AdS/CFT prescription for angledeficit space and winding geodesics
HJE
prescription for anglede cit space and winding geodesics
Irina Ya. Aref'eva 0 1
Mikhail A. Khramtsov 0 1
0 Gubkina str. 8, 119991, Moscow , Russia
1 Steklov Mathematical Institute, Russian Academy of Sciences
We present the holographic computation of the boundary twopoint correlator using the GKPW prescription for a scalar eld in the AdS3 space with a conical defect. Generally speaking, a conical defect breaks conformal invariance in the dual theory, however we calculate the classical bulkboundary propagator for a scalar eld in the space with conical defect and use it to compute the twopoint correlator in the boundary theory. We compare the obtained general expression with previous studies based on the geodesic approximation. They are in good agreement for short correlators, and main discrepancy comes in the region of long correlations. Meanwhile, in case of Zrorbifold, the GKPW result coincides with the one obtained via geodesic images prescription and with the general result for the boundary theory, which is conformal in this special case.
AdSCFT Correspondence; Gaugegravity correspondence

AdS/CFT
1 Introduction
2
Setup
3
4
5
2.1
2.2
2.3
2.4
2.5
Scalar eld on AdS3 space with particle
The GKPW prescription for boundary correlators in global Lorentz AdS
Boundary dual to the conical defect and AdS3 orbifolds
Extrapolation BDHM dictionary and geodesics approximation
Geodesics image method for AdSde cit spacetime
GKPW prescription for AdS3 with static particles
Comparison of GKPW prescription for AdS3cone with geodesic image
method. Integer 1=A case
Comparison of GKPW prescription for AdS3cone with geodesic image
method. Noninteger 1=A case
5.1
Equal time correlators
5.1.1
5.1.2
Small de cit
Large de cit
5.2
Nonequal time correlators
6
Conclusion
1
Introduction
AdS/CFT and holography [1{4] have been proving to be very fruitful tools in providing a
computational framework for stronglycoupled systems, as well as giving new insights into
the underlying structures of string and conformal eld theories. They have demonstrated to
be very useful for description of strong interacting equilibrium and nonequilibrium system
in high energy physics, in particular, heavyion collisions and formation of QGP [5{7],
as well as in the condensed matter physics [8, 9]. The frameworks of these applications
are set up essentially through consideration of di erent modi cations of the basic AdS
background, in particular, backgrounds which break asymptotic conformal symmetry of
the boundary [10{14].
In the paper we consider deformations of AdS3 by conical defects. There are several
reasons to consider this problem. First of all, AdS3/CFT2 allows to probe fundamental
theoretical problems, such as the thermalization problem [15{21], entanglement problem
and information paradoxes [22{25], chaos in QFT [26] using simple toy models. The second
reason is that in this case one can distinguish the peculiar features of several approximations
{ 1 {
that are widely used in AdS/CFT correspondence. The prime example of such
approximation is the holographic geodesic approximation [27]. It plays a very important role in
holographic calculations. Many physical e ects have been described within this
approximation, in particular, behaviour of physical quantities such as entanglement and mutual
entropies, Wilson loops during thermalization and quench are studied mainly within this
approximation [15{25, 28, 29]. Recent developments in the 2D CFT bootstrap techniques
show the deep relation between the geodesic approximation and semiclassical limit of the
conformal eld theory [30, 31].
Recently, geodesic approximation has been used extensively to study the structure
of the twodimensional CFT and its deformations which are dual to various locally AdS3
backgrounds, such as BTZ black holes or DeserJackiw pointparticle solutions. The latter
is the subject of study of the present paper. The point particles in AdS3 [32{35] produce
conical singularities, cutting out wedges from the space, but leaving it locally AdS3. We will
focus on the case of the static massive particle. The recent work [36{39] was devoted to the
study of the twopoint correlation function and the entanglement entropy in the boundary
dual to the AdS3de cit spacetime in the framework of geodesic approximation. The main
feature observed therein is a nontrivial analytical structure of correlators, which is caused
by the fact that identi cation of the faces of the wedge cut out by the particle allows to
have, generally speaking, multiple geodesics connecting two given points at the boundary.
Since this is true only for some regions of the boundary, naturally, the geodesic result for
the twopoint function may be discontinuous and can exhibit some peculiar behaviour in
the long range region.
