#### The broken string in Anti-de Sitter space

HJE
The broken string in Anti-de Sitter space
David Vegh 0 1 2
0 CH-1211 Geneva 23 , Switzerland
1 Theory Group, Physics Department , CERN
2 Princeton , NJ 08540 , U.S.A
This paper describes an e cient method for solving the classical string equations of motion in (2+1)-dimensional anti-de Sitter spacetime. Exact string solutions are identi ed that are the analogs of piecewise linear strings in to approximate any smooth string motion to arbitrary accuracy. Cusps on the string move with the speed of light and their collisions are described by a re ection formula. Explicit examples are shown with the string ending on two boundary quarks. The technique is ideally suited for numerical simulations.
Bosonic Strings; Integrable Field Theories; Long strings; AdS-CFT Corre-
1 Introduction
2 The basic solution
3 Cusps on the string
4 Collision of cusps
4.1
4.2
Collisions in at space
Collisions in AdS3
5 Examples
6 Discussion
A Details of the Mathematica code
boundary. According to the AdS/CFT correspondence [1{3], the dual boundary gauge
theory contains a Wilson loop on which the string ends [4, 5]. It is useful to think of the
Wilson loop as the path of an in nitely heavy quark-antiquark pair. The string in the
bulk is the holographic dual to the color
ux tube that connects them. The motion of
the quarks can be speci ed. Generic motion creates non-linear waves on the string that
propagate toward the other endpoint. The aim of this paper is to describe the collisions of
these waves and therefore calculate the string motion in the most e cient way.
The canonical embedding of AdS3 into R2;2 is given by the universal covering space of
the surface
Y~ Y~
Y 21
Y02 + Y12 + Y22 =
1:
(1.1)
Time corresponds to the phase on the Y 1, Y0 plane. Since on the surface this time
dimension is compact, the surface itself only covers a part of global AdS.
Coordinates on the Poincare patch will also be used, for which the metric is
The coordinate transformation is given by
ds2 =
dt2 + dx2 + dz2
z2
(t; x; z) =
Y2
Y0
;
Y 1 Y2
Y1
;
Y 1 Y2
1
Y 1
:
{ 1 {
The string motion is described by a classically integrable system. The string equations
of motion in conformal gauge are
(1.2)
These are supplemented by the Virasoro constraints
The above system can be reduced to a generalized sinh-Gordon theory [6{11] by de ning
HJEP02(18)45
e2 (z;z) =
2
p =
2
Note that N~
Y~ = N~
N~ = 1. Furthermore, p = p(z) and
p = p(z) are holomorphic and antiholomorphic functions, respectively. Then, the potential
satis es the generalized sinh-Gordon equations
scattering problem where
for numerical calculations.
Given a solution, the string embedding can be computed by solving an auxiliary fermion
appears as a potential. This is feasible1 but not very practical
In this paper, we follow a di erent approach. Highly symmetric string pieces will be
glued together; these are the AdS analogs of straight lines. This way any generic solution
can be approximated. The corresponding potential satis es (1.3) with p(z) = 0, i.e. the
Liouville equation. At the attachment points the string will have cusps. Information about
the original sinh-Gordon equation is condensed to these points.
In the next section, the basic symmetric solution is described. In section 3, the gluing
procedure is explained. Section 4 discusses what happens when two cusps collide. Section
5 contains various examples. Details about the numerical calculations are presented in
the appendix.
2
The basic solution
A simple solution to the string equation of motion is obtained by setting N to be a constant
unit length vector.
This is the AdS3 analog of a static in nite straight string in
at
spacetime. By applying an appropriate transformation from the SO(2; 2) isometry group
of AdS3, N~ can be rotated such that
1For an analytical solution to the scattering problem with N singular solitons, see [9].
N~ (t) = (0; 0; 0; 1)
{ 2 {
quarks (q and q) on the boundary. The Poincare patch is bounded by the two orange surfaces.
axes are the x and z coordinates, respectively. The top of the
gure is z = 0, the AdS boundary.
The quark and the antiquark move on this line in opposite directions and are connected by a circular
string.
The corresponding spacetime solution is an in nitely long string: AdS2 embedded into
AdS3. Figure 1 shows the worldsheet in global AdS spacetime. The string in these
coordinates is static and ends on two antipodal points on the boundary S1. Figure 2 shows
the string on the Poincare patch at a given time t. The string is a contracting/expanding
semi-circle centered at (x; z) = (0; 0) with radius R(t) =
1 + t2. On the boundary of
p
AdS, the two quarks move on hyperbolae: x1;2(t) =
R(t), see gure 3.
N~ will be called the normal vector, or the vector perpendicular to the string patch.
