Virasoro algebras, kinematic space and the spectrum of modular Hamiltonians in
CFT2

Suchetan Das1,2, Bobby Ezhuthachan2, Somnath Porey2, Baishali Roy2

1Department of Physics,

Indian Institute of Technology Kanpur,

Kanpur 208016, India.

2Ramakrishna Mission Vivekananda Educational and Research Institute,

Belur Math,

Howrah-711202, West Bengal, India.

suchetan[at]iitk.ac.in, bobby.ezhuthachan[at]rkmvu.ac.in,

somnathhimu00[at]gm.rkmvu.ac.in, baishali.roy025[at]gm.rkmvu.ac.in

We construct an infinite class of eigenmodes with integer eigenvalues for the
Vacuum Modular Hamiltonian of a single interval $N$ in 2d CFT and study some
of its interesting properties, which includes its action on OPE blocks as well
as its bulk duals. Our analysis suggests that these eigenmodes, like the OPE
blocks have a natural description on the so called kinematic space of CFT2 and
in particular realize the Virasoro algebra of the theory on this kinematic
space. Taken together, our results hints at the possibility of an effective
description of the CFT2 in the kinematic space language.

###### Contents

  1. 1 Introduction
  2. 2 Modular Hamiltonian in 2D CFT and its spectrum
    1. 2.1 OPE Blocks
    2. 2.2 A new class of modular eigenmodes and its properties
    3. 2.3 Action on the OPE blocks
    4. 2.4 MVA and the Kinematic space
  3. 3 The global subalgebra of the MVA
    1. 3.1 Symmetries of the CFT2 causal diamonds
    2. 3.2 g-MVA and modular inclusions
  4. 4 Pulling the $\mathbb{L}_{n}$ into the bulk
  5. 5 Discussion
  6. A Modular inclusion in CFT2 and finite dimensional system
    1. A.1 Modular inclusion
    2. A.2 Modular inclusion in vacuum CFT2
    3. A.3 Modular inclusion in finite dimensional Hilbert space
  7. B Commutation relation of modular generators and Virasoro algebra

##  1 Introduction

Research over the past several years has made it abundantly clear that Quantum
information/entropy related ideas play a crucial role in developing a deeper
understanding of Quantum Field Theory and Quantum Gravity. The algebraic
formulation of QFT (AQFT) in terms of algebra of observables associated to
causal domains of spatial subregions [1],[2], seems to be particularly well
suited for such entropic studies. The many successes of this approach include
formulating a precise version of various entropy bounds in QFT [3]-[5],
developing a deeper understanding of RG flows in terms of relative entropy of
states, [6]-[10], proofs of various null energy conditions in QFT [11]-[14],
developing a more precise understanding of bulk Reconstruction [15]-[30] among
others.

A key role in most of these studies is played by the (total) modular
hamiltonian [31]111Total modular hamiltonian is defined as the difference of
modular hamiltonians of a given subregion and it’s complement. In the rest of
the note, we refer modular hamiltonian as the total modular hamiltonian.. In
the AQFT formulation, the modular hamiltonian operator $K^{\psi}_{\Sigma}$,
for a particular state $|\psi\rangle$ generates an automorphism of the algebra
of the operators localized in the causal diamond $\mathcal{D}(\Sigma)$
associated with the spatial subregion $\Sigma$. Under this automorphism,
operators localized within $\mathcal{D}(\Sigma)$ transform into each other,
thus generating a flow called the modular flow 222 Under this flow, an
operator $\mathcal{O}\rightarrow\mathcal{O}(s)$, where $\mathcal{O}(s)\equiv
e^{isK}\mathcal{O}e^{-isK}$. Both
$\mathcal{O}\;\textrm{and}\;\mathcal{O}_{s}\;\textrm{have support
within}\;\mathcal{D}_{\Sigma}$. In applications to holography, the importance
of the modular hamiltonian operator comes from its identification, at leading
order in the inverse bulk newton’s constant($\frac{1}{G_{N}}$), with the
corresponding bulk modular hamiltonian operator, where the corresponding bulk
region is the bulk causal diamond associated with the region bounded by the RT
surface and $\Sigma$ [15]. These modular flows play an important role in the
entanglement wedge reconstruction program333 Recently, a different, but
related, notion of the connes co-cycle flows also have been discussed in the
context of extracting bulk information from the entanglement wedge region
which is casually disconnected from the boundary [32],[33]. It has also been
argued that the emergence of a semiclassical bulk spacetime might itself be
understood from the algebra of modular hamiltonians of all subregions in the
boundary QFT [30].

Given its relevance, particularly in the context of bulk reconstruction
program alluded to above, it would be a useful endeavor to study the modular
hamiltonian operator in detail both in general QFTs as well more specifically
in simple but concrete examples. One way to characterize these operators would
be through the spectrum of its eigenstates. It is reasonable to expect that
this spectrum would encode the entanglement content of the QFT. The modular
hamiltonian $K^{\psi}_{\Sigma}$ for a state $|\psi\rangle$ and a spatial
region $\Sigma$ annihilates the state, ie $K^{\psi}_{\Sigma}|\psi\rangle=0$.
One may then construct its eigenstates by acting on $|\psi\rangle$ with a
special class of operators ($\mathcal{O}_{\omega}$) which has the following
commutation relation with the modular hamiltonian
$[K,\mathcal{O}_{\omega}]=\omega\mathcal{O}_{\omega}$. These are referred to
as the modular eigenmodes. It’s easy to see that the fourier transform of the
modular evolved operators (ie: $\int dse^{is\omega}\mathcal{O}_{\omega}$) are
modular eigenmodes. These eigenmodes, in particular the zero modes play a
crucial role in reconstruction of bulk fields inside the entanglement wedge
[17],[19]. The zero modes, which commute with the modular hamiltonian may be
thought of as local symmetries of the corresponding state, in the sense that
correlation function of operators inside the region $\mathcal{D}(\Sigma)$,
would be invariant under transformations (of the operators) generated by the
zero modes. These are local because for the same given state, but for a
different region, the modular hamiltonian and therefore the zero modes would
be different. It has been argued that in the bulk these local symmetries
generated by the zero modes corresponding to large diffeomorphisms which are
not trivial on the RT surface [22]. Thus the modular eigenmodes seem to have a
very important bearing on the emergence of bulk geometry itself.

Given the above motivations, a detailed study of the modular eigenmodes in
these theories would be interesting. While for generic states and regions the
modular hamiltonian as well its eigenmodes are nonlocal operators, there are a
few examples, where they take a simple form as an integral of local fields.
The simplest example of which is the case of the single interval in the vacuum
state of a $\textrm{CFT}_{2}$. In [37], two of us had shown that OPE blocks of
primary fields of different spins are nonzero modular eigenmodes of the
modular hamiltonian of the single interval, in the vacuum of CFT2, where the
endpoints of the interval corresponds to the location of the two primary
fields whose expansion define the OPE block. This generalizes known results in
the literature that scalar OPE blocks are modular zero eigenmodes [20],[17].

In this note, we continue with the study initiated in [37], of the eigenmodes
of the vacuum modular hamiltonian for a single interval (labelled as $N$) in
2D CFT. We find a new infinite class of modular eigenmodes with integer
eigenvalues and discuss some of their interesting properties. The key point we
want to make here is that like the OPE blocks, these new eigenmodes we
construct here have a natural description on the so-called kinematic
space(k-space)[34], which is essentially the space of causal diamonds in CFT2.
In particular, they realize the virasoro algebra of the CFT2 on this k-space.
As evidence of this fact we show that OPE blocks, which are local fields in
the k-space description, transform as modes of a primary field under this
‘k-space virasoro algebra’, which we refer to as the modular virasoro
algebra(MVA) in the bulk of the text. Moreover, as we show, a subset of the
new modular eigenmodes, which generate the global subalgebra of the MVA
representation can be directly identified with the modular hamiltonians of
subregions of $N$. We believe that these observations, taken together, hint at
the possibility of an equivalent effective description of the CFT2 in the
k-space language, a detailed study of which we leave for future work.

This draft is organized as follows. In the next section, after presenting a
brief summary of the known examples of modular eigenmodes- the OPE blocks, and
their bulk duals, we present the new class of eigenmodes and discuss its
interesting properties. Specifically, we show that these eigenmodes together
with the modular hamiltonian satisfy the virasoro algebra. The details of this
calculation are presented in appendix B. We also compute the commutator of
these new class of modular eigenmodes with the OPE blocks, and show that the
result is same as that of the usual (local) virasoro generators with the modes
of a primary field in CFT2. This fact suggests that the OPE blocks transform
as modes of a primary field under conformal transformations generated by the
MVA. Since the OPE blocks can be described naturally as fields living on the
so-called kinematic space(k-space), this also suggest that the new eigenmodes
have a natural action on the k-space. We explore the kinematic aspects of this
question in subsection 2.4.

In section 3, we focus on the global subalgebra of the MVA. Interestingly, we
show that it is isomorphic to the algebra of the modular hamiltonians of $N$
as well as its subregions $N^{\prime}$ and $N^{\prime\prime}$ and that they
implement the so-called modular inclusion within the lightcone of $N$. For
completeness we review the definition of the modular inclusion and, as an
example, discuss the modular inclusion for finite dimensional Hilbert space in
appendix A.

In section 4, we discuss the bulk dual of our construction. In particular we
show the emergence of the RT geodesic very naturally from our constructions.
We conclude with a summary of our results as well a discussion on some of the
questions and directions opened up in the light of these results, in section
5.

Note Added: While we were in the final stages of preparing this article, [58]
was posted on the arxiv, in which the authors construct Virasoro algebra from
generators constructed out of light-ray operators. Although the expression for
those operators are similar to what we refer here as ‘modular Virasoro
generators’, the context and motivation of the approaches seem to us to be
different.

##  2 Modular Hamiltonian in 2D CFT and its spectrum

The Modular Hamiltonian of a single interval with endpoints ($z_{2}$, $z_{3}$)
on a constant time slice ($t=0$) in the vacuum of 2D CFT is given by the
following integral expression:

$K=\int^{\infty}_{-\infty}d\zeta\frac{(z_{2}-\zeta)(\zeta-
z_{3})}{z_{2}-z_{3}}T_{\zeta\zeta}(\zeta)+\int^{\infty}_{-\infty}d\bar{\zeta}\frac{(z_{2}-\bar{\zeta})(\bar{\zeta}-z_{3})}{z_{2}-z_{3}}T_{\bar{\zeta}\bar{\zeta}}(\bar{\zeta})$
(2.1)

Modular eigenmodes are operators which satisfy the following commutation
relation with the modular hamiltonian $[K,\mathcal{O}]=\lambda\mathcal{O}$. In
[17],[37], it was shown that global OPE blocks in CFT2, are eigen-modes of
vacuum modular Hamiltonian. We therefore begin this section, with a brief
review of the OPE blocks in $\textrm{CFT}_{2}$.

