1. Introduction and Summary
With the help of radial quantization, local operators in a conformal field theory in d dimensions (CFT) can be mapped in a one-to-one fashion to states in the same CFT on , with labelling the time direction. This takes a somewhat trivial form in . Since is a collection of two points, the states live in the Hilbert space of two copies of the CFT. On the other hand the absence of spatial separation makes all operators in the Hilbert space of a single copy of the CFT local. Thus the state-operator correspondence reduces to the standard map between operators in and states in the tensor product of two copies of . In particular the identity operator gets mapped to the maximally entangled state between the two copies of .
This picture takes a geometric form for a class of CFT
which are dual to string theory on AdS
for some compact manifold
. These geometries typically arise as the near horizon geometries of black holes in the extremal limit [
1,
2]. In this case
contains the compactification manifold as well as the angular coordinates of the asymptotic space-time. When we represent global AdS
as an infinite strip, the two copies of the CFT
live on the two boundaries of the strip. Furthermore as argued in [
3,
4] and reviewed in
Section 2, a single copy of the dual CFT
just consists of a finite number (
N) of degenerate states representing the ground states of the black hole in a given charge sector. Thus two copies of the CFT
living on the two boundaries of AdS
will contain
states. By AdS/CFT correspondence [
5] we expect the dual string theory on global AdS
to also contain
states. One of these states is easy to identify—the Hartle-Hawking vacuum of string theory on AdS
[
6]. This is dual to the identity operator in CFT
and hence represents the maximally entangled state between the two copies of the CFT
. Our goal in this note will be to identify possible origin of the other states in string theory on AdS
which are expected to exist according to the AdS
/CFT
correspondence.
It is generally expected that AdS
cannot support any finite energy excitation since this will destroy the asymptotic boundary condition [
7]. (This argument assumes that
is compact. If
contains a non-compact piece, e.g.,
, then there is no gap in the spectrum and hence in the infrared limit we can get finite energy excitations. We can use local fields to generate the corresponding states in string theory in AdS
, leading to non-trivial correlators [
8,
9,
10].) This is not a problem for us since in CFT
all states are of the same energy (which we can take to be zero by a shift) and hence we need to look only for zero energy excitations in AdS
. However, this rules out the usual procedure for constructing excitations in AdS
using local fields in the bulk [
11,
12], since this typically produces finite energy excitations. Some suggestions for constructing zero energy excitations in AdS
were made in [
7]. However the fragmented geometries of the type discussed in [
7] will be absent if the charge carried by the black hole is primitive, since this prevents the total flux to be split into multiple aligned fluxes each through one AdS
throat. This still leaves open the possibility of contribution from the scaling solutions described in [
13,
14,
15,
16] involving three or more throats, with fluxes through different throats aligned along different directions in the charge lattice. But given that the phase space associated with these configurations has finite volume preventing the centers to come arbitrarily close to each other in the quantum theory [
17,
18,
19], it is more natural to count their effect as part of multi-centered black holes rather than as part of a single AdS
throat. In any case in
supersymmetric string theories there is reasonable evidence that solutions with three or more centers do not contribute to the index [
20,
21], and hence we must look for different states.
To look for clues for where the zero energy states might come from, let us examine the state-operator correspondence in the dual CFT
. A linearly independent basis of operators in the CFT
is provided by the set of all
Hermitian matrices. We shall find it more convenient to work with
unitary matrices instead; if we have sufficiently large number of these matrices then any other matrix can be expressed as linear combinations of these matrices. The correlation functions in CFT
on
are then traces of products of these matrices. Furthermore, since all the
N states in CFT
are degenerate, these
transformations generate exact symmetries of the theory. By AdS
/CFT
correspondence this symmetry must be present in the dual string theory as well. Thus to compute these correlation functions in the dual string theory on AdS
we represent Euclidean global AdS
as a disk so that the boundary on which CFT
lives becomes a circle, and then compute a
twisted partition function in which we require the fields to satisfy a twisted boundary condition along the boundary of AdS
[
22,
23], the twist being related to the product of the matrices in CFT
whose correlation function we wish to compute. This suggests that when we represent AdS
as a strip, we can construct the states in string theory on AdS
via Euclidean path integral as in the case of Hartle-Hawking state, albeit with a twisted boundary condition in the asymptotic past. This way the matrix elements between these states naturally produces the twisted partition function.