Since N~ 2 = 1, the AdS2 spaces form a 3-parameter set. The string solutions are still
shrinking/expanding semicircles, but they are now shifted in the x and t directions, and
their minimal radii are also di erent. Let us denote the center of the circle by x0, the
minimal radius by R0, and the time when the string reaches the minimal radius by t0. The
relationship between N~ and these parameters is given by
(t0; x0; R0) =
N0
N 1 + N2
;
N1
;
N 1 + N2 jN 1 + N2j
:
Let us brie y discuss how this solution is connected to the
= 0 \ground state" of
the sinh-Gordon theory. This trivial solution corresponds to a rotating string [12] in the
in nite angular momentum limit. The endpoints are on the boundary of AdS and they
move with the speed of light. In order to stop the rotation, two anti-solitons can be added
to the potential. In the center-of-mass frame [8],
1
2
(z; z) =
log
1
2
v0 cosh X
cosh T
v0 cosh X + cosh T
{ 3 {
(2.1)
the quarks is constant and they are out of causal contact as indicated by the light rays (dashed lines).
where X = 2 pz+z , T = 2 vp0(z z) , and v0 is the initial speed of the anti-solitons.
! 1 at the location of the anti-solitons. Thus, the
string touches the boundary at these points. On the worldsheet, the anti-solitons approach
each other and after reaching a d minimal coordinate distance, they turn around.
coordinates z !
2
In the v0 ! 1 limit, d
/
1
v0. By a simultaneous rescaling of the worldsheet
1 v0 z0, we can zoom in on the collision point. Then, the anti-solitons
will move on the hyperbola de ned by z0z0 = 1 and the potential becomes
1
2
(z0; z0) =
log (1
z0z0)2
log
1
2
v0
+ O(1
v0)
The rst term is a renormalized potential that satis es (1.3) with p = 0. The corresponding
string embedding is the circular string with a constant normal vector.
3
Cusps on the string
In order to describe more general string con gurations, the circular string solution from
the previous section must be perturbed. Waves moving in a single direction may easily
be obtained following [13]. The motion of one of the string endpoints on the boundary
is speci ed by giving a one-parameter family of lightlike vectors ~l( ) 2 R2;2. The string
solution is given by
Y~ ( ; ) =
_
~l + ~l( )
{ 4 {
String embedding if one of the quark accelerations is suddenly changed. This creates
a cusp that propagates on the string with the speed of light. The string consists of two circular
pieces. Their normal vectors are not arbitrary; they satisfy N~1 N~2 = 1.
The induced metric on the string is
HJEP02(18)45
g =
a( )2
1
of string on the Poincare patch that propagates with the speed of light (see gure 4 and
also [14]). No perturbation can propagate through this piece, because the speed of the
perturbation parallel to the string will be zero. The solution may be interpreted as a string
that ends here.
If the acceleration of the quark changes continuously, then the string worldsheet will be
smooth. Generic infalling (left-moving) waves are produced this way. However, collisions
between left- and right-moving waves are not easily described. Therefore, in the following
we will be interested in elementary building blocks whose shape survives the collisions
without any deformations. These solitonic excitations are cusps on the string produced by
letting the acceleration of the boundary quark jump in time.2 This is shown in
gure 6.
The resulting cusp moves on the string either left or right with the speed of light and
connects two string patches, see gure 5.
If the worldline of the cusp is given by X~ ( ) and the normal vector of the left patch is
denoted N~ L, then the following equations are satis ed
The worldline parameter
possible, because (1.1) is a doubly ruled surface. One can prove that on the x
z plane (on
the Poincare patch) the cusp moves on a straight line with the speed of light. The vector
perpendicular to the right patch can be expressed as
2In principle, one could consider adding extra solitons on top of the double anti-soliton background
of (2.1). However, the corresponding cusps on the string disappear in the v ! 1 limit.
{ 5 {
accelerations is suddenly changed. The quark worldline now asymptotes to a di erent light ray.
Worldsheet Penrose diagram corresponding to the collision of two waves (in red) that
were sent in from the boundary in the far past. The string embedding in the gray region is to
be computed.
where
parametrizes the \strength" of the cusp. If N~ L is xed, then the possible N~ R's
form a 2-parameter subset of the 3-parameter family of solutions which have a constant
normal vector. In summary, they satisfy
evitably collide. The worldsheet Penrose diagram of gure 7 illustrates the situation: the
string patch between two cusps (red lines) disappears and after the collision a new patch
is created (in gray).
