1204 . We accordingly will interpret $T_{\mu\nu}$ in the Banados
geometry as a CFT operator, following e.g. the interpretation in Kaplan1708 .
The Banados metric allows us to read off the specific relation between the
metric perturbations $h_{\mu\nu}$ and the CFT stress tensor $T_{\mu\nu}$.

Using the Banados relation between $h_{\mu\nu}$ and $T_{\mu\nu}$, we find
$\epsilon_{1}$ and $\epsilon_{\bar{1}}$ are given by

$\displaystyle\epsilon_{1}$
$\displaystyle=\frac{Z^{4}\frac{6}{c}T_{++}}{1-Z^{4}\frac{36}{c^{2}}T_{--}T_{++}}\partial_{-}-\frac{Z^{6}\frac{36}{c^{2}}T_{--}T_{++}}{1-Z^{4}\frac{36}{c^{2}}T_{--}T_{++}}\partial_{+}$
(D.10) $\displaystyle\epsilon_{\bar{1}}$
$\displaystyle=\frac{Z^{4}\frac{6}{c}T_{--}}{1-Z^{4}\frac{36}{c^{2}}T_{--}T_{++}}\partial_{+}-\frac{Z^{6}\frac{36}{c^{2}}T_{--}T_{++}}{1-Z^{4}\frac{36}{c^{2}}T_{--}T_{++}}\partial_{-}\,.$
(D.11)

The function $c_{1}$ in (D.6) is fixed to zero by imposing that the Banados
form (D.9) of the geometry is preserved. The resulting ‘new’ $\xi_{1}$ in
holographic gauge, i.e. respecting $\mathcal{L}_{\xi_{1}}h_{Z\mu}=0$, is given
by the combination $\xi_{1}+\epsilon_{1}\rightarrow\xi_{1}$,

$\displaystyle\xi_{1}$
$\displaystyle=Zx^{-}\partial_{Z}+\left((x^{-})^{2}+\frac{Z^{4}\frac{6}{c}T_{++}}{1-Z^{4}\frac{36}{c^{2}}T_{--}T_{++}}\right)\partial_{-}-\frac{Z^{2}}{1-Z^{4}\frac{36}{c^{2}}T_{--}T_{++}}\partial_{+}.$
(D.12)

The expression for $\xi_{\bar{1}}$ is obtained from the one for $\xi_{1}$ by
changing all $\pm$ indices to $\mp$. Equation (D.12) was previously written
down in Kaplan1712  and in Roberts:2012aq , where a more general expression
for $\xi_{n}$ transformations ($n$ integer) can be found, corresponding to
conformal transformations generated by Virasoro generators $L_{n}$ on the
boundary.

Now let us go back to the action of the modular Hamiltonian on the
gravitationally interacting scalar field $\phi$. Under a diffeomorphism
$\chi$, the scalar field $\phi$ transforms to
$\phi+\mathcal{L}_{\chi}\phi=\phi+\chi^{M}\partial_{M}\phi$. It follows that
the compensating gauge transformations $\epsilon_{1}$ and $\epsilon_{\bar{1}}$
will give rise to a contribution $\epsilon^{M}\partial_{M}\phi$ to the modular
Hamiltonian action on $\phi$ through the $\frac{1}{2R}(\bar{L}_{1}-L_{1})$
term in (2.7):

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi+\epsilon^{M}\partial_{M}\phi.$ (D.13)

Written out,

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi+\epsilon^{+}\partial_{+}\phi+\epsilon^{-}\partial_{-}\phi$
(D.14)

with $\epsilon^{+}=\frac{1}{2R}(\epsilon_{\bar{1}}^{+}-\epsilon_{1}^{+})$ and
$\epsilon^{-}=\frac{1}{2R}(\epsilon_{\bar{1}}^{-}-\epsilon_{1}^{-})$ given by

$\displaystyle\epsilon^{+}=\frac{1}{2R}\frac{\frac{6}{c}Z^{4}T_{--}+\frac{36}{c^{2}}Z^{6}T_{--}T_{++}}{1-\frac{36}{c^{2}}Z^{4}T_{--}T_{++}},\qquad\epsilon^{-}=-\frac{1}{2R}\frac{\frac{6}{c}Z^{4}T_{++}+\frac{36}{c^{2}}Z^{6}T_{--}T_{++}}{1-\frac{36}{c^{2}}Z^{4}T_{--}T_{++}}.$
(D.15)

We have found that the corrections $\Lambda\phi$ in (D.1) take the form of
compensating gauge transformation contributions $\mathcal{L}_{\epsilon}\phi$,
which express the fact that physical bulk field operators are not local.