Formally this prescription gives a complete map between the CFT
operators and correlation functions and the corresponding quantities in string theory on AdS
. The main problem of realizing this idea is that at present we do not know of any explicit construction of such
symmetries in string theory on AdS
. However there are special cases where we can realize a small part of this symmetry. Typically as we move around in the moduli spaces of a supersymmetric string theory, we encounter special points at which there are enhanced discrete symmetries (not to be confused with enhanced continuous symmetries). Since typically the black hole microstates get transformed into each other under this discrete symmetry, this has a non-trivial embedding in
. The dual string theory on AdS
also has this symmetry manifest and we can use this to construct the twisted states in AdS
. While this is far from providing a complete construction of all the states of string theory on AdS
, this at least demonstrates that it is possible to construct non-trivial states in AdS
without destroying the asymptotic boundary conditions. To this end we note that even if the near horizon geometry possesses an enhanced discrete symmetry, it need not be a symmetry of the asymptotic theory where the moduli can take different values. Thus our ability to construct these special states is not tied to the existence of some symmetry at infinity that allows us to distinguish different black holes trivially by doing appropriate scattering experiments, e.g., in [
24,
25].
This picture also incorporates naturally the dual interpretation of the entropy of an extremal black hole. It has been known since the work of Bekenstein and Hawking that black holes carry entropy. One natural explanation of this entropy is that a single black hole represents a large collection of quantum states, and the black hole entropy is given by the logarithm of the degeneracy of microstates the black hole represents. Indeed one of the major successes of string theory has been to reproduce the black hole entropy from the counting of states in the microscopic description of the black hole [
26,
27]. On the other hand the geometry of the black hole, which includes a horizon, suggests an alternate interpretation: the black hole entropy represents the result of entanglement between the degrees of freedom living outside the horizon and the degrees of freedom living inside the horizon [
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40]. (For a different viewpoint on the relationship between black hole entropy and entanglement see [
41] and references therein.) In the framework of AdS
/CFT
correspondence we see that both interpretations are equally good. The black hole entropy
can be interpreted as the logarithm of the degeneracy of a single copy of the CFT
living on one of the boundaries of AdS
, or as the entanglement entropy between the two copies of CFT
living on the two boundaries of AdS
in the Hartle-Hawking vacuum. Since the latter corresponds to a maximally entangled state, its entanglement entropy is given by
.
This observation of course is not new—it is the zero temperature version of the well-known connection between black holes and thermofield dynamics. Given any thermal system, there is a standard doubling trick that allows us to express the thermal averages as quantum mechanical expectation values in an auxiliary system containing two copies of the original Hilbert space [
42,
43,
44,
45,
46,
47,
48], and the thermal entropy of the original system can be regarded as the entanglement entropy of the auxiliary system. This correspondence was exploited in [
33,
49,
50,
51,
52] to identify the two copies of the Hilbert space as being associated with the two boundaries of the extended space-time for a black hole solution. In a related development it was observed in [
40] that if we take the global AdS
space-time that arises in the near horizon geometry of a black hole in the extremal limit and calculate the entanglement entropy between the quantum theories living on the two boundaries of this global AdS
, then in the classical limit the result agrees with the Wald entropy. The argument given above shows that this must continue to hold in the full quantum theory. While we have a prescription for computing the degeneracy of states in the full quantum theory as a partition function of string theory in AdS
[
3], the prescription of [
40] for the holographic computation of the entanglement entropy in CFT
involves evaluating the partition function of string theory on a space-time with conical defect. At the classical level the two entropies calculated using these two apparently different computations give the same result, but it is not clear that this equality will continue to hold in the full quantum theory. In
Section 7 we suggest a different approach to computing the entanglement entropy of CFT
holographically that does not entail any conical defect and makes the equality of statistical and entanglement entropy manifest even in the quantum theory.