###  2.1 OPE Blocks

In CFT, a global OPE block $B^{ij}_{k}$ is defined as the contribution of a
conformal family (ie a given primary field $\mathcal{O}_{k}$ of dimension
$h_{k}$, $\bar{h}_{k}$ and all it’s global descendants) to the OPE of two
primary operators ($\mathcal{O}_{i}$, $\mathcal{O}_{j}$) of dimensions
($h_{i}$, $\bar{h}_{i}$) and ($h_{j}$, $\bar{h}_{j}$) respectively [34].
Mathematically,

$\displaystyle\mathcal{O}_{i}(z_{1},\bar{z}_{1})\mathcal{O}_{j}(z_{2},\bar{z}_{2})=z_{12}^{-(h_{i}+h_{j})}\bar{z}_{12}^{-(\bar{h}_{i}+\bar{h}_{j})}\sum_{k}C_{ijk}B_{k}^{ij}(z_{1},\bar{z}_{1};z_{2},\bar{z}_{2})$
(2.2)

Here, $C_{ijk}$ is the OPE coefficient, which is dynamical input of the
theory. The above equation tells us how the OPE block transforms under global
conformal transformations and this is enough to fix the form of the OPE
blocks. Indeed, $B^{ij}_{k}$ has an integral expression which can be derived
[34], [38] using the shadow operator formalism [39], [40] and takes the
following form,

$\displaystyle B^{ij}_{k}(z_{1},\bar{z}_{1};z_{2},\bar{z}_{2})=$
$\displaystyle\int_{z_{1}}^{z_{2}}d\zeta\int_{\bar{z}_{1}}^{\bar{z}_{2}}d\bar{\zeta}\left(\frac{(\zeta-
z_{1})(z_{2}-\zeta)}{z_{2}-z_{1}}\right)^{h_{k}-1}\left(\frac{z_{2}-\zeta}{\zeta-
z_{1}}\right)^{h_{ij}}\times$
$\displaystyle\left(\frac{(\bar{\zeta}-\bar{z}_{1})(\bar{z}_{2}-\bar{\zeta})}{\bar{z}_{2}-\bar{z}_{1}}\right)^{\bar{h}_{k}-1}\left(\frac{\bar{z}_{2}-\bar{\zeta}}{\bar{\zeta}-\bar{z}_{1}}\right)^{\bar{h}_{ij}}\mathcal{O}_{k}(\zeta,\bar{\zeta})$
(2.3)

One can now show444See appendix A of [37], for the details of the proof.,
using the OPE of $T$, $\bar{T}$ with the primary field $\mathcal{O}$, that
these OPE blocks are indeed eigenmodes of $K$, with eigenvalue proportional to
the spin difference ($l_{ij}$)of the two operators.

$\Big{[}K,B^{ij}_{k}\Big{]}=2\pi il_{ij}B^{ij}_{k}$ (2.4)

The scalar zero-modes have been shown to be dual to the so-called geodesic
operators [34], which are essentially smeared geodesic integrals of the
appropriate bulk dual field of $\mathcal{O}_{k}$. The geodesic endpoints being
the location of the two primary fields, whose OPE defines the specific OPE
block in question.

$B^{ij}_{k}=\int_{\lambda}ds\;e^{-s\Delta_{ij}}\phi(x(s),z(s),t(s))$ (2.5)

Here $\phi$ is the dual scalar field to $\mathcal{O}_{k}$. ($x,\;t$) are the
boundary coordinates while $z$ is the bulk coordinate, and the integral is
over a geodesic with end points on the boundary. The smearing function is
$e^{-s\Delta_{ij}}$, with $\Delta_{ij}=\Delta_{i}-\Delta_{j}$ being the
difference in scaling dimensions of the two operators. This can be generalized
to non zero scalar modes [37], where now the integral is over a Lorentzian
cylinder. The cylindrical surface is generated by $K$ and $P_{D}$ which
generate boosts in the plane normal to the geodesic and translations along the
geodesic respectively.

$B^{ij}_{k}=c_{k}\int_{cylinder}d\tilde{t}ds\;e^{-s(\theta)\Delta_{ij}}e^{-\tilde{t}(\rho,\theta)l_{ij}}\phi(x(\rho,\theta),z(\rho,\theta),t(\rho,\theta))$
(2.6)

Here $l_{ij}$ is the spin difference between the two operators, $\tilde{t}$
and $s$ labels the coordinates on the cylinder and the $c_{k}$ is a
normalization constant which can be fixed by the appropriate boundary
condition. See [37] for the details of the derivation. In the next section, we
introduce the new class of eigenmodes which are smeared integrals of $T$ and
$\bar{T}$ and discuss their bulk duals.

###  2.2 A new class of modular eigenmodes and its properties

We now present a new class of integrated operators, which are all eigenmodes
of the modular hamiltonian. Unlike the OPE blocks, these exist in any 2d CFT
and are not theory dependent. As advertised earlier, these also satisfy a
virasoro algebra. For this reason, we label them as $\mathbb{L}_{n}$ and
$\bar{\mathbb{L}}_{n}$. The explicit expressions are given below.

$\displaystyle\mathbb{L}_{n}=a_{n}\int^{\infty}_{-\infty}d\zeta\;\frac{(z_{2}-\zeta)^{-n+1}(\zeta-
z_{3})^{n+1}}{z_{2}-z_{3}}T(\zeta)$ (2.7)
$\displaystyle\bar{\mathbb{L}}_{n}=\bar{a}_{n}\int^{\infty}_{-\infty}d\bar{\zeta}\;\frac{(z_{2}-\bar{\zeta})^{n+1}(\bar{\zeta}-z_{3})^{-n+1}}{z_{2}-z_{3}}\bar{T}(\bar{\zeta})$
(2.8)

Note that naively the integrand in the above formulae, blow up at $z_{2}$ and
$z_{3}$, however we can regulate the integral, by choosing a deformed contour
such that it doesn’t pass through $z_{2}$ and $z_{3}$. Equivalently, we can
give both $z_{2}$ and $z_{3}$ a small imaginary component. The $a_{n}$ are
arbitrary normalization constants555If we choose the normalization constant to
be independent of the endpoints $z_{2}$ and $z_{3}$, then it is easy to see
that the $L_{n}$ and $\bar{L}_{n}$ are really only a function of
($z_{2}$-$z_{3})$, however if the normalization constants are non trivial
functions of the endpoints, then the eigenmodes are bi-local ($z_{2}$ and
$z_{3}$). In this notation, $\mathbb{L}_{0}+\bar{\mathbb{L}}_{0}$, with
$a_{0}=\bar{a}_{0}=1$ is the modular hamiltonian of the single interval with
endpoints $z_{2}$ and $z_{3}$. It can be shown that they satisfies,

$[\mathbb{L}_{0},\mathbb{L}_{n}]=-n\mathbb{L}_{n},\;\;[\bar{\mathbb{L}}_{0},\bar{\mathbb{L}}_{n}]=-n\bar{\mathbb{L}}_{n}$
(2.9)

It follows that the $\mathbb{L}_{n}$ and $\bar{\mathbb{L}}_{n}$ for ($n\neq
0$) are indeed modular eigenmodes. Furthermore, if we normalize the
$\mathbb{L}_{n}$ suitably, which can be done without any loss of generality,
such that the $a_{n}=r^{n}$ and $\bar{a}_{n}=\bar{r}^{n}$ where $r$ and
$\bar{r}$ are two arbitrary constants, then in fact these eigenmodes satisfy
the virasoro algebra, with the correct central charge term.

$\displaystyle[\mathbb{L}_{m},\mathbb{L}_{n}]=(m-n)\mathbb{L}_{m+n}+\frac{c}{12}n(n^{2}-1)\delta_{m+n,0}$
(2.10)

For this reason, we refer to this, as the modular virasoro algebra (MVA). As
we explain in detail in section 3, there is a nice geometric interpretation of
the global $SO(2,2)$ subalgebra of the MVA. In particular, the generators of
this global subalgebra ie : $\mathbb{L}_{0,\pm}$ and
$\bar{\mathbb{L}}_{0,\pm}$ are linear combinations of the holomorphic and
antiholomorphic components of the modular hamiltonians corresponding to the
subregions $N^{\prime}(z_{1},z_{2})$ and $N^{\prime\prime}(z_{3},z_{1})$ of
$N$. See figure 1. This is particularly transparent if one parameterizes the
normalization constant as follows:
$r=\frac{1}{\bar{r}}=\left(\frac{z_{2}-z_{1}}{z_{3}-z_{1}}\right)$. As is
clear from figure 1, the $z_{1}$ in this parametrization is the point within
the line segment $N$, which divides it into $N^{\prime}$ and
$N^{\prime\prime}$. For this reason, in the remainder of the note, we use this
normalization for the $\mathbb{L}_{n}$ and $\bar{\mathbb{L}}_{n}$.

Finally we note that there is an interesting ’duality’ between the standard
generators of the CFT, which we denote as ${\bf L}_{n}$, and the
$\mathbb{L}_{n}$ we construct here. In particular, there exists a conformal
transformation which interchanges the two. The explicit map between the two
conformal frames is given in equation 4.2. Under this transformation,
$\mathbb{L}_{n}\Longleftrightarrow{\bf L}_{n}$. In particular, this means that
the modular hamiltonian gets interchanged with the usual CFT hamiltonian.

As we will argue in the rest of the note, it is natural to interpret the
$\mathbb{L}_{n}$ as realizing the virasoro algebra on the space of causal
diamonds in the CFT2, which is termed as the kinematic space (k-space).
Evidence of this is provided in the following sections, where we analyze the
action of $\mathbb{L}_{n}$ on the OPE blocks which are bilocal operators in
the CFT2 but have a simple local description in the k-space, and later when we
understand the geometric meaning of the global subalgebra of the MVA.

###  2.3 Action on the OPE blocks

We can compute the commutator of the “modular” $\mathbb{L}_{n}$ operators with
the OPE blocks, by using the commutator of $T$ with primary operators, which
can be obtained from the $T\mathcal{O}$ OPE.

$\displaystyle[T(\omega),\mathcal{O}_{k}(\zeta,\bar{\zeta})]=2\pi
i(h\partial_{\zeta}\delta(\zeta-\omega)+\delta(\zeta-\omega)\partial_{\zeta})\mathcal{O}_{k}(\zeta,\bar{\zeta}),$
(2.11)
$\displaystyle[\bar{T}(\bar{\omega}),\mathcal{O}_{k}(\zeta,\bar{\zeta})]=-2\pi
i(h\partial_{\bar{\zeta}}\delta(\bar{\zeta}-\bar{\omega})+\delta(\bar{\zeta}-\bar{\omega})\partial_{\bar{\zeta}})\mathcal{O}_{k}(\zeta,\bar{\zeta})$
(2.12)

One then needs to evaluate the action of $\mathbb{L}_{n}$ on primary field
$\mathcal{O}_{k}(\zeta,\bar{\zeta})$. Using (2.7), (2.11) we get

$\displaystyle[\mathbb{L}_{n},\mathcal{O}_{k}(\zeta,\bar{\zeta})]$
$\displaystyle=\frac{2\pi i}{z_{2}-z_{3}}(z_{2}-\zeta)^{-n}(\zeta-
z_{3})^{n}\left(\frac{z_{21}}{z_{31}}\right)^{n}$
$\displaystyle\times\Big{[}h_{k}\left(n(z_{2}-z_{3})+(z_{2}+z_{3}-2\zeta)\right)+(z_{2}-\zeta)(\zeta-
z_{3})\partial_{\zeta}\Big{]}\mathcal{O}(\zeta,\bar{\zeta})$ (2.13)
$\displaystyle[\bar{\mathbb{L}}_{n},\mathcal{O}_{k}(\zeta,\bar{\zeta})]$
$\displaystyle=\frac{2\pi
i}{z_{2}-z_{3}}(z_{2}-\bar{\zeta})^{n}(\bar{\zeta}-z_{3})^{-n}\left(\frac{z_{21}}{z_{31}}\right)^{-n}$
$\displaystyle\times\Big{[}\bar{h}_{k}\left(-n(z_{2}-z_{3})+(z_{2}+z_{3}-2\bar{\zeta})\right)+(z_{2}-\bar{\zeta})(\bar{\zeta}-z_{3})\partial_{\bar{\zeta}}\Big{]}\mathcal{O}(\zeta,\bar{\zeta})$
(2.14)

Using (2.1) and (2.3), we now present the commutator of the $\mathbb{L}_{n}$
and the $B^{ij}_{k}$.