The result (D.14) is consistent with the first order in $1/N$ expression (6.1)
and the order by order expressions for $[H_{mod},\phi]$ obtained from a CFT
point of view in section 6. While we have used the Banados geometries to
deduce the action of $H_{mod}$ on $\phi$, one has to keep in mind that this is
$H_{mod}$ of the CFT vacuum state with zero expectation values for
$T_{\mu\nu}$. All the terms in the $\frac{1}{N}$ expansion involving
$\epsilon^{\pm}$ are not due to corrections to the modular Hamiltonian but due
to corrections to the bulk operator $\phi$.

We note also that one can of course rewrite the $H_{mod}$ action as

$\displaystyle\frac{1}{2\pi i}[H_{mod},\phi]=\xi_{H}^{M}\partial_{M}\phi$
(D.16)

in terms of a vector $\xi_{H}$ that is composed of $\xi$ in (A.20) and (D.15)
treated as some classical functions. This vector is an asymptotic Killing
vector of the Banados geometry (D.9): it preserves holographic gauge per
construction $\mathcal{L}_{\xi_{H}}g_{ZZ}=\mathcal{L}_{\xi_{H}}g_{Z\mu}=0$,
keeping the metric of the form (D.2). However, $\xi_{H}$ is not a Killing
vector of the full Banados geometry since $\mathcal{L}_{\xi_{H}}g_{\pm\pm}\neq
0$ and $\mathcal{L}_{\xi_{H}}g_{+-}\neq 0$. In particular,

$\displaystyle\mathcal{L}_{H}g_{--}=\frac{6}{c}[H_{mod},T_{--}],\quad\mathcal{L}_{H}g_{++}=\frac{6}{c}[H_{mod},T_{++}],\quad\mathcal{L}_{H}g_{+-}=\frac{36}{c^{2}}Z^{2}[H_{mod},T_{++}T_{--}]$
(D.17)

such that the Banados metric is transformed under $H_{mod}$ into a new Banados
metric with transformed stress tensors. The action of $H_{mod}$ on the stress
tensors in (D.17) follows immediately from (F.22)-(F.23).

##  Appendix E  $\phi^{(1)}$ comparison to charge distribution approach

In this Appendix we show how one gets the bulk field to first non-trivial
order in $\frac{1}{N}$ from the charge distribution condition Chen:2019hdv

$[j_{-}(x^{-}),(\phi^{(0)}+\frac{1}{N}\phi^{(1)})(Z,y^{-},y^{+})]=-\frac{q}{N}\delta(x^{-}-y^{-})\phi^{(0)}.$
(E.1)

The known CFT representation for the free bulk field $\phi^{(0)}$ can be
written either as a smearing integral expression (A.1) or as an expansion in
the holographic radius $Z$ Nakayama:2015mva ; CarneirodaCunha:2016zmi ;
Kaplan1708 :

$\phi^{(0)}=\sum_{n=0}^{\infty}\alpha_{n}Z^{2n}(\partial_{-}\partial_{+})^{n}{\cal
O}$ (E.2)

with

$\alpha_{n}=\frac{\Gamma(\Delta)}{2(\Delta-1)}Z^{\Delta}(-1)^{n}\frac{1}{\Gamma(n+1)\Gamma(\Delta+n)}.$
(E.3)

Further, we know the CFT commutation relations

$[j_{-}(x^{-}),{\cal O}(y^{-},y^{+})]=-\frac{q}{N}\delta(x^{-}-y^{-}){\cal O}\
,\ \ [j_{-}(x^{-}),j_{-}(y^{-})]=ik\partial_{x^{-}}\delta(x^{-}-y^{-}).$ (E.4)