The string embedding is shown in gure 8 (thick line). Very close to the collision point,
spacetime is approximately
at and the three string pieces are straight lines labeled N1,
A and N2 that move with constant perpendicular velocities. The corresponding normal
vectors will be denoted by N~ 1, A~ and N~ 2. There exists an SO(2; 2) transformation (or an
analogous boost in at spacetime) that puts us in a frame where N1 and N2 are static and
A moves with the velocity v. The con guration is highly symmetric and the only thing
that can happen is that the collision switches ~v !
~v. This will be shown in the following
Collision of cusps on the string (thick line). In an appropriate frame the lines N1 and
N2 are static, whereas A moves with a perpendicular velocity ~v. The collision changes ~v !
~v.
through a at space example where the piecewise linear string is exhibited as a limit of
smooth strings.
4.1
Collisions in at space
In (2+1)-dimensional at space, an explicit string solution X(z; z) that illustrates this
behavior is given by3
f (z) = a1 tanh (z
g(z) = a2 tanh (z
z0)
z0)
f (z)2~e2
g(z)2~e1
f (z)~e3
g(z)~e3
1
2
1
2
1
2
1
2
1
2
where
~e1 = p (1; 1; 0); ~e2 = p (1; 1; 0); ~e3 = p (0; 0; 1):
The spacetime signature is ( 1; 1; 1). The tanh functions can be replaced by any other
smooth step functions.
The equations are easily integrated and smooth string solutions are obtained for
> 0
values. The
! 0 limit is a piecewise linear string: two cusps collide on the worldsheet
at (z; z) = (z0; z0). The strengths of the cusps are given by a1 and a2. Additional
leftor right-moving cusps can be created by adding extra tanh functions to f (z) and g(z),
respectively. For instance, the solution will have two left-moving cusps if we change
f (z) ! a1 tanh (z
z0) + a3 tanh (z
z00)
A
at space con guration that reproduces the string dynamics in
gure 8 can be
obtained by setting z0 = z0 = 0 and a1 = a2 = a. The string consists of two static hal ines
and an interval in between that moves with a constant velocity
3See section 2 of [9] for related solutions.
~v =
2p2a
2 + a2 ~e3:
{ 7 {
Patches on the worldsheet. The dashed lines in the middle are lightlike worldlines
of two colliding cusps. The normal vectors are labeled A~, A~0, N~1, and N~2. The collision formula
computes any one of these vectors from the other three.
The pieces are connected in a smooth way and the smoothing is parametrized by . After
the collision of the two cusps, the velocity of the middle piece changes direction: ~v !
~v.
4.2
Collisions in AdS3
In AdS3, the situation on the worldsheet is shown in
gure 9. The two dashed lines are
the worldlines of the cusps. Before the collision, the string consists of three pieces that are
characterized by three normal vectors: N~ 1, A~ and N~ 2. Note that
HJEP02(18)45
This is required so that the string does not contain badly behaved pieces (see gure 4).
After the collision, A~ changes to A~0 given by the collision formula
A~ N~ 1 = A~ N~ 2 = 1:
A~0 =
A~ + 4
N~ 1 + N~ 2
(N~ 1 + N~ 2)2
(4.1)
The formula can be justi ed by the following facts. It preserves the scalar product: A~0 N~ 1 =
A~0 N2 = 1. If specialized to the case in
~
gure 8, the formula reproduces the change in
velocity v !
covariant under SO(2; 2). Thus, it gives the correct A~0 in any generic frame. In fact, it
computes any one of the (A~, A~0, N~ 1, N~ 2) vectors from the other three by an appropriate
v for the line A. Furthermore, the formula is invariant under N~ 1 $ N~ 2 and
permutation of the labels. For instance,
N~ 1 =
N~ 2 + 4
A~ + A~0
(A~ + A~0)2
The collision formula can be cast in a Picard-Lefschetz form
A~0 =
A~ + (A~ N~ )N~
with
N~ = p
N~ 1 + N~ 2
2 jN~ 1 + N~ 2j
In numerical computations with many cusps, this version introduces exponentially growing
numerical errors to the equation A~ N~ i = 1. For such calculations, (4.1) is preferable (or a
projection has to be performed).
Note that one can exchange A~ and A~0 and the constraints on the scalar products of
neighbors in gure 9 are still satis ed. This transformation produces a string embedding
{ 8 {
that looks like the one in gure 10: the string is now longer and folded. After the collision,
the cusps move away from the open ends of N1 and N2. Thus, swapping the normal vectors
reduces the number of future collisions that happen on the Poincare patch.
Finally, for in nitely many weak and tightly spaced cusps, the formula reduces to a
di erential equation for the normal vector N~ (z; z)
This is precisely the same equation as (1.2) for Y~ . This is no coincidence: the similarity
follows from an internal SO(2; 2) symmetry that acts on the Y~ ; e
HJEP02(18)45
variables [10].