$\displaystyle[\mathbb{L}_{n},B^{ij}_{k}(z_{2},z_{3})]$
$\displaystyle=\int_{z_{3}}^{z_{2}}d\zeta\int_{\bar{z}_{3}}^{\bar{z}_{2}}d\bar{\zeta}\left(\frac{(\zeta-
z_{3})(z_{2}-\zeta)}{z_{2}-z_{3}}\right)^{h_{k}-1}\left(\frac{z_{2}-\zeta}{\zeta-
z_{3}}\right)^{h_{ij}}\times$
$\displaystyle\left(\frac{(\bar{\zeta}-\bar{z}_{3})(\bar{z}_{2}-\bar{\zeta})}{\bar{z}_{2}-\bar{z}_{3}}\right)^{\bar{h}_{k}-1}\left(\frac{\bar{z}_{2}-\bar{\zeta}}{\bar{\zeta}-\bar{z}_{3}}\right)^{\bar{h}_{ij}}\frac{2\pi
i}{z_{2}-z_{3}}(z_{2}-\zeta)^{-n}(\zeta-
z_{3})^{n}\left(\frac{z_{21}}{z_{31}}\right)^{n}$
$\displaystyle\times\Big{[}h_{k}\left(n(z_{2}-z_{3})+(z_{2}+z_{3}-2\zeta)\right)+(z_{2}-\zeta)(\zeta-
z_{3})\partial_{\zeta}\Big{]}\mathcal{O}_{k}(\zeta,\bar{\zeta})$
$\displaystyle=(\text{T.D})+2\pi
i(nh_{k}-n+h_{ij})\left(\frac{z_{21}}{z_{31}}\right)^{n}\int_{z_{3}}^{z_{2}}d\zeta\int_{\bar{z}_{3}}^{\bar{z}_{2}}d\bar{\zeta}\left(\frac{(\zeta-
z_{3})(z_{2}-\zeta)}{z_{2}-z_{3}}\right)^{h_{k}-1}$
$\displaystyle\times\left(\frac{z_{2}-\zeta}{\zeta-
z_{3}}\right)^{h_{ij}}\left(\frac{(\bar{\zeta}-\bar{z}_{3})(\bar{z}_{2}-\bar{\zeta})}{\bar{z}_{2}-\bar{z}_{3}}\right)^{\bar{h}_{k}-1}\left(\frac{\bar{z}_{2}-\bar{\zeta}}{\bar{\zeta}-\bar{z}_{3}}\right)^{\bar{h}_{ij}}\left(\frac{z_{2}-\zeta}{\zeta-
z_{3}}\right)^{-n}\mathcal{O}_{k}(\zeta,\bar{\zeta})$ $\displaystyle=2\pi
i\left(\frac{z_{21}}{z_{31}}\right)^{n}[n(h_{k}-1)+h_{ij}]B^{ij-n}_{k}\;;\;\text{for}\;\Big{(}h_{ij}-h_{k}\leq
n\leq h_{ij}+h_{k}\Big{)}$ (2.15)

Here (T.D) is the total derivative term which vanishes for
$\Big{(}h_{ij}-h_{k}\leq n\leq h_{ij}+h_{k}\Big{)}$.

Equation (2.3) is identical to the action of usual Virasoro generator $l_{n}$
in CFT2 on the modes $\phi_{m}$ of a primary field $\phi$,

$\displaystyle[l_{n},\phi_{m}]=[n(h-1)-m]\phi_{n+m}$ (2.16)

with the identification
$\phi_{m}=\left(\frac{z_{21}}{z_{31}}\right)^{-h_{ij}}B^{ij}_{k}$. Thus we see
that the OPE blocks play the role of modes of some highest weight primary
field representation of the MVA.

To summarize, the key results of this section are:

  * a.

The integrated stress tensor operators $\mathbb{L}_{n}$ and
$\bar{\mathbb{L}}_{n}$ form an infinite class of modular eigenmodes of the
modular hamiltonian corresponding to an single interval $N$ with endpoints
($z_{1},z_{2}$).

  * b.

These modular eigenmodes are bi-local, similar to the OPE blocks, in that they
are a function of the end points ($z_{1},z_{2}$) of the interval ($N$).

  * c.

Their commutators satisfy the virasoro algebra. For obvious reasons we refer
to this representation of the virasoro algebra as the modular virasoro algebra
(MVA). Moreover under a conformal transformation given in equation4.2, the
${\mathbb{L}}_{n}\Longleftrightarrow{\bf L}_{n}$. In particular, the modular
hamiltonian is interchanged with the usual CFT hamiltonian.

  * d.

Under the transformations generated by the $\mathbb{L}_{n}$ and
$\bar{\mathbb{L}}_{n}$, the OPE blocks transforms as should the modes of a
primary operator in CFT2.

Now, the OPE blocks which are bi-local fields in the CFT2 have a natural
description as local fields on the so-called kinematic space (k-space)
[34]-[36]. In the light of [c] and [d], it is natural to wonder whether there
exists an “effective CFT” description in the k-space itself, with the OPE
blocks being the primary fields in this “k-space cft”. We explore the
kinematical aspects of this question in the next section.

###  2.4 MVA and the Kinematic space

The kinematic space of CFT2 is defined as the space of a ‘pair of space like
points’ in the CFT. Thus its a four dimensional space, with coordinates given
by the coordinates of the two points($t_{2},x_{2};t_{3},x_{3}$) with signature
($+,-,+,-$). One can fix the metric on this space by demanding its invariance
under conformal transformations of both points. This leads to a unique metric.

$ds^{2}_{kspace}=2\Big{[}\frac{dz_{2}dz_{3}}{(z_{2}-z_{3})^{2}}+\frac{d\bar{z}_{2}d\bar{z}_{3}}{(\bar{z}_{2}-\bar{z}_{3})^{2}}\Big{]}\textrm{with}\;\;z_{i}=t_{i}+x_{i},\;\bar{z}_{i}=t_{i}-x_{i}$
(2.17)

Thus the 4d space factorizes into two 2d conformally flat space times, spanned
by the two sets of k-space light cone coordinates $(z_{2},\;z_{3})$ and
$(\bar{z}_{2},\;\bar{z}_{3})$ respectively.

The k-space formalism allows us to visualize the OPE blocks, which are bi-
local fields in the CFT2 as local fields living on this k-space. Moreover it
geometrizes the conformal kinematic properties of these OPE blocks. In
particular, this means that the conformal casimir equation which the OPE
blocks satisfies, derived from its conformal transformation properties as
obtained from its definition in 2.2, translates in the k-space terminology
into an ‘equations of motion’ to be satisfied by the OPE blocks on the k-space
[34],[35]. For scalar OPE blocks, this is just the Klein-Gordon (KG) equation,
while the spinning OPE blocks satisfy a slightly modified KG equation [38].
The short distance behaviour of the OPE blocks 666The short distance limit of
an ope implies the ope block behaves like a single primary, i.e:
$\lim_{z_{2},\bar{z}_{2}\rightarrow
z_{3},\bar{z}_{3}}B_{k}^{ij}(z_{2},\bar{z}_{2};z_{3},\bar{z}_{3})\sim
z_{23}^{h_{k}}\bar{z}_{23}^{\bar{h}_{k}}\mathcal{O}_{h_{k},\bar{h}_{k}}(z_{3},\bar{z}_{3})$.
is now interpreted as a boundary condition to be imposed along the k-space
coordinates ($z_{23},\;\bar{z}_{23}\rightarrow 0$)

####  2.4.1  Realizing $\mathbb{L}_{n}$ on k-space

Points [c] and [d] of the last section seem to hint at the possibility of an
effective CFT description in the k-space with the OPE blocks being modes of
the highest weight representation of the corresponding MVA, which in this
interpretation would act locally on the k-space. Now from the explicit form of
the $\mathbb{L}_{n}$, it is clear that they are functions of ($z_{2}-z_{3}$).
Now in our case, we had chosen the interval to be on a constant time slice, so
that $z_{2}=\bar{z}_{2}$ and $z_{3}=\bar{z}_{3}$. If, on the other hand, had
we chosen an arbitrary spacelike interval, then indeed $\mathbb{L}_{n}$,
$\bar{\mathbb{L}}_{n}$ are function of ($z_{23}=z_{2}-z_{3}$) and
($\bar{z}_{23}=\bar{z}_{2}-\bar{z}_{3}$) respectively. As is clear from the
k-space metric 2.17, the $z_{23}$, and $\bar{z}_{23}$ are spatial coordinates
on the two decoupled spaces and not light cone coordinates. Thus the
$\mathbb{L}_{n}$ and $\bar{\mathbb{L}}_{n}$ and should be thought of as
generating independent spatial diffeomorphisms along $z_{23}$ and
$\bar{z}_{23}$ directions rather than generating conformal transformations in
a 2d space.

It is still possible that there is a useful effective k-space description in
terms of product of two CFT1’s, with a 1d stress tensor which we denote as
$\mathbb{T}(\zeta)$ and $\mathbb{T}(\bar{\zeta})$ given by

$\mathbb{T}(\zeta)=\sum_{n}\frac{\mathbb{L}_{n}(z_{23})}{(\zeta-
z_{23})^{n+2}},\;\;\textrm{and}\;\;\bar{\mathbb{T}}(\bar{\zeta})=\sum_{n}\frac{\mathbb{L}_{n}(\bar{z}_{23})}{(\bar{\zeta}-\bar{z}_{23})^{n+2}}$
(2.18)

such that the OPE blocks are modes of a field $\Phi(\zeta,\bar{\zeta})$ with
respect to both of the 1d cfts. Where the field $\Phi$ could be formally mode-
expanded in terms of the OPE blocks as follows:

$\Phi(\zeta,\bar{\zeta})=\sum_{h_{ij},\bar{h}_{ij}}\frac{\mathcal{B}^{ij}_{k}(z_{2},z_{3};\bar{z}_{2},\bar{z}_{3})}{(\zeta-
z_{23})^{h_{k}-h_{ij}}(\bar{\zeta}-\bar{z}_{23})^{\bar{h}_{k}-\bar{h}_{ij}}}$
(2.19)

However to establish whether a consistent description of this type can be
constructed, would involve proving that it satisfies consistent crossing
equations among other things. We do not attempt to answer this question here.
The only point we want to make here, is that the algebra of the OPE blocks
with the $\mathbb{L}_{n}$ is consistent with the existence of such an
effective k-space description.

Of course, if such an effective description does exist in k-space, it would
only be a reformulation of the original 2d cft in the k-space language, and
the stress tensors defined via equation 2.18, would also be related to the CFT
stress tensor components. Nevertheless, we know that the k-space description
is a useful intermediary between the AdS and CFT descriptions, because the
k-space has the advantage of being directly identified as the space of bulk
geodesics which end on the boundary [36]. This fact has been used to derive a
very simple proof of the identification 2.5,2.6 of OPE blocks as geodesic
operators in AdS [34]. For these reasons, a way to incorporate cft dynamics in
k-space language would be interesting from the AdS/CFT perspective. We hope to
come back to this issue in the near future.

##  3 The global subalgebra of the MVA

In this section, we focus on the global subalgebra of the MVA, which is
spanned by $\mathbb{L}_{0,\pm 1}$ and $\mathbb{\bar{L}}_{0,\pm 1}$. We point
out that this subset of $\mathbb{L}_{n}$ has a nice geometric interpretation
as modular hamiltonians of $N$ itself as well as its subparts, labelled as
$N^{\prime}$ and $N^{\prime\prime}$. This in turn realizes the action of
’modular inclusion’ [50]-[53], within $\mathcal{D}_{N}$. For brevity, we refer
to this subalgebra as the g-MVA in the rest of the note. We begin with a short
discussion on the symmetries of causal diamonds in CFT2.