The goal of the present investigation is to study the twopoint boundary correlator from
the point of view of the onshell action for the scalar eld via GKPW prescription [2, 3] on
AdS3 with a conical defect, and compare the result to the one obtained from the geodesic
prescription. As an interesting special case, we formulate the images prescription for the
correlator in case when the space is an orbifold AdS3=Zr and compare it with the image
method based on the geodesic approximation [36]. In the general case we illustrate that
the discontinuities in the geodesic result correspond to the nonconformal regime.
We
emphasize though that since we generally deal here with conformal symmetry breaking,
our study, being based on the original AdS/CFT prescription, indicates the need for caution
when applying holographic methods. Although in some cases it also justi es the application
of techniques based either on geodesic approximation or computation of the onshell action,
and it provides some limited evidence for a possibility of modi cation of AdS3/CFT2
prescription which could take into account nonconformal deformations of the holographic
correspondence.
The paper is organized as follows. Section 2 contains a brief overview of the geometry
of AdS3 with a massive static particle in the bulk and shortly describes the Lorentzian
GKPW prescription in case of the empty AdS3 space. We also review the e ect of the
conical defect on the boundary
eld theory from the symmetry point of view and the
geodesics prescription for de citangle in the bulk and its relation to the general holographic
dictionary. We then proceed to generalize the GKPW approach to the case of AdSde cit
spaces in section 3. In the section 4 we consider the special case of Zrorbifold when
{ 2 {
we have a conformal theory on the boundary and compare the general result with the
images prescription for geodesics. Then in section 5 we consider general nonconformal
deformations in case of small and large de cit angle, as well as their e ect on the temporal
dependence of correlators in GKPW and geodesic prescriptions.
originally in the at space [32] and generalized to the case of constant curvature in [33].
The AdS3 space with a conical defect is such solution with negative cosmological constant.
It represents a static massive particle sitting in the origin of the empty AdS space. This is
the only place in which the particle can be at the mechanical equilibrium because any small
deviation from the center get suppressed by the quadratic gravitational potential caused
by the negative cosmological constant. The metric in global coordinates can be written as
follows (in the present paper we set AdS radius to 1):
ds2 =
1
cos2
dt2 + d 2 + sin2
d#2 ;
where we have
2 [0; 2 ) as the holographic coordinate, AdS boundary is located at =2;
and # 2 [0; 2 A) is the angular coordinate. We parametrize the conical defect as
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
where
is the mass of the particle, and G is the threedimensional Newtonian constant.1
It is clear that the above metric indeed has the de cit angle of value
The case of A = 0 is the BTZ black hole threshold.
We will consider the real scalar eld on the background (2.1) with action2
S =
The scalar equation of motion in the metric, similarly to the empty AdS case [47], has
the form
+
cos2
sin2
sin
+
1
2
m2
cos2
= 0 ;
1In the case when the living space angle is 2 times an integer, i.e. when A = s, s 2 Z+, the spacetime
has an angle excess. This particular case is a solitonic topological solution of the pure 3D gravity [40], s
representing the winding number.
2Classical and quantum theories of the scalar eld on a cone on AdS3 have been considered in [33] and in
the at case [34, 41, 42]. Recently there have been interesting developments concerning correlation functions
and conformal symmetry on spaces with conical defects [43, 44]. QFT on the cone presents interest also in
context of cosmic strings applications [45, 46].
A = 1
4G ;
lar momentum, which factorizes from equation (2.5). Thus we have
Y (#) = ei Al # ;
l 2 Z ;
Substituting the ansatz into (2.5), we obtain a Schroedingertype eigenproblem for the
radial component (here the prime symbol denotes the
derivative):
tal in construction of boundary correlators. The case of A = 1 is the case of pure AdS3,
which we discuss in the following subsection.
2.2
The GKPW prescription for boundary correlators in global Lorentz AdS
Our goal is to obtain the expression for a twopoint correlation function of a scalar operator
on the boundary of AdS3 with a conical defect,3 described by the metric (2.1), using
the GubserKlebanovPolyakov/Witten (GKPW) holographic prescription [2, 3]. Since
we are interested in realtime correlation functions, we take the bulk (and, consequently,
boundary) metric signature to be Lorentzian. To take into account a particular choice of
boundary conditions for the Green's function in order to get a concrete realtime correlator
(i.e. retarded, Wightman or causal), we will use the prescription in the form of Skenderis
and van Rees [50]. In the present subsection we brie y review the prescription in the case
of empty AdS3, i.e. A = 1. We write
e
D i R dtd# '0OE
CFT
= eiSon shell[ ]j jbd='0 ;
The variables are separated via the usual ansatz
(t; ; #) = ei!tY (#)R( ) :
where as usual, the equality is supposed to hold after renormalization.