5
Examples
In this section, a few string solutions are presented. The solutions are based on the circular
string in
gure 2. Cusps are sent in from the boundary by perturbing both endpoints.
The gures have been generated by a Mathematica code that is available to the reader.
The rst example is shown in gure 11. There is one left-moving and one right-moving
cusp on the worldsheet. They are indicated by red and gray ticks. Dashed circle indicates
the original patch. Without the cusps, the string would lie on this circle. The third
gure
shows the moment of collision. The cusps move through each other. The patch between
the two cusps is re ected using the collision formula.
Another example is shown in gure 12. In this case, the cusps have opposite momentum
and their presence makes the string longer. After their collision, the string folds and the
two cusp angles become large.4
The third example is shown in gure 13. A smooth string is approximated by letting
70 weak cusps enter the string on both sides. The cusp locations are shown in red and gray.
So far, the examples have been open strings that end on the boundary of AdS. However,
closed strings can also be built in a similar fashion. A
at space example is presented in
gure 14. The string (thick line) has two left-moving and two right-moving cusps that
move on the dashed square. After the collision, the cusps start moving in the opposite
direction. At any given time, the shape of the string is a rectangle and the string oscillates.
4Folding is also observed for smooth strings. Cusps are created even if the string was initially smooth.
{ 9 {
x z halfplane at various times. The string ends on the boundary of AdS on the top of the gures.
string pieces. The cusps move on the sides of the dashed square with the speed of light.
Discussion
The string embeddings constructed in this paper can be used to approximate any smooth
string motion on the Poincare patch. However, they are also interesting by themselves,
because they are exact solutions to the classical equations of motion. Consequently, this type
of discretization does not introduce any numerical errors that otherwise might accumulate
over time and would lead to various numerical instabilities.
Motion of the string endpoints on the boundary has been speci ed through the third
time derivative of their positions. Cusps on the string were created by adding delta
functions to x000(t). The cusps can be smoothed by resolving the delta functions.
The results can be generalized in various ways. In higher dimensional backgrounds,
the collision of cusps can be reduced to the (3+1)-dimensional case. In
at space, the
collision event is a deformation of gure 8 where N1 and N2 do not lie in the same plane.
It would also be interesting to study brane dynamics based on the techniques presented in
this paper.
Another generalization can be the inclusion of an emblackening factor in the
background geometry. Integrability of the theory will presumably be lost, but approximate
solutions similar to the ones in the present paper may still be of use. An idea is to exhibit
the background geometry as a sum of thin AdS3 slices with a given
z thickness. Then, as
the tiny string patches travel in the z direction, they need to interact with the AdS3 domain
walls in some way. One can hope to satisfy the equations of motion in the
z ! 0 limit.
Finally, the implications of the results for the holographic [15] ER=EPR
correspondence [16] will be discussed elsewhere [17].
Acknowledgments
The author would like to thank Douglas Stanford for discussions and helpful comments on
the manuscript, and the University of Leiden for hospitality.
A
Details of the Mathematica code
In this appendix, we discuss some details of the attached Mathematica code. The code
generates plots of the string on the Poincare patch by sending in SOLNUM cusps from both
the left and right endpoints.
First it computes the normal vectors and stores them in the SOLNUM SOLNUM matrix
vectable. In order to do this, it starts with the string corresponding to the normal vector
~
N0 = (0; 0; 0; 1). At increasing Poincare time it adds cusps to the left side of the strings
and then to the right side of the string and computes the corresponding N~ vectors. They
are stored in the rst row and rst column of vectable. The \strengths" of the cusps are
taken from the prede ned lists lambda1 and lambda2. Other elements of vectable are
computed by means of the collision formula (4.1).
Collision times are computed and stored in the timetable matrix. Sometimes a
collision is calculated to happen earlier than previous collisions. This means that the collision
gure 13. The rows and the
columns label left- and right-moving cusps on the string, respectively. Colors indicate the time of
collision (brighter colors correspond to later times and white means that the collision never takes
place on the Poincare patch). The chain of patches corresponding to the 4th plot in
gure 13 is
shown as a white line. It extends from the top-left corner to the bottom-right corner which means
that all 140 cusps have already entered the string.
event actually takes place on the next Poincare patch and therefore must be discarded. In
this case, the corresponding time in timetable is set to CUTOFF (a large number).
Finally, TIMESTEPS plots are generated for Poincare times between MINTIME and
MAXTIME. For a
xed time, the code computes a path in vectable that goes through
the relevant string patches. This list of points is stored in path. A sample path is shown
in gure 15 in white. Along the path, the code computes the arcs of strings corresponding
to the normal vector of each patch. They are drawn and stored in plots.
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