###  3.1 Symmetries of the CFT2 causal diamonds

CFT2 causal diamonds associated with intervals on a constant time slice are
preserved by a $SO(1,1)\times SO(1,1)$ subset of global conformal symmetry
group $SO(2,2)$. Due to chiral structure of symmetry algebras, one can find a
right moving and a left moving conformal killing vector (CKV) which stabilizes
the diamond. For a causal diamond with upper and lower tips at ($v,\bar{v}$)
and ($u,\bar{u}$) respectively (in the light cone coordinates), the CKVs take
the following form,

$\displaystyle
K^{\zeta}\partial_{\zeta}=\frac{(v-\zeta)(\zeta-u)}{v-u}\partial_{\zeta}\;,\;K^{\bar{\zeta}}\partial_{\bar{\zeta}}=\frac{(\bar{v}-\bar{\zeta})(\bar{\zeta}-\bar{u})}{\bar{v}-\bar{u}}\partial_{\bar{\zeta}}$
(3.1)

Here $\zeta(=X+T),\bar{\zeta}(=X-T)$ are the lightcone coordinates. One can
similarly define the corresponding conserved charges as:

$\displaystyle
K^{R}=\int^{\infty}_{-\infty}d\zeta\frac{(v-\zeta)(\zeta-u)}{u-v}T_{\zeta\zeta}(\zeta)\;,\;K^{L}=\int^{\infty}_{-\infty}d\bar{\zeta}\frac{(\bar{v}-\bar{\zeta})(\bar{\zeta}-\bar{u})}{\bar{v}-\bar{u}}T_{\bar{\zeta}\bar{\zeta}}(\bar{\zeta})$
(3.2)

If we take an interval on $T=0$ slice with endpoints $(z_{2},z_{3})$, the
upper and lower tips of the corresponding causal diamond(say $N$) are located
at $y^{\mu}=(\frac{z_{2}-z_{3}}{2},\frac{z_{2}+z_{3}}{2})$ and
$x^{\mu}=(\frac{z_{3}-z_{2}}{2},\frac{z_{2}+z_{3}}{2})$ respectively. In this
case: $(u,\bar{u})\equiv(x^{1}-x^{0},x^{1}+x^{0})=(z_{2},z_{3})$ and
$(v,\bar{v})\equiv(y^{1}-y^{0},y^{1}+y^{0})=(z_{3},z_{2})$. The total modular
Hamiltonian $K_{N}$ for the interval $N$ is the sum of $K^{R}_{N}$ and
$K^{L}_{N}$, i.e

$\displaystyle
K=K^{R}_{N}+K^{L}_{N}=\int^{\infty}_{-\infty}d\zeta\frac{(z_{2}-\zeta)(\zeta-
z_{3})}{z_{2}-z_{3}}T_{\zeta\zeta}(\zeta)+\int^{\infty}_{-\infty}d\bar{\zeta}\frac{(z_{2}-\bar{\zeta})(\bar{\zeta}-z_{3})}{z_{2}-z_{3}}T_{\bar{\zeta}\bar{\zeta}}(\bar{\zeta})$
(3.3)

This can be derived from the expression of modular Hamiltonian of Rindler half
space by a conformal transformation from Rindler wedge to CFT2 causal diamond.
One can similarly define $P_{D}$ as the antisymmetric combination of $K^{R}$
and $K^{L}$, i.e. $P_{D}=K^{R}-K^{L}$ [20]. Together, $K$ and $P_{D}$
generates a geometrical flow which preserve the diamond. $K$ generate flows
from lower tip to the upper tip, while $P_{D}$ generates flows from left to
the right tip of the diamond.

Consider a CFT2 interval $N(z_{3},z_{2})$ on a time slice $T=0$777i.e.
$z_{3,2}=\bar{z}_{3,2}$. Divide this line segment into two parts
$N^{\prime}(z_{1},z_{2})$ and $N^{\prime\prime}(z_{3},z_{1})$ around a point
$z_{1}$. The corresponding causal diamonds of $N^{\prime}$ and
$N^{\prime\prime}$ divides the causal diamond of $N$ into four parts such that
($N\supset{N^{\prime},N^{\prime\prime},U,L}$), where $U$ and $L$ are the upper
and lower diamond as shown in the Figure(1)888We will be using the same labels
interchangeably for the line segments as well as the corresponding causal
diamonds..

Z3Z2Z1 Figure 1: Causal diagram of different regions on a $T=0$ slice.

Using the $TT$ OPE, it can be shown that the
$K_{(N^{\prime},N,N^{\prime\prime})}^{R}$ satisfy the following algebra:

$[K_{N^{\prime}}^{R},K_{N}^{R}]=2\pi
i\left(K_{N}^{R}-K_{N^{\prime}}^{R}\right)$ (3.4)
$[K_{N}^{R},K_{N^{\prime\prime}}^{R}]=2\pi
i\left(K_{N}^{R}-K_{N^{\prime\prime}}^{R}\right)$ (3.5)
$[K_{N^{\prime}}^{R},K_{N^{\prime\prime}}^{R}]=-2\pi
i\left(K_{N^{\prime}}^{R}+K_{N^{\prime\prime}}^{R}\right)$ (3.6)

This is isomorphic to the holomorphic $SO(2,1)$ subsector of the full
$SO(2,2)$ conformal algebras, with the following identifications:999We have
absorbed the $2\pi i$ factor by redefining $K^{R,L}$ as $\frac{1}{2\pi
i}K^{R,L}$.

$K_{N^{\prime\prime}}^{R}-K_{N}^{R}=\mathbb{L}_{1}\;;\;K_{N}^{R}=\mathbb{L}_{0}\;;\;K_{N^{\prime}}^{R}-K_{N}^{R}=\mathbb{L}_{-1}$
(3.7)

Similarly, $K_{(N^{\prime},N,N^{\prime\prime})}^{L}$ can be shown to satisfy
the following commutation relations:

$[K_{N^{\prime}}^{L},K_{N}^{L}]=2\pi
i\left(K_{N^{\prime}}^{L}-K_{N}^{L}\right)$ (3.8)
$[K_{N}^{L},K_{N^{\prime\prime}}^{L}]=2\pi
i\left(K_{N^{\prime\prime}}^{L}-K_{N}^{L}\right)$ (3.9)
$[K_{N^{\prime}}^{L},K_{N^{\prime\prime}}^{L}]=2\pi
i\left(K_{N^{\prime}}^{L}+K_{N^{\prime\prime}}^{L}\right)$ (3.10)

This is again isomorphic to the anti-holomorphic $SO(2,1)$ sub-algebra, with
the identification:

$K_{N^{\prime}}^{L}-K_{N}^{L}=\bar{\mathbb{L}}_{1}\;;\;K_{N}^{L}=\bar{\mathbb{L}}_{0}\;;\;K_{N^{\prime\prime}}^{L}-K_{N}^{L}=\bar{\mathbb{L}}_{-1}$
(3.11)

The remaining right and left chiral generators of the diamond $U$ and $L$ can
be expressed in terms of $K^{R,L}_{N,N^{\prime},N^{\prime\prime}}$ as follows:

$\displaystyle
K^{R}_{U}=K^{R}_{N^{\prime\prime}}\;,\;K^{L}_{U}=K^{L}_{N^{\prime}}\;,\;K^{R}_{L}=K^{R}_{N^{\prime}}\;,\;K^{L}_{L}=K^{L}_{N^{\prime\prime}}$
(3.12)

Thus, the six modular generators of the three diamonds, ($K^{R(L)}_{N,U,L}$s)
also satisfy the $SO(2,1)\times SO(2,1)$ global conformal algebra.

We denote the CKV’s associated with these generators $\mathbb{L}_{0,\pm 1}$ as
$L_{0,\pm}$. It’s easy to see that these are simply linear combinations of the
standard representations of the global conformal generators $l_{1,0,-1}$
defined earlier. The exact relation between them are given by:

$\displaystyle
l_{1}=\frac{2z_{2}z_{3}}{z_{3}-z_{2}}L_{0}+\frac{z_{3}^{2}(z_{1}-z_{2})}{(z_{1}-z_{3})(z_{3}-z_{2})}L_{-1}+\frac{z_{2}^{2}(z_{1}-z_{3})}{(z_{1}-z_{2})(z_{3}-z_{2})}L_{1}$
(3.13) $\displaystyle
l_{0}=\frac{(z_{2}+z_{3})}{z_{2}-z_{3}}L_{0}+\frac{z_{3}(z_{1}-z_{2})}{(z_{1}-z_{3})(z_{2}-z_{3})}L_{-1}+\frac{z_{2}(z_{1}-z_{3})}{(z_{1}-z_{2})(z_{2}-z_{3})}L_{1}$
(3.14) $\displaystyle
l_{-1}=\frac{2}{z_{3}-z_{2}}L_{0}+\frac{(z_{1}-z_{2})}{(z_{1}-z_{3})(z_{3}-z_{2})}L_{-1}+\frac{(z_{1}-z_{3})}{(z_{1}-z_{2})(z_{3}-z_{2})}L_{1}$
(3.15)

With a similar set of relations between $\bar{l}_{1,0,-1}$ and
$\bar{L}_{1,0,-1}$. As is clear from the definitions, in our set-up,
$K_{N}=\mathbb{L}_{0}+\bar{\mathbb{L}}_{0}$ and $\mathbb{L}_{1,-1}$ and
$\bar{\mathbb{L}}_{1,-1}$ are it’s $\pm 1$ eigenmodes. These eigenmodes are
constructed by the $K$s and $P_{D}$s of the regions which reside inside the
$N$ itself. This set-up exhibits some other features like modular inclusion as
we discuss in the next subsection.

###  3.2 g-MVA and modular inclusions

Out of the left and right moving CKVs, one can notice that $K_{N}$, $K_{U}$,
$K_{L}$ and $P_{D,N}$, $P_{D,N^{\prime}}$, $P_{D,N^{\prime\prime}}$ are closed
under $SO(2,1)$ subalgebra separately101010However these two $SO(2,1)$ don’t
commute with each other..

$\displaystyle[P_{D,N},P_{D,N^{\prime}}]=2\pi i(P_{D,N^{\prime}}-P_{D,N})$
(3.16) $\displaystyle[P_{D,N},P_{D,N^{\prime\prime}}]=2\pi
i(P_{D,N}-P_{D,N^{\prime\prime}})$ (3.17)
$\displaystyle[P_{D,N^{\prime\prime}},P_{D,N^{\prime}}]=2\pi
i(P_{D,N^{\prime\prime}}+P_{D,N^{\prime}})$ (3.18)

Hence, $P_{D,N^{\prime\prime}}-P_{D,N}$, $P_{D,N}$ and
$P_{D,N^{\prime}}-P_{D,N}$ satisfy the $SO(2,1)$ sub-algebra.

In a similar fashion, one could also obtain the following

$\displaystyle[K_{N},K_{U}]=K_{N}-K_{U}$ (3.19)
$\displaystyle[K_{N},K_{L}]=K_{L}-K_{N}$ (3.20)
$\displaystyle[K_{U},K_{L}]=K_{U}+K_{L}$ (3.21)

Here $K_{U}-K_{N}$, $K_{N}$ and $K_{L}-K_{N}$ construct the another $SO(2,1)$
sub-algebra.

The above commutation relations (3.16) and (3.19) has the structure of modular
inclusion as we discuss in detail in the appendix (A). In particular, (3.16)
gives an unitary geometric operation using which one can find a map of algebra
of observable between the nested diamonds $N$, $N^{\prime}$ and
$N^{\prime\prime}$. Similarly, using (3.19) we have a map of algebra of
observables between $N$, $U$ and $L$. See appendix (A), for further details.
Using the inclusion properties and the fact that $K$ and $P_{D}$ of any
diamond can be constructed in the basis of modular generators of $N$,
$N^{\prime}$, $N^{\prime\prime}$ as in (3.13), we could in principle construct
algebra of observables of any region (diamond) or provide a map to any diamond
in the spacetime from $N$. However, since here the modular generators in
vacuum are constructed out of conformal symmetries, this inclusion property
i.e.: the map from different causal domains is just an artefact of the global
conformal symmetry.

