Wilson loops T-dual to Short Strings

0.55 MB·English

by  M. Kruczenski

Imperial-TP-AAT-2012-08
January 22, 2013
Wilson loops T-dual to Short Strings
M. Kruczenskia,1 and A.A. Tseytlinb,2
3
1
0 a Department of Physics, Purdue University, W. Lafayette, IN 47907, USA
2
b Blackett Laboratory, Imperial College, London SW7 2AZ, U.K.
n
a
J

1
2

Abstract
]
h
t We show that closed string solutions in the bulk of AdS space are related by T-duality
-
p to solutions representing an open string ending at the boundary of AdS. By combining
e
the limit in which a closed string becomes small with a large boost, we find that the near-
h
[ flat space short string in the bulk maps to a periodic open string world surface ending on

a wavy line at theboundary. This openstring solution was previouslyfound byMikhailov
2
v and corresponds to a time-like near BPS Wilson loop differing by small fluctuations from
6
a straight line. A simple relation is found between the shape of the Wilson loop and
8
the shape of the closed string at the moment when it crosses the horizon of the Poincare
8
4 patch. As a result, the energy and spin of the closed string are encoded in properties of
.
2 theWilsonloop. Thissuggeststhatclosedstringamplitudeswithoneoftheclosedstrings
1 fallingintothePoincarehorizonshouldbedualtogaugetheorycorrelatorsinvolvinglocal
2
operators and a Wilson loop of the T-dual (“momentum”) theory.
1
:
v
i
X
r
a
[email protected]
2Also at Lebedev Institute, Moscow. [email protected]
Contents
1 Introduction 2
2 Definition of T-duality transformation and some examples 5
3 T-duality relation between short closed strings and long wavy open strings 10
3.1 T-duality transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Area of the open string surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Energy of the wavy line open string . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Including string motion in S5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Folded spinning string case 19
5 Comments on interpretation of the T-duality relation 23
6 Discussion 28
A Elliptic Integrals and Jacobi elliptic functions 29
B Spiky string case 30
C T-duality in a non-compact direction on a cylinder 31
1 Introduction
There are many known relations between closed and open strings. For example, in flat space
a world sheet representing a propagation of a closed string may be viewed, after τ ↔ σ inter-
change, as describing also propagation of an open string in periodic time; an open string disc
diagram for open strings may be viewed as a closed string emission by a D-brane into vacuum;
KLT relations express closed string scattering amplitudes in terms of open string scattering
amplitudes, etc.
In the gauge-string duality context, one recent example is the relation between the coefficient
of the leading logarithmic term in the large spin (long string) asymptotics of the closed folded
spinning string energy [1] and the cusp anomalous dimension of the Wilson loop defined by a
light-like cusp [2, 3, 4]. There is also a (KLT-like) relation between a correlator of closed string
vertex operators at null separations and a square of a Wilson loop defined by the corresponding
null polygon [5].
These are cases of far-from-BPS configurations, but there are other examples that hint to-
wards a possible closed-open string relation also for “short” strings or “small” deviations from
the BPS limit. According to [6], for a straight line Wilson loop with a small cusp of angle π−φ,
2
φ (cid:28) 1 in AdS one finds1
5
(cid:104) (cid:105)
(cid:104)W (cid:105) = exp −Γ (φ,λ)lnΛ+... , (1.1)
cusp cusp
Γ (φ,λ) = −B(λ)φ2 +... , (1.2)
cusp
√φ→0 √
λI ( λ) 1 (cid:16)√ 3 3 (cid:17)
B(λ) = 2 √ = λ− + √ +... . (1.3)
4π2I ( λ) 4π2 2 8 λ
1
In (1.3) we gave the expansion of the coefficient B(λ) at strong coupling or for large string
tension, i.e. λ (cid:29) 1.2 The same expression is found for a cusp of angle π − θ in S5 (with
φ2 → −θ2). It was observed in [8] that switching on an orbital momentum J in S5, the
expression for the leading λ (cid:29) 1 term in the corresponding Γ (θ,λ;J) appears to be related
cusp
to the small-spin limit of the energy of a folded spinning string in S3.
Moreover, there is a striking similarity [6] between the small angle coefficient B in (1.2) and
the slope function [9] found in the small spin limit of the AdS folded string energy. According
5
to [9, 10, 11], the dimension of the sl(2) sector gauge-theory operator with spin S and twist J
