# Time dependent field correlators from holographic EPR pairs

Shoichi Kawamoto<EMAIL_ADDRESS>Department of Physics, National
Tsing Hua University, Hsinchu, Taiwan, R.O.C.  Center for High Energy Physics,
Chung Yuan Christian University, Taoyuan, Taiwan, R.O.C.  College of Arts and
Sciences, J. F. Oberlin University, Tokyo, Japan  Da-Shin Lee
<EMAIL_ADDRESS>Department of Physics, National Dong Hwa University,
Hualien, Taiwan, R.O.C.  Chen-Pin Yeh<EMAIL_ADDRESS>Department
of Physics, National Dong Hwa University, Hualien, Taiwan, R.O.C.

###### Abstract

We study the correlators of the fields that couple to the quark and anti-quark
EPR pair in the super Yang-Mills theory using the holographic description,
which is a string in AdS space with its two ends anchoring on the boundaries.
We consider the cases that the endpoints of the string are static and that the
endpoints are uniformly accelerated in opposite directions where the exact
solutions for the string’s profiles are available. In both cases, the two-
point correlators of the boundary field, described by the linearized
perturbations in the worldsheet, can also be derived exactly where we obtain
the all-time evolution of the correlators. In the case of the accelerating
string, the induced geometry on the string worldsheet has the causal structure
of a two-sided AdS black hole with a wormhole connecting two causally
disconnected boundaries, which can be a realization of the ER=EPR conjecture.
We find that causality plays a crucial role in determining the nature of the
dispersion relation of the particle and the feature of the induced mutual
interaction between two particles from the field. In the case that two
boundaries of the worldsheet are causally disconnected, the induced effect
from the field gives the dissipative dynamics of each particle with no
dependence on the distance between two particles, and the induced mutual
coupling between them vanishes in the late times, following a power law. When
two ends are causally connected, the induced dispersion relation becomes non-
dissipative in the late times. Here, we will also comment on the implications
of our findings to the entangled particle dynamics and the ER=EPR conjecture.

##  I introduction

Time-dependent field correlators provide a qualitative description of how fast
the information spreads out in a quantum theory. This is also related to how
fast two entangled subsystems get disentangled. In the content of the AdS/CFT
correspondence, ER = EPR is a conjecture stating that two entangled particles,
the so-called Einstein-Podolsky-Rosen or EPR pair, can be associated in a
gravity theory by a wormhole background called Einstein-Rosen or ER bridge
ER=EPR . One realization is the eternal AdS black hole, which is dual to the
thermofield double state Maldacena_01 . In this case, the correlators for the
operators separately inserted at two causally disconnected boundaries decay
exponentially in the boundary time, which is in tension with the information
conservation for a quantum black hole Maldacena_01 . Another realization of
the ER=EPR conjecture is a model consisting of a string in AdS space with its
two ends located at the AdS boundary, which is dual to the entangled pair of
the quark and anti-quark in the super Yang-Mills theory. As the two ends are
uniformly accelerating in the opposite directions, the exact worldsheet
profile was found in Xiao_08  and had the same causal structure as the eternal
AdS black hole. From this exact solution, it was proposed in Jensen_13  and
Sonner_13  that the EPR pair of the quark and anti-quark is holographically
encoded in the induced wormhole geometry on the string worldsheet. For
example, the entanglement entropy of the EPR pair is dual to the entropy in
the two-sided black hole of the worldsheet, and the gravity action is dual to
the EPR pair production rate.

How fast the information spreads out is also crucial in quantum information
processing. The viability of quantum computation depends on how long we can
maintain the system in coherent or entangled states without losing information
to environments. Thus, it is interesting to study how quantum fields influence
the entanglement of the system. We may start with the initial density matrix
of the system and environment. The effective theory for the system is
described by the reduced density matrix, which is obtained by tracing out
environmental variables in the full density matrix. It is an essential
quantity for studying the dynamics of entanglement WC . The references SY1 ;
SY2  study the entanglement of two objects interacting with the common free
quantum field when two objects either undergo uniform acceleration or are at
rest. They found that the entanglement between two objects can be created
through the coupling with the quantum field and then decays to vanishing in
some finite duration (called the “sudden death” of quantum entanglement YU ).

The idea of the holographic duality has also been applied to study strong-
coupling problems in condensed matter systems and the hydrodynamics of the
quark-gluon plasma (see Hartnoll_09 ; Rangmanai_09  for reviews). There are
considerable efforts to employ the holographic duality to explore the
dissipation behavior of a particle moving in a strongly coupled environment.
In these studies, the string’s endpoint on the boundary of the AdS black hole
serves as a probe particle. The reference Holographic QBM  reviews the
application to non-equilibrium Brownian motion. We have employed this approach
to study various behaviors of Brownian particles in the strongly coupled
fields, which potentially can be verified experimentally Lee_13 ; Lee_15 ;
Lee_16 .

In this work, we would like to adopt the holographic approach to obtain the
time-dependent correlators of the strongly coupled fields that couple to two
entangled particles of either undergoing uniform acceleration or being static.
The dual gravity description is a probe string in AdS space with its two ends
anchoring at the boundary. In the case of the uniform accelerating pair, the
setting is the same as in Xiao_08 . In both cases, we obtain the exact time-
dependent field correlators by solving the linearized equation of motion for
the perturbations of the string Nambu-Goto actions. Compared to the one-end
string, the solutions here are fixed by the boundary conditions at the two
ends of the string. To see the implication of the single-particle dynamics, we
also implement the infalling boundary condition to the one branch of our
solutions to obtain the exact retarded Green’s function as in Son_09 ; Lee_19
. We interpret this Green’s function as a response function for a single
particle resulting from integrating out the degrees of freedom of both the
field and the other particle. In what follows, the dispersion relation of the
particle of the entangled pair can presumably be decoded from the exact
retarded Green’s function through the Langevin equations. In future work, we
will derive these Langevin equations by taking the variation of the effective
action of the particles in this paper. We may use the detailed dynamics from
these equations to study, for instance, the time evolution of the entanglement
entropy between two particles as in Lee_19 . In this paper, we focus on the
exact results of the force-force correlators and the retarded Green’s function
with their implications for particle dynamics. Based on the interpretations
above, we find that the causality in the string worldsheet plays a crucial
role in determining the nature of the dispersion relation of the particles and
also the feature of the induced mutual interaction between them. In the case
of an accelerating string with the causally disconnected two ends, the induced
effect from the field gives the dissipative dynamics for the particles with no
dependence on the distance between them. As a comparison, in the case of a
static string, the particle dynamics is dissipative at the early enough times
when two ends are not in causal contact with each other. The induced
dispersion relation of each particle becomes non-dissipative when two ends are
causally connected at late times. Moreover, in both cases, at late times, the
induced mutual coupling between two ends, which comes from the field
correlator or the force-force correlator, follows a power law to vanish. It
will also be interesting to see whether this different time dependence of the
field correlators and causality play any role in the black hole information
problem proposed in Maldacena_01 .

The remaining part of the paper is organized as follows. Section II reviews
the exact solution of the accelerating string and then solve the linearized
equation for the perturbations in this worldsheet background. We obtain the
all-time two-point correlators of the fields that the quark and anti-quark
pair couple to, in contrast to the earlier works where only the late-time
behavior of the correlators was found Xiao_08 . As a comparison, we also find
the exact solution of the linearized equation of the string worldsheet in the
case when its two ends are static in section III. We then conclude and comment
in section IV.

##  II fluctuations of transverse modes in the accelerating string

We first review the fundamental equations and their exact solutions for the
accelerating string in $AdS_{d+1}$ space Xiao_08 . Then we exactly solve the
linearized equation for the perturbations in these string backgrounds and give
a prescription to obtain the generating functional for the boundary fields
coupled to the string endpoints.

###  II.1 The exact solutions of the string background and perturbations

We consider the $AdS_{d+1}$ metric in the Poincare coordinates given by

$ds^{2}=\frac{R^{2}}{z^{2}}(-dt^{2}+dz^{2}+dx^{2}+\sum_{i=1}^{d-2}dy_{i}^{2})\,,$
(1)

where $R$ is the curvature radius. The string in this background is described
by the Nambu-Goto action

$S=-T_{0}\int d\tau d\sigma\sqrt{-h}\,,$ (2)

where $T_{0}$ is the string tension and $h=\text{det}\,h_{ab}$ is the
determinant of the induced metric. We choose a static gauge
$(\tau,\sigma)=(t,z)$ and the embedding of the string as
$X^{\mu}(t,z)=(t,z,x(t,z),0,\cdots,0)$ with $\mu=1,2,3,\cdots,d$. Then,
$\sqrt{-h}=\frac{R^{2}}{z^{2}}\sqrt{1-\dot{x}^{2}+x^{\prime 2}}$. The
classical equation of motion is given by

$\frac{\partial}{\partial
t}\Big{(}\frac{\dot{x}}{z^{2}\sqrt{1-\dot{x}^{2}+x^{\prime
2}}}\Big{)}-\frac{\partial}{\partial
z}\Big{(}\frac{x^{\prime}}{z^{2}\sqrt{1-\dot{x}^{2}+x^{\prime
2}}}\Big{)}=0\,,$ (3)

where ${}^{\prime}=\frac{\partial}{\partial z}$ and
$\cdot=\frac{\partial}{\partial t}$. This equation has an exact solution
Xiao_08 ,

$x_{b}(t,z)=\pm\sqrt{t^{2}+b^{2}-z^{2}}\,,$ (4)

where $b$ is a real constant. In the solution, the trajectory of the end-
points at $z=0$ describes the motion of two particles along the $x$-direction
with uniform deceleration $\frac{1}{b}$. They head toward each other from
$x=\pm\infty$ when $t=-\infty$, subsequently stop at $x=\pm b$ when $t=0$ and
then turn around in the opposite directions, moving away from each other. We
label the position of the particle moving in the positive (negative) $x$
region by $q_{R}(q_{L})$, respectively. The induced metric on the worldsheet
with the embedding coordinates is

$ds_{ws}^{2}=\frac{R^{2}}{(t^{2}+b^{2}-x^{2})^{2}}((x^{2}-b^{2})dt^{2}-2tx\,dt\,dx+(t^{2}+b^{2})dx^{2})\,.$
(5)

As noticed in Jensen_13 , this metric has the same casual structure as that of
the eternal two-sided $AdS$ back hole (see Fig. 1) and has the bifurcating
horizons located at $t=\pm x$, the time-like boundaries at the particle
trajectories $x^{2}-t^{2}=b^{2}$, and the space-like “singularity” at
$x^{2}-t^{2}\rightarrow-\infty$. However, this space-like singularity is not a
physical singularity and just reflects the coordinate singularity in the
background AdS as $z\rightarrow\infty$ to approach the Poincare Killing
horizon. Due to the analog between the EPR pair and the wormhole geometry
ER=EPR , it was proposed in Jensen_13 ; Sonner_13  that this metric (5) gives
the gravity description of the entangled quark and anti-quark pair.

Figure 1: Few sample light rays (solid black lines) in the worldsheet(5). Here
$x$ and $t$ are in unit of $b$. The hyperbolic curves (blue) are trajectories
of quark and anti-quark. Two quarks are causally disconnected with the
horizons at $t=\pm x$ (dash lines).

We now consider the fluctuations of a string in the direction transverse to
the $x$ coordinate, say $y_{1}$, with the embedding in (1) as
$X^{\mu}=(t,z,x_{b}(t,z),q(t,z),0,\cdots,0)$ where $q$ is small as compared to
$x_{b}$. The quadratic action for $q$ is obtained as

$S_{q}=-R^{2}T_{0}\int dt\,dz\frac{1}{z^{2}}\left(\frac{q^{\prime
2}-\dot{q}^{2}-\dot{x}_{b}^{2}q^{\prime 2}-x^{\prime
2}_{b}\dot{q}^{2}+2\dot{x}_{b}x^{\prime}_{b}\dot{q}q^{\prime}}{2\sqrt{1-\dot{x}_{b}^{2}+x_{b}^{\prime
2}}}\right)\,.$ (6)

As seen in the action, in this coordinate system, there is no separable
solution in the variables $t$ and $z$. However, in the comoving frame of one
of the accelerating particles, say the $R$ branch in (4), defined as,

$\displaystyle x=\sqrt{b^{2}-r^{2}}e^{\frac{\alpha}{b}}\cosh\frac{\tau}{b}\,,$
(9) $\displaystyle
t=\sqrt{b^{2}-r^{2}}e^{\frac{\alpha}{b}}\sinh\frac{\tau}{b}\,,$ $\displaystyle
z=re^{\frac{\alpha}{b}}\,,$

with $0<r<b$, the metric of (1) becomes

$ds^{2}=\frac{R^{2}}{r^{2}}\left(-\Big{(}1-\frac{r^{2}}{b^{2}}\Big{)}d\tau^{2}+\frac{dr^{2}}{1-\frac{r^{2}}{b^{2}}}+d\alpha^{2}+e^{-\frac{2\alpha}{b}}\sum_{i=1}^{d-2}dy_{i}^{2}\right)\,.$
(10)

As in Xiao_08 , this metric has the event horizon at $r=b$ with the Hawking
temperature $T=\frac{1}{2\pi b}$, which shows no causal connection between the
two particles. However, on the worldsheet (5), it can be seen that a wormhole
connects them. We also expect that in the particle rest frame, the particles
will experience quantum fluctuations at finite temperature $T$ to be verified
later. Note that this temperature $T$ can also be interpreted as the Unruh
temperature that an accelerating particle observes. In this coordinate system,
the background worldsheet solution in (4) is

$\alpha(\tau,r)=0\,.$ (11)

And the quadratic action for the perturbations in the $y_{1}$ direction on the
worldsheet becomes

$S_{qc}=-\frac{R^{2}T_{0}}{2}\int d\tau
dr\frac{1}{r^{2}}\left(\Big{(}1-\frac{r^{2}}{b^{2}}\Big{)}q^{\prime
2}-\frac{\dot{q}^{2}}{1-\frac{r^{2}}{b^{2}}}\right)\,$ (12)

with ${}^{\prime}=\frac{\partial}{\partial r}$ and
$\cdot=\frac{\partial}{\partial\tau}$. In terms of the Fourier modes

$q(\tau,z)=\int\frac{d\omega}{2\pi}\,y_{\omega}(z)\,e^{-i\omega\tau}\,,$ (13)

the equation of motion reads

$y^{\prime\prime}_{\omega}-\frac{2}{z(1-\frac{z^{2}}{b^{2}})}y^{\prime}_{\omega}+\frac{\omega^{2}}{(1-\frac{z^{2}}{b^{2}})^{2}}y_{\omega}=0\,.$
(14)

On the worldsheet, the comoving coordinate $r$ is equal to the static
coordinate $z$, and now ${}^{\prime}=\frac{\partial}{\partial z}$. To make the
action finite, we introduce a cutoff scale by putting the boundary brane at
the finite $z=z_{m}$. Then, with the normalization condition
$y_{\omega}(z_{m})=1$, the equation (14) has two independent solutions

$Y_{\omega}(z)=\frac{(1-i\omega z)e^{i\omega
b\tanh^{-1}\frac{z}{b}}}{(1-i\omega z_{m})e^{i\omega
b\tanh^{-1}\frac{z_{m}}{b}}}~{}~{}~{}\mbox{and}~{}~{}~{}Y_{\omega}^{*}(z)\,,$
(15)

where $Y_{\omega}(z)$ ($Y^{*}_{\omega}(z)$) is the incoming (outgoing) wave
near the horizon $z=b$. We would like to stress that these are the exact
solutions, whereas in Xiao_08  only the approximate solutions in the
small-$\omega$ expansion are given.