##  4 Pulling the $\mathbb{L}_{n}$ into the bulk

Given the explicit form of the $\mathbb{L}_{n}$ and $\bar{\mathbb{L}}_{n}$,
one can read off the corresponding CKV’s ( $L_{n}$) from equations 2.7 and
3.1. Their explicit forms are as follows.

$\displaystyle L_{n}=\frac{(z_{2}-\zeta)^{-n+1}(\zeta-
z_{3})^{n+1}}{z_{3}-z_{2}}\left(\frac{z_{2}-z_{1}}{z_{3}-z_{1}}\right)^{n}\partial_{\zeta},$
$\displaystyle\bar{L}_{n}=-\frac{(z_{2}-\bar{\zeta})^{n+1}(\bar{\zeta}-z_{3})^{-n+1}}{z_{3}-z_{2}}\left(\frac{z_{3}-z_{1}}{z_{2}-z_{1}}\right)^{n}\partial_{\bar{\zeta}}$
(4.1)

Following [42], in this section we will extend the CKV’s into the bulk where
they would generate bulk diffeomorphisms. From our previous discussion of
section 3, we already know what these are for the special case of $n=0,\pm 1$.
Since the $\mathbb{L}_{0,\pm 1}$ and $\bar{\mathbb{L}}_{0,\pm 1}$ generate
isometries of the causal diamond of the subregions $N^{\prime}$ and
$N^{\prime\prime}$, their duals would generate boosts around and translations
along the respective RT geodesics for $N^{\prime}$ and $N^{\prime\prime}$.

For the generic $n$ case, we proceed as follows. We first use the following
transformation
$(\zeta,\bar{\zeta})\rightarrow(\zeta^{\prime},\bar{\zeta^{\prime}})$ to
transform $(L_{n},\bar{L}_{n})\rightarrow(l_{n},\bar{l}_{n})$.

$\zeta^{\prime}=\frac{1}{\beta}(\frac{\zeta-
z_{3}}{z_{2}-\zeta}),\;\;\;\bar{\zeta}^{\prime}=\beta\frac{z_{2}-\bar{\zeta}}{\bar{\zeta}-z_{3}},\;(\textrm{with}\;\beta=\frac{z_{13}}{z_{21}})$
(4.2)

One can then extend these transformations into the bulk. ($y$, $\zeta$,
$\bar{\zeta}$) $\rightarrow$ ($y^{\prime}$, $\zeta^{\prime}$,
$\bar{\zeta^{\prime}}$). Working in the Fefferman Graham gauge [41], the
corresponding dual bulk transformations are given by:

$\displaystyle\zeta^{\prime}=\frac{1}{\beta}[\frac{\zeta-
z_{3}}{z_{2}-\zeta}-\frac{z_{23}}{z_{2}-\zeta}\frac{y^{2}}{y^{2}-(z_{2}-\zeta)(\bar{\zeta}-z_{3})}]$
$\displaystyle\bar{\zeta}^{\prime}=\beta[\frac{z_{2}-\bar{\zeta}}{\bar{\zeta}-z_{3}}-\frac{z_{23}}{\bar{\zeta}-z_{3}}\frac{y^{2}}{y^{2}-(z_{2}-\zeta)(\bar{\zeta}-z_{3})}]$
$\displaystyle y^{\prime}=\frac{y}{y^{2}-(z_{2}-\zeta)(\bar{\zeta}-z_{3})}$
(4.3)

In the $y\rightarrow 0$ limit, this equation reduces to equation (4.2), as it
should. From the above transformations, we can now obtain the expression for
the bulk counterpart of the $L_{n}$’s. In the primed coordinates, the action
of the bulk $l_{n}$, is given by [42]

$l^{(b)}_{n}=\delta_{n}\zeta^{\prime}\partial_{\zeta^{\prime}}+\delta_{n}\bar{\zeta^{\prime}}\partial_{\bar{\zeta^{\prime}}}+\delta_{n}y^{\prime}\partial_{y^{\prime}}$
(4.4)

where:

$\delta_{n}\zeta^{\prime}=(-\zeta^{\prime})^{n+1},\;\delta_{n}\bar{\zeta^{\prime}}=-n(n+1)y^{\prime
2}(-\zeta^{\prime})^{n-1},\;\delta_{n}y=\frac{1}{2}(n+1)y^{\prime}(-\zeta^{\prime})^{n}$
(4.5)

Similar expression may be obtained for the $\bar{l}^{b}_{n}$. By using eqn(4)
and eqn(4.5) in eqn(4.4), we can obtain the explicit expression for
$L^{b}_{n}$

$L^{b}_{n}=\frac{{(-\zeta^{\prime})}^{n+1}}{z_{21}.A.RT}[U\partial_{\zeta}+V\partial_{\bar{\zeta}}+W\partial_{y}]$
(4.6)

Here, the expression for $\zeta^{\prime}$ is given by the first of the
equations(4) and the explicit expressions of $U$, $V$, $W$, $A$ and $RT$ are
given below.

$\displaystyle U=y^{4}n(n+1)-A(A+z_{21}(n+1)y^{2}),$ $\displaystyle
V=-A(A+z_{21}(n+1)(z_{2}-\zeta)(\bar{\zeta}-z_{3}))+n(n+1)(\bar{\zeta}-z_{3})^{2}(z_{2}-\zeta)^{2},$
$\displaystyle
W=A(2A+z_{21}((\bar{\zeta}-z_{3})(z_{2}-\zeta)+y^{2}))+2ny^{2}(\bar{\zeta}-z_{3})(n-z_{2}+\zeta)$
$\displaystyle+nz_{21}[(z_{2}-\zeta)^{2}(\bar{\zeta}-z_{3})^{2}(\zeta-
z_{3})+y^{2}(y^{2}(z_{2}+z_{3}-2\zeta)+(z_{2}-\zeta)^{2}(\bar{\zeta}-z_{3}))],$
$\displaystyle A=(\bar{\zeta}-z_{3})(\zeta-
z_{3})(z_{2}-\zeta)+y^{2}(z_{2}+z_{3}-2\zeta),$ $\displaystyle
RT=(y^{2}-(z_{2}-\zeta)(\bar{\zeta}-z_{3}))$ (4.7)

An interesting feature of the above formulae is the emergence of the RT
geodesic. For instance, notice the appearance of the RT geodesic expression
($y^{2}-(z_{2}-\zeta)(\bar{\zeta}-z_{3})$) in the RHS of eqn(4). Thus these
equations blow up on the RT surface. This means that the bulk coordinates
$(\zeta,\bar{\zeta},y)$ cover only the region between the geodesic and the
boundary. Thus they provide a natural set of coordinates for the entanglement
wedge associated to $N$. The RT geodesic also appears in the expressions for
the bulk counterparts of $L_{n}$’s given in equation (4.4). The fact that the
bulk coordinates and the bulk extensions of the $L_{n}$ ‘know’ about the RT
geodesic is not surprising. It is simply a reflection of the fact that the
boundary $L_{n}$ are modular eigenmodes by construction and thus has
information about the boundary causal diamond of $N$.

##  5 Discussion

In summary, we have constructed an infinite class of modular eigenmodes
($\mathbb{L}_{n}$) for the single interval in the vacuum of CFT2. These are
expressed as smeared integrals of the stress tensor components and thus exist
in any CFT2 111111Such smeared intergrals of the stress tensor has appeared in
several different contexts recently, for instance in the study of the light
ray operators [54]-[58] as well in the context of the so-called dipolar
quantization of CFT2 as discussed in [59],[60]. We thank Bartek Czeck for
bringing these works on the dipolar quantization to our attention.. Our
construction of these eigenmodes are intimately tied to the causal diamond of
$N$. This fact manifests itself in many of its interesting features. For
instance, one way in which this connection to the causal diamond manifests
itself is in the way $\mathbb{L}_{n}$ acts on OPE blocks. We showed that this
action is identical to the action of conformal generators on local primary
fields in CFT2. Coupled with the fact that the OPE blocks have a local
description as fields living on the k-space, which is the space of causal
diamonds of the CFT2, this hints at the possibility of finding an equivalent
effective description of the CFT on k-space. We argued that on this k-space,
the $\mathbb{L}_{n}$ seem to generate 1d diffeomorphisms along two independent
directions. Unfortunately our discussions are only at a kinematic level, and
it would be nice if these ideas can be made more concrete.

The connection to the causal diamonds is even more transparent, in the
subclass of the eigenmodes corresponding to $n=0,\pm 1$. In fact, as we
showed, these generators are essentially linear combinations of the modular
hamiltonians of the causal diamonds for the subregions $N^{\prime}$ and
$N^{\prime\prime}$ of $N$. We further showed how this structure of the g-MVA
realizes modular inclusions within in this setup. The half sided modular
inclusion has been studied previously in some examples like certain regions on
null plane in higher dimensions [47] and it has been used to show that in
certain special situation, black hole interior could be reconstructed from the
algebra of exterior region [48]. In our example, the inclusion structure
emerges quite naturally due to the rich symmetric structure of the
vacuum121212As a testing ground of such algebraic structure or modular
properties it is always very useful to study them in quantum mechanical system
having finite dimensional Hilbert space[2]. With this motivation in mind, in
appendix A we study inclusion properties in an example of finite dimensional
Hilbert space where inclusion algebras are satisfied trivially..

Finally we also discussed the action of the bulk counterparts of the
$\mathbb{L}_{n}$ on the bulk spacetime. We saw that these dual descriptions
already ‘know’ about the bulk RT geodesic, which is again a reflection of the
close connection of our construction with the causal diamond.

A natural question that arises is whether one can extend this construction of
algebra and its representation beyond the vacuum in CFT for at least some
class of excited states131313For locally excited states in CFT2 which are
connected to vacuum by local conformal transformation, we do have local
expression for modular Hamiltonian in single interval [19], [45]. However, we
expect this case to be almost identical to the vacuum case.. Perhaps a more
tractable direction to pursue would be to find the extension of such algebras
for disconnected multi-interval cases where analytic expression of modular
Hamiltonian are known[43],[44].141414Recently analytic expression of modular
Hamiltonian for intervals in BMS invariant field theories has been discussed
where we could study similar construction to study algebra[46].

We hope to return to some of these questions in the near future.

Acknowledgment: SD would like to acknowledge the support provided by the Max
Planck Partner Group grant MAXPLA/PHY/2018577. The work of SP and BR was
supported by a Junior Research Fellowship(JRF) from UGC.

##  Appendix A Modular inclusion in CFT2 and finite dimensional system

###  A.1 Modular inclusion

The Reeh-Schileder theorem states 151515The readers may look at [2] for a
recent review on algebraic QFT and modular theory that an algebra
$\mathcal{A}_{V}$, made out of bounded operators restricted to an arbitrary
small open set $V$ in spacetime (flat), is enough to generate (by acting on
vacuum) the full vacuum sector of the Hilbert space. Due to this property the
vacuum state is said to be ‘cyclic’ w.r.t the algebra of operators
$\mathcal{A}_{V}$ in that small open region $V$. Incorporating microcausality,
an obvious conclusion can be drawn that such a state is also separating w.r.t
$\mathcal{A}_{V}$, which means that, there exists no operator in $V$ which
annihilate the vacuum. In algebraic QFT, some useful quantum information
quantities like Relative entropy, total modular Hamiltonian can be rigorously
constructed for such cyclic and separating states of the QFT Hilbert space. In
particular, a self-adjoint ‘modular operator’ $\Delta$ ($=e^{-K}$, K is the
total modular Hamiltonian) and an antiunitary operator ‘modular conjugation’
$J$ are the central objects of ‘Tomita-Takesaki theory’, which lies at the
foundation of modular theory or modular algebra. The main result of the
Tomita-Takesaki theory is that $\Delta$ defines an automorphism which maps an
algebra of a region to itself while $J$ defines an isomorphism from the
algebra to its commutant $\mathcal{A}^{\prime}_{V}$.