To specify a concrete realtime twopoint correlator of the operator O
with conformal
dimension
obtained via functional di erentiation of the CFT generating functional, we
deform the contour of integration over time into a contour C lying in the complex time
plane. This is a generalization of imposing standard Feynman radiation boundary
conditions on the path integral, which is used to get the causal correlator [51]. The contour C
is deformed in such a way that it goes through the elds required by the chosen boundary
conditions at t =
T (t being the parameter of the complex curve,
T are the corner
points of the contour), and the endpoints, corresponding to vacuum states in Z = h j i
are either at imaginary in nity in the zerotemperature case, or at nite identi ed points,
when the temperature is nite. In the current paper we consider the zerotemperature case.
3The AdS/CFT correspondence for the case of presence of defects on the boundary is a subject of
numerous investigations and applications, see for example [48, 49].
{ 4 {
(2.6)
(2.7)
(2.8)
(2.9)
notation
so that the 2h+ =
O , and h+ + h
consider only the case of
To construct the bulk dual, we deform the integration contour in the bulk onshell
action as well. As a result, we have the contributions from several onshell actions: those
which correspond to vertical segments are e ectively Euclidean actions, and those that
correspond to integration over horizontal segments, correspond to Lorentzian action. The
sources '0 are set to zero on all Euclidean segments, and satisfy the condition '0( T; #) =
0. Thus, while the Euclidean pieces do not contribute directly into the boundary term of
the onshell action, they determine the contour in the complex frequency plane, which is
used to de ne the bulkboundary propagator, through the condition of smoothness of the
scalar eld on the contour C.
The bulkboundary propagator is de ned in the boundary momentum representation
as a solution R!;l( ) of the radial equation (2.8) (since we consider the empty AdS case
here, we set A = 1 in this subsection), which is regular at the origin and has the leading
behaviour R!;l( ) = "2h + : : : near the boundary, where " = 2
. Here we introduce a
h =
1
2
1 p
2
1 + m2 ;
corresponds to the conformal dimension of the boundary operator
= 1 . Also, we de ne
= h+
h , so that
= 1 + . In this paper we
Note, however, that in general R consists of two pieces [47]: the nonnormalizable piece
with leading behaviour "2h , which grows near the boundary, and the normalizable piece
with the leading behaviour (!; l) (!; l)"2h+, where
h+ + 2 (jlj + !) +
h+ + 12 (jlj
h + 12 (jlj
1
h+ + 2 (jlj
!)
!)
;
!)
where by dots we denote the terms which are analytical in !. The digamma functions in
are nonanalytic and have poles at
!nl = (2h+ + 2n + jlj) ;
scalar eld equation in the bulk can be written as
Thus normalizable modes are quantized, and while they clearly don't change the leading
asymptotic nearboundary behaviour of R, they de ne the complex contour C in the
frequency space around these poles. By adding or removing extra normalizable modes, we
can deform C to obtain a concrete i prescription for the boundary correlator, and this is
indeed happening via accounting for the smoothness conditions on the corners of the time
contour C.
{ 5 {
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
hO(t; #)i = "li!m0 i p
"
( ; t; #)
d x
=
subtr
;
(2.15)
To obtain the twopoint correlator, one rst obtains the onepoint function, de ned by
where all divergences are subtracted from the action, and
= tan
1=" is the
determinant of the induced metric on the slices of constant . Note that, generally speaking, we
would have also contributions from corners of the contour C, but they all vanish by virtue
of smoothness conditions for the solution . The twopoint correlator is then obtained by
hO (t; #)O (0; 0)i =
(
1
)
for cos t
cos # < 0
for cos t
cos # > 0
i Z
{ 6 {
0 '0(t0; #0) hO (t; #)i ;
(2.16)
where 0 is the boundary metric determinant, which is just 1 in our case.