2. CFT and Its State-Operator
Correspondence
We shall begin by reviewing the properties of the CFT
dual to string theory on AdS
that arises as the near horizon geometry of some extremal black hole. By the usual rules of AdS/CFT correspondence this CFT
must be given by the infrared limit of the quantum mechanics describing the dynamics of the brane system producing the black hole. In known examples, e.g., the D1-D5-p system producing a five dimensional black hole [
26,
27], or the D1-D5-p-KK monopole system producing a four dimensional black hole [
53,
54,
55,
56], the spectrum of the underlying quantum system has a gap separating the BPS ground states from the first excited states in a fixed charge sector. The gap is small when the charges are large, but is nevertheless non-zero. As a third example consider a BPS black hole in type IIB string theory compactified on a Calabi-Yau 3-fold
, described as a configuration of 3-brane wrapped on an appropriate supersymmetric three cycle inside
. The quantum mechanics describing the system is a (0 + 1) dimensional sigma model with the moduli space of supersymmetric 3-cycles as target space. Again as long as this moduli space is compact we expect the spectrum of the quantum theory to be discrete, and there will be a gap between the supersymmetric ground states and the first excited state. We shall assume that this is always the case for the quantum system describing an extremal black hole. Then in the infrared limit only the ground states of this quantum mechanics will survive, and the CFT
will consist of a finite number
N of degenerate states.
The usual state-operator correspondence in a
d dimensional conformal field theory relates every local operator in the conformal field theory to a state in the conformal field theory on
. For
this is usually achieved by the standard map from
to
via the coordinate transformation that takes the north and the south poles of
to
of
. In this case local operators inserted at the south pole of
create the corresponding states at
on
. The state-operator correspondence in
works in a more or less similar way. First the map from
to
is achieved via the coordinate transformation
Indeed this takes the circle labelled by
θ to a pair of lines
where
corresponds to the pair of points
and
is labeled by
τ. The points
are mapped to
, the segment
is mapped to the line at
and the segment
is mapped to the line at
. Thus CFT
on
actually corresponds to two copies of the CFT
. On the other hand since for
there is no notion of spatial separation, every operator acting on the Hilbert space
of a single copy of the CFT
can be regarded as a local operator. Thus we are looking for a map between the set of operators acting on a single copy of
to the set of states living on two copies of
at the two boundaries
. It is straightforward to construct such a map—the operator
inserted at
on
creates the state
on
at
. Here
denotes a complete set of orthonormal basis states in
, the subscripts
and
denote the two copies of
, and
denotes a state in
. For this state the density matrix in the Hilbert space of the first copy of CFT
, obtained by tracing over the states in the second copy, is given by
Given two such states
and
, we have:
This can be interpreted as the two point function of
and
in CFT
on
, in accordance with the usual rules of state-operator correspondence in conformal field theories.
A special state corresponding to the identity operator in the CFT
is
We shall refer to this state as the vacuum state although all states have equal energy. The corresponding density matrix is
, showing that it is a maximally entangled state. This however is not the only maximally entangled state—it follows from (
3) that for any unitary operator
the corresponding state
has density matrix
, and hence describes a maximally entangled state. Furthermore (
4) shows that for unitary operators
W and
V, the inner product
is given by
.
Note that in CFT
,
may be expressed as
where
denotes the operator
W acting on the first copy of the Hilbert space. Thus CFT
correlation functions can be interpreted as the expectation values of the operators acting on the first copy of the CFT
in the vacuum state.
Our goal will be to seek possible representation of these states in dual string theory on AdS.
3. AdS Space in Different Coordinates
In this section we shall review some facts about the near horizon geometry of black holes in the extremal limit. For higher dimensional branes one usually takes a brane solution at zero temperature and then takes the near horizon limit to get an AdS space-time. This corresponds to looking at excitations whose energies are small from the point of view of the asymptotic observer but large compared to the temperature of the brane. This is not a sensible limit for a black hole since, as reviewed in
Section 2, the black hole quantum mechanics has a gap that separates the ground state from the first excited state, and so the only low energy excitations are zero energy excitations. So the sensible infrared limit is to take the energy scale to zero as we take the temperature to zero. (I wish to thank Hong Liu for a discussion on this point.) This can be achieved by taking the extremal limit in an appropriately rescaled coordinate system in which the two horizons remain finite coordinate distance away from each other [
3,
57]. In this limit part of the near horizon geometry of the black hole involving the time and the radial coordinates takes the form [
7,
58,
59]
where
a is some constant. Here, up to a rescaling,
r and
t can be identified as the radial and the time variables of the full black hole solution. The inner and the outer horizons are at
. The metric (
7) describes a locally AdS
space-time. This can be extended to global AdS
with the help of the coordinate transformation [
7]:
In this coordinate system the metric takes the form:
The range of
can be taken to be
,
). This space has two boundaries, at
and at
. These two boundaries lie on opposite sides of the horizon
of the original metric (
7). The asymptotic boundary
in the original metric (
7) lies at
.