$\displaystyle\Delta^{is}A\Delta^{-is}=\tilde{A}\;;\;JAJ=A^{\prime},\;\;\;(A,\tilde{A})\in\mathcal{A}_{V},\;A^{\prime}\in\mathcal{A}^{\prime}_{V},\;\forall
s\in\mathbb{R}$ (A.1)

The $\Delta$ generates a modular flow w.r.t total modular Hamiltonian $K$.
Here, the algebra $\mathcal{A}_{V}$ is considered to be a type of Von-Neumann
algebra such that, $\mathcal{A}_{V}=\hat{\mathcal{A}}_{V}$. Where,
$\hat{\mathcal{A}}_{V}$ is the algebra of the causal domain of the region $V$.

Within the context of the Tomita-Takesaki theory, a notion of inclusion of
algebras has been discussed -the so-called ‘half-sided modular inclusion’
(hsmi)[49]-[51]. Take two Von-Neumann algebra of observables $M,M^{\prime}$,
such that the vacuum $\Omega$ is a common cyclic and separating state for both
of them. We can define $\tilde{M}\subset M$ as the +hsmi if it satisfies the
condition that $M^{\prime}$ is preserved under the modular flow of $M$, i.e

$\displaystyle\Delta^{-it}_{M}\tilde{M}\Delta^{it}_{M}\subset\tilde{M},\;\;\;\forall
t\geq 0$ (A.2)

Here $\Delta_{M},\Delta_{\tilde{M}}$ are modular operator of $M,\tilde{M}$.
161616The corresponding modular conjugation operators are
$J_{M},J_{\tilde{M}}$. However, in the present context, we won’t need the
properties of $J$s and we only focus on modular flows generated by $\Delta$s.
For further details, we refer the readers to the following references
[50],[51]. Once the above condition is satisfied, one can construct an one-
parameter unitary group $U(a)$ on the Hilbert space such that,

$\displaystyle
U(a)=e^{iap};\;p\equiv\frac{1}{2\pi}(\ln\Delta_{\tilde{M}}-\ln\Delta_{M})\geq
0;\;\forall a\in\mathbb{R}$ (A.3)

The generator $p$ is a positive operator. In such settings, the following
properties hold:

$\displaystyle\Delta_{M}^{it}U(a)\Delta_{M}^{-it}$
$\displaystyle=\Delta_{\tilde{M}}^{it}U(a)\Delta_{\tilde{M}}^{it}=U(e^{-2\pi
t}a);\;\forall a,t\in\mathbb{R}$ (A.4) $\displaystyle\Delta_{\tilde{M}}^{it}$
$\displaystyle=U(1)\Delta^{it}_{M}U(-1);\;\forall t\in\mathbb{R}$ (A.5)
$\displaystyle\tilde{M}$ $\displaystyle=U(1)MU(-1)$ (A.6)
$\displaystyle\Delta_{M}^{it}\Delta_{\tilde{M}}^{-it}$
$\displaystyle=e^{i\left(-1+e^{-2\pi t}\right)p}$ (A.7)

One can see that the first two relations are solved by

$\displaystyle[K_{M},K_{\tilde{M}}]=2\pi
ip;\;K_{M,\tilde{M}}=-\ln\Delta_{M,\tilde{M}}$ (A.8)

We get the last two relations from the first two. Hence, if $\tilde{M}\subset
M$ is a modular inclusion, then (A.8) must be satisfied. When the condition of
inclusion (A.2) is satisfied for $t\leq 0$, it is called -hsmi. For that case,
the commutation relation of modular Hamiltonian is given by
$[K_{M},K_{\tilde{M}}]=-2\pi ip$. Using this $\pm$hsmi, a representation of
the $SL(2,\mathbb{R})$ could be constructed in the following way [52],[53]
Theorem:
Let $M,M_{1},M_{2}$ be Von-Neumann algebras on a Hilbert space $\mathcal{H}$
and $\Omega$ is a cyclic and separating state $\Omega\in\mathcal{H}$. Assume:

  * •

1 $M_{1}\subset M$ is a -hsmi

  * •

2 $M_{2}\subset M$ is a +hsmi

  * •

3 $M_{2}\subset M^{\prime}_{1}$ is a -hsmi

(where $M^{\prime}_{1}$ is the commutant of $M_{1}$.) Then
$\Delta_{M}^{it},\Delta_{M_{1}}^{ir},\Delta^{is}_{M_{2}}$ ,
$t,r,s\in\mathbb{R}$ generate a representation of $SL(2,\mathbb{R})$ where,

$\displaystyle
P\equiv\frac{1}{2\pi}\left(\ln\Delta_{M_{1}}-\ln\Delta_{M}\right);\;K\equiv\frac{1}{2\pi}\left(\ln\Delta_{M_{2}}-\ln\Delta_{M}\right);\;D\equiv\frac{1}{2\pi}\ln\Delta_{M}$
(A.9)

In this way, the algebraic structure of modular inclusion provides an
interesting way to construct chiral part of 2D conformal algebra.

###  A.2 Modular inclusion in vacuum CFT2

Within the set up of section 3, we can explicitly see (3.16) and (3.19)
exhibits both $\pm$hsmi structure. However, (3.16) is constructed out of
$P_{D}$ which is not the modular Hamiltonian. However, we can see $P_{D}$s of
$N$, $N^{\prime}$, $N^{\prime\prime}$ satisfies all the criterion of hsmi.
Hence, in CFT2 vacuum, we define two types of inclusion structure which we
call ‘$K$-inclusion’ and ‘$P_{D}$-inclusion’. $P_{D}$-inclusion
Let us first consider the two nested diamonds $N^{\prime}$ and $N$ where
$N^{\prime}\subset N$. Since $P_{D}$ generates a geometric flow from the left
tip to the right tip of a diamond, the algebra of smaller nested diamond
$N^{\prime}$ remain invariant under the flow of $P_{D}$ of the larger diamond
$N$ i.e. $e^{-iP_{D,N}t}N^{\prime}e^{iP_{D,N}t}\subset N^{\prime}$. In such
case, we call such inclusion $N^{\prime}\subset N$ as the ‘$P_{D}$-inclusion’
which satisfies (A.2). From the algebra of (3.16) we have seen that
$P_{D,N^{\prime}}$ and $P_{D,N}$ indeed satisfy half sided modular inclusion
algebra which is $[P_{D,N},P_{D,N^{\prime}}]=2\pi
i(P_{D,N^{\prime}}-P_{D,N})$. Since $P_{D}$ is self adjoint, we could
construct a self-adjoint $p\equiv P_{D,N}-P_{D,N}$. Using $U(a)$, we can check
the following inclusion property $N^{\prime}=U(1)NU(-1)$, where
$U(a)=e^{iap}$.

Here in the spacetime representation,

$\displaystyle
p_{(N,N^{\prime})}=\frac{z_{31}(z_{2}-\zeta)^{2}}{z_{12}z_{32}}\partial_{\zeta}+\frac{z_{31}(z_{2}-\bar{\zeta})^{2}}{z_{12}z_{32}}\partial_{\bar{\zeta}}$
(A.10)

Hence the action of $U(1)$ on spacetime point $(\zeta,\bar{\zeta})$ gives

$\displaystyle e^{p_{(N,N^{\prime})}}(\zeta,\bar{\zeta})=\left(\frac{\alpha
z_{2}(\zeta-z_{2})+\zeta}{\alpha(\zeta-z_{2})+1},\frac{\alpha
z_{2}(\bar{\zeta}-z_{2})+\bar{\zeta}}{\alpha(\bar{\zeta}-z_{2})+1}\right)\;;\;\alpha=\frac{z_{31}}{z_{12}z_{32}}$
(A.11)

Here this particular $SL(2,\mathbb{R})$ transformation
$\zeta\rightarrow\frac{(\alpha z_{2}+1)\zeta-\alpha
z_{2}^{2}}{\alpha\zeta+1-\alpha z_{2}}$ gives the map from the larger diamond
$N$ to smaller diamond $N$. For instance, the left tip $(z_{3},z_{3})$ maps to
$(z_{1},z_{1})$, upper tip $(z_{2},z_{3})$ maps to that of $N^{\prime}$ i.e
$(z_{2},z_{1})$ and so on. Using the reverse transformation $U(-1)$ one could
construct $N$ from $N^{\prime}$. Similarly we could treat
$N^{\prime\prime}\subset N$ as a -half sided $P_{D}$ inclusion as the
commutators gives an overall minus sign. In the same way, one can define

$\displaystyle
p_{(N,N^{\prime\prime})}=\frac{z_{12}(z_{3}-\zeta)^{2}}{z_{32}z_{31}}\partial_{\zeta}+\frac{z_{12}(z_{3}-\bar{\zeta})^{2}}{z_{32}z_{31}}\partial_{\bar{\zeta}}$
(A.12)

Here the action of $U(-1)$ is given by

$\displaystyle
e^{-p_{(N,N^{\prime\prime})}}(\zeta,\bar{\zeta})=\left(\frac{(\beta
z_{3}-1)\zeta-\beta z_{3}^{2}}{\beta\zeta-\beta z_{3}-1},\frac{(\beta
z_{3}-1)\bar{\zeta}-\beta z_{3}^{2}}{\beta\bar{\zeta}-\beta
z_{3}-1}\right)\;;\;\beta=\frac{z_{12}}{z_{32}z_{31}}$ (A.13)

In this map $z_{2}\rightarrow z_{1}$ and $z_{3}$ remains unchanged and thus it
transforms $N$ to $N^{\prime\prime}$. Hence using $p_{N,N^{\prime}}$,
$p_{N,N^{\prime\prime}}$ consecutively we can map $N^{\prime}$ to
$N^{\prime\prime}$ and vice versa. In this way, $P_{D}$ inclusion gives a
natural way to map between diamonds with the structures like $N$,
$N^{\prime}$, $N^{\prime\prime}$. $K$-inclusion
Let us look at the another set of algebra described in (3.19) which provides
another notion of modular inclusion which we call ‘$K$-modular inclusion’. It
consists of three diamonds $N$, $U$, $L$ such that $U$, $L\subset N$. Since
the total modular Hamiltonian $K$ generates a flow from lower to upper tip,
the algebra of $U$ and $L$ left unchanged under the flow of $K^{N}$, i.e
$e^{-iK^{N}t}(U,L)e^{iK^{N}t}\subset(U,L)$. In a similar way of
$P_{D}$-inclusion, here we can define a self adjoint $p\equiv K^{N}-K^{U,L}$.
For instance, considering the inclusion $U\subset N$, we have

$\displaystyle\frac{z_{12}(z_{3}-\zeta)^{2}}{z_{32}z_{31}}\partial_{\zeta}+\frac{z_{13}(z_{2}-\bar{\zeta})^{2}}{z_{12}z_{32}}\partial_{\bar{\zeta}}$
(A.14)

Hence the action of $U(1)$ gives,

$\displaystyle e^{p_{(N,U)}}(\zeta,\bar{\zeta})=\left(\frac{(\beta
z_{3}-1)\zeta-\beta z_{3}^{2}}{\beta\zeta-\beta z_{3}-1},\frac{\alpha
z_{2}(\bar{\zeta}-z_{2})+\bar{\zeta}}{\alpha(\bar{\zeta}-z_{2})+1}\right)$
(A.15)

In this map, one could see the left tip of $N$ i.e $(z_{3},z_{3})$ maps to
left tip of $U$ i.e $(z_{3},z_{1})$, the right tip $(z_{2},z_{2})$ of $N$ maps
to that of $U$ i.e $(z_{1},z_{2})$, the lower tip $(z_{2},z_{3})$ maps to the
same $(z_{1},z_{1})$ and the upper tip remains unchanged for both diamonds. In
the similar fashion, we could obtain the map from $N$ to $L$ using -half-sided
$K$-inclusion of $L\subset N$.