###  II.2 Single-particle dynamics

Using the holographic prescription, the retarded Green’s function of the
fields coupled to one end of the string is

$\displaystyle
G_{R}(\omega)=\frac{R^{2}T_{0}}{b^{2}}\left(\frac{b^{2}}{z_{m}^{2}}-1\right)Y^{\prime}_{\omega}(z_{m})Y^{*}_{\omega}(z_{m})$
$\displaystyle=$
$\displaystyle\frac{R^{2}T_{0}}{z_{m}b^{2}}\frac{1}{1+\omega^{2}z_{m}^{2}}\left(\omega^{2}(b^{2}-z_{m}^{2})+i(z_{m}\omega+b^{2}z_{m}\omega^{3})\right)$
(16) $\displaystyle=$ $\displaystyle
i\gamma_{T}\,\omega+M_{T}\omega^{2}+\mathcal{O}(\omega^{3})\,,$

which is analytic in the upper half complex $\omega$-plane. Its small $\omega$
limit agrees with the results in Xiao_08  and also our previous papers Lee_13
, giving the effects to the particle of the temperature-dependent damping term
$\gamma_{T}=R^{2}T_{0}/b^{2}$ and the effective mass term $M_{T}=M_{0}+\Delta
M_{T}$. $M_{T}$ consists of the large temperature-independent mass
$M_{0}=R^{2}T_{0}/z_{m}$ and the small temperature-dependent mass correction
$\Delta M_{T}=-R^{2}T_{0}z_{m}/b^{2}$ in the limit of $z_{m}/b\rightarrow 0$.
This picture of the particle dynamics is captured in the Langevin equation
Son_09 ; Lee_19 ,

$\int_{0}^{\tau}G_{R}(\tau-\tau^{\prime})q(\tau^{\prime})d\tau^{\prime}=\eta(\tau)$
(17)

where $q(\tau)$ is the position of the particle and $\eta(\tau)$ is the noise
force with $\langle\eta(\tau)\eta(\tau^{\prime})\rangle$ given by the Hadamard
function of the field, $G_{H}(\tau-\tau^{\prime})$. This can also be used to
obtain the particle correlator, which in frequency space is given by

$\langle\tilde{q}(\omega)\tilde{q}(-\omega)\rangle=\frac{G_{H}(\omega)}{G_{R}(\omega)G_{R}(-\omega)}\,.$
(18)

This equation incorporates not only the dissipation effect given by the
retarded Green’s function $G_{R}$ of the field in (16) but also the force-
force correlation function in terms of the Hadamard function that satisfies
the fluctuation-dissipation theorem, namely
$G_{H}(\omega)=\coth(\omega/T)\,{\rm{Im}}G_{R}(\omega)$ (this relation was
shown to be true in very general holographic setups Dimitrio_18 ). To estimate
the late time behavior of the particle correlator, we consider the retarded
Green’s function in the small $\omega$ expansion in (16). Through the
fluctuation-dissipation theorem, the Hadamard function $G_{H}\simeq
T\gamma_{T}$. So we have
$\langle\tilde{q}(\omega)\tilde{q}(-\omega)\rangle\propto\omega^{-2}$. Thus,
to obtain the finite particle correlator, we need to introduce an infrared
cutoff, for example, by adding a potential trap $\Omega^{2}q^{2}(\tau)$ to the
end of the string111Another way to introduce the cutoff is by deforming the
integral to the complex $\omega$-plane, where the double pole contribution at
$\omega=0$ also gives the correlator linear in $\tau$. This prescription can
be naturally incorporated in defining different type of Green’s functions by
the contour integrals. Here, the potential trap cut-off is more natural as we
want to make the saturated entanglement entropy finite as in Lee_19 ., and
this gives in the small $\omega$ limit,

$\langle
q(\tau)q(0)\rangle\propto\int_{-\infty}^{\infty}\frac{e^{-i\omega\tau}}{\omega^{2}+\Omega^{2}}d\omega=\pi\Omega^{-1}e^{-\Omega\tau}\,.$
(19)

When taking away the cutoff, $\Omega\rightarrow 0$, we have the cutoff
independent term that grows linearly in $\tau$. This signals the late-time
growth of uncertainty in the particle position for observers moving together
with the particle. Notice that here $\omega$ is the frequency corresponding to
the proper time for the particle. For the particle correlator in the inertial
time, $t$, we can see from (38) and the fact that in the late time
$\tau=b\ln(t/b)$,

$\langle q(t)q(0)\rangle_{cutoff-independent}\propto t^{-1}\ln(t/b)\,.$ (20)

In Lee_19 , we have used this particle correlator, in the setup of strings
with only one end on the boundary, to calculate the entanglement entropy
between particles and environment fields in the late times, and in that case
the divergent piece in $\langle q(t)q(0)\rangle$ is absorbed into the
renormalized saturated entropy. Here, we are more interested in the dynamics
of two entangled particles in the current paper, but we will leave the
detailed study for future works. For the moment, we can qualitatively see the
rate for information exchange between a single particle and the field to which
it couples.

Furthermore, as far as we know, via a bilinear coupling between a particle and
environmental fields, the temperature-dependent damping term can only be
achieved from the holographic approach, whereas in the free field theory, the
same type of the coupling leads to the state-independent back reaction from
the field to the particle JT_1 . It is also worth mentioning that when two
ends of the string undergo the acceleration, they are never in causal contact
during their journeys. In this case, the energy of the particle transferred to
the field does not have a chance to come back, which gives the dissipative
dynamics of the particles. Moreover, due to the causality consideration, the
retarded Green’s function of the particle has no dependence on the distance
between two particles. In contrast, in the static string case, two ends are
initially causally disconnected and then become connected at the time scale of
the distance between two ends. It thus induces a somewhat different dispersion
relation of the particle after two ends reach the causal contact.

###  II.3 The generating functional of fields

To obtain the force-force correlations between two ends of the string, which
encode the information of the wormhole, we need to find the generating
functional for the field that couples to them. For this purpose, we transform
the solution in (15) back to the coordinates in (1). The $R(L)$ branch
corresponds to the $+(-)$ background solution in (4). Also, notice that the
$R$ and $L$ branches have different relations between the proper time $\tau$
and the coordinate time $t$. Thus, from (9)
$\tau_{R/L}=\mp\tanh^{-1}\frac{t}{\sqrt{b^{2}+t^{2}-z^{2}}}$. Then we can
write the general solutions as

$q^{R/L}(t,z)=\int\frac{d\omega}{2\pi}e^{\mp i\omega
b\tanh^{-1}\frac{t}{\sqrt{b^{2}+t^{2}-z^{2}}}}\left(f^{R/L}(\omega)Y_{\omega}(z)+g^{R/L}(\omega)Y^{*}_{\omega}(z)\right)\,.$
(21)

The boundary values of $q^{R/L}(t,z)$ are interpreted holographically as the
sources for the quark and anti-quark, $q^{R/L}(t,z_{m})\equiv q_{0}^{R/L}(t)$.
And their “Fourier transform” is defined as

$q_{0}^{R/L}(t)=\int\frac{d\omega}{2\pi}e^{\mp i\omega
u(t)}\tilde{q}_{0}^{R/L}(\omega)\,,$ (22)

where $u(t)\equiv b\tanh^{-1}\frac{t}{\sqrt{b^{2}+t^{2}-z_{m}^{2}}}$. Thus we
have two boundary conditions at $z=z_{m}$ as

$f^{R/L}(\omega)+g^{R/L}(\omega)=\tilde{q}_{0}^{R/L}(\omega)\,.$ (23)

The other two boundary conditions for $q^{R/L}(t,z)$ are imposed at the
horizon $z=b$. As in Son_03 , the analyticity properties of the boundary
conditions for the mode functions cross the horizon of a maximally extended
black hole gives the Schwinger-Keldysh correlators for a single particle. In
our previous work Lee_15 , we also obtained the same set of correlators by
imposing the constraints from the unitarity and periodicity of the finite
temperature boundary correlators. In the setting of this paper, the maximally
extended black hole does not describe the double copy of a single particle
action giving the Schwinger-Keldysh correlators, but the action of the two
coupled particles. To achieve it, we will modify the definition of the Kruskal
coordinates in Son_03 . We will also give the prescription from the field
theory constraints similar to the ones in Lee_15 .

Near the horizon in (10), the mode functions can be simplified when the
Kruskal coordinates, say in the $R$ branch, are introduced as

$V=-e^{-\frac{\tau+r^{*}}{b}},~{}~{}~{}U=e^{\frac{\tau-r^{*}}{b}}\,,$ (24)

where $r^{*}=b\tanh^{-1}\frac{r}{b}$. Then the mode function can be written as

$q^{R}\simeq f^{R}(\omega)(U)^{-ib\omega}+g^{R}(\omega)(-V)^{ib\omega}\,.$
(25)

Notice that $UV=\frac{r-b}{r+b}$. Therefore, we can even define the mode
functions inside the horizon when $UV>0$. A similar definition of the mode
functions for the $L$ branch is adopted, where now
$r^{*}=-b\tanh^{-1}\frac{r}{b}$. So, the value of $r^{*}$ is negative in the
$L$ branch and becomes positive in the $R$ branch. In the $L$ branch,

$V=e^{\frac{\tau+r^{*}}{b}},~{}~{}~{}U=-e^{-\frac{\tau-r^{*}}{b}}\,.$ (26)

Note that $\tau$ in the $L$ branch has an opposite sign relative to the $R$
branch. Then the mode function in the $L$ branch can be written down as

$q^{L}\simeq f^{L}(\omega)(V)^{ib\omega}+g^{L}(\omega)(-U)^{-ib\omega}\,.$
(27)

To match $q^{R}$ with $q^{L}$, we require them to be analytic on the real $U$
and $V$ axes and the lower half of complex $U$ and $V$ planes as in Son_03 .
Thus, the matching conditions are given by

$f^{R}(\omega)=e^{-\pi b\omega}g^{L}(\omega),~{}~{}~{}g^{R}(\omega)=e^{\pi
b\omega}f^{L}(\omega)\,,$ (28)

and together with (23), they uniquely fix the bulk mode functions
$q^{R/L}(t,z)$ with

$\displaystyle
f^{R}(\tilde{\omega})=\frac{\tilde{q}^{R}_{0}(\omega)}{1-e^{2b\pi\omega}}-\frac{e^{b\pi\omega}\tilde{q}^{L}_{0}(\omega)}{1-e^{2b\pi\omega}}\,,$
(29) $\displaystyle
g^{R}(\tilde{\omega})=\frac{e^{b\pi\omega}\tilde{q}^{L}_{0}(\omega)}{1-e^{2b\pi\omega}}-\frac{e^{2b\pi\omega}\tilde{q}^{R}_{0}(\omega)}{1-e^{2b\pi\omega}}\,,$
(30) $\displaystyle
f^{L}(\tilde{\omega})=\frac{\tilde{q}^{L}_{0}(\omega)}{1-e^{2b\pi\omega}}-\frac{e^{b\pi\omega}\tilde{q}^{R}_{0}(\omega)}{1-e^{2b\pi\omega}}\,,$
(31) $\displaystyle
g^{L}(\tilde{\omega})=\frac{e^{b\pi\omega}\tilde{q}^{R}_{0}(\omega)}{1-e^{2b\pi\omega}}-\frac{e^{2b\pi\omega}\tilde{q}^{L}_{0}(\omega)}{1-e^{2b\pi\omega}}\,.$
(32)

The bulk on-shell action for this solution, which contains only the boundary
terms, is identified as a generating functional for the boundary field
correlators. Leaving out the contact terms that should be related to the
renormalization of the particle self-energy, we have

$\displaystyle S[\tilde{q}^{R}_{0},\tilde{q}^{L}_{0}]$ $\displaystyle=$
$\displaystyle S_{q}(\tilde{q}_{0}^{R})+S_{q}(\tilde{q}_{0}^{L})$ (33)
$\displaystyle=$
$\displaystyle\frac{1}{2}\int\frac{d\omega}{2\pi}\\{\tilde{q}^{R}_{0}(\omega)[\mbox{Re}G_{R}(\omega)+\frac{e^{\frac{\omega}{T}}+1}{e^{\frac{\omega}{T}}-1}i\mbox{Im}G_{R}(\omega)]\tilde{q}^{R}_{0}(-\omega)$
(37)
$\displaystyle+\tilde{q}^{R}_{0}(\omega)[\frac{-2ie^{\frac{\omega}{2T}}}{e^{\frac{\omega}{T}}-1}\mbox{Im}G_{R}(\omega)]\tilde{q}^{L}_{0}(-\omega)$
$\displaystyle+\tilde{q}^{L}_{0}(\omega)[\frac{-2ie^{\frac{\omega}{2T}}}{e^{\frac{\omega}{T}}-1}\mbox{Im}G_{R}(\omega)]\tilde{q}^{R}_{0}(-\omega)$
$\displaystyle+\tilde{q}^{L}_{0}(\omega)[\mbox{Re}G_{R}(\omega)+\frac{e^{\frac{\omega}{T}}+1}{e^{\frac{\omega}{T}}-1}i\mbox{Im}G_{R}(\omega)]\tilde{q}^{L}_{0}(-\omega)\\}$

with $G_{R}(\omega)$ given in (16). Then we can read off the correlators
$G^{ij}(\omega)=\frac{\delta}{\delta\tilde{q}^{i}_{0}}\frac{\delta}{\delta\tilde{q}^{j}_{0}}S[\tilde{q}^{R}_{0},\tilde{q}^{L}_{0}]$
with $i,j=R,L$, and the real-time correlators:

$G^{ij}(t,t^{\prime})=\frac{(-1)^{i}b}{\sqrt{b^{2}+t^{2}-z_{m}^{2}}}\frac{(-1)^{j}b}{\sqrt{b^{2}+t^{\prime
2}-z_{m}^{2}}}\int\frac{d\omega}{2\pi}G^{ij}(\omega)e^{-i\omega(\tau^{i}(t)-\tau^{j}(t^{\prime}))}\,.$
(38)

Here, $\tau^{R}(t)=u(t)$ and $(-1)^{R}=1$, while $\tau^{L}(t)=-u(t)$ and
$(-1)^{L}=-1$.