We start with

$[j_{-},\phi^{(0)}]=-\frac{q}{N}\delta(x^{-}-y^{-})\phi^{(0)}-\frac{q}{N}\sum_{n=0}^{\infty}\alpha_{n}Z^{2n}\sum_{m=1}^{n}\frac{\Gamma(n+1)}{\Gamma(m+1)\Gamma(n-m+1)}\partial^{m}_{y^{-}}\delta(x^{-}-y^{-})\partial^{n-m}_{-}\partial_{+}^{n}{\cal
O}$ (E.5)

and see that we need to cancel the second term on the right hand side, which
contains derivatives of delta functions of increasing order. In fact from this
point of view, the purpose of the correction term in the bulk operator is to
cancel unwanted parts of the charge distribution of $\phi^{(0)}$.

The most general objects that we can put into $\phi^{(1)}$ which are boundary
scalars and have the right scaling dimensions under dilatations are of the
form

$\phi^{(1)}=\sum_{l=0}^{\infty}Z^{2l+2+\Delta}\sum_{r=0}^{l}a_{lr}\partial_{-}^{r}j_{-}\partial_{-}^{l-r}\partial_{+}^{l+1}{\cal
O}.$ (E.6)

Now to order $\frac{1}{N}$ the contribution of $[j_{-},\frac{1}{N}\phi^{(1)}]$
comes from the $[j_{-},j_{-}]$ commutator888 The $[j_{-},{\cal O}]$ commutator
is down by $\frac{1}{N}$ compared to the current-current commutator., thus we
find to this order

$[j_{-},\frac{1}{N}\phi^{(1)}]=-\frac{ik}{N}\sum_{l=0}^{\infty}Z^{2l+2}\sum_{r=0}^{l}a_{lr}\partial^{r+1}_{y^{-}}\delta(x^{-}-y^{-})\partial^{l-r}_{-}\partial_{+}^{l+1}{\cal
O}+O(\frac{1}{N^{2}}).$ (E.7)

To get (E.1) one needs

$a_{lr}=i\frac{q}{k}\frac{\Gamma(\Delta)}{2(\Delta-1)}(-1)^{l+1}\frac{1}{\Gamma(\Delta+l+1)\Gamma(r+2)\Gamma(l-r+1)}.$
(E.8)

This agrees with computations in Chen:2019hdv 999 In particular, their
definition of charge $q$ maps to $-\frac{iq}{k}$ in our notation, and our
normalization of $\phi^{(0)}$ in (A.1) differs by a factor $2(\Delta-1)$ from
the normalization in Chen:2019hdv . .

Note that since
$[j_{-}(x^{-}),j_{-}(y^{-})]=ik\partial_{x^{-}}\delta(x^{-}-y^{-})$ one cannot
cancel the first term in equation (E.5), which does not have any derivative on
the delta function. This means that one cannot construct a bulk operator
$\phi$ such that $[j_{-},\phi]=0$. This is the source of the bulk Gauss law in
this formulation.

Let us also show that the obtained double sum expression for $\phi^{(1)}$ in
(E.6) matches the smearing integral expression for $\phi^{(1)}$ in
(A.2)-(A.3), as obtained in Kabat:2020nvj . This proceeds in the same way as
showing the equivalence between expressions (A.1) and (E.2) for $\phi^{(0)}$,
which involves expanding the operator
$\mathcal{O}(x^{+}+iy^{\prime}+t^{\prime},x^{-}+iy^{\prime}-t^{\prime})$ in
the integrand around $\mathcal{O}(x^{+},x^{-})$ and subsequently working out
the integral over $y^{\prime}$ and $t^{\prime}$ explicitly (making use of
$\int_{0}^{1}dx\,(1-x)^{p}x^{q}=\frac{\Gamma(1+q)\Gamma(1+p)}{\Gamma(2+q+p)}$).
This is detailed for example in CarneirodaCunha:2016zmi .