Also both $K$-inclusion and $P_{D}$-inclusion of the form (3.19) and (3.16),
satisfy $SL(2,\mathbb{R})$ algebra which we describe above as a theorem.

Using the fact that any modular generators of any diamond can be constructed
from the modular algebra of $N^{\prime}$, $N$, $N^{\prime\prime}$ and using
the above mentioned $K$ and $P_{D}$ inclusion, we can now reproduce all causal
diamonds and the fields of them from the modular conformal generators.

###  A.3 Modular inclusion in finite dimensional Hilbert space

Let us consider a finite dimensional quantum system and divide it into four
subsystems $A,A^{\prime},B$and $B^{\prime}$, such that the dimensions of
subsystems are related in the following way:

$\displaystyle H_{tot}=H_{A}\otimes H_{A^{\prime}}\otimes H_{B}\otimes
H_{B^{\prime}};\;d_{A}=d_{A^{\prime}}=N,d_{B}=d_{B^{\prime}}=N^{\prime}.$
(A.16)

Without any loss of generality, we also assume that the total Hilbert space
can be factorized as $H_{tot}=H_{AA^{\prime}}\otimes H_{BB^{\prime}}$, such
that, there exists the state vectors $\ket{\psi}\in H_{tol}$, $\ket{\phi}\in
H_{AA^{\prime}}$ and $\ket{\chi}\in H_{BB^{\prime}}$ which satisfy

$\ket{\psi}=\ket{\phi}_{AA^{\prime}}\otimes\ket{\chi}_{BB^{\prime}}$

We will first show that for such construction of the state $\ket{\psi}$, the
modular inclusion criterion (A.2) will be automatically satisfied. Here we
take $M$ to be the system $AB$ and $\tilde{M}$ to be $A$. To show this, we
first define the corresponding density matrices and reduced density matrices
as follows:

$\displaystyle\rho_{\psi}=\ket{\phi}\bra{\phi}\otimes\ket{\chi}\bra{\chi}$
(A.17)
$\displaystyle\rho_{\\!{}_{AB}}=tr_{\\!{}_{A^{\prime}B^{\prime}}}\rho=tr_{A^{\prime}}\ket{\phi}\bra{\phi}\otimes
tr_{B^{\prime}}\ket{\chi}\bra{\chi}$ (A.18)
$\displaystyle\rho_{\\!{}_{A^{\prime}B^{\prime}}}=tr_{\\!{}_{AB}}\rho=tr_{A}\ket{\phi}\bra{\phi}\otimes
tr_{B}\ket{\chi}\bra{\chi}$ (A.19)
$\displaystyle\rho_{\\!{}_{A}}=tr_{\\!{}_{A^{\prime}BB^{\prime}}}\rho=tr_{A^{\prime}}\ket{\phi}\bra{\phi}$
(A.20)
$\displaystyle\rho_{\\!{}_{A^{\complement}}}=tr_{\\!{}_{A}}\rho=tr_{A}\ket{\phi}\bra{\phi}\otimes\ket{\chi}\bra{\chi}$
(A.21)

The total modular Hamiltonians $K_{M,\tilde{M}}$ or the modular operator
$\Delta_{M,\tilde{M}}$ for the regions $M$ and $\tilde{M}$ are defined as:

$\Delta_{M}\equiv\Delta_{AB}=\rho_{\\!{}_{AB}}\otimes{\rho_{\\!{}_{A^{\prime}B^{\prime}}}}^{-1}$
(A.22)
$\Delta_{N}\equiv\Delta_{A}=\rho_{\\!{}_{A}}\otimes{\rho_{\\!{}_{A^{\complement}}}}^{-1}$
(A.23)

To begin with, we use Schmidt decomposition of $\ket{\phi}_{AA^{\prime}}$ and
$\ket{\chi}_{BB^{\prime}}$ as following:

$\ket{\phi}_{AA^{\prime}}=\sum\limits_{i=1}^{N}C_{i}\ket{i}_{A}\otimes\ket{i}_{A^{\prime}}$
(A.24)

And

$\ket{\chi}_{BB^{\prime}}=\sum\limits_{k=1}^{N^{\prime}}D_{k}\ket{k}_{B}\otimes\ket{k}_{B^{\prime}}$
(A.25)

Hence in this basis, we get

$\rho_{\\!{}_{AB}}=\sum\limits_{i,k=1}^{N,N^{\prime}}\mathinner{\\!\left\lvert
C_{i}\right\rvert}^{2}\mathinner{\\!\left\lvert
D_{k}\right\rvert}^{2}\ket{i}_{A}\ket{k}_{B}\bra{i}_{A}\bra{k}_{B}$
$\rho_{\\!{}_{A^{\prime}B^{\prime}}}=\sum\limits_{i,k}\mathinner{\\!\left\lvert
C_{i}\right\rvert}^{2}\mathinner{\\!\left\lvert
D_{k}\right\rvert}^{2}\ket{i}_{A^{\prime}}\ket{k}_{B^{\prime}}\bra{i}_{A^{\prime}}\bra{k}_{B^{\prime}}$

Using the definition of (A.22), we get:

${\Delta_{AB}}^{it}=(\,\sum\limits_{i,j,k,l}\frac{{\mathinner{\\!\left\lvert
C_{i}\right\rvert}^{2}\mathinner{\\!\left\lvert
D_{j}\right\rvert}^{2}}}{\mathinner{\\!\left\lvert
C_{k}\right\rvert}^{2}\mathinner{\\!\left\lvert
D_{l}\right\rvert}^{2}})\,^{it}(\ket{i}_{A}\ket{j}_{B}\bra{i}_{A}\bra{j}_{B})(\ket{k}_{A^{\prime}}\ket{l}_{B^{\prime}}\bra{k}_{A^{\prime}}\bra{l}_{B^{\prime}}).$
(A.26)

To show the inclusion condition (A.2), we need to define an operator which has
support only in the region A, as :

$\sum_{m,n}\mathcal{O}_{m,n}\ket{m}_{A}\bra{n}_{A}\otimes\mathbb{I}_{\\!{}_{A^{\prime}}}\otimes\mathbb{I}_{\\!{}_{B}}\otimes\mathbb{I}_{\\!{}_{B^{\prime}}}$
(A.27)

Using the definition of $\Delta$, it is straightforward to show that,

${\Delta_{AB}}^{-it}(\sum_{m,n}\mathcal{O}_{m,n}\ket{m}_{A}\bra{n}_{A}\otimes\mathbb{I}_{\\!{}_{A^{\prime}}}\otimes\mathbb{I}_{\\!{}_{B}}\otimes\mathbb{I}_{\\!{}_{B^{\prime}}}){\Delta_{AB}}^{it}=\sum_{m,n}(\frac{{\mathinner{\\!\left\lvert
C_{m}\right\rvert}^{2}}}{\mathinner{\\!\left\lvert
C_{n}\right\rvert}^{2}})^{it}\mathcal{O}_{m,n}\ket{m}_{A}\bra{n}_{A}\otimes\mathbb{I}_{\\!{}_{A^{\prime}}}\otimes\mathbb{I}_{\\!{}_{B}}\otimes\mathbb{I}_{\\!{}_{B^{\prime}}}$
(A.28)

From the above equation, it is clear that the state $\psi$ satisfy the
equation (A.2). With this, we want to check explicitly if it satisfies the
condition of (A.8). To do so, we need to evaluate
$\rho_{\\!{}_{A^{\complement}}}$. However,
$\rho_{\\!{}_{A^{\\!{}_{\complement}}}}^{-1}$ may not be defined. Since we are
calculating $ln\Delta_{A}$, this won’t matter. We can write,

$\displaystyle\ln\Delta_{AB}=\ln\rho_{\\!{}_{AB}}\otimes{\mathbb{I}_{\\!{}_{A^{\prime}B^{\prime}}}}-\mathbb{I}_{\\!{}_{AB}}\otimes{\ln\rho_{\\!{}_{A^{\prime}B^{\prime}}}}$
(A.29)
$\displaystyle\ln\Delta_{A}=\ln\rho_{\\!{}_{A}}\otimes{\mathbb{I}_{\\!{}_{A^{\complement}}}}-\mathbb{I}_{\\!{}_{A}}\otimes{\ln\rho_{\\!{}_{A^{\complement}}}}$
(A.30)

To calculate the commutation, we act $\ln\Delta_{AB}$ and $\ln\Delta_{A}$
consecutively on a basis state
$\ket{i}_{A}\ket{j}_{A^{\prime}}\ket{i}_{B}\ket{j}_{B^{\prime}}$ . One can see
that,
$\ln\Delta_{AB}\ket{i}_{A}\ket{j}_{A^{\prime}}\ket{i}_{B}\ket{j}_{B^{\prime}}=0$
So,

$\bra{j^{\prime}}_{B^{\prime}}\bra{i^{\prime}}_{B}\bra{j^{\prime}}_{A^{\prime}}\bra{i^{\prime}}_{A}(\ln\Delta_{A}\ln\Delta_{AB})\ket{i}_{A}\ket{j}_{A^{\prime}}\ket{i}_{B}\ket{j}_{B^{\prime}}=0$
(A.31)

In the similar manner we can also check that,

$\ln\Delta_{A}\ket{i}_{A}\ket{j}_{A^{\prime}}\ket{i}_{B}\ket{j}_{B^{\prime}}=(\ln\mathinner{\\!\left\lvert
C_{i}\right\rvert}^{2}-\ln\mathinner{\\!\left\lvert
C_{j}\right\rvert}^{2})\ket{i}_{A}\ket{j}_{A^{\prime}}\ket{i}_{B}\ket{j}_{B^{\prime}}$

Since
$\ln\Delta_{AB}\ket{i}_{A}\ket{j}_{A^{\prime}}\ket{i}_{B}\ket{j}_{B^{\prime}}=0$,
it follows from above that,

$\bra{j^{\prime}}_{B^{\prime}}\bra{i^{\prime}}_{B}\bra{j^{\prime}}_{A^{\prime}}\bra{i^{\prime}}_{A}\ln\Delta_{AB}\ln\Delta_{A}\ket{i}_{A}\ket{j}_{A^{\prime}}\ket{i}_{B}\ket{j}_{B^{\prime}}=0$
(A.32)

So from the above equations, we finally get

$\bra{j^{\prime}}_{B^{\prime}}\bra{i^{\prime}}_{B}\bra{j^{\prime}}_{A^{\prime}}\bra{i^{\prime}}_{A}[\ln\Delta_{AB},\ln\Delta_{A}]\ket{i}_{A}\ket{j}_{A^{\prime}}\ket{i}_{B}\ket{j}_{B^{\prime}}=0$
(A.33)

Similarly, one can easily check that

$\bra{j^{\prime}}_{B^{\prime}}\bra{i^{\prime}}_{B}\bra{j^{\prime}}_{A^{\prime}}\bra{i^{\prime}}_{A}\left(\ln\Delta_{AB}-\ln\Delta_{A}\right)\ket{i}_{A}\ket{j}_{A^{\prime}}\ket{i}_{B}\ket{j}_{B^{\prime}}=0$
(A.34)

Therefore in such example we get the desired inclusion properties (since it is
true for any basis state)

$[\ln\Delta_{AB},\ln\Delta_{A}]=\ln\Delta_{AB}-\ln\Delta_{A}$

Thus for such finite dimensional quantum system modular inclusion still holds.