Thus, for the Wightman correlator one gets
We can sum the series for any integer . Note that i prescription here serves as a regulator
to conduct the summation over n. The result for the twopoint correlator of a scalar
operator of dimension
=
+ 1 is
The
= 1 case has slightly di erent coe cient in front of the normalizable piece of the
bulkboundary propagator [47], and the result in this case is
GW (t; #) = hO (t; #)O (0; 0)i =
are de ned by the expression (2.21), and the retarded/advanced Green's function is equal
to zero.
Boundary dual to the conical defect and AdS3 orbifolds
The theory on the boundary, which is dual to the AdSde cit space, is a eld theory on a
cylinder of circumference 2 A. To understand its relation to the \covering" CFT, i.e. the
one dual to the empty AdS, we recall that the algebra of asymptotic symmetries, which
has the Virasoro form for empty AdS, for the AdSde cit case has to be replaced by its
subalgebra, whose generators ln are de ned as [25, 52]:
A L n ;
A
where w = t + . This subalgebra only has the Virasoro form as well if A = 1r , r 2 Z+.
In this case the bulk spacetime is the AdS3=Zr orbifold, and the boundary theory is a
CFT with central charge c = rc~ (we denote quantities from the covering CFT by tilde).
Its operator algebra can be constructed from that of the covering CFT by symmetrizing
operators with respect to the identi cation map, see [25] up to a normalization factor:
This allows us to express matrix elements through those of the covering CFT as well. In
particular, for a twopoint correlator we have
hO(t1; #1)O(t2; #2)i =
(2.23)
(2.24)
(2.25)
Hence we've obtained the expression for the correlator as a sum over images, which is
what we expect for orbifoldlike spaces.4 For general A we emphasize that the boundary
algebra of symmetries does not have Virasoro form, and thus the theory is not conformally
invariant. As we will demonstrate, this can be seen directly from the holographic expression
for the twopoint function obtained from geodesic approximation.
4The similar known applications of the images method other than the AdS3=Zr orbifold case are thermal
AdS case [50], the BTZ black hole case [50, 53] and multiboundary AdS orbifold constructions [54].
{ 7 {
O(t; #) =
=
=
=
1 Xr1 r 1
r2 a=0 b=0
1 Xr1 r 1
X
r2 a=0 b=0
1 Xr1 r 1
X
r2 a=0 b=0
Since we are also interested in comparison of the GKPW prescription and geodesic
approximation for purposes of calculation of twopoint functions in the AdSde cit space, it will
be useful to brie y recall the general relation between these prescriptions and the results
given by the geodesics prescription.
Originally [27], the geodesic prescription was suggested as an approximation to the
boundary propagator in the Euclidean AdS space, which is obtained from the bulk
propagator using the dictionary which extrapolates the bulk elds
( ; t; #) to the boundary, or
BDHM dictionary [55]. In coordinates (2.1), this is expressed in de ning boundary elds as
The bulk propagator for a scalar eld can be written in the worldline representation as
a path integral over particle trajectories and approximated using the leading order of the
steepest descent expansion:
Gbulk(A; B) =
Z B
A
DPe m R d px_ 2
X
saddles
e mL(A;B) ;
where DP is the measure on the space of particle trajectories between points A and B which
includes the FaddeevPopov determinant originating from the worldline reparametrization
invariance,
is the parameter of a trajectory and L(A; B) is the geodesic length between A
and B. It is implied that m
is large. The BDHM dictionary then leads to consideration
of geodesics between two boundary points, with divergences subtracted from their lengths.
It was conjectured in [55] and proven in [56] that the BDHM dictionary in case of
(locally) asymptotically AdS spacetime is equivalent to the GKPW dictionary. Therefore,
in our case we can consider the geodesic boundary correlator as the approximation to the
full GKPW expression, i.e. in Euclidean case
GGKPW = GBDHM
X
saddles
e mL :
Thus the geodesic approximation is given in the leading order of the WKB approximation
to the full GKPW expression.
One can formulate the geodesics prescription in the Lorentzian space, that would be
valid for points A and B that are spacelike separated. However, for timelike separated
boundary points there are no connecting geodesics. If the spacetime allows for Euclidean
analytic continuation, one can obtain the Lorentzian correlator which would be valid for
the entire Lorentzian plane of the boundary by making the reverse transition from the
Euclidean case. However, for more general backgrounds (for example nonstationary ones)
one does not have this opportunity, so for timelike separated points the geodesics
prescription has to be formulated using di erent considerations. As an example, below we
brie y discuss possible continuation into the timelike region for geodesic prescription on
the background of the moving particle.