The integral expression for $\phi^{(1)}$ takes the form (A.2)-(A.3),

$\displaystyle\phi^{(1)}=\sum_{l=0}^{\infty}\sum_{n=0}^{l}a_{l}d_{ln}\,\int_{t^{\prime
2}+y^{\prime
2}<Z^{2}}dt^{\prime}dy^{\prime}\,\frac{1}{2\pi}\left(\frac{Z^{2}-t^{\prime
2}-y^{\prime
2}}{Z}\right)^{\Delta+2l}\,\partial_{-}^{n}j_{-}\partial_{-}^{l-n}\partial_{+}^{l+1}\mathcal{O},$
(E.9)

where the operator in the integrand is evaluated at the location
$(x^{+}+t^{\prime}+iy^{\prime},x^{-}-t^{\prime}+iy^{\prime})$, and $d_{ln}$ is
defined in (2.6). The coefficients $a_{l}$ can be determined either by solving
the bulk equation of motion or by imposing the behavior under the modular
Hamiltonian action given in (2.18), leading to

$\displaystyle a_{l}$
$\displaystyle=\frac{iq}{k}\frac{\Gamma(\Delta)}{\Gamma(l+2)\Gamma(\Delta+l+1)}\frac{\sum_{k=0}^{l}d_{lk}(2)_{k}(\Delta)_{l-k}(\Delta+l-k)_{l}}{\sum_{k=0}^{l}\sum_{p=0}^{l}d_{lk}d_{lp}(2)_{p}(2+p)_{k}(\Delta)_{l-p}(\Delta+l-p)_{l-k}}$
(E.10)

written in terms of Pochhammer symbols.

We can first rewrite $\phi^{(1)}$ in terms of two infinite sums by applying
the sum switching identity
$\sum_{l=0}^{\infty}\sum_{n=0}^{l}=\sum_{n=0}^{\infty}\sum_{l=n}^{\infty}$ and
then setting $l=m+n$:

$\displaystyle\phi^{(1)}=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}a_{m+n}d_{(m+n)n}\,\int_{t^{\prime
2}+y^{\prime
2}<Z^{2}}dt^{\prime}dy^{\prime}\,\frac{1}{2\pi}\left(\frac{Z^{2}-t^{\prime
2}-y^{\prime
2}}{Z}\right)^{\Delta+2(m+n)}\,\partial_{-}^{n}j_{-}\partial_{-}^{m}\partial_{+}^{m+n+1}\mathcal{O}.$
(E.11)

Next, we expand the operator in the integrand around the location
$(x^{+},x^{-})$ and explicitly evaluate the integral over $t^{\prime}$ and
$y^{\prime}$. In the expansion, derivatives on the composite operator are
worked out using
$\partial^{i}(f\,g)=\sum_{j=0}^{i}\frac{\Gamma(i+1)}{\Gamma(j+1)\Gamma(i-j+1)}\partial^{i-j}f\partial^{j}g$.
Redefining indices $N$ and $M$ that count the powers of $\partial_{-}$
derivatives in the composite operator, the integral expression for
$\phi^{(1)}$ is rewritten as a four-sum expression:

$\displaystyle\phi^{(1)}=\sum_{N=0}^{\infty}\sum_{M=0}^{\infty}c_{NM}Z^{2N+2M+2+\Delta}\partial_{-}^{N}j_{-}\partial_{-}^{M}\partial_{+}^{N+M+1}\mathcal{O}(x^{+},x^{-})$
(E.12)

with

$\displaystyle
c_{NM}=\sum_{n=0}^{N}\sum_{m=0}^{M}a_{m+n}d_{(m+n)n}\frac{\Gamma(\Delta+2n+2m+2)(-1)^{M+N+1}}{\Gamma(N+M+n+m+2+\Delta)(M-m)!(N-n)!}.$
(E.13)

It can be checked numerically that $c_{r,l-r}=a_{lr}$ so that we find
agreement with expression (E.6) for $\phi^{(1)}$ after relabeling the indices
$N\rightarrow r$ and $M\rightarrow l-r$. This concludes the demonstration of
equivalence of the smearing integral and double sum expressions for
$\phi^{(1)}$.