From the field theory point of view, we consider the particles
$\tilde{q}_{0}^{R/L}$ coupled to the common quantum field at finite
temperature through a bilinear coupling with the same coupling strength.
Integrating out the degrees of freedom of the field leads to the effective
action of the particles. Thus, the generating function of the field requires
to be symmetric under the exchange of $R$ and $L$, leading to $G^{RL}=G^{LR}$
and $G^{RR}=G^{LL}$. Also, the Green’s functions $G^{RR}$ and $G^{LL}$ should
have the form of the finite temperature time-ordered Green’s functions at
temperature $T=\frac{1}{2\pi b}$. The above requirements turn out to be
equivalent to (28) from the bulk analyticity condition.

Notice that $G^{RR}(\omega)=G^{LL}(\omega)$ has the same form as the finite-
temperature time-order correlators. Even though we have not studied the
coupled Langevin equations, the correlation between two entangled particles
is, in principle, encoded in the cross correlators, $G^{LR}(\omega)$ and
$G^{RL}(\omega)$, just like in the case of the single-particle dynamics (18).
$G^{LR}(t,0)=G^{RL}(0,t)$ is plotted numerically in Fig. 2.

Figure 2: The time evolution of the $G^{LR}(t,0)=G^{RL}(0,t)$ correlator. $t$
is in the unit of $b$ and $G^{LR}$ is in the unit of
$\frac{4R^{2}T_{0}i}{b^{4}\sqrt{1-\frac{z_{m}^{2}}{b^{2}}}}$. The curves
correspond to the parameters $\frac{z_{m}}{b}=0.9$(blue), $0.7$(orange) and
$0.1$(green), respectively.

We turn to the early-time and late-time analytic behaviors of the
$G^{LR}(t,0)=G^{RL}(0,t)$ correlators. Note that the retarded Green’s
function, $G_{R}(\omega)$ in (16), has the pole at $\omega=-\frac{i}{z_{m}}$,
and the Boltzmann factor gives the poles at $\omega=\frac{in}{b}$ with non-
zero integers. We take $b\gg z_{m}$, where $b$ plays a role of an IR cut-off
and $z_{m}$ a UV cut-off. Thus, in the limit $t\ll z_{m}$ with the proper time
$u(t)\simeq t$, the time evolution of $G^{LR}(t,0)=G^{RL}(t,0)$ is dominated
by the pole at $\omega=-\frac{i}{z_{m}}$ giving

$G^{LR}(t,0)=G^{RL}(t,0)\propto e^{-\frac{t}{z_{m}}}\simeq
1-\frac{t}{z_{m}}+\cdots\,.$ (39)

As for $t\gg b$ with $u(t)\simeq b\ln(t/b)$, the pole at $\omega=\frac{-i}{b}$
dominates the correlators $G^{ij}(t,0)$ resulting in the power-law decay

$G^{LR}(t,0)=G^{RL}(t,0)\propto\left(\frac{t}{b}\right)^{-2}\,.$ (40)

This can be compared with the correlators of the free scalar fields on the
spacetime points of two observers following the trajectories, say along the
$x$-direction with the uniform acceleration $1/b$. The worldlines of two
observers are specified by
$x_{L}^{\mu}=(b\sinh\frac{\tau_{L}}{b},-b\cosh\frac{\tau_{L}}{b},0,0)$ and
$x_{R}^{\mu}=(b\sinh\frac{\tau_{R}}{b},b\cosh\frac{\tau_{R}}{b},0,0)$. In
particular, the correlator of the free scalar field at the spacetime point
$x_{L}^{\mu}$ and $x_{R}^{\mu}$ is given by

$G(x_{L}^{\mu},x_{R}^{\mu})=\frac{1}{4\pi^{2}}\frac{1}{|\vec{x}_{R}-\vec{x}_{L}|^{2}-|x_{R}^{0}-x_{L}^{0}|^{2}}\,.$
(41)

Thus, when $t\ll b$ giving the proper time $\tau\simeq t$,

$G(t,0)\propto\left(1-\frac{t^{2}}{2b^{2}}+...\right)\,,$ (42)

whereas for $t\gg b$ giving $\tau\simeq b\ln(t/b)$,

$G(t,0)\propto\left(\frac{t}{b}\right)^{-1}\,$ (43)

instead. By comparing with (40), we see that for strongly coupled fields the
correlation between fields at two spacelike-separate points dies out more
quickly than the one for free fields, and this potentially induces the
interaction between two spacelike-separate particles and gives a shorter time
scale for the particles to sustain quantum entanglement between them.

##  III fluctuations of transverse modes in the static string

This section considers the static string in the same $AdS_{d+1}$ background in
(1). We also solve the linearized equation for the perturbations in this
string background exactly and obtain the generating functional for the field
that couples to the two ends of the string. In the same gauge choice as in
(3), where the coordinate $x$ is independent of time, the classical equation
of motion for the static string with its two ends at the boundary $z=0$ has a
solution with the two-branch joint at $z=z_{0}$ Maldacena:1998im ,

$x_{0}(z)=\pm\int_{z_{0}}^{z}\sqrt{\frac{y^{4}}{z_{0}^{4}-y^{4}}}dy\,.$ (44)

The induced metric on this worldsheet parametrized by $t$ and $x$ has the
causal structure as in Fig. 3. As compared with the case of the accelerating
string (in Fig. 1), we see that two ends are in causal contact in this case.

Figure 3: Few sample light rays (black curves) in the worldsheet(44). Here $x$
and $t$ are in the unit of $z_{0}$. Two vertical lines (blue) are trajectories
of quark and anti-quark. Light rays emitted from one quark reach the other in
some finite times. So two quarks are causally connected.

Similar to the accelerating string case, we consider the linearized
perturbations in the direction transverse to the worldsheet, with the
embedding in (1) as $X^{\mu}=(t,z,x_{0}(z),p(t,z),0,0...)$ where $p$ is small
as compared to $x_{0}$. We define the Fourier transform of the transverse
modes as

$p(t,z)=\int\frac{d\omega}{2\pi}h_{\omega}(z)e^{-i\omega t}\,,$ (45)

which has the linearized equation of motion on both $R$ and $L$ branches as,

$h^{\prime\prime}_{\omega}-\frac{2}{\rho(1-\rho^{4})}h^{\prime}_{\omega}+\frac{\tilde{\omega}^{2}}{1-\rho^{4}}h_{\omega}=0\,,$
(46)

with the dimensionless variables $\rho=\frac{z}{z_{0}}$ and
$\tilde{\omega}=z_{0}\omega$. There are two exact independent solutions, which
are normalized in the boundary at $\rho=\rho_{m}\equiv\frac{z_{m}}{z_{0}}$,

$F_{\tilde{\omega}}(\rho)=e^{A_{\tilde{\omega}}(\rho)-A_{\tilde{\omega}}(\rho_{m})}\,,\quad\quad
H_{\tilde{\omega}}(\rho)=e^{B_{\tilde{\omega}}(\rho)-B_{\tilde{\omega}}(\rho_{m})}\,$
(47)

where

$\displaystyle
A_{\tilde{\omega}}(\rho)=\frac{1}{2}\ln(1+\tilde{\omega}^{2}\rho^{2})+\frac{\sqrt{1-\tilde{\omega}^{4}}}{\tilde{\omega}}(E_{0}(\rho)-E_{\tilde{\omega}}(\rho))\,,$
(48) $\displaystyle
B_{\tilde{\omega}}(\rho)=\frac{1}{2}\ln(1+\tilde{\omega}^{2}\rho^{2})-\frac{\sqrt{1-\tilde{\omega}^{4}}}{\tilde{\omega}}(E_{0}(\rho)-E_{\tilde{\omega}}(\rho))\,,$
(49)

with

$E_{0}(\rho)=\int_{0}^{\rho}\frac{dy}{\sqrt{1-y^{4}}}\,,\quad\quad
E_{\tilde{\omega}}(\rho)=\int_{0}^{\rho}\frac{dy}{(1+\tilde{\omega}^{2}y^{2})\sqrt{1-y^{4}}}\,.$
(50)

Then the general solutions of the perturbations on the $R$ and $L$ branches
are written as

$p^{R/L}(t,\rho)=\int\frac{d\omega}{2\pi}\left(C_{1}^{R/L}(\tilde{\omega})F_{\tilde{\omega}}(\rho)+C_{2}^{R/L}(\tilde{\omega})H_{\tilde{\omega}}(\rho)\right)e^{-i\omega
t}\,.$ (51)

The boundary values of the perturbations are interpreted as the sources for
the quark and anti-quark pair,

$p^{R/L}(t,\rho_{m})=p^{R/L}_{0}(t)=\int\frac{d\omega}{2\pi}\,\tilde{p}^{R/L}_{0}(\omega)\,e^{-i\omega
t}\,.$ (52)

Thus, we have the boundary conditions

$C_{1}^{R/L}(\tilde{\omega})+C_{2}^{R/L}(\tilde{\omega})=\tilde{p}^{R/L}_{0}(\tilde{\omega})\,.$
(53)

The other two boundary conditions are imposed at the tip of the string
($\rho=1$). We require the perturbations to be smooth across the tip, namely
$p^{R}(t,1)=p^{L}(t,1)$ and
$\partial_{\rho}p^{R}(t,1)=-\partial_{\rho}p^{L}(t,1)$, which lead to

$\displaystyle
C_{1}^{R}(\tilde{\omega})F_{\tilde{\omega}}(1)+C_{2}^{R}(\tilde{\omega})H_{\tilde{\omega}}(1)=C_{1}^{L}(\tilde{\omega})F_{\tilde{\omega}}(1)+C_{2}^{L}(\tilde{\omega})H_{\tilde{\omega}}(1)\,,$
(54) $\displaystyle
C_{1}^{R}(\tilde{\omega})F_{\tilde{\omega}}(1)-C_{2}^{R}(\tilde{\omega})H_{\tilde{\omega}}(1)=-C_{1}^{L}(\tilde{\omega})F_{\tilde{\omega}}(1)+C_{2}^{L}(\tilde{\omega})H_{\tilde{\omega}}(1)\,.$
(55)

Note that to obtain (55), we have factored out a common divergence factor.
Together with (53), the unique solution can be obtained as

$\displaystyle
C^{R}_{1}(\tilde{\omega})=\frac{\tilde{p}^{R}_{0}(\tilde{\omega})}{1-e^{2\kappa(\tilde{\omega})}}-\frac{e^{\kappa(\tilde{\omega})}\tilde{p}^{L}_{0}(\tilde{\omega})}{1-e^{2\kappa(\tilde{\omega})}}\,,$
(56) $\displaystyle
C^{R}_{2}(\tilde{\omega})=\frac{e^{\kappa(\tilde{\omega})}\tilde{p}^{L}_{0}(\tilde{\omega})}{1-e^{2\kappa(\tilde{\omega})}}-\frac{e^{2\kappa(\tilde{\omega})}\tilde{p}^{R}_{0}(\tilde{\omega})}{1-e^{2\kappa(\tilde{\omega})}}\,,$
(57) $\displaystyle
C^{L}_{1}(\tilde{\omega})=\frac{\tilde{p}^{L}_{0}(\tilde{\omega})}{1-e^{2\kappa(\tilde{\omega})}}-\frac{e^{\kappa(\tilde{\omega})}\tilde{p}^{R}_{0}(\tilde{\omega})}{1-e^{2\kappa(\tilde{\omega})}}\,,$
(58) $\displaystyle
C^{L}_{2}(\tilde{\omega})=\frac{e^{\kappa(\tilde{\omega})}\tilde{p}^{R}_{0}(\tilde{\omega})}{1-e^{2\kappa(\tilde{\omega})}}-\frac{e^{2\kappa(\tilde{\omega})}\tilde{p}^{L}_{0}(\tilde{\omega})}{1-e^{2\kappa(\tilde{\omega})}}\,,$
(59)

where
$\kappa(\tilde{\omega})=\frac{2\sqrt{1-\tilde{\omega}^{4}}}{\tilde{\omega}}(E_{0}(1)-E_{\tilde{\omega}}(1)-E_{0}(\rho_{m})+E_{\tilde{\omega}}(\rho_{m}))$.
Moreover, the on-shell action, which is identified as the generating
functional of the field, becomes