{ 8 {
The geodesic prescription for particles in AdS3 has been considered in [36{39, 57]. it is based
around the fact that there can be several geodesics between two points in general, which
di er in number of windings around the defect. It is proven that the lengths of winding
geodesics can be expressed through lengths of geodesics connecting certain auxiliary points
at the boundary. These points are images of the correlator arguments with respect to
the isometry corresponding to the identi cation of faces of the wedge. The prescription is
formulated in the Lorentzian signature. Thankfully, in the case of the static AdSde cit
spacetime, there is a straightforward Euclidean analytic continuation, so one can obtain the
Lorentzian geodesics prescription for the entire boundary spacetime by making the reverse
Wick rotation from the Euclidean case. For the Wightman correlator, that is expressed in
the transition
! it + . The resulting correlator obtained via the geodesics prescription
on the conical defect is written as a sum over images.
For small de cit, 12 < A < 1, the correlator is:
GW (t; #) = (
#)
12 , the correlator is given by
GW (t; #) =
1
i )
where (square brackets represent the integer part):
1
#
The angular dependence in thetafunctions and in limits of summation says that these
functions are generally discontinuous. We will discuss the analytic structure of the
geodesic correlators more thoroughly when we compare them to the result of GKPW
calculation.
In the general case of moving particle, there is no straightforward way to perform
Euclidean analytic continuation, and one has to formulate a separate prescription for the
timelike region, for example based on quasigeodesics method [19, 38] and symmetry
properties [38, 57]. The quasigeodesics method accurately captures the behaviour of the
preexponential factor in the expressions (2.20) for every term in the sum over images in the
timelike region. It yields that the correlator for timelike separated points (t1; #1) and
{ 9 {
(t2; #2) in empty AdS can be computed using the spacelike geodesic between the points
(t1 + ; #1 +
) and (t2; #2) [57]. Keeping it in mind, for integer conformal weights one
can formulate the geodesic images prescription for the entire correlator on the AdSde cit
space taking into account its causal structure, without relying on the Euclidean analytic
continuation. As seen in (2.21), in the case of integer
, the expressions for correlation
functions are signi cantly simpli ed.
GKPW prescription for AdS3 with static particles
Now we consider the scalar eld equation in the space with metric (2.1) for arbitrary
A 2 (0; 1). It is clear that the form of the equation is the same as in the case of empty
AdS. The only di erence is that now the angular eigenfunctions are de ned by (2.7).
Therefore, the radial wave equation is the same as in pure AdS3, only with l divided by A.
Consequently, the general solution of the scalar EOM on the anglede cit AdS3 space is
obtained from that on pure AdS3 by transition l ! l=A. Tracing this replacement through
the GKPW computation scheme outlined above, we infer that it will lead to the change of
location of poles of digamma functions, which now are at
Expressions (2.12){(2.13) now read
!~nl =
(2(h+ + n) + j j ) ;
l
A
Therefore, the resulting expression in the form of series over residues in the frequency space
for the Wightman twopoint function will now read
In the form analogous to the (2.17) the resulting expression is:
where we have omitted the prescription. We can sum the series for
= 0, which gives
the result for
= 1:
h+ +
+
h+ +
cos A# :
!
n
l
j j
A
l
j j
A
e i!~n+l(t i )+i Al #:
e i(1+ +2n)t i jlj t+i Al # ;
A
(3.1)
(3.2)
(3.4)
(3.5)
(3.6)
Thus, the result for arbitrary integer
=
1 can be obtained using the di erentiation
under the sum and formally written as
sin At
Comparison of GKPW prescription for AdS3cone with geodesic image
method. Integer 1=A case
Consider the case when 2 is an integer number of the angle de cits, i.e. A = 1=r and r is
an integer, and the space is the AdS3=Zr orbifold. We have from the general formula (3.5):
(
2
X X1 (n + )! (n + rjlj + )!
To prove (4.2), consider the sum over l:
X (n + rjlj + )!