We note that a straightforward generalization of the above procedure to the
spin $2$ case leads to agreement between the first order gravitationally
interacting field $\phi^{(1)}$ as a smearing integral Kabat:2020nvj  and as a
double sum expression Kaplan1708 .

##  Appendix F Summary of used operator actions

Scalar CFT operator $\mathcal{O}$ with conformal dimensions $(h_{-},h_{+})$
transforms as

$\displaystyle[\bar{L}_{1},\mathcal{O}]$
$\displaystyle=\left((x^{+})^{2}\partial_{+}+2x^{+}h_{+}\right)\mathcal{O},\qquad[\bar{L}_{0},\mathcal{O}]=x^{+}\partial_{+}\mathcal{O}+h_{+}\mathcal{O},\qquad[\bar{L}_{-1},\mathcal{O}]=\partial_{+}\mathcal{O}$
(F.1) $\displaystyle[L_{1},\mathcal{O}]$
$\displaystyle=\left((x^{-})^{2}\partial_{-}+2x^{-}h_{-}\right)\mathcal{O},\qquad[L_{0},\mathcal{O}]=x^{-}\partial_{-}\mathcal{O}+h_{-}\mathcal{O},\qquad[L_{-1},\mathcal{O}]=\partial_{-}\mathcal{O}.$
(F.2)

###  F.1 Chern-Simons gauge field

Scalar bulk field $\phi$ and CFT current $j$ transform as

$\displaystyle[\bar{L}_{1},\phi]$
$\displaystyle=(Zx^{+}\partial_{Z}+(x^{+})^{2}\partial_{+}-Z^{2}\partial_{-})\phi+\frac{\beta}{N}Z^{2}j_{-}\phi$
(F.3) $\displaystyle[L_{1},\phi]$
$\displaystyle=(Zx^{-}\partial_{Z}+(x^{-})^{2}\partial_{-}-Z^{2}\partial_{+})\phi+\frac{\beta}{N}Z^{2}j_{+}\phi$
(F.4) $\displaystyle[\bar{L}_{0},\phi]$
$\displaystyle=(\frac{1}{2}Z\partial_{Z}+x^{+}\partial_{+})\phi,\qquad[\bar{L}_{-1},\phi]=\partial_{+}\phi$
(F.5) $\displaystyle[L_{0},\phi]$
$\displaystyle=(\frac{1}{2}Z\partial_{Z}+x^{-}\partial_{-})\phi,\qquad[L_{-1},\phi]=\partial_{-}\phi$
(F.6)

$\displaystyle\frac{1}{2\pi
i}[H_{mod},\phi]=\xi^{M}\partial_{M}\phi+\frac{\beta}{N}\frac{Z^{2}}{2R}\,j_{-}\phi-\frac{\beta}{N}\frac{Z^{2}}{2R}\,j_{+}\phi$
(F.7) $\displaystyle C_{2}\circ\phi$
$\displaystyle=\nabla^{2}\phi-\frac{2\beta
Z^{2}}{N}j_{+}\partial_{-}\phi-\frac{2\beta Z^{2}}{N}j_{-}\partial_{+}\phi$
(F.8) $\displaystyle[\bar{L}_{1},j_{+}]$
$\displaystyle=\left((x^{+})^{2}\partial_{+}+2x^{+}\right)j_{+},\qquad[\bar{L}_{0},j_{+}]=x^{+}\partial_{+}j_{+}+j_{+},\qquad[\bar{L}_{-1},j_{+}]=\partial_{+}j_{+}$
(F.9) $\displaystyle[L_{1},j_{+}]$
$\displaystyle=0,\qquad[L_{0},j_{+}]=0,\qquad[L_{-1},j_{+}]=0$ (F.10)
$\displaystyle[L_{1},j_{-}]$
$\displaystyle=\left((x^{-})^{2}\partial_{-}+2x^{-}\right)j_{-},\qquad[L_{0},j_{-}]=x^{-}\partial_{-}j_{-}+j_{-},\qquad[L_{-1},j_{-}]=\partial_{-}j_{-},$
(F.11) $\displaystyle[\bar{L}_{1},j_{-}]$
$\displaystyle=0,\qquad[\bar{L}_{0},j_{-}]=0,\qquad[\bar{L}_{-1},j_{-}]=0$
(F.12)