$\displaystyle
S[\tilde{p}^{R}_{0},\tilde{p}^{L}_{0}]=-\frac{R^{2}T_{0}}{2z^{3}_{0}}\lim_{\rho\rightarrow\rho_{m}}\int
dt\frac{\sqrt{1-\rho^{4}}}{\rho^{2}}\left(p^{R}(t,\rho)\partial_{\rho}p^{R}(t,\rho)+p^{L}(t,\rho)\partial_{\rho}p^{L}(t,\rho)\right)$
(60)
$\displaystyle=-\frac{R^{2}T_{0}}{2z^{3}_{0}}\frac{\sqrt{1-\rho_{m}^{4}}}{\rho_{m}^{2}}\int\frac{d\omega}{2\pi}\Big{\\{}\tilde{p}_{0}^{R}(\tilde{\omega})\left[\frac{1}{1-e^{2\kappa(\tilde{\omega})}}(A^{\prime}_{\tilde{\omega}}(\rho_{m})-e^{2\kappa(\tilde{\omega})}B^{\prime}_{\tilde{\omega}}(\rho_{m}))\right]\tilde{p}_{0}^{R}(-\tilde{\omega})$
(61)
$\displaystyle+\tilde{p}_{0}^{R}(\tilde{\omega})\left[\frac{e^{\kappa(\tilde{\omega})}}{1-e^{2\kappa(\tilde{\omega})}}(B^{\prime}_{\tilde{\omega}}(\rho_{m})-A^{\prime}_{\tilde{\omega}}(\rho_{m}))\right]\tilde{p}_{0}^{L}(-\tilde{\omega})$
(62)
$\displaystyle+\tilde{p}_{0}^{L}(\tilde{\omega})\left[\frac{e^{\kappa(\tilde{\omega})}}{1-e^{2\kappa(\tilde{\omega})}}(B^{\prime}_{\tilde{\omega}}(\rho_{m})-A^{\prime}_{\tilde{\omega}}(\rho_{m}))\right]\tilde{p}_{0}^{R}(-\tilde{\omega})$
(63)
$\displaystyle+\tilde{p}_{0}^{L}(\tilde{\omega})\left[\frac{1}{1-e^{2\kappa(\tilde{\omega})}}(A^{\prime}_{\tilde{\omega}}(\rho_{m})-e^{2\kappa(\tilde{\omega})}B^{\prime}_{\tilde{\omega}}(\rho_{m}))\right]\tilde{p}_{0}^{L}(-\tilde{\omega})\Big{\\}}\,.$
(64)

Based on the analyticity property of the mode functions, we can identify the
retarded Green’s function as

$\displaystyle
G^{(0)}_{R}(\omega)=-\frac{R^{2}T_{0}}{z^{3}_{0}}\frac{\sqrt{1-\rho_{m}^{4}}}{\rho_{m}^{2}}F^{\prime}_{\tilde{\omega}}(\rho_{m})F_{\tilde{\omega}}(\rho_{m})$
(65)
$\displaystyle=-\frac{R^{2}T_{0}}{z^{3}_{0}}\frac{\sqrt{1-\rho_{m}^{4}}}{\rho_{m}^{2}}A^{\prime}_{\tilde{\omega}}(\rho_{m})=-\frac{R^{2}T_{0}}{z^{3}_{0}}\frac{\sqrt{1-\rho_{m}^{4}}}{\rho_{m}^{2}}\,\,\frac{\tilde{\omega}\rho_{m}(\tilde{\omega}+\rho_{m}\sqrt{\frac{1-\tilde{\omega}^{4}}{1-\rho_{m}^{4}}})}{1+\tilde{\omega}^{2}\rho_{m}^{2}}\,,$
(66)

which is analytic in the upper half complex $\omega$ plane. Similarly, the
advanced Green’s function, which is analytic in the lower half complex
$\omega$ plane, can also be identified as

$G^{(0)}_{A}(\omega)=-\frac{R^{2}T_{0}}{z^{3}_{0}}\frac{\sqrt{1-\rho_{m}^{4}}}{\rho_{m}^{2}}B^{\prime}_{\tilde{\omega}}(\rho_{m})=-\frac{R^{2}T_{0}}{z^{3}_{0}}\frac{\sqrt{1-\rho_{m}^{4}}}{\rho_{m}^{2}}\,\,\frac{\tilde{\omega}\rho_{m}(\tilde{\omega}-\rho_{m}\sqrt{\frac{1-\tilde{\omega}^{4}}{1-\rho_{m}^{4}}})}{1+\tilde{\omega}^{2}\rho_{m}^{2}}\,.$
(67)

Thus, from the generating functional, we can read off the field correlators
arising from the field vacuum fluctuations as

$\displaystyle
G^{(0)RR}(\omega)=G^{(0)LL}(\omega)=\frac{1}{1-e^{2\kappa(\tilde{\omega})}}(G^{(0)}_{R}(\omega)-e^{2\kappa(\tilde{\omega})}G^{(0)}_{A}(\omega))\,,$
(68) $\displaystyle
G^{(0)RL}(\omega)=G^{(0)RL}(\omega)=\frac{e^{\kappa(\tilde{\omega})}}{1-e^{2\kappa(\tilde{\omega})}}(G^{(0)}_{A}(\omega)-G^{(0)}_{R}(\omega))\,.$
(69)

To see that these Green’s functions are reasonable, we consider the limits of
$\omega^{-1}\ll z_{0}$ and $z_{0}\gg z_{m}$; they lead to $\tilde{\omega}\gg
1$ and $\rho_{m}\ll 1$, where the time-order correlators on the single
particle are recovered in the limit of $z_{0}\rightarrow\infty$ as expected.
With $\kappa(\tilde{\omega})\simeq 2i\gamma_{1}\tilde{\omega}$ and
$\gamma_{1}\equiv
E_{0}(1)=\frac{\sqrt{\pi}\Gamma(\frac{5}{4})}{\Gamma(\frac{3}{4})}$,
$G^{(0)}_{R}(\omega)=\big{(}G^{(0)}_{A}(\omega)\big{)}^{*}$ gives the Green’s
functions of the form

$G^{(0)RR}(\omega)=G^{(0)LL}(\omega)\simeq\mbox{Re}G^{(0)}_{R}(\omega)+\frac{1+e^{4i\gamma_{1}\tilde{\omega}}}{1-e^{4i\gamma_{1}\tilde{\omega}}}i\mbox{Im}G^{(0)}_{R}(\omega)\,.$
(70)

Note that this Green’s function is not well defined since there are poles on
the real $\omega$ values. We thus employ the $i\varepsilon$ prescription. We
replace $\omega$ by $\omega(1-i\varepsilon)$ and take $\varepsilon\rightarrow
0$ only at the end of the calculations. Then, in the large $\tilde{\omega}$
limit, the Green’s functions boil down to

$G^{(0)RR}(\omega)=G^{(0)LL}(\omega)\simeq\mbox{Re}G^{(0)}_{R}(\omega)+\mbox{sign}(\omega)i\mbox{Im}G^{(0)}_{R}(\omega)\,,$
(71)

which is precisely the Feynman Green’s function at zero temperature. In the
limit of $\omega\gg z_{0}^{-1}$ with the relatively large distance between two
particles but $\omega\ll z_{m}^{-1}$ for this small $\omega$ compared with the
UV cutoff scale $z_{m}^{-1}$, the retarded Green’s function is approximated by

$\displaystyle G^{(0)}_{R}(\omega)$ $\displaystyle\simeq$
$\displaystyle-\frac{R^{2}T_{0}}{z_{m}^{3}}\frac{z_{m}^{2}\omega^{2}+iz_{m}^{3}\omega^{3}}{1+z_{m}^{2}\omega^{2}}$
(72) $\displaystyle\simeq$ $\displaystyle-
M_{0}\,\omega^{2}-i\gamma\,\omega^{3}+\cdots\,.$

This gives the induced mass $M_{0}\simeq R^{2}T_{0}/z_{m}$, which has the same
form as in the accelerating string case. The damping term of the $\omega^{3}$
dependence is a typical self-force effect for the Brownian motion with the
cutoff-independent damping coefficient $\gamma\simeq R^{2}T_{0}$, which also
agrees with our previous results Lee_13 . Furthermore, this damping term can
be compared with the one in the Abraham-Lorentz-Dirac equation of the charged
particle with the self-force effects from the coupling to the electromagnetic
field Guijosa_09 ; Guijosa_10 ; JT_1 . On top of that, as anticipated, the
$z_{0}$ dependence drops off at this limit where two particles are far apart.
Moreover, with the $i\epsilon$ prescription, the cross correlators
$G^{(0)RL}(\omega)$ and $G^{(0)LR}(\omega)$ vanish in this limit.

In the other limits of $\tilde{\omega}\ll 1$ and $\rho_{m}\ll 1$, the
wavelength of the perturbations is longer than the $z_{0}$ scale. The late-
time behavior, when two ends are causally connected, is probed. In this case,
we have

$\kappa(\tilde{\omega})=2\gamma_{2}\tilde{\omega}$ (73)

with $\gamma_{2}=\frac{\sqrt{\pi}\Gamma(\frac{7}{4})}{3\Gamma(\frac{5}{4})}$.
As ${\omega}<z_{0}^{-1}$, both the retarded and advanced Green’s functions
become real-valued. This implies that there exists the threshold time scale
$z_{0}^{-1}$, after which the particle dynamics become non-dissipative and
depend on $z_{0}$. Thus, the strongly coupled field induces strikingly
different effects on the particle compared to those given by the free field
theory JT . In this late-time limit, the retarded and advanced Green’s
functions can be approximated by

$\displaystyle G^{(0)}_{R}(\omega)$ $\displaystyle\simeq$
$\displaystyle-\frac{R^{2}T_{0}}{z_{0}^{3}}\left(\tilde{\omega}+\frac{\tilde{\omega}^{2}}{\rho_{m}}-\rho_{m}^{2}\tilde{\omega}^{3}+\mathcal{O}(\tilde{\omega}^{4})\right)\,,$
(74) $\displaystyle G^{(0)}_{A}(\omega)$ $\displaystyle\simeq$
$\displaystyle-\frac{R^{2}T_{0}}{z_{0}^{3}}\left(-\tilde{\omega}+\frac{\tilde{\omega}^{2}}{\rho_{m}}+\rho_{m}^{2}\tilde{\omega}^{3}+\mathcal{O}(\tilde{\omega}^{4})\right)\,.$
(75)

Then the correlators become

$\displaystyle G^{(0)RR}(\omega)=G^{(0)LL}(\omega)$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}(G^{(0)}_{R}(\omega)+G^{(0)}_{A}(\omega))+\frac{1+e^{4\gamma_{2}\tilde{\omega}}}{1-e^{4\gamma_{2}\tilde{\omega}}}\frac{1}{2}(G^{(0)}_{R}(\omega)-G^{(0)}_{A}(\omega))$
(76) $\displaystyle\simeq$ $\displaystyle-
M_{z_{0}}\omega^{2}+\frac{R^{2}T_{0}}{2\gamma_{2}}\frac{1}{z_{0}^{3}}+\mathcal{O}(\omega^{4})\,,$
(77)

and also

$\displaystyle G^{(0)RL}(\omega)=G^{(0)LR}(\omega)$ $\displaystyle\simeq$
$\displaystyle\frac{e^{2\gamma_{2}\tilde{\omega}}}{1-e^{4\gamma_{2}\tilde{\omega}}}(G^{(0)}_{A}(\omega)-G^{(0)}_{R}(\omega))$
(78) $\displaystyle\simeq$
$\displaystyle-\frac{R^{2}T_{0}}{2\gamma_{2}}\left(\frac{1}{z_{0}^{3}}-\frac{z_{m}^{2}}{z_{0}^{3}}\omega^{2}+\mathcal{O}(\omega^{4})\right)\,.$
(79)

Thus, we find that the induced mass $M_{z_{0}}=M_{0}+\Delta M_{z_{0}}$ now has
a $z_{0}$ dependent minor correction $\Delta
M_{z_{0}}=\frac{R^{2}T_{0}}{2\gamma_{2}}\frac{z_{m}^{2}}{z_{0}^{3}}$. The
cross correlators, $G^{(0)RL}(\omega)$ and $G^{(0)RL}(\omega)$, are dominated
by the constant term of $1/z_{0}^{3}$, giving the decay rate in time as
$t^{-1}$. Compared to the accelerating string case where the decay rate of the
cross correlators is $\propto t^{-2}$, this potentially gives the higher
disentanglement rate of the quark and anti-quark pair. We will explore it in
our future work.

##  IV conclusion

In this paper, we study the holographic dual of the EPR pair of the quark and
anti-quark in the super Yang-Mills theory. We calculate the two-point field
correlators exactly, which are described by the linearized perturbations in
the worldsheet of the string with two ends either undergoing uniform
acceleration or being static. We find that the causality between two ends
crucially determines the nature of the induced dispersion relation of the
particles anchoring at the endpoints and also the feature of the mutual
correlation between fields at the positions of two particles. For the causally
disconnected two ends, the induced dispersion relation of each particle is
dissipative, and the mutual correlation between fields dies out in the late
times in the power law of time as $t^{-2}$. In contrast, for the casually
connected two ends, the dispersion relation becomes non-dissipative in the
late times, and the mutual correlation of fields decays in the power law of
time as $t^{-1}$. There are some interesting problems to further understand
the entanglement dynamics between two particles from the mutual field
correlators. For example, we plan to explore how the time evolution of the
entanglement entropy of the Brownian particles anchoring at two ends of a
static string depends on the distance $z_{0}$ between two particles using the
method in the paper Lee_19 .