(n + rjlj)!
l2Z
=
=
l= 1
l= 1
(n + rjlj)! r p=0
(n + jlj)! r p=0
X1 (n + rjlj + )! 1 Xr1 e ijljrt+irl(#+ 2rp )
X1 (n + jlj + )! 1 Xr1 e ijljt+il(#+ 2rp )
cos t cos # + 2 kr
1
r
1)2 Xr1
k=0
r 1 X1 (n + rjlj + q + )! 1 Xr1 e i(jljr+q)t+i(rl+q)(#+ 2rp )
X
e i(1+ +2n)t ijljteil(#+ 2rk ) ;
HJEP04(216)
(4.6)
(4.7)
(4.8)
The summation over p in the last term can be conducted:
Therefore, the entire qsum vanishes, and we have
hO1+ (t; #)O1+ (0; 0)i =
which, in analogy to the empty AdS result (2.17), is precisely the sum over images (4.3).
A particular case of the formula (4.3) was obtained in [58] in case of a massless scalar eld
(i.e.
= 2) by using the images prescription for the bulkboundary propagator itself.
The answer for geodesic correlator in the orbifold case is given by the formula (2.30),
where we set A = 1r :
hO (t; #)O (0; 0)i
The normalization factor dependent on the conformal dimension is scheme dependent and
is not reproduced by the geodesic approximation, however the GKPW result (4.3) has a
factor 1=r as well, which generally does not come from a saddle point expansion. However,
it is required from the point of the boundary CFT, as seen in (2.25).
Thus, the twopoint correlator on the boundary CFT dual to the AdS3=Zr orbifold is
precisely reproduced by the GKPW prescription, and also by the geodesic approximation
up to a numerical factor.
5
Comparison of GKPW prescription for AdS3cone with geodesic image
method. Noninteger 1=A case
for the geodesic correlator equal to 2 2 A.
In this case there is no obvious way of rewriting the sum (3.5) in terms of the geodesic
contributions. We are going to compare it with the geodesic result in some special cases.
Before we proceed, note that since the geodesic prescription does not x the overall
numerical factor, we have to choose it manually. In the orbifold case we have seen that the GKPW
result gives an extra factor of 1r = A, so it is natural for us to propose the normalization
5.1
5.1.1
Equal time correlators
Small de cit
Here by \small" we mean that
< . In this case there is always a region where 2 geodesics
contribute instead of only one in the remaining part of the living space. In this case the
geodesic approximation predicts the correlator in the form (2.30), which for convenience
we rewrite as (recall that
A) is the angle removed by the defect):
G(t; #) =
): the only contribution is the direct geodesic from 0 to #.
]: the only contribution is the image geodesic from 2
to #.
; ): both direct and image geodesics contribute.
At the endpoints of these intervals we have discontinuities, which are re ected by Heaviside
functions in the above formula. However, the general GKPW result (3.5) does not have
these discontinuities. We can observe that for higher
the size of discontinuities
diminishes, and at
! 1 the geodesic result approaches the GKPW expression. Examples,
illustrating this point, are presented in
gure 1. Note that in the small de cit case the
GKPW value is between two values of the geodesic correlator at points of discontinuity.
Also, we see that the most signi cant discrepancy happens in the zone of longest
correlations, which suggests that geodesic approximation apparently obtains some subleading
corrections which are prominent in the the longrange correlations region. A similar e ect
was observed in [59] in the context of Vaidya model for thermalization. We leave the issue
of longrange corrections in the geodesic approximation for the future study.
5.1.2
Large de cit
In the case when the de cit angle is more than , or equivalently when A < 1=2, the geodesic
correlator is given by (2.30). For now we focus on the case of equaltime geodesics. Because
of the angular dependence in the limits of summation, the equal time correlator in the large
angle case has three zones as well  transition from one zone to another corresponds to the
change in the number of geodesic images given by (2.31). These constraints come from the
fact that spacelike basic geodesics connecting two points at the boundary of AdS cannot
revolve around the origin. Two sums express the two sets of images obtained by acting with
the wedge identi cation isometry like a rotation clockwise or counterclockwise with respect
to the origin in the equaltime section of AdS cylinder. Naturally, depending on the position
of the correlator endpoints and the value of the de cit angle, the number of clockwise and
counterclockwise images will change, which is what expressed by (2.31). In the orbifold
case discussed in 4 the identi cation isometry is a Zr group of rotations, so any image can
be obtained by acting on the basic geodesic (or any other image) for any value of the angular
12 G1
35 G1
10
5
30
25
20
15
10
5
1
1
2
2
Δ=5
3
3
Geodesics
4
4
θ
method for A = 34 for di erent conformal weights. Contributions of discontinuities in the geodesic
result (represented by the brown line) diminish as
increases.
variable. Therefore the correlator obtained from geodesics approximation can be written
as a single sum (4.8). It coincides with the GKPW expression and is smooth in the entire
living space. The answer for the inverse equal time correlator compared with the GKPW
expression given by 3 is shown in
gure 2. We see that contributions of discontinuities
also diminish with the increase of the conformal weight, but the sign of corrections to the
geodesic approximation is opposite to the small angle case, the GKPW value of the inverse
correlator is sightly lower than that of the geodesic expression, and correction contributions
in the long range region are much smaller than in the small de cit case.