Figure 1 shows AdS
in the
coordinate system where the locations of the horizons at
have been shown by the dashed line [
7,
60].
Figure 1.
Global AdS and the location of the horizon(s). The two vertical solid lines label the two boundaries of AdS at (left) and (right). The dashed lines label the locations of the event horizons of the original black hole.
Figure 1.
Global AdS and the location of the horizon(s). The two vertical solid lines label the two boundaries of AdS at (left) and (right). The dashed lines label the locations of the event horizons of the original black hole.
Let us now consider the Euclidean version of the metrics (
7) and (
9). The Euclidean version of the metric (
7) is obtained by replacing
t by
. This gives the metric
Introducing new coordinate
we can express the metric as
In this coordinate it is clear that absence of conical singularity at
(
) requires
θ to be a periodic coordinate with period
. The resulting two dimensional space spanned by
with
,
describes a unit disk.
Euclidean version of the metric (
9) is obtained by replacing
T by
. This gives
This time there is no periodicity requirement of
τ. This describes an infinite strip spanned by
with
,
.
Even though for the Lorentzian signature the coordinates
for
cover only a patch of the global AdS
spanned by the coordinates
in (
9), the Euclidean spaces (
10) and (
12) have an exact one-to-one map:
This is the standard one-to-one conformal map between the unit disk and the infinite strip as shown in
Figure 2. The segment of the boundary of the disk
gets maps to the
boundary of the strip, and the segment
gets mapped to the
boundary of the strip. In fact for
this reduces to the standard map from
to
described in (
1).
Figure 2.
Conformal map from unit disk to the strip. The left boundary of the strip is at and the right boundary is at .
Figure 2.
Conformal map from unit disk to the strip. The left boundary of the strip is at and the right boundary is at .
4. AdS/CFT Correspondence
Next we shall review some aspects of the AdS
/CFT
correspondence proposed in [
3]. Consider an AdS
geometry arising as the near horizon limit of an extremal black hole carrying a fixed charge. Then this is dual to the CFT
obtained as the infrared limit of the brane system describing the dynamics of the black hole. If CFT
has
N states, then the black hole entropy, identified as the logarithm of the number of states of the CFT
, is given by
. (If the black hole solution has hair modes then we must remove their contribution while counting
N [
61,
62].)
An algorithm for computing this entropy using the bulk description was given in [
3,
4]. For this we consider the Euclidean AdS
geometry given in (
10) and denote by
the partition function of string theory in AdS
, computed with the natural boundary condition that requires us to fix the electric fields at infinity and integrate over the
r independent modes of the gauge fields. (As discussed in [
3], this requires introducing a Wilson loop operator along the boundary of AdS
while computing
.) Due to infinite size of the Euclidean AdS
space this partition function is divergent, so we need to regularize the divergence by putting a cut-off on
r, say
or equivalently
. This makes AdS
have a finite volume and the boundary of AdS
, situated at
, have a finite length which we shall call
L. Now by AdS/CFT correspondence
should be given by the partition function of the CFT
living on the boundary circle at
. The latter in turn is given by
, where
H is the Hamiltonian of CFT
and
is the energy of the
N degenerate states of CFT
. Thus we have
This suggests that in order to calculate
N, we first calculate
and then extract its finite part by expressing it as
for some finite constants
and
C in the
limit. In that case
C can be identified with
and
, called the quantum entropy function, can be identified as the ground state degeneracy
N of the black hole. This gives a complete prescription for computing the black hole entropy in the bulk theory. In the classical limit
defined this way reproduces the exponential of the Wald entropy [
3]. In principle quantum corrections may be computed directly [
63,
64,
65], or, for supersymmetric black holes, using localization [
66]. Significant advances towards computing
using localization techniques have been made recently [
67].
By adjusting the boundary terms in the action describing string theory on AdS we can make the constant C vanish, so that can be directly identified as the degeneracy of CFT. This corresponds to a constant shift in the definition of the energy of CFT to make vanish. For simplifying the notation we shall proceed with this convention although we can always include the explicit dependence in all the equations below if so desired.