Another interesting problem is considering the effects of quantum field
fluctuations on the entanglement entropy between the entangled quark and anti-
quark as found in Jensen_13  with the accelerating string solution. Compared
to the weak coupling analysis in SY1 ; SY2 , we would obtain some insights
into the time evolution of entanglement in strongly coupled theories. In
particular, it is of interest to know whether or not the sudden death of the
entanglement entropy also occurs in strongly coupled theory. If so, since the
entanglement is geometrically realized as a fundamental string connecting
quark-antiquark pair, the time evolution may involve some nonperturbative
effects leading to the change of the background solutions. For this purpose,
we should extend the formula of the holographic influence functional in Lee_15
and Son_03  for finding the dynamics of a pair of the entangled particles.

In Maldacena_01 , the exponential decay of the correlation between two
entangled boundary regions in the eternal AdS black hole implies the loss of
information about the black hole behind the horizon. In the case of the
accelerating string, the induced worldsheet background also has the causal
structure the same as an eternal AdS black hole. However, the decay of the
correlation between two boundary particles is expected as they couple to the
super Yang-Mills fields, and the information can leak into the environments.
If the ER=EPR conjecture also works in this case, the disentanglement of two
particles can lead to the breakdown of the wormhole geometry, which is beyond
the probe string approach we consider in this paper. We do not have a solution
for this problem yet. How this solution can be related to the black hole
information problem is also an exciting problem to pursue.

## Acknowledgments

The work of S. K. was supported in part by MOST109–2811–M–007–558 and
MOST109–2112–M–007–018–MY2. The work of D.S. L. was supported in part by
MOST-110-2112-M-259 -003. The work of C.P. Y. was supported in part by
MOST-110-2112-M-259 -004 and MOST-109-2112-M-259 -006.

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AMSbUmsbmn

# Reconstructing curves from their Hodge classes

Maria Gioia Cifani, Gian Pietro Pirola and Enrico Schlesinger  Dipartimento di
matematica e fisica, Università di Roma 3, Largo San Leonardo Murialdo, 1 –
00146 Roma<EMAIL_ADDRESS>Department of Mathematics ’F.
Casorati’, University of Pavia, via Ferrata 5, 27100 Pavia, Italy
<EMAIL_ADDRESS>Dipartimento di Matematica ’F. Brioschi’,
Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
<EMAIL_ADDRESS>
###### Abstract.

Let $S$ be a smooth algebraic surface in $\mathbb{P}^{3}(\mathbb{C})$. A curve
$C$ in $S$ has a cohomology class $\eta_{C}\in
H^{1}\left(\Omega^{1}_{S}\right)$. Define $\alpha(C)$ to be the equivalence
class of $\eta_{C}$ in the quotient of $H^{1}\left(\Omega^{1}_{S}\right)$
modulo the subspace generated by the class $\eta_{H}$ of a plane section of
$S$. In the paper ”Reconstructing subvarieties from their periods” the authors
Movasati and Sertöz pose several interesting questions about the
reconstruction of $C$ from the annihilator $I_{\alpha(C)}$ of $\alpha(C)$ in
the polynomial ring $R=H^{0}_{*}(\mathcal{O}_{\mathbb{P}^{3}})$. It contains
the homogeneous ideal of $C$, but is much larger as $R/I_{\alpha(C)}$ is
artinian. We give sharp numerical conditions that guarantee $C$ is
reconstructed by forms of low degree in $I_{\alpha(C)}$. We also show it is
not always the case that the class $\alpha(C)$ is perfect, that is, that
$I_{\alpha(C)}$ could be bigger than the sum of the Jacobian ideal of $S$ and
of the homogeneous ideals of curves $D$ in $S$ for which
$I_{\alpha(D)}=I_{\alpha(C)}$.

##  1\. Introduction

The Hodge conjecture, one of the most challenging and interesting open
questions in algebraic geometry, can be regarded as a reconstruction problem.
Even when the Hodge conjecture is known, as for curves on surfaces, there are
a series of somewhat related problems that might shed a new light on some
aspects of the cycle map.

A good example is given by [GH83, Theorem 4.b.26] where it is proven that,
given a smooth surface $S\subset\mathbb{P}^{3}$ and an integral class $\gamma$
in $H^{1}\left(\Omega^{1}_{S}\right)$ with the same numerical properties as
the fundamental class of a curve $C\subset S$, then $\gamma$ is itself the
fundamental class of an effective divisor $D\subset S$ provided $\deg(S)$ is
large relative to the self-intersection of $\gamma$ and to $\deg(C)$.

In a similar vein, very interesting recent work by Movasati and Sertöz [MS19]
concerns the reconstruction of subvarieties of $\mathbb{P}^{N}$ from their
periods. Our purpose is to give an answer, in the special case of curves lying
on a smooth algebraic surface $S$ in complex projective space, to two
questions raised in [MS19] that we now illustrate. A curve $C$ in $S$ has a
fundamental cohomology class $\eta_{C}\in H^{1}\left(\Omega^{1}_{S}\right)$.
We denote by $\alpha(C)$ the equivalence class of $\eta_{C}$ in the quotient
of $H^{1}\left(\Omega^{1}_{S}\right)$ modulo the subspace generated by the
class $\eta_{H}$ of a plane section of $S$: the class $\alpha(C)$ depends on
the embedding of $S$ in $\mathbb{P}^{3}$, and can be seen as a linear form on
the primitive cohomology $H^{1}\left(\Omega^{1}_{S}\right)^{\perp_{H}}$.
Following [MS19] we focus our analysis on the annihilator $I_{\alpha(C)}$ of
$\alpha(C)$ in the polynomial ring
$R=H^{0}_{*}(\mathcal{O}_{\mathbb{P}^{3}})$. Note - see Proposition 2.3 \-
that $I_{\alpha(C)}=I_{\alpha(D)}$ for two curves $C$ and $D$ in $S$ if and
only if $mC+nD+pH$ is linearly equivalent to zero for some choice of integers
$m$, $n$ and $p$ with $m$ and $n$ non zero and relatively prime. Thus the
annihilator $I_{\alpha(C)}$, which contains the homogeneous ideal $I_{C}$ of
$C$ and the Jacobian ideal $J_{S}$ of $S$, in general it is much larger than
$I_{C}+J_{S}$, as it contains the ideal $I_{D}$ for any curve $D$ for which
there is a relation $mC+nD+pH\sim 0$ as above. Still, one can ask whether $C$
can be reconstructed from $I_{\alpha(C)}$ when $\deg(S)$ is large with respect
to the degree or other invariants of $C$, and Movasati and Sertöz in [MS19]
investigate, in a more general context than ours, the following questions:

  1. (1)

under which conditions $I_{\alpha(C)}$ reconstructs $C$, in the sense that
forms of low degree in $I_{\alpha(C)}$ cut out the curve $C$ scheme-
theoretically? To be precise, we will say that $C$ is reconstructed at level
$m$ by $I_{\alpha(C)}$ if its homogeneous ideal $I_{C}$ is generated over $R$
by $I_{\alpha(C),\leq m}$ \- that is, by forms of degree $\leq m$ in
$I_{\alpha(C)}$.

  2. (2)

as the example of complete intersection suggests, they define a class
$\alpha\in H^{1}\left(\Omega^{1}_{S}\right)/{\mathbb{C}\,\eta_{H}}$ to be
_perfect at level $m$_ if there exist effective divisors $D_{1},\ldots,D_{q}$
in $S$ such that $I_{\alpha(D_{i})}=I_{\alpha}$ for every $i=1,\ldots q$ and

$I_{\alpha,j}=\sum_{i=1}^{q}I_{{D_{i}},j}+J_{S,j}\quad\mbox{for every $j\leq
m$}$

where $J_{S}$ denotes the Jacobian ideal of $S$. The question is under which
conditions the class $\alpha(C)$ is perfect at level $m$, and whether all
classes $\alpha(C)$ are perfect at every level $m$.

In this paper we prove two theorems that give partial answers to these
questions. Our first theorem extends known results on complete intersections
[Dan17, MS19] to arithmetically Cohen-Macaulay curves (ACM curves for short).
The tools we need for this are provided by a very nice paper by Ellingsrud and
Peskine [EP93] which unfortunately seems to be little known. In [EP93] the
authors were interested in the study of the Noether-Lefschetz locus, and the
invariant $\alpha(C)$ plays a prominent role in their work because it vanishes
if and only if the curve is a complete intersection of $S$ and another
surface. Their paper connects the class $\alpha(C)$ to the normal sequence
arising from the inclusions $C\subset S\subset\mathbb{P}^{3}$ and gives an
effective tool for computing its annihilator $I_{\alpha(C)}$ \- see Lemma 2.5.
To state our first theorem, given a curve $C$ in $\mathbb{P}^{3}$, we let
$s(C)$ be the minimum degree of a surface containing $C$, and $e(C)$ the index
of speciality of $C$, that is, the maximum $n$ such that $\mathcal{O}_{C}(n)$
is special, that is, $h^{1}(\mathcal{O}_{C}(n)))>0$.

######  Theorem 1.1.

Suppose $C$ is an ACM curve on the smooth surface
$S\subseteq\mathbb{P}^{3}(\mathbb{C})$. Let $s$ denote the degree of $S$. Then

  1. (1)

if $s\geq 2e(C)+8-s(C)$, the curve $C$ is reconstructed at level $e(C)+3$ by
$I_{\alpha(C)}$;

  2. (2)

the class $\alpha(C)$ is perfect at level $m$ for every $m$.

For example, let $C$ be a twisted cubic curve: $C$ is then ACM with invariants
$s(C)=2$ and $e(C)=-1$. By Theorem 1.1, if $S$ is a quartic surface containing
$C$, then $C$ is cut out scheme-theoretically by the quadrics whose equations
lie in $I_{\alpha(C,S)}$. This was suggested and verified for thousands of
randomly chosen quartic surfaces containing $C$ in [MS19, Sections 2.3 and
3.2].

Our second theorem provides a first example of a non-perfect algebraic class
$\alpha(C)$, giving a negative answer to Question 2.12 in [MS19].

######  Theorem 1.2.

Let $C\subset\mathbb{P}^{3}$ be a smooth rational curve of degree $4$
contained in a smooth surface $S$ of degree $s=4$. The class $\alpha(C)$ in
$S$ is not perfect at level $3$.

It would be very interesting to determine conditions for a class $\alpha(C)$
to be perfect, and we don’t know whether ACM curves form the largest set of
curves $C$ whose classes $\alpha(C)$, in any smooth surface $S$ containing
$C$, are perfect.

Finally, we note that Movasati and Sertöz pose their questions of
reconstruction and perfectness in a more general context, namely for classes
in $H^{n}(\Omega_{X}^{n})$ of varieties of dimension $n$ in smooth
hypersurfaces $X$ in $\mathbb{P}^{2n+1}$. An interesting and challenging
problem is trying to answer those question for every $n$, generalizing as far
as it is possible the results of this paper to higher dimension.

The paper is structured as follows. In section 2 we collect some well known
facts we need, and, for the benefit of the reader, we recall in some detail
the constructions from [EP93] we will need in the sequel of the paper. In
section 3 we prove Proposition 3.1 \- a numerical criterion that guarantees,
when the degree of $S$ is large to respect to that of $C$, that the curve $C$
is reconstructed at a certain level $m$ by $I_{\alpha(C)}$. As an example we
prove in Corollary 3.2 a reconstruction result for a general rational curve of
degree $d$. In section 4 we prove Theorem 1.1, which is split in Theorems 4.1
and 4.8. In section 5 we prove Theorem 1.2.

The authors are members of GNSAGA of INdAM. The authors were partially
supported by national MIUR funds, PRIN 2017 Moduli and Lie theory, and by
MIUR: Dipartimenti di Eccellenza Program (2018-2022) - Dept. of Math. Univ. of
Pavia.

##  2\. Preliminaries

We work in the projective space $\mathbb{P}^{3}$ over the field $\mathbb{C}$
of complex numbers. Given a coherent sheaf $\mathscr{F}$ on $\mathbb{P}^{3}$
and $i\in\mathbb{N}$, we define

$H^{i}_{*}(\mathcal{F})=\bigoplus_{n\in\mathbb{N}}H^{i}\left(\mathbb{P}^{3},\,\mathcal{F}(n)\right).$

These are graded module over the polynomial ring

$R=H^{0}_{*}(\mathcal{O}_{\mathbb{P}^{3}})\cong\mathbb{C}[x,y,z,w].$

Given a subscheme $X$ of $\mathbb{P}^{3}$, we will denote by $\mathscr{I}_{X}$
its sheaf of ideals, and by $I_{X}=H^{0}_{*}(\mathscr{I}_{X})$ its saturated
homogeneous ideal in $R$. We will write $I_{X,n}$ to denote its $n^{th}$
graded piece $H^{0}(\mathbb{P}^{3},\mathscr{I}_{X}(n))$.