###  F.2 Gravity

Scalar bulk field $\phi$ and CFT stress tensor $T_{\mu\nu}$ transform as

$\displaystyle[\bar{L}_{1},\phi]$
$\displaystyle=(Zx^{+}\partial_{Z}+(x^{+})^{2}\partial_{+}-Z^{2}\partial_{-})\phi+[\bar{L}_{1},\phi]_{\text{c.g.}}$
(F.13) $\displaystyle[L_{1},\phi]$
$\displaystyle=(Zx^{-}\partial_{Z}+(x^{-})^{2}\partial_{-}-Z^{2}\partial_{+})\phi+[L_{1},\phi]_{\text{c.g.}}$
(F.14) $\displaystyle[\bar{L}_{0},\phi]$
$\displaystyle=(\frac{1}{2}Z\partial_{Z}+x^{+}\partial_{+})\phi,\qquad[\bar{L}_{-1},\phi]=\partial_{+}\phi$
(F.15) $\displaystyle[L_{0},\phi]$
$\displaystyle=(\frac{1}{2}Z\partial_{Z}+x^{-}\partial_{-})\phi,\qquad[L_{-1},\phi]=\partial_{-}\phi$
(F.16)

with compensating gauge transformation contributions

$\displaystyle[\bar{L}_{1},\phi]_{\text{c.g.}}$
$\displaystyle=\lambda_{-}\partial_{+}\phi-\omega\,\partial_{-}\phi$ (F.17)
$\displaystyle[L_{1},\phi]_{\text{c.g.}}$
$\displaystyle=\lambda_{+}\partial_{-}\phi-\omega\,\partial_{+}\phi$ (F.18)
$\displaystyle\lambda_{-}$
$\displaystyle=\frac{Z^{4}T_{--}}{\mathcal{N}},\qquad\lambda_{+}=\frac{Z^{4}T_{++}}{\mathcal{N}},\qquad\omega=\frac{Z^{6}T_{--}T_{++}}{\mathcal{N}},\qquad\mathcal{N}=1-Z^{4}T_{--}T_{++}$
(F.19)

$\displaystyle\frac{1}{2\pi i}[H_{mod},\phi]$
$\displaystyle=\xi^{M}\partial_{M}\phi+\frac{1}{2R}[\bar{L}_{1}-L_{1},\phi]_{\text{c.g.}}=\xi^{M}\partial_{M}\phi+\frac{\lambda_{-}+\omega}{2R}\partial_{+}\phi-\frac{\lambda_{+}+\omega}{2R}\partial_{-}\phi$
(F.20)

$\displaystyle
C_{2}\circ\phi=\nabla^{2}\phi-2\partial_{+}\left(\lambda_{-}\partial_{+}\phi-\omega\,\partial_{-}\phi\right)-2\partial_{-}\left(\lambda_{+}\partial_{-}\phi-\omega\,\partial_{+}\phi\right)$
(F.21)

$\displaystyle[\bar{L}_{1},T_{++}]$
$\displaystyle=\left((x^{+})^{2}\partial_{+}+4x^{+}\right)T_{++},\qquad[\bar{L}_{0},T_{++}]=x^{+}\partial_{+}T_{++}+2T_{++},\qquad[\bar{L}_{-1},T_{++}]=\partial_{+}T_{++}$
$\displaystyle[L_{1},T_{++}]$
$\displaystyle=0,\qquad[L_{0},T_{++}]=0,\qquad[L_{-1},T_{++}]=0$ (F.22)
$\displaystyle[L_{1},T_{--}]$
$\displaystyle=\left((x^{-})^{2}\partial_{-}+4x^{-}\right)T_{--},\qquad[L_{0},T_{--}]=x^{-}\partial_{-}T_{--}+2T_{--},\qquad[L_{-1},T_{--}]=\partial_{-}T_{--},$
$\displaystyle[\bar{L}_{1},T_{--}]$
$\displaystyle=0,\qquad[\bar{L}_{0},T_{--}]=0,\qquad[\bar{L}_{-1},T_{--}]=0$
(F.23)

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