or the energy of the corresponding dual spinning string state has the following expansion in the
formal S (cid:28) 1 limit3
E2 = J2 +h(λ,J)S+O(S2) , (1.4)
√
√ √ I ( λ)
h = 2J +h¯(λ,J) , h¯( λ,J) = 2 λ J+1√ . (1.5)
I ( λ)
J
If one formally sets here J = 1 then the slope function in (1.5) becomes directly related to B
in eq.(1.3), i.e.4
√ 1
h(λ,1) = 2+8π2B(λ) = 2 λ−1+O(√ ) . (1.6)
λ
A motivation behind the present work is to try to find a possible relation between small (nearly
point-like) closed strings in AdS and long open strings ending at the boundary (and thus
5
corresponding to nearly-straight Wilson lines).
Let us recall that the coefficient B(λ) in (1.2),(1.3) has also other interpretations. As was
argued in [6], for a euclidean Wilson loop defined by a curve which is a small transverse
1Here we quote the coefficient function in the planar approximation only (λ is the ’t Hooft coupling). Λ is a
UV cutoff and I is a modified Bessel function.
n
2The coefficient B is also proportional to the correlator of the circular Wilson loop [7] with the integrated
√
dilaton operator, i.e. B = 4πλ2d√dλlog(cid:104)Wcircle(cid:105).
3This result follows directly from the asymptotic Bethe ansatz expression for the string spectrum. It is
not sensitive to the dressing phase, which should have to do with its “near-BPS” nature avoiding non-trivial
order-of-limitsorrenormalizationissues, see[12]. NotethatthecomputabilityofthecoefficientB in(1.2)is, in
turn, due to supersymmetry allowing one to use localization techniques to compute certain correlators of BPS
Wilson loop with particular local operators.
4Heuristically, the J =1 choice may be thought of as required to represent an open string “half” of a closed
string state dual to sl(2) sector operator with the minimal possible value of J =2.
3
deviation from a straight line [14, 15] one has
(cid:90) [∂ (cid:126)x (σ)−∂ (cid:126)x (σ(cid:48))]2
(cid:104)W (cid:105) = 1+ 1B(λ) dσdσ(cid:48) σ ⊥ σ ⊥ +... , (1.7)
wavy 2 (σ −σ(cid:48))2
whereitisassumedthat|∂ (cid:126)x | (cid:28) 1. Furthermore,thesameB(λ)appearsalsointheexpression
σ ⊥
for the energy emitted [16] by a slowly moving “quark” [6] (see also [13])
(cid:90)
E = 2πB(λ) dτ v˙2 +... , (cid:126)v =(cid:126)x˙(τ) (cid:28) 1 , (1.8)
where (cid:126)x(τ) is a spatial deviation of the quark’s trajectory from a straight line. As was shown in
[16], the Minkowski open-string world surface ending on the quark’s trajectory at the boundary
√
of the Poincare patch leads to the following expression for the classical ( λ (cid:29) 1) string energy
√ √
λ (cid:90) v˙2 −[(cid:126)v ×(cid:126)v˙]2 λ (cid:90)
E = dτ ≈ dτ v˙2 +... , (1.9)
0 2π (1−v2)3 2π
which is a relativistic generalization of the integral in (1.8).
The derivation of the classical energy formula (1.9) is straightforward in the limit of small
deviations from a straight line [16]. Using the Poincare patch coordinates ds2 = 1 (dz2−dx2+
z2 0
dx2) and choosing the static gauge
i
x = τ, z = σ , (1.10)
0
the string (Nambu) action for small fluctuations in x -directions is
i
√
(cid:90)
λ dτdσ(cid:104) 1 (cid:105)
S = 1+ (x˙2 −x(cid:48)2)+O(∂x3) . (1.11)
2π σ2 2 i i
The corresponding equation of motion x¨ − x(cid:48)(cid:48) + 2σ−1x(cid:48) = 0 has solution x (τ,σ) which is
i i i i
uniquely determined by the two boundary data functions5 – x+(τ) and x−(τ), or, equivalently,
the shape of the boundary curve x (τ) ≡ x (τ,0) = x+(τ)+x−(τ) and its 3-rd normal derivative
i i
[16]. Explicitly, one finds6
x (τ,σ) = x (τ,σ)−σ∂ x (τ,σ)
i i σ i
(cid:104) (cid:105)
= x+(τ +σ)+x−(τ −σ)−σ x˙+(τ +σ)−x˙−(τ −σ) +O(x2) , (1.12)
i i i i
x (τ,0) ≡ x (τ) = x+(τ)+x−(τ) , ∂3x (τ,0) = −2∂3[x+(τ)−x−(τ)] , (1.13)
i i i i σ i τ i i
where x (τ,σ) = x+(τ +σ)+x−(τ −σ) is generic flat-space solution (harmonic function). The
i i i
string energy associated with translations in the x = τ direction
0
√
λ (cid:90) ∞ dσ(cid:104) 1 (cid:105)
E = 1+ (x˙2 +x(cid:48)2)+O(∂x3) , (1.14)
0 2π σ2 2 i i
0
5ThisisaMinkowski,notaEuclideanWilsonlinestationarysurface;itisthelatterwhichis(atleastlocally)
uniquely determined by the boundary curve.
6 Note that the near-boundary (small σ) expansion is x (τ,σ) = x (τ)− 1σ2x¨ (τ)+O(σ3) making it clear
i i 2 i
that the third σ-derivative is an independent boundary data.
4
then leads, after renaming the integration variable σ → τ, to the same expression as in (1.9),
√
(cid:82)
i.e. λ dτ x¨2.7
2π i
In this paper we will show that T-duality along the boundary directions of AdS relates the
5