If $M=\bigoplus_{n\in\mathbb{Z}}M_{n}$ a graded $R$-module, the graded
$\mathbb{C}$-dual module $M^{*}$ of $M$ is defined by setting
$(M^{*})_{m}=\operatorname{Hom}_{\mathbb{C}}(M_{-m},\mathbb{C})$ with
multiplication $R_{n}\times(M^{*})_{m}\to(M^{*})_{m+n}$ defined by

$g\lambda(v)=\lambda(gv),\ \ \forall g\in R_{n},\lambda\in(M^{*})_{m},v\in
M_{-m-n}.$

By Serre’s duality, if $X\subseteq\mathbb{P}^{N}$ is an equidimensional Cohen-
Macaulay subscheme of dimension $d$, then for any locally free sheaf
$\mathcal{F}$ on $X$ there is an isomorphism of graded $R$-modules

$\left(H^{i}_{*}(X,\mathcal{F})\right)^{*}\cong
H^{d-i}_{*}(X,\mathcal{F}^{\vee}\otimes\omega_{X})$

Let $S$ be a smooth algebraic surface of degree $s$ in $\mathbb{P}^{3}$, and
$C\subset S$ a curve, that is, an effective Cartier divisor in $S$. The curve
$C$ has a cohomology class $\eta_{C}\in H^{1}\left(S,\Omega^{1}_{S}\right)$.
It can be defined as follows: the curve $C$ defines a linear form
$\lambda_{C}$ on the set of $(1,1)$ forms by integration; abstractly one can
define this linear form as the image of the trace map
$H^{1}\left(C,\Omega^{1}_{C}\right)\rightarrow\mathbb{C}$ under the transpose
of the morphism $H^{1}\left(S,\Omega^{1}_{S}\right)\rightarrow
H^{1}\left(C,\Omega^{1}_{C}\right)$ obtained by restricting differentials on
$S$ to $C$ [Har77, Chapter III Ex. 7.4]. The cohomology class $\eta_{C}$ is
the image of $\lambda_{C}$ under the Serre’s duality isomorphism
$H^{1}\left(\Omega^{1}_{S}\right)^{*}\cong H^{1}\left(\Omega^{1}_{S}\right)$.

If $\mathcal{O}_{S}(C)$ denotes the invertible sheaf on $S$ corresponding to
$C$, then $\eta_{C}=c(\mathcal{O}_{S}(C))$ where $c$ denotes the first Chern
class homomorphism

(1) $c:\mbox{Pic}(S)\to H^{1}\left(S,\Omega^{1}_{S}\right).$

The perfect pairing $\langle\quad,\quad\rangle$ of Serre’s duality is
compatible with the intersection product of divisor classes [Har77, Chapter V
Ex. 1.8] in the sense that for every pair of Cartier divisors $D$ and $E$ on
$S$

$\langle c(\mathcal{O}_{S}(D)),c(\mathcal{O}_{S}(E))\rangle=D\cdot E.$

Since $S$ is a surface in $\mathbb{P}^{3}$, numerically equivalent divisors on
$S$ are linearly equivalent, and the first Chern class map $\mbox{Pic}(S)\to
H^{1}\left(S,\Omega^{1}_{S}\right)$ is injective.

The cotangent bundles of $S$ and $\mathbb{P}^{3}$ are related by the exact
sequence

(2)
$0\to\mathcal{O}_{S}(-s)\to\Omega^{1}_{\mathbb{P}^{3}}\otimes\mathcal{O}_{S}\to\Omega^{1}_{S}\to
0.$

It is well known (see e.g. [EGPSm85]) that
$H^{1}(\Omega^{1}_{\mathbb{P}^{3}}\otimes\mathcal{O}_{S})\cong\mathbb{C}$ and
that its image in $H^{1}(\Omega^{1}_{S})$ is the class $\eta_{H}$ of a plane
section $H$ of $S$. We look at a portion of the long cohomology sequence
arising from (2)

(3)
$H^{1}\left(\Omega^{1}_{S}\right)\stackrel{{\scriptstyle\delta}}{{\to}}H^{2}\left(\mathcal{O}_{S}(-s)\right)\stackrel{{\scriptstyle\epsilon}}{{\to}}H^{2}\left(\Omega^{1}_{\mathbb{P}^{3}}\otimes\mathcal{O}_{S}\right)$

Dualizing and using Serre’s duality we get an exact sequence

(4)
$H^{0}\left(\mathcal{T}_{\mathbb{P}^{3}}(s-4)\right)\stackrel{{\scriptstyle\epsilon^{*}}}{{\to}}H^{0}\left(\mathcal{O}_{S}(2s-4)\right)\to\mbox{Im}(\delta)^{*}\to
0$

We denote by $J_{S}$ the Jacobian ideal of $S$, that is, the ideal of $R$
generated by the partial derivatives of an equation of $S$. Then the above
discussion is summarized by Griffith’s theorem: the primitive first cohomology
group of $S$ is isomorphic to the $(2s-4)$-graded piece of the Jacobian ring
of $S$:

$H^{1}\left(\Omega^{1}_{S}\right)^{\perp_{H}}\cong\mbox{Im}(\delta)^{*}\cong\dfrac{H^{0}\left(\mathcal{O}_{\mathbb{P}^{3}}(2s-4)\right)}{J_{S,2s-4}}$

######  Definition 2.1.

Given a curve $C$ in $S$, we will denote by
$\alpha(C)=\alpha(C,S,\mathbb{P}^{3})$ the image of its cohomology class
$\eta_{C}$ under the map

$H^{1}\left(\Omega^{1}_{S}\right)\stackrel{{\scriptstyle\delta}}{{\to}}H^{2}\left(\mathcal{O}_{S}(-s)\right)\cong
H^{0}\left(\mathcal{O}_{S}(2s-4)\right)^{*}$

Thus $\alpha(C)$ is a linear form on $H^{0}\left(\mathcal{O}_{S}(2s-4)\right)$
that vanishes on $J_{S,2s-4}$.

Given $\alpha\in H^{0}(\mathcal{O}_{S}(2s-4))^{*}$, we denote by $I_{\alpha}$
the annihilator of $\alpha$ in the polynomial ring $R$: it is the homogeneous
ideal in $R$ whose $n^{th}$ graded piece is

$I_{\alpha,n}=\\{f\in R_{n}\ |\ \alpha(fg)=0,\ \forall g\in
H^{0}(\mathcal{O}_{S}(2s-4-n))\ \\}.$

######  Remark 2.2.

When writing the paper, we decided to take all ideals in the polynomial ring
$R=H^{0}_{*}(\mathcal{O}_{\mathbb{P}^{3}})$: thus $J_{S}$ and $I_{\alpha}$ are
for us ideals of $R$, and $J_{S}\subset I_{\alpha(C)}$. Our motivation is that
we would like to compare $I_{\alpha}$ with the ideal of $C$ as a curve in
$\mathbb{P}^{3}$. In [MS19] the author’s denote by $I_{\alpha}$ the
annihilator of $\alpha$ in the Jacobian ring and by $\tilde{I_{\alpha}}$ its
preimage in $R$.

Let $T=R/I_{S}=H^{0}_{*}(\mathcal{O}_{S})$. Then $\alpha\in(T_{2s-4})^{*}$,
and the ideal $I_{\alpha}$ is determined by $\mbox{Ker}(\alpha)\subseteq
T_{2s-4}$; conversely, one can recover $\mbox{Ker}(\alpha)$ as the image of
$I_{\alpha,2s-4}$ via the quotient map $R_{2s-4}\rightarrow T_{2s-4}$. The
perfect pairing

$R_{n}/I_{\alpha,n}\times\left(R_{2s-4-n}/I_{\alpha,2s-4-n}\right)^{*}\to\mathbb{C}$

shows $A:=R/I_{\alpha}=\bigoplus_{n=0}^{2s-4}A_{n}$ is an artinian Gorenstein
ring of socle $2s-4$ [EP93, Prop 1.3].

In [EP93] the authors were interested in the study of the Noether-Lefschetz
locus, and the invariant $\alpha(C)$ plays a prominent role in their work
because it vanishes if and only if the curve is a complete intersection of $S$
and another surface. More generally, a Lefschetz type theorem about the Picard
group of $S$ (see [SGA73, B7̆8, Voi02], ) implies the following fact:

######  Proposition 2.3.

Let $C$ and $D$ be effective divisors on a smooth surface
$S\in\mathbb{P}^{3}$, and let $H$ denote a plane section of $S$. Then
$I_{\alpha(C)}=I_{\alpha(D)}$ if and only if there exist $m,n,p\in\mathbb{Z}$,
$m,n\neq 0$ and relatively prime, such that $mC+nD+pH$ is linearly equivalent
to zero.

###### Proof.

Suppose $mC+nD+pH$ is linearly equivalent to zero and $m$ and $n$ are nonzero.
The cotangent complex (2) gives rise to an exact sequence in cohomology

(5)
$H^{1}\left(\Omega^{1}_{\mathbb{P}^{3}}\otimes\mathcal{O}_{S}\right)\cong\mathbb{C}\stackrel{{\scriptstyle\gamma}}{{\to}}H^{1}\left(\Omega^{1}_{S}\right)\stackrel{{\scriptstyle\delta}}{{\to}}H^{2}\left(\mathcal{O}_{S}(-s)\right)\simeq
H^{0}\left(\mathcal{O}_{S}(2s-4)\right)^{*}$

and one knows that $\gamma(1)=\eta_{H}$, so that the kernel of $\delta$ is the
$\mathbb{C}$-line spanned by $\eta_{H}$. From $mC+nD+pH\sim 0$ we then deduce
$m\alpha(C)=-n\alpha(D)$. Since $m$ and $n$ are nonzero, the linear forms
$\alpha(C)$ and $\alpha(D)$ have the same kernel, hence
$I_{\alpha(C)}=I_{\alpha(D)}$.

In the other direction, suppose $I_{\alpha(C)}=I_{\alpha(D)}$, that is,
$\alpha(C)$ and $\alpha(D)$ have the same kernel. Then
$\alpha(C)=c\,\alpha(D)$ for a nonzero complex number $c$. Using (5) and the
intersection pairing we deduce that are integers $m$, $n$, $p$, with $m$ and
$n$ nonzero,such that $mC+nD+pH$ is linearly equivalent to zero. Finally, $m$
and $n$ can be taken relatively prime because $Pic(S)/\mathbb{Z}H$ has no
torsion (see for example [B7̆8, Theorem B]). In particular, when $D=0$, one
can take $m=1$. ∎

As noted in [EP93] and [MS19, Lemma 2.3], the ideal $I_{\alpha(C)}$ contains
the ideal of $C$ in $S$. This follows from the remark of [EP93] that
$\alpha(C)\in H^{0}\left(\mathcal{O}_{S}(2s-4)\right)^{*}$ is the pull-back of
a linear form $\beta(C)\in H^{0}\left(\mathcal{O}_{C}(2s-4)\right)^{*}$. For
the benefit of the reader and for later use, we give a proof of this fact. The
linear form $\beta(C)$ arises from the normal bundles exact sequence:

(6)
$0\to\mathcal{N}_{C/S}\cong\omega_{C}(4-s)\to\mathcal{N}_{C/\mathbb{P}^{3}}\to\mathcal{N}_{S/\mathbb{P}^{3}}\otimes\mathcal{O}_{C}\cong\mathcal{O}_{C}(s)\to
0.$

Tensoring (6) with $\mathcal{O}_{C}(-s)$ and taking cohomology we obtain a map
$H^{0}(\mathcal{O}_{C})\to H^{1}(\omega_{C}(4-2s))$ and we let

$\beta(C)\in H^{0}\left(\mathcal{O}_{C}(2s-4)\right)^{*}\cong
H^{1}(\omega_{C}(4-2s))$

denote the image of $1\in H^{0}(\mathcal{O}_{C})$.

######  Proposition 2.4.

[EP93, Construction 1.8] The linear form $\alpha(C)$ is the pull-back of
$\beta(C)$ to $S$, that is, $\alpha(C)=\rho^{*}(\beta(C))$ where $\rho^{*}$ is
the transpose of the natural map
$\rho:H^{0}\left(\mathcal{O}_{S}(2s-4)\right)\rightarrow
H^{0}\left(\mathcal{O}_{C}(2s-4)\right)$.

###### Proof.

Observe that $\Omega^{1}_{S}$ is a rank two vector bundle with determinant
$\omega_{S}$, hence the tangent bundle
$\mathcal{T}_{S}=(\Omega^{1}_{S})^{\vee}$ is isomorphic to
$\Omega^{1}_{S}\otimes\omega_{S}^{-1}=\Omega^{1}_{S}(4-s)$. The tangent
complex of $S\subseteq\mathbb{P}^{3}$ and the normal bundle sequence (6) give
rise to a commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{S}\cong\mathcal{T}_{S}(s-4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}_{\mathbb{P}^{3}}\otimes\mathcal{O}_{S}(s-4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{S}(2s-4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\omega_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{N}_{C/\mathbb{P}^{3}}(s-4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{C}(2s-4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Taking cohomology and dualizing one sees that $\alpha(C)$ is the pull back of
$\beta(C)$ to $S$. ∎

The following Lemma in [EP93] gives an effective method to compute
$I_{\alpha}$ in many cases.

######  Lemma 2.5.

[EP93, Lemma 1.10] Let $N(C)$ denote the image of the map
$H^{0}_{*}\mathcal{N}_{C/\mathbb{P}^{3}}(-s)\to H^{0}_{*}(\mathcal{O}_{C})$
arising from the normal bundle sequence (6). Let
$\pi:R=H^{0}_{*}(\mathcal{O}_{\mathbb{P}^{3}})\to H^{0}_{*}(\mathcal{O}_{C})$
be the natural map. Then, for every integer $n$,

$\pi^{-1}\left(N(C)_{n}\right)\subseteq I_{\alpha(C),n}$

with equality if $\pi_{2s-4-n}$ is surjective.

###### Proof.

The exact sequence

$\textstyle{H^{0}_{*}\mathcal{N}_{C/\mathbb{P}^{3}}(-s)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}_{*}(\mathcal{O}_{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad
1\mapsto\beta}$$\textstyle{\left(H^{0}_{*}(\mathcal{O}_{C}(2s-4))\right)^{*}}$

shows $N(C)=\operatorname{Ann}_{H^{0}_{*}(\mathcal{O}_{C})}(\beta)$.