Can we use the bulk description to calculate other observables in CFT
? Since CFT
consists of a finite number of degenerate states, the only observables are
matrices
M acting on this
N dimensional vector space. Let us focus on the cases of unitary matrices which generate
transformations in this
N dimensional vector space. Since the all
N states in CFT
are degenerate,
is an exact symmetry of CFT
. Hence it must also exist as an exact symmetry of the dual string theory on AdS
, and corresponding to any
element
W there must be a corresponding transformation (also denoted by
W) acting on the variables in the dual string theory. (In AdS/CFT correspondence global symmetries in the boundary theory arise as local symmetries of the bulk theory. We shall not try to make this distinction here since we shall use the
transformations to twist the boundary condition, and for this only the global part of the group is relevant anyway. However it is important that there should not be any dynamical
gauge fields in the bulk since this will force the black hole to carry fixed charges under the Cartan generators of
[
3].) Computing
in CFT
will then correspond to evaluating the partition function of string theory on AdS
with a
W twisted boundary condition on the bulk fields under
.
While this gives a way to relate
in the boundary theory to a specific quantity in the bulk theory, in general the action of
W on the bulk fields is not known. But in special cases, e.g., when
W represents a known
symmetry generator of the theory, there is a natural lift of the action of
W to the bulk fields. In this case
in CFT
can be identified as a
twisted partition function in the bulk theory. Since the boundary circle of the Euclidean AdS
space described by the metric (
11) is contractible in the interior, a
W twisted boundary condition is not allowed there. Thus the original AdS
geometry does not contribute to this amplitude. But a
orbifold of the AdS
geometry does contribute and gives a non-zero answer for
[
22]. This prescription has passed non-trivial tests in a class of
supersymmetric string theories where an independent microscopic computation of
is possible [
22,
23]. (Although the computation of
in the bulk theory will be the same as the one described in [
22], the spirit in which we want to use this is different. In [
22,
23] the asymptotic moduli were adjusted to also have the unbroken
symmetry so that we could compute
microscopically and compare with the macroscopic result. In contrast, here we want to interpret
as an observable in the near horizon theory irrespective of whether or not it is a symmetry of the asymptotic theory. For this we only need to adjust the asymptotic moduli to be in a certain subspace which under attractor flow [
68,
69,
70] approaches the point of enhanced discrete symmetry at the horizon.) Note that for
we recover the original partition function
on the bulk side and
on the CFT
side.
5. States in String Theory on AdS
The result of the previous sections suggests a way of associating the states in two copies of CFT
with states in string theory on AdS
. Let us for definiteness work with states in CFT
on
of the form
with unitary operators
W. To the states
and
we want to associate wave-functions
and
in string theory such that the inner product of
and
generates the CFT
two point function
. The latter in turn is described by the path integral of string theory in AdS
, with the boundary condition that as
near the boundary, the fields are twisted by
. This can be achieved by defining
to be generated by the result of string theory path integral over the half disk (or semi-infinite strip) with a cut corresponding to the transformation
W that reaches the boundary of the half disk (see
Figure 3). The inner product
will then be obtained by gluing the two half disks, with cuts
W and
V along the boundaries, along their common diameter. This is given by the path integral over the whole disk with a twisted boundary condition by
along the boundary, as required.
Figure 3.
Generating a state in string theory on AdS from a state in the two copies of the Hilbert space of CFT. The thick semi-circular line is the boundary of AdS whereas the thin diameter is the line on which the string fields, appearing in the argument of , live. The dashed line reaching the boundary of AdS denotes a cut which relates the field configurations on the right of the cut to those on the left of the cut by a transformation by W.
Figure 3.
Generating a state in string theory on AdS from a state in the two copies of the Hilbert space of CFT. The thick semi-circular line is the boundary of AdS whereas the thin diameter is the line on which the string fields, appearing in the argument of , live. The dashed line reaching the boundary of AdS denotes a cut which relates the field configurations on the right of the cut to those on the left of the cut by a transformation by W.
Note that in
Figure 3 the cut can either end in the interior of the half disk if it is an allowed configuration in string theory, or reach the diameter on which the wave-function
is defined. This is reminiscent of the sum over geometries in [
33]. The necessity for allowing the cut to reach the diameter can be seen in the computation of
. The dominant contribution to this amplitude comes from the configuration where the cut extends all the way across the disk, representing the usual AdS
geometry without any twist.