world sheets of small closed strings in the bulk of AdS to the open string surfaces ending on
wavy lines representing small-velocity “quark” trajectories at the boundary. Our starting point
will be the observation that T-duality along the boundary directions in the Poincare patch of
AdS space maps a massless geodesic in the bulk of AdS into a straight line Wilson loop surface.
As was first observed in [17], the formal T-duality along all xµ = (x0,xi) boundary directions
maps AdS into AdS space provided it is combined with a simple coordinate transformation –
the inversion of the radial direction, z → 1/z (which effectively interchanges the boundary and
the horizon).
This T-duality was used (in Euclidean world sheet case) in [18] to relate (imaginary) world
sheet solution that dominate a semiclassical path integral for open-string (gluon) scattering
amplitudes to (real) solutions describing the corresponding null polygon Wilson loops in the
dual momentum space. Here we will be considering the case of the Minkowski signature in both
the target space and the world sheet, so that the T-duality transformations will be mapping
real solutions into real solutions.
We shall start in section 2 with a review of the basic definitions and properties of T-duality
transformation in AdS and discuss some simple examples.
In section 3 we shall describe in detail the map between small closed strings and long open
strings ending on a nearly straight line at the boundary of AdS and thus corresponding to the
wavy line Wilson loops.
In section 4 we shall consider an example of application of the T-duality to a closed string
of finite size – folded string with an arbitrary spin. Section 5 will contain some comments on a
physical interpretation of the T-duality relation and implications for a process of a small closed
string crossing a horizon. Some concluding remarks will be made in section 6.
InAppendixAweshallgiveourdefinitionsofellipticfunctionsusedinsection4. InAppendix
B we shall discuss the application of the T-duality transformation to the small spiky string
solution. Appendix C contains a discussion of implementation of T-duality on a cylinder a
closed string insertion.
2 Definition of T-duality transformation and some ex-
amples
Let us start with describing the basic sets of coordinates in AdS we will be using. The
5
embedding coordinates are defined by
ds2 = dX dXM , −X XM = X2 +X2 −X2 −X2 −X2 −X2 = 1 . (2.1)
M M −1 0 1 2 3 4
AdS5
7Thederivationrequiresregularizingneartheboundaryz =σ =(cid:15)→0anddroppingthesingularfree-motion
and boundary terms, see section 3.3 below for more details.
5
It is convenient to introduce also the light-like coordinates
X = X ±X , X X −X Xµ = 1 , X = (X ,X ) . (2.2)
± −1 4 + − µ µ 0 i
The global coordinates (t,ρ,nˆ ) are related to X by
r M
X +iX = coshρ eit, X = sinhρ nˆ , r = 1,...,4 , (2.3)
0 −1 r r
ds2 = −cosh2ρ dt2 +dρ2 +sinh2ρ dΩ2 , (2.4)
[3]
where nˆ is a 4-component unit vector parametrising an S3. The Poincare coordinates are
r
defined by8
1
Z = , X = X −X = coshρ sint−sinhρnˆ , (2.5)
− −1 4 4
X
−
X coshρ cost
0
X ≡ T = = , (2.6)
0
X X
− −
X sinhρ nˆ
i i
X = = , i = 1,2,3 , (2.7)
i
X X
− −
1
ds2 = (dZ2 +dX dXµ) , X = (T ,X ), µ = 0,1,2,3. (2.8)
Z2 µ µ i
√
WeshallusetheconformalgaugeinwhichtheclassicalbosonicstringactionS = λS¯,equations
2π
of motion and the conformal-gauge constraints written in the embedding coordinates are
(cid:90) (cid:90)
1 1
S¯ = dσdτ (∂ X ∂aX −∂ X ∂aXµ)+ dσdτΛ(X X −X Xµ −1) , (2.9)
a + − a µ + − µ
2 2
∂ ∂aX = ΛX , ∂ ∂aXµ = ΛXµ ,
a ± ± a
Λ = ∂ X ∂aX −∂ X ∂aXµ , X X −X Xµ = 1 , (2.10)
a + − a µ + − µ
∂ X ∂ X +∂ X ∂ X −∂ X ∂ Xµ −∂ X ∂ Xµ = 0 ,
σ + σ − τ + τ − σ µ σ τ µ τ
∂ X ∂ X +∂ X ∂ X −2∂ X ∂ Xµ = 0 . (2.11)
σ + τ − τ + σ − σ µ τ
In what follows we shall map one class of solutions of these equations (small closed strings)
into another class of their solutions (open-string Wilson-loop type solutions) using the formal
T-duality (2-d scalar-scalar duality) transformation that maps AdS into AdS if combined with
the Z → 1/Z coordinate transformation [17]. For brevity, we shall often refer to this combined
transformation simply as T-duality.
Using the Minkowski signature in both the target space and the world sheet the T-duality
transformation rule is then (Z,X ) → (Z˜,X˜ ) where ∂ X˜ = (cid:15) Z−2∂bX (with (cid:15) = −1), i.e.
µ µ a µ ab µ τσ
1 1 1
˜ ˜ ˜
∂ X = − ∂ X , ∂ X = − ∂ X , Z = = X . (2.12)
τ µ Z2 σ µ σ µ Z2 τ µ Z −
8In contrast to Introduction, in what follows