##  Appendix B Commutation relation of modular generators and Virasoro algebra

Here we will reproduce the Virasoro algebra (2.10) from the expression of
$L_{n}$s which is of the form (2.7), using the commutation relations of stress
energy tensors. In CFT2, $TT$ OPE takes the following form,

$\displaystyle
T(z)T(\omega)=\frac{c/2}{(z-\omega)^{4}}+\frac{2T(\omega)}{(z-\omega)^{2}}+\frac{\partial
T(\omega)}{z-\omega}+\text{regular terms}$ (B.1)

Using the Sokhotski-Plemelj formula, after analytically continuing to
lightcone coordinate by $i\epsilon$ prescription, we get the following stress
tensor commutators which we need to evaluate the $L_{n}$ commutators.

$\displaystyle[T(\zeta),T(\omega)]=2\pi
i[-\frac{c}{12}\partial^{3}_{\omega}\delta(\omega-\zeta)+\delta(\omega-\zeta)\partial_{\omega}T(\omega)+2\partial_{\omega}\delta(\omega-\zeta)T(\omega)]$
(B.2) $\displaystyle[\bar{T}(\bar{\zeta}),\bar{T}(\bar{\omega})]=-2\pi
i[-\frac{c}{12}\partial^{3}_{\bar{\omega}}\delta(\bar{\omega}-\bar{\zeta})+\delta(\bar{\omega}-\bar{\zeta})\partial_{\bar{\omega}}\bar{T}(\bar{\omega})+2\partial_{\bar{\omega}}\delta(\bar{\omega}-\bar{\zeta})\bar{T}(\bar{\omega})]$
(B.3)

Inserting this into the commutator of $\mathbb{L}_{m}$, we have

$\displaystyle[\mathbb{L}_{m},\mathbb{L}_{n}]$
$\displaystyle=\int^{\infty}_{-\infty}d\zeta\int^{\infty}_{-\infty}d\omega\frac{(z_{2}-\zeta)^{-m+1}(\zeta-
z_{3})^{m+1}}{(z_{2}-z_{3})^{2}}\left(\frac{z_{21}}{z_{31}}\right)^{m+n}(z_{2}-\omega)^{-n+1}(\omega-
z_{3})^{n+1}$ $\displaystyle\times 2\pi
i[-\frac{c}{12}\partial^{3}_{\omega}\delta(\omega-\zeta)+\delta(\omega-\zeta)\partial_{\omega}T(\omega)+2\partial_{\omega}\delta(\omega-\zeta)T(\omega)]$
(B.4)

First let us consider the last two term (ignoring the $c$ term) of the
$[T,T]$, we have

$\displaystyle\frac{1}{2\pi i}[\mathbb{L}_{m},\mathbb{L}_{n}]^{(1)}$
$\displaystyle=-\int^{\infty}_{-\infty}d\zeta\frac{(z_{2}-\zeta)^{-m-n+2}(\zeta-
z_{3})^{m+n+2}}{(z_{2}-z_{3})^{2}}\left(\frac{z_{21}}{z_{31}}\right)^{m+n}\partial_{\zeta}T(\zeta)+2\text{(T.D)}_{1}$
$\displaystyle-2\int^{\infty}_{-\infty}d\zeta\frac{(z_{2}-\zeta)^{-m+1}(\zeta-
z_{3})^{m+1}}{(z_{2}-z_{3})^{2}}\left(\frac{z_{21}}{z_{31}}\right)^{m+n}T(\zeta)$
$\displaystyle\times\left[(n-1)(z_{2}-\zeta)^{-n}(\zeta-
z_{3})^{n+1}+(n+1)(z_{2}-\zeta)^{-n+1}(\zeta-z_{3})^{n}\right]$
$\displaystyle=2\text{(T.D)}_{1}-\text{(T.D)}_{2}$
$\displaystyle+\int^{\infty}_{-\infty}d\zeta\frac{(z_{2}-\zeta)^{-m-n+1}(\zeta-
z_{3})^{m+n+1}}{(z_{2}-z_{3})^{2}}\left(\frac{z_{21}}{z_{31}}\right)^{m+n}\left[(m+n-2)(\zeta-
z_{3})+(m+n+2)(z_{2}-\zeta)\right]T(\zeta)$
$\displaystyle-\int^{\infty}_{-\infty}d\zeta\frac{(z_{2}-\zeta)^{-m-n+1}(\zeta-
z_{3})^{m+n+1}}{(z_{2}-z_{3})^{2}}\left(\frac{z_{21}}{z_{31}}\right)^{m+n}\left[(2n-2)(\zeta-
z_{3})+(2n+2)(z_{2}-\zeta)\right]T(\zeta)$
$\displaystyle=2\text{(T.D)}_{1}-\text{(T.D)}_{2}+(m-n)\int^{\infty}_{-\infty}d\zeta\frac{(z_{2}-\zeta)^{-m-n+1}(\zeta-
z_{3})^{m+n+1}}{z_{2}-z_{3}}\left(\frac{z_{21}}{z_{31}}\right)^{m+n}T(\zeta)\zeta$
(B.5)

We can identify the last term as the $(m-n)\mathbb{L}_{m+n}$. Here
$(T.D)_{1,2}$ are two total derivative terms coming from the intermediate
steps of the partial integration. Here,

$\displaystyle\text{(T.D)}_{1}=\int^{\infty}_{-\infty}d\zeta\frac{(z_{2}-\zeta)^{-m+1}(\zeta-
z_{3})^{m+1}}{(z_{2}-z_{3})^{2}}\left(\frac{z_{21}}{z_{31}}\right)^{m+n}\bigg{[}(z_{2}-\omega)^{-n+1}(\omega-
z_{3})^{n+1}\delta(\omega-\zeta)T(\omega)\bigg{]}^{\omega=\infty}_{\omega=-\infty}$
(B.6)

Due to the Dirac delta function, the total derivative term inside the bracket,
vanishes. Hence this term vanishes. The other term is,

$\displaystyle\text{(T.D)}_{2}=\left(\frac{z_{21}}{z_{31}}\right)^{m+n}\left[\frac{(z_{2}-\zeta)^{-m-n+2}(\zeta-
z_{3})^{m+n+2}}{(z_{2}-z_{3})^{2}}T(\zeta)\right]^{\zeta=\infty}_{\zeta=-\infty}$
(B.7)

To analyze this term, we need to look at the behavior of the stress tensor
near spacetime infinity. From the transformation property of stress tensor we
know, $T^{\prime}(\zeta^{\prime})=\left(\frac{\partial z^{\prime}}{\partial
z}\right)^{-2}+$ Schwarzian derivative term. We choose a global
transformation171717Hence we can get rid of the Schwarzian derivative term.
$\zeta^{\prime}=\frac{a\zeta+b}{c\zeta+d}$ at
$\zeta=\zeta_{0}=-\frac{d}{c}+\epsilon$, such that
$\zeta^{\prime}_{0}\sim\frac{1}{\epsilon}$. For such choice, we get the
transformation of stress tensor in the following way,

$\displaystyle
T^{\prime}(\zeta^{\prime}_{0})=(c\zeta_{0}+d)^{4}T(\zeta_{0})\sim\frac{1}{\epsilon^{4}}T(\zeta_{0})$
(B.8)

Hence, for $\epsilon\rightarrow 0$, we get the behavior of the stress tensor
near infinity as $T(\zeta)|_{\zeta\rightarrow\infty}\sim\frac{1}{\zeta^{4}}$.
Using this, if we look at the term $(T.D)_{2}$, we get,

$\displaystyle\text{(T.D)}_{2}=\left(\frac{z_{21}}{z_{31}}\right)^{m+n}\lim_{\Lambda\rightarrow\infty}\left[\frac{(z_{2}-\Lambda)^{-m-n+2}(\Lambda-
z_{3})^{m+n+2}}{(z_{2}-z_{3})^{2}}\frac{1}{\Lambda^{4}}-\Big{(}\Lambda\rightarrow-\Lambda\Big{)}\right]=0$
(B.9)

Hence, both the total derivative terms vanishes. Let us now see the
contribution coming from the central charge($c$) part of the stress tensor
commutator.

$\displaystyle\frac{1}{2\pi i}[\mathbb{L}_{m},\mathbb{L}_{n}]^{(2)}$
$\displaystyle=-\frac{c}{12}\left(\frac{z_{21}}{z_{31}}\right)^{m+n}\int^{\infty}_{-\infty}d\zeta\int^{\infty}_{-\infty}d\omega\frac{(z_{2}-\zeta)^{-m+1}(\zeta-
z_{3})^{m+1}}{(z_{2}-z_{3})^{2}}(z_{2}-\omega)^{-n+1}(\omega-
z_{3})^{n+1}\partial^{3}_{\omega}\delta(\omega-\zeta)$ (B.10)

Let us denote the constant term
$\frac{c}{12}\left(\frac{z_{21}}{z_{31}}\right)^{m+n}\frac{1}{(z_{2}-z_{3})^{2}}\equiv
A$. In a similar fashion of the previous calculation, after some simple
algebraic steps the final integration is of the following form

$\displaystyle\frac{1}{2\pi i}[\mathbb{L}_{m},\mathbb{L}_{n}]^{(2)}$
$\displaystyle=\text{(T.D)}_{3}+\text{(T.D)}_{4}+\text{(T.D)}_{5}+n(n^{2}-1)A(z_{2}-z_{3})^{3}\int^{\infty}_{-\infty}d\zeta(z_{2}-\zeta)^{-m-n-1}(\zeta-
z_{3})^{m+n-1}$ (B.11)

Here, the total derivative terms $(\text{T.D})_{3,4,5}$ are getting vanished
due to the presence of dirac delta function and it’s derivatives as we argued
before. After carefully choosing a contour, we get the final result of the
complex integration as (we choose $\text{Re}[z_{2}]>0,\text{Re}[z_{3}]<0$)

$\displaystyle\frac{1}{2\pi
i}[\mathbb{L}_{m},\mathbb{L}_{n}]^{(2)}=\frac{c}{12}n(n^{2}-1)\left(\frac{z_{21}}{z_{31}}\right)^{m+n}\frac{\left(\frac{-z_{2}}{z_{2}}\right)^{m+n}-\left(\frac{-z_{3}}{z_{3}}\right)^{m+n}}{m+n};$
(B.12)

This term vanishes for any $m+n\neq 0,\in\mathbb{Z}$. To extract the
contribution for $m+n=0$, we can perform an analytic continuation by choosing
$m+n=\epsilon$, taking $\epsilon\rightarrow 0$. This gives,

$\displaystyle\frac{1}{2\pi
i}[\mathbb{L}_{m},\mathbb{L}_{n}]^{(2)}=\frac{c}{12}n(n^{2}-1)\lim_{\epsilon\rightarrow
0}\left(\frac{z_{21}}{z_{31}}\right)^{\epsilon}\frac{(-1)^{\epsilon}-(-1)^{-\epsilon}}{\epsilon}=\frac{c}{12}n(n^{2}-1)2\pi
i$ (B.13)

Hence, combining $[\mathbb{L}_{m},\mathbb{L}_{n}]^{(1)}$ and
$[\mathbb{L}_{m},\mathbb{L}_{n}]^{(2)}$, we finally have

$\displaystyle[\mathbb{L}_{m},\mathbb{L}_{n}]=(m-n)\mathbb{L}_{m+n}+\frac{c}{12}n(n^{2}-1)\delta_{m+n,0}$
(B.14)

Here, $\mathbb{L}_{m,n}$s are redefined as
$\mathbb{L}_{m,n}\rightarrow\frac{1}{2\pi i}\mathbb{L}_{m,n}$.

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