5.2
Nonequal time correlators
Here we examine the di erences between the time dependencies of the GKPW
answer (3.6){(3.7) and the geodesic expression (2.30). In
gure 3A{C we trace the increase
of discrepancy between the two prescriptions when we slightly increase the value of the
de cit parameter starting from A = 13 . In this point the two prescriptions coincide:
sin 3t
=
1
3
sin t cos 3t
cos 3#
1
1
cos t
cos #
cos t
+
1
cos # + 23
+
cos t
1
cos # + 43
!
:
(5.2)
As we begin to deform the orbifold, we observe that some di erences in the analytic
structure of GKPW and geodesic expressions start to evolve.
HJEP04(216)
1.0
Δ=5
θ
G1
1.5
0.5
1.5
0.5
for A = 0:3. The living space angle in this case equals 35 , which cannot t into 2 integer number
of times, so we have discontinuities in the geodesic result, which bring signi cant discrepancy with
the GKPW result. However, this discrepancy also diminishes as we increase the conformal weight.
First, consider zeros of the reverse correlator. These correspond to the singularities
in the correlator itself. The geodesic correlator (2.30) has singularities at lightcones
corresponding to image points, and GKPW expressions (3.6){(3.7) obtain their singularities
from the (cos t=A
cos #=A) 1 and its derivatives in case of higher weights. In the general
case these two sets do not coincide, but in the orbifold case one can derive trigonometric
formulae similar to the above. We see in
gure 3B that when we move from the orbifold
point, the GKPW reverse correlator obtains an additional zero. This happens because the
change in the de cit parameter was too small to eliminate or add a term in the geodesic
expression, but it was enough to have an impact on the denominator of the GKPW
formula. This e ect does not depend on the value of angle and, in general, on the number of
geodesic images: we see in gure 3C that the number of images has decreased, and GKPW
expression re ects this as well, but still keeps its extra zero.
Second e ect is related to the singularities of the reverse correlator, or zeros of the
correlator itself. For GKPW expression, these come from the factor sin At in (3.6) (and,
again, its derivatives for higher weights in (3.7)). In the geodesic prescription (2.30) these
come from the sum of all image denominators if one tries to bring the sum to a common
denominator. Again, in the general case these two sets are di erent. In
gure 3C we see
that when the number of terms in the sum is lowered by the constraint (2.31), the number
of singularities of the reverse correlator decreases as well  in other words, the number of
singularities is basically equal to the number of image geodesics, and thus to the number
of lightcone zeros (on the interval t 2 [0; ]). This is, however, not the case for the GKPW
10 G1
G1
6
t
6
t
5
5
10
0.5
0.5
1.0
GKPW
Geodesics
A= 13.3 ; Δ=3
1
2
3
4
5
1
2
3
4
5
t
6
6
t
HJEP04(216)
10 G1
10 G1
5
5
5
10
5
10
A= 13.3 ; Δ=1
1
2
3
4
5
1
2
3
4
5
tions. Plots AC show the increase of discrepancy between two prescriptions in case of
= 1 when
the de cit parameter is close to the orbifold value A = 13 . Plot D shows the discrepancy for
= 3.
The value of angular variable is xed # = 6 .
expression  the number of singularities is still the same, whereas the number of zeros has
decreased compared to the gure 3B case.
This di erence in the region between dashed lines between the geodesic and GKPW
expressions is similar in its nature to the longrange contributions in the angular dependence
of correlators discussed above. The comparison of plots C and D in
gure 3, that show
the cases of di erent conformal dimensions at A = 13:3 , illustrates that the increase of
the conformal dimension makes the geodesic prescription approach the GKPW expression
where it has an extra zero, but the singularities between the dashed lines for certain values of
the de cit parameters are still unique to the GKPW expression for general . Thus, unlike
the longrange equal time case, the analytic structure of the reverse GKPW expression is
not completely reproduced even in the large
limit.