The map $\pi:R\to H^{0}_{*}(\mathcal{O}_{C})$ factors through
$\rho:H^{0}_{*}(\mathcal{O}_{S})\to H^{0}_{*}(\mathcal{O}_{C})$. To simplify
notation, write $T=H^{0}_{*}(\mathcal{O}_{S})$ and $e=2s-4$. As $\alpha$ is an
element of the $T$-module $T^{*}$, the ideal $I_{\alpha}$, which by definition
is the annihilator of $\alpha$ in $R$, is the inverse image of
$\operatorname{Ann}_{T}(\alpha)$ under the surjective map $R\to T$. Hence what
we have to prove is that
$\rho^{-1}\left(N(C)_{n}\right)\subseteq\operatorname{Ann}_{T}(\alpha)_{n}$
for every integer $n$, with equality holding when $\rho_{e-n}$ is surjective.
Now

$\operatorname{Ann}_{T}(\alpha=\rho^{*}(\beta))_{n}=\left\\{g\in
T_{n}:g\rho^{*}(\beta)(v)=\beta(\rho(g)\rho(v))=0\quad\forall v\in
T_{e-n}\right\\}$

while the inverse image $\rho^{-1}\left(N(C)_{n}\right)$ of the $n^{th}$
graded piece of the annihilator of $\beta(C)$ in $H^{0}_{*}(\mathcal{O}_{C})$
is equal to

$\left\\{g\in T_{n}:(\rho(g)\beta)(w)=\beta(\rho(g)w)=0\quad\forall w\in
H^{0}(\mathcal{O}_{C}(e-n))\right\\}.$

The thesis is now evident. ∎

######  Corollary 2.6.

The annihilator $I_{\alpha(C)}$ of $\alpha(C)$ contains both the homogeneous
ideal of $C$ and the Jacobian ideal of the surface $S$.

To exemplify the scope of this construction, we remark that it immediately
yields the following well known corollary (originally due to Griffiths and
Harris, see [EGPSm85] for more details).

######  Corollary 2.7.

Suppose $S$ is a smooth surface in $\mathbb{P}^{3}$ and $C$ is an effective
divisor on $S$. Then $C$ is a complete intersection of $S$ and another surface
if and only if the sequence 6 of normal bundles splits.

###### Proof.

If $C$ is a complete intersection of $S$ and another surface, it is clear that
the sequence splits. Conversely,if the sequence splits, then $\beta(C)=0$.
Therefore $\alpha(C)=0$, and the thesis follows from Proposition 2.3. ∎

##  3\. Reconstruction of the ideal

Motivated by [MS19], we want to compare $I_{C}$ and $I_{\alpha(C)}$. The
following proposition gives rather sharp sufficient conditions for the curve
$C$ to be reconstructed at level $p$ by $I_{\alpha(C)}$.

######  Proposition 3.1.

Let $S$ be a smooth surface of degree $s$ in $\mathbb{P}^{3}$, and let $C$ be
an effective Cartier divisor on $S$. Assume that the homogeneous ideal $I_{C}$
is generated by its forms of degree $\leq p$ and that the following vanishing
conditions are satisfied

  1. (1)

$h^{1}(\mathscr{I}_{C}(2s-4-p))=0$

  2. (2)

$h^{0}(\mathcal{N}_{C/\mathbb{P}^{3}}(p-s))=0$

then $I_{\alpha(C),p}=I_{C,p}$, therefore $C$ is reconstructed at level $p$ by
$I_{\alpha(C)}$.

###### Proof.

Since $h^{0}\left(\mathcal{N}_{C/\mathbb{P}^{3}}(p-s)\right)=0$, the
annihilator of $\beta(C)$ in degree $p$ vanishes. Since
$\pi_{2s-4-p}:R_{2s-4-p}\to H^{0}\mathcal{O}_{C}(2s-4-p))$ is surjective, by
Lemma 2.5

$I_{\alpha(C),p}=\pi_{p}^{-1}(\operatorname{Ann}(\beta_{C})_{p})=I_{C,p}.$

∎

We can now answer a question raised in [MS19, Section 2.3.1] about twisted
cubics contained in quartic surfaces: if $C$ is a twisted cubic contained in a
smooth quartic surface $S\subset\mathbb{P}^{3}$, then $C$ is cut out by
quadrics in $I_{\alpha(C)}$. More generally:

######  Corollary 3.2.

Suppose $C\subset\mathbb{P}^{3}$ is a general rational curve of degree $d\geq
3$ and let $n_{0}$ be the round up of $\sqrt{6d-2}-3$, that is, the smallest
positive integer $n$ such that $\binom{n+3}{3}-nd-1\geq 0$. If $C$ is
contained in a smooth surface $S$ of degree $s\geq n_{0}+3$, then $C$ is
reconstructed at level $n_{0}\\!+\\!1$ by $I_{\alpha(C,S)}$.

###### Proof.

By [HH85] a general rational curve is a curve of maximal rank, that is,
$h^{0}(\mathscr{I}_{C}(n))=0$ for $n\leq n_{0}-1$ and
$h^{1}(\mathscr{I}_{C}(n))=0$ for $n\geq n_{0}$. Hence $C$ is $n_{0}+1$
regular in the sense of Castelnuovo-Mumford, and $I_{C}$ is generated by its
forms of degree $\leq n_{0}+1$. Furthermore, by [EVdV81] the normal bundle of
the immersion $\mathbb{P}^{1}\to C\subset\mathbb{P}^{3}$ is isomorphic to
$\mathcal{O}_{\mathbb{P}^{1}}(2d\\!-\\!1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(2d\\!-\\!1)$.
Hence $h^{0}(\mathcal{N}_{C/\mathbb{P}^{3}}(-m))=0$ for every $m\leq-2$. Thus
we can apply Proposition 3.1 with $p=n_{0}\\!+\\!1$. ∎

######  Remark 3.3.

If $C$ is a smooth irreducible curve of degree $d$, then
$h^{1}\left(\mathscr{I}_{C}(n)\right)=0$ for every $n\geq d\\!-\\!3-e$ (see
[GP87] and [Han00]), where $e:=e(C)=\max\\{n\ |\
h^{1}\left(\mathcal{O}(n)\right)>0\\}$ is the index of speciality of $C$.

######  Corollary 3.4.

Let $S$ be a smooth surface of degree $s$ in $\mathbb{P}^{3}$, and let $C$ be
an effective Cartier divisor on $S$. Suppose $\mathscr{I}_{C}$ is $r$-regular
in the sense of Castelnuovo-Mumford. If $s\geq 2r+1$, then $C$ is
reconstructed at level $r$ by $I_{\alpha(C)}$.

###### Proof.

Since $\mathscr{I}_{C}$ is $r$-regular, the ideal $I_{C}$ is generated by its
forms of degree $\leq r$ and $H^{1}(\mathscr{I}_{C}(n))=0$ for every $n\geq
r-1$. As $s\geq 2r+1$ and $r\geq 1$, the first condition
$h^{1}\mathscr{I}_{C}(2s-4-r)=0$ in Proposition 3.1 is satisfied for $p=r$.

We are left to check that $h^{0}\mathcal{N}_{C/\mathbb{P}^{3}}(r-s)=0$.

By [PS74, Prop 4.1], there are two surfaces $S_{1}$ and $S_{2}$ of degree $r$
meeting properly in a complete intersection

$X=S_{1}\cap S_{2}=C\cup D$

so that $C$ and $D$ have no common component. Consider the exact sequence

$0\to\mathscr{I}_{X}\to\mathscr{I}_{C}\to\mathscr{I}_{C,X}\to 0.$

Applying $\operatorname{Hom}(-,\mathcal{O}_{C})$ we get

$0\to\operatorname{Hom}(\mathscr{I}_{X},\mathcal{O}_{C})\to\mathcal{N}_{C}\to{\mathcal{N}_{X}}_{|C}$

and $\operatorname{Hom}(\mathscr{I}_{X},\mathcal{O}_{C})=0$ since $C$ and $D$
have no common component. Therefore, there is an inclusion

$\mathcal{N}_{C}\hookrightarrow(\mathcal{N}_{X})_{|C}=\mathcal{O}_{C}(r)\oplus\mathcal{O}_{C}(r)$

hence $h^{0}\mathcal{N}_{C}(m)=0$ for $m\leq-r-1$. In particular
$h^{0}\mathcal{N}_{C}(r-s)=0$ because $s\geq 2r+1$. ∎

##  4\. Arithmetically Cohen-Macaulay curves

In this section we explain how Example 1.15.3 in [EP93] extends the result
about the perfection of complete intersections to the much larger class of
arithmetically Cohen-Macaulay curves (from now on, ACM curves). Recall that a
curve $C\subset\mathbb{P}^{3}$ is called ACM if its homogeneous ring
$R_{C}=R/I_{C}$ is Cohen-Macaulay, or, equivalently, if $C$ is locally Cohen-
Macaulay of pure dimension $1$ and $H^{1}_{*}(\mathscr{I}_{C})=0$. A smooth
ACM curve is what classically was referred to as a projectively normal curve.
We refer the reader to [HS11] for a detailed study of ACM curves on a surface
in $\mathbb{P}^{3}$.

If $C\subset\mathbb{P}^{3}$ is an ACM curve, then $I_{C}$ has a free graded
resolution of the form

(7) $0\to
E=\bigoplus_{j=0}^{r}R(-b_{j})\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}F=\bigoplus_{i=0}^{r+1}R(-a_{i})\to
I_{C}\to 0$

and $I_{C}$ coincides with the ideal generated by the $r\times r$ minors of
$\phi$ by the Hilbert-Burch theorem - cf. [MDP90, Proposition II.1.1 p. 37].

Applying the functor $\operatorname{Hom}_{R}(\bullet,R/I_{C})$ to (7) as in
[Ell75, p. 428] one obtains a long exact sequence

(8) $0\to
H^{0}_{*}(\mathcal{N}_{C})\longrightarrow\bigoplus_{i=0}^{r+1}R_{C}(a_{i})\longrightarrow\bigoplus_{j=0}^{r}R_{C}(b_{j})\longrightarrow
H^{0}_{*}(\omega_{C}(4))\to 0$

The importance of this sequence for our purposes is that it allows to compute
the Hilbert function $n\mapsto h^{0}(\mathcal{N}_{C,\mathbb{P}^{3}}(n))$ of
$\mathcal{N}_{C,\mathbb{P}^{3}}$ as a function of the Hilbert function
$n\mapsto h^{0}(\mathcal{O}_{C}(n))$ of $C$; we can then compute the dimension
of $\operatorname{Ann}(\beta(C))_{n}$ and of $I_{\alpha(C),n}$ in terms solely
of the Hilbert function of $C$ and of the degree $s$ of $S$. To justify our
assertion, one needs to observe that to compute
$h^{0}(\mathcal{N}_{C,\mathbb{P}^{3}}(n))$ out of (8) one does not need to
know the numbers $a_{i}$’s and $b_{j}$’s, but only for each $n$ the difference

$\\#\\{i:\;a_{i}=n\,\\}-\\#\\{j:\;b_{j}=n\,\\}$

which depends only on the Hilbert function of $C$.

As an application of this argument, we can give for ACM curves a sharp bound
for the smallest integer $n$ such that $I_{\alpha(C),n}=I_{C,n}$. For this we
will not need the full Hilbert function of $C$, but just its index of
speciality $e:=e(C)=\max\\{n\ |\
h^{1}\left(\mathcal{O}(n)\right)=h^{2}\left(\mathscr{I}_{C}(n)\right)\ >0\\}$
and the minimum degree $s(C)$ of a surface containing $C$:
$s(C)=\operatorname{min}\\{n\ |\ h^{0}\left(\mathscr{I}_{C}(n)\right)\ >0\\}$.
For an ACM curve $C$, the ideal $\mathscr{I}_{C}$ is $e+3$-regular because
$H^{1}_{*}(\mathscr{I}_{C})=0$. In particular, the ideal $I_{C}$ is generated
in degrees $\leq e+3$, and $s(C)\leq e+3$.

######  Theorem 4.1.

Let $S$ be a smooth surface of degree $s$ in $\mathbb{P}^{3}$. Let $C\subset
S$ be an ACM curve, let $s(C)$ be the minimum degree of a surface containing
$C$ and let $e(C)$ be the index of speciality of $C$.

If $s\geq 2e(C)+8-s(C)$ then $I_{\alpha(C),(e+3)}=I_{C,(e+3)}$. Therefore $C$
is reconstructed at level $e+3$ by $I_{\alpha(C)}$.

###### Proof.

The statement follows from Proposition 3.1 with $p=e+3$ provided we can show
that $h^{0}\left(\mathcal{N}_{C/\mathbb{P}^{3}}(e+3-s)\right)=0$. For this we
use the exact sequence (8), which shows that the maximum $n$ for which
$h^{0}\mathcal{N}_{C,\mathbb{P}^{3}}(n)=0$ is $n=s(C)-e(C)-5$. ∎

######  Remark 4.2.

A twisted cubic curve $C$ is ACM with invariants $s(C)=2$ and $e(C)=-1$. Hence
from Theorem 4.1 it follows once more that, if $C$ is contained in a smooth
quartic surface $S$, then $C$ is cut out by quadrics in $I_{\alpha(C,S)}$.

######  Remark 4.3.

Theorem 4.1 improves for ACM curves the bound of Corollary 3.4 because, since
$r=e+3$, then $2e+8-s(C)=2r+2-s(C)$.

In [MS19, Sec 2.3], motivated by the case of complete intersections, formulate
the notion of a perfect class:

######  Definition 4.4.

Let $S$ be a smooth surface of degree $s$ in $\mathbb{P}^{3}$. A class
$\alpha\in H^{1}(\Omega_{S})/\mathbb{C}\eta_{H}\subseteq
H^{0}(\mathcal{O}_{S}(2s-4))^{*}$ is _perfect at level $m$_ if there exist
effective divisors $D_{1},\ldots,D_{q}$ in $S$ such that
$I_{\alpha(D_{i})}=I_{\alpha}$ for every $i=1,\ldots,q$ and

$I_{\alpha,j}=\sum_{i=1}^{q}I_{{D_{i}},j}+J_{S,j}\quad\mbox{for every $j\leq
m$.}$

We say the class is _perfect_ if
$I_{\alpha}=\displaystyle\sum_{i=1}^{q}I_{D_{i}}+J_{S}$. We make the
convention that the zero class is perfect - geometrically, this amounts to
consider the empty set as a (empty) curve, and is consistent with regarding
the zero divisor as an effective divisor.