While this gives an abstract prescription for associating the states living in two copies of the Hilbert space of CFT to states of string theory on AdS, in general we cannot explicitly construct these states since the action of W on the bulk fields is not known. However for the special cases when W can be associated with some known symmetry generator of string theory on AdS, we can explicitly construct the corresponding state; the path integral will be over all field configurations whose boundary values jump by the action of this discrete symmetry transformation W at some point on the boundary. Even though this is a special case, this at least demonstrates that it is possible for global AdS to admit non-trivial quantum states.
It is worth emphasizing that this discrete symmetry need not be a symmetry away from the horizon—the asymptotic moduli could be in a configuration that breaks this symmetry while the attractor mechanism pulls the near horizon geometry towards a configuration that is invariant under this symmetry. From this point of view this discrete symmetry is on the same footing as the rest of the proposed symmetry, in that this symmetry is present in the near horizon geometry but not necessarily present away from the horizon.
7. Entanglement vs. Statistical
Entropy
When
W is the identity operator then there is no cut on the disk, and the corresponding state in string theory is simply the Hartle-Hawking state obtained by string theory path integral over the half disk without any cut. In the boundary theory this represents the state
defined in (
5). (This is precisely the structure of the entangled state found in [
40] for the special case of extremal BTZ black holes by starting with a finite temperature system and then taking its zero temperature limit.) Thus the Hartle-Hawking state is the maximally entangled state in the two copies of the CFT
living on the two boundaries of global AdS
. The corresponding entanglement entropy is given by
where
N is the dimension of the Hilbert space of CFT
. This agrees with the statistical entropy defined as the logarithm of the degeneracy of states in the CFT
. This is a special case of the general result of [
33], and confirms the explicit finding of [
40] that the entropy of an extremal black hole can be interpreted as the entanglement entropy between the two copies of CFT
living on the two boundaries of the global AdS
. In fact not only the state (
5) has entanglement entropy
, any twisted state
introduced earlier has density matrix
and hence entanglement entropy
.
The equality between the statistical and entanglement entropy is obvious in the CFT
. This can also be seen in the bulk theory as follows. The standard procedure for computing the entanglement entropy in a CFT uses the definition:
Thus for computing
using the bulk description we need to find a way of computing
holographically. This can be done as follows. Since the CFT
lives on a pair of points and since the state we are interested in is the Hartle-Hawking state, we can construct this as a path integral in the CFT
on a line of length
with boundary conditions
and
at the two ends labelling the states of the two CFT
’s. Eventually we want to take
(although since all states have the same energy the
L dependence is trivial). Now to compute the unnormalized density matrix
by taking the trace over states of the second system we take another copy of this line segment with boundary conditions
and
and glue the second ends of the two segments after identifying
with
. This leaves us with a line segment of length
L with boundary conditions
and
at the two ends. For computing
we simply take
n copies of this line segment and glue the primed end of the
i-th segment with the unprimed end of the
-th segment for
. This gives a line segment of length
with boundary conditions
and
at the two ends. Finally to calculate
we join the two ends of this line segment by identifying
with
and carry out the path integral over the fields of the CFT
on this circle. Thus the holographic computation of
will involve calculating the partition function of string theory over all spaces, each of which having a boundary circle of length
and approaching the AdS
geometry asymptotically. If we denote this contribution by
then we have
Hence from (
16) we get
can be identified with the
given in (
14). To compute
we note that the leading contribution to this partition function comes from the Euclidean AdS
geometry (
10) itself with the cut-off on
r adjusted to produce the appropriate boundary length
. Even the full quantum contribution to the partition function will be given by the quantum contribution to
with a different infrared cut-off so that the boundary has length
. Thus we have from (
10)
Substituting this into (
18) gives
which is manifestly equal to the statistical entropy.
Note that this approach differs from the holographic prescription of [
40,
74] for the computation of
. In the latter approach while computing
we compute the partition function of of string theory on
n-fold cover of AdS
. This has a conical defect at the center. On the other hand in our approach we first take an
n-fold cover of the boundary circle and then integrate over all possible bulk space-time with this boundary condition. The leading saddle point is the AdS
space itself with a different infrared cut-off. Although classically the two approaches give the same result, it is not clear
a priori if the agreement will continue to hold in quantum string theory. Indeed at present we do not know how to define string theory on a space with conical defect for an arbitrary defect angle. Clearly understanding the relationship between these two computations will be desirable since it might also given us a clue as to when, why and how the holographic prescription [
40,
74,
75,
76] for computing entanglement entropy works.