6
Conclusion
We have calculated the twopoint boundary correlator in the AdS space with a static conical
defect using the GKPW prescription for a scalar
eld. The version of the holographic
prescription that we have used is formulated in the Lorentzian signature and is based on
the deformation of the temporal integration contour in the bulk partition function. It
does not require continuation from the Euclidean case. Generally speaking, the resulting
correlator does not retain the conformal symmetry on the boundary.
In the case of integer de cit, when the angle de cit equals to
shown that twopoint correlator is equal to the CFT correlator. It can be represented as
a sum over images. Each image contribution is expressed in this case by an expression for
the correlator in the empty AdS3.
Comparing these GKPW correlators with correlators obtained through geodesic
approximation, we observe that in general case with increasing
the geodesic approximation
reproduces the GKPW expression more precisely. However, for equaltime correlators we
see that correlators obtained via the geodesic approximation exhibit nontrivial generally
discontinuous behaviour in the region of large spatial separations, which signi cantly di ers
from the behaviour of the GKPW correlators. The situation is slightly di erent for cases
of small and large defects.
For small angle de cit, 12 < A < 1, we observe that the geodesic correlator exhibits
discontinuities between regions of contributions of image and basic geodesic. We see
that the value of the reverse correlator is signi cantly lower around the point # =
A
than that of the GKPW expression. The latter depends on angles continuously
everywhere in the living space.
For large angle de cit, A < 12 , we observe that the geodesic correlator exhibits
discontinuities between regions of contributions of di erent sets of images. Contrary to
the previous case, we see that the value of the reverse correlator is higher around the
point # =
A than that of the GKPW expression.
In general, we see that longrange corrections have higher impact in the spacetimes with
small de cit angles.
We also have examined the temporal behaviour of correlators obtained from GKPW
and geodesic prescriptions, and we see that there is a di erence in temporal dependence
of the geodesic correlator and the GKPW one, which is similar to the largeseparation
discrepancy, in both cases of large and small de cits. Notice that the large
limit does
not reproduce the GKPW result completely in some temporal regions. Indeed, the number
of singularities in the GKPW expression is higher than the number of singularities in the
geodesic correlator, which corresponds to the total number of geodesics involved in the
images prescription, unless the de cit parameter is A = 1r , r 2 Z, i.e. the space is a
Zrorbifold. In that case, as we have observed, the geodesic approximation gives the exact
answer for the CFT correlator, which coincides with the GKPW expression in this case as
well, and the images method for calculating the Green's function in GKPW prescription
coincides with the geodesic images prescription.
The presence of nontrivial longrange corrections in the general case appears to be a
common property of geodesic approximation in the various locally AdS backgrounds with
broken asymptotic conformal symmetry. In order to try to give a physical interpretation to
this fact, we recall from the discussion in 2.4 that in our case the geodesic approximation
can be thought of as the leading order of the WKB approximation to the full GKPW
expression, so we can interpret the discrepancy in the longrange region as inapplicability of
the WKB approximation. The failure of the WKB approximation could be related with the
essential role of quantum corrections to the geodesics correlators for longest geodesics. This
could take place because in that region the longestwave excitations which give signi cant
contribution to the path integral (2.27) near its saddle points begin to interact in a purely
quantum way with the static particle located at the origin (recall that the longest geodesics
in AdS3 pass closest to the origin). This interpretation suggests that the geodesic
prescription can exhibit distinction from GKPW expression for other multiconnected locally AdS
backgrounds. Another interesting point is that the longrange region the saddle points of
the path integral (2.27) are closest to each other (as illustrated, for example, in (5.1)  for
longrange correlations the lengths of basic and image geodesics are closest to each other).
This hints us that a viable improvement of the geodesic prescription on multiconnected
spaces perhaps could be constructed by accounting for quantum corrections from adjacent
saddles in every term of the right hand side in (2.27), and it is plausible that those
corrections could be interpreted as the e ect of quantum scattering of particles on the defect.
We hope to obtain some further insight on this issue and its connection to the conformal
symmetry breaking in the future studies.
Acknowledgments
The authors are grateful to Dmitrii Ageev and Andrey Bagrov for useful discussions. This
work is supported by the Russian Science Foundation (project 145000005, Steklov
Mathematical Institute).
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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