######  Example 4.5.

If $C\subset S$ is the complete intersection of two surfaces meeting properly,
then $\alpha(C)$ is perfect (see [MS19, Ex 2.11], [EP93, Ex 1.15.2], [Dan17,
Prop. 2.14] ). If one does not agree that the zero class is perfect, then one
needs to add the condition that $C$ is cut out by two surfaces of degrees
$<s=\deg(S)$.

We now wish to generalize the previous example to the class of ACM curves
showing that, if $C$ is ACM, then the class $\alpha(C)$ is perfect. For this
we need to recall more facts from [EP93]. Suppose the ACM curve $C$ is
contained in a smooth surface $S$ of degree $s$ and equation $f=0$. Then the
polynomial $f$ can be written in the form

$f=\sum_{i=1}^{r+1}g_{i}h_{i}$

where the $h_{i}$’s are the images of the generators of the free module $F$ in
the resolution (7) of $I_{C}$, Since the $h_{i}$’s are the signed $r\times r$
minors of $\phi$, then polynomial $f$ is the determinant of the morphism
$\psi:E\oplus R(-s)\to F$ obtained adding the column
$[g_{1},\ldots,g_{r+1}]^{T}$ to the matrix of $\phi$: in other words, $\psi$
coincides with $\phi$ on $E$, and sends $1\in R(-s)$ to
$\displaystyle\sum_{i=1}^{p+1}g_{i}e_{i}$, where the $e_{i}$’s are the
generators of $F$. We thus obtain a resolution of $I_{C}/I_{S}$:

(9) $0\to E\oplus R(-s)\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}F\to
I_{C}/I_{S}\to 0$

Since $S$ is smooth, the curve $C$ is Cartier on $S$ so that $I_{C}/I_{S}$ can
locally be generated by one element. It follows that the ideal $I_{r}(\psi)$
generated by the $r\times r$ minors of $\psi$ is irrelevant, that is, its
radical is the irrelevant maximal ideal $(x,y,z,w)$ of the polynomial ring
$R$.

######  Proposition 4.6.

[EP93, Prop. 1.16] Let $C\subset\mathbb{P}^{3}$ be an ACM curve contained in
the smooth surface $S$. Suppose $I_{C}$ has the resolution (7). Then

  1. (1)

if $\psi$ is as in exact sequence (9) the presentation of $I_{C,S}$, then

$I_{\alpha(C)}=I_{r}(\psi)$

is the ideal generated by the $r\times r$ minors of $\psi$;

  2. (2)

the $n^{th}$-graded piece $\operatorname{Ann}(\alpha(C))_{n}$ of the
annihilator of $\alpha(C)$ in $H^{0}_{*}(\mathcal{O}_{S})$ is the image of the
natural map

$\bigoplus_{m\in\mathbb{Z}}H^{0}\mathcal{O}_{S}(C+(n+m)H)\otimes
H^{0}\mathcal{O}_{S}(-C-mH)\longrightarrow H^{0}(\mathcal{O}_{S}(n))$

######  Remark 4.7.

Note that $\operatorname{Ann}(\alpha(C))=I_{\alpha(C)}/I_{S}$. The equality
$I_{\alpha(C)}=I_{r}(\psi)$ is a non trivial fact that is not given a full
proof in [EP93]; a complete proof can be found in [KMR09, Proposition 4.3 and
p. 382].

We can now prove that the class $\alpha(C)$ of an ACM curve in a smooth
surface $S$ is perfect:

######  Theorem 4.8.

Let $C\subset S$ be an ACM curve and let $S$ be a smooth surface. Then the
class $\alpha(C)$ of $C$ in $S$ is perfect.

###### Proof.

Fix an integer $n$. By Proposition 4.6 $\operatorname{Ann}(\alpha(C))_{n}$ is
the image of the natural map

$\bigoplus_{m\in\mathbb{Z}}H^{0}\mathcal{O}_{S}(C+(n+m)H)\otimes
H^{0}\mathcal{O}_{S}(-C-mH)\longrightarrow H^{0}(\mathcal{O}_{S}(n)).$

Note the sum on the left hand side is finite, and consists of those $m$ for
which the linear systems $|C+(m+n)H|$ and $|-C-mH|$ are both non-empty. For
such an $m$ we pick a basis $g_{1},\ldots,g_{r_{m}}$ of
$H^{0}\mathcal{O}_{S}(-C-mH)$ and corresponding effective divisors
$D_{k}=(g_{k})_{0}\in|-C-mH|$. The image of

$H^{0}\mathcal{O}_{S}(C+(n+m_{k})H)\otimes g_{k}$

in $H^{0}(\mathcal{O}_{S}(n))$ is $H^{0}\mathscr{I}_{D_{k}/S}(n)$. (if $C\sim
tH$ is a complete intersection of $S$ and another surface, taking $m_{k}=-t$
and $n=0$ we get $D_{k}$ the empty curve, and in this case $\alpha(C)=0$ is
perfect by our definition). Note that
$\mathbb{Q}\alpha(D_{k})=\mathbb{Q}\alpha(C)$ by Proposition 2.3. Now letting
$k$ and $m$ vary we see that $\alpha(C)$ is perfect at level $n$, for every
$n$. Since $\operatorname{Ann}(\alpha(C))$ is finitely generated, we can let
$n$ vary up to the maximum degree of a generator of
$\operatorname{Ann}(\alpha(C))$, and recover the whole
$\operatorname{Ann}(\alpha(C))$ as the sum of finitely many $I_{D_{k}/S}$ with
$D_{k}\sim C+(n+m)H$ for some $m$ and $n$. Therefore $\alpha_{C}$ in $S$ is
perfect.

∎

##  5\. Example of a non perfect class

######  Theorem 5.1.

Let $C\subset\mathbb{P}^{3}$ be a smooth rational curve of degree $4$
contained in a smooth surface $S$ of degree $s=4$. The class $\alpha(C)$ in
$S$ is not perfect at level $3$.

###### Proof.

A smooth rational quartic curve $C\subset\mathbb{P}^{3}$ is contained in a
unique quadric surface $Q$, and $Q$ is necessarily smooth (all curves on the
quadric cone are arithmetically Cohen-Macaulay by [Har77, Chapter V Ex. 2.9
]). We may assume $C$ is a divisor of type $(3,1)$ on $Q$. The ideal sheaf of
$C$ is $3$-regular, hence $I_{C}$ generated by quadrics and cubics.

Suppose $C$ is contained in a smooth quartic surface $S$. Then $Q\cap S$ is
the union of $C$ and an effective divisor $D_{0}$ of type $(1,3)$ on $Q$. Note
that $D_{0}$ is a curve of degree $4$ and arithmetic genus $0$; as the divisor
class of $D_{0}$ is different from that of $C$ and $C$ is irreducible, we
conclude that $C$ and $D_{0}$ have no common component.

The curves $C$ and $D_{0}$ don’t move in their linear system on the quartic
surface $S$: for $C$ this follows from $C^{2}=-2$, and in any case for both
$D_{0}$ and $C_{0}$ one might argue that

$h^{0}(\mathcal{O}_{S}(D_{0}))=h^{0}(\mathcal{O}_{S}(2H-C))=h^{0}(\mathscr{I}_{C}(2))=1.$

Having established the geometric set-up, we proceed to show that
$I_{\alpha(C)}$ contains too many cubics for $\alpha(C)$ to be perfect at
level $3$. To compute the dimension of $I_{\alpha(C),3}$, we use the fact that
$R/I_{\alpha(C)}$ is a Gorenstein ring with socle in degree $2s-4=4$, hence

$\operatorname{dim}I_{\alpha(C),3}=\operatorname{dim}I_{\alpha(C),1}+\operatorname{dim}R_{3}-\operatorname{dim}R_{1}=\operatorname{dim}I_{\alpha(C),1}+16\geq
16.$

This estimate is good enough for us to prove the theorem, but let us show
anyway that $\operatorname{dim}I_{\alpha(C),3}=16$: as $C$ is a divisor of
type $(3,1)$ on $Q$, $h^{1}\left(\mathscr{I}_{C}(3)\right)=0$ hence by Lemma
2.5 $I_{\alpha(C),1}$ is the pull back to $R_{1}$ of $N(C)_{1}$, the image of
$H^{0}\left(\mathcal{N}_{C/\mathbb{P}^{3}}(-3)\right)$ in
$H^{0}\left(\mathcal{O}_{C}(1)\right)$; as the normal bundle of $C$ pulls-back
on $\mathbb{P}^{1}$ to
$\mathcal{O}_{\mathbb{P}^{1}}(7)\oplus\mathcal{O}_{\mathbb{P}^{1}}(7)$ by
[EVdV81, Proposition 6]), we conclude that $I_{\alpha(C),1}=0$, hence
$\operatorname{dim}I_{\alpha(C),3}=16.$ The same argument shows that
$I_{\alpha(C),2}=I_{C,2}$ as well.

To check whether $I_{\alpha,3}$ is perfect, we need to determine curves $D$ in
$S$ with $I_{\alpha(D)}=I_{\alpha(C)}$ and
$h^{0}\left(\mathscr{I}_{D}(3)\right)\geq 1$ so that $D$ can contribute to
$I_{\alpha,3}$. Thus suppose $D$ is such a curve. By Proposition 2.3, there
exist $m,n,p\in\mathbb{Z}$, $m,n\neq 0$ and relatively prime, such that
$pH+mC+nD$ is linearly equivalent to zero. By assumption $3H-D$ is effective;
as $C$ is not linearly equivalent to $tH$ for any $t$, neither is $D$, hence
$1\leq\deg(D)=D\cdot H\leq 11$. Replacing $D$ with $D^{\prime}=3H-D$ we can
even assume $D\cdot H\leq 6$.

Now consider the matrix

$M=\begin{bmatrix}H^{2}&C\cdot H&H\cdot D\\\ C\cdot H&C^{2}&C\cdot D\\\ H\cdot
D&C\cdot D&D^{2}\end{bmatrix}=\begin{bmatrix}4&4&x\\\ 4&-2&y\\\
x&y&z\end{bmatrix}$

As $pH+mC+nD$ is linearly equivalent to zero, the vector $v=[p,m,n]^{T}$ is in
the kernel of $M$. Set $x=H\cdot D$, $y=C\cdot D$ and $z=D^{2}$. Note that
$z=D^{2}=2(p_{a}(D)-1)=2q$ is even.

The determinant of $M$ must vanish, so

$x^{2}+4xy-2y^{2}-24q=0$

From this we deduce first that $x$ and $y$ must be even, and then that $4$
divides $x$. As $1\leq x\leq 6$, we must have $x=4$. Thus $D$ is a curve of
degree $4$, and either $D=C$ or $C$ is not a component of $D$, hence $y=C\cdot
D\geq 0$. Assume that $D\neq C$. Writing $y=2t$ with $t\geq 0$, we obtain the
equation

$t^{2}-4t+3q-2=0$

Looking at the discriminant of this quadratic equation in $t$ we deduce $6-3q$
is a perfect square, so that $q=2-3a^{2}$ for an integer $a\geq 0$. Then
solving for $t$ and imposing $t\geq 0$ we obtain $t=2+3a$. So $H\cdot D=x=4$,
$C\cdot D=y=4+6a$ and $D^{2}=4-6a^{2}$. Then solving the linear system $Mv=0$
for $v=[p,m,n]^{T}$ we find $m=an$ and $p=-(a+1)n$. Since $m$ and $n$ are
relatively prime and non zero and $a\geq 0$, the only possibility is that
$a=1$. Then we can take $m=n=1$ and conclude $C+D\sim 2H$, so that $C+D$ is
the complete intersection of the unique quadric $Q$ containing $C$ with $S$,
and $D=D_{0}$ is the residual to $C$ in the complete intersection $Q\cap S$.

We conclude that the only curves $D$ in $S$ that are contained in a cubic
surface and satisfy $I_{\alpha(D)}=I_{\alpha(C)}$ are $C$, the residual
$D_{0}$ to $C$ in the complete intersection $Q\cap S$, and the effective
divisors linearly equivalent to either $3H-C$ or $3H-D_{0}$. But observe that,
if $D^{\prime}\sim 3H-D_{0}\sim C+H$ is effective, then

$h^{0}\mathscr{I}_{D^{\prime}}(3)=h^{0}\mathscr{I}_{C}(2)=1.$

Therefore there is a unique cubic containing $D^{\prime}$, whose equation is
contained in the ideal of $D_{0}$. Similarly, if $D^{{}^{\prime\prime}}\sim
3H-C$ is effective, there is a unique cubic containing
$D^{{}^{\prime\prime}}$, whose equation is contained in the ideal of $C$.
Hence any cubic form that belongs to the ideal of a curve $D$ on $S$
satisfying $I_{\alpha(D)}=I_{\alpha(C)}$ is in the vector space spanned by
$I_{C,3}$ and $I_{D_{0},3}$.

To show $\alpha(C)$ is not perfect at level $3$ it is now enough to show that
cubics containing either $C$ or $D_{0}$ plus the cubics in the Jacobian ideal
$J_{S}$ do not span $I_{\alpha(C),3}$.

To this end, note that cubic surfaces that contain both $C$ and $D_{0}$ are in
the ideal of the complete intersection of $S$ and $Q$, and so form a vector
space of dimension $4$. By Grassmann’s formula

$\operatorname{dim}I_{C,3}+\operatorname{dim}I_{D_{0},3}=7+7-4=10$

There are four independent cubics in the Jacobian ideal, so

$\operatorname{dim}I_{C,3}+\operatorname{dim}I_{D_{0},3}+\operatorname{dim}J_{S,3}\leq
14<16=\operatorname{dim}I_{\alpha(C),3}$

and this shows that $\alpha(C)$ in $S$ is not perfect at level 3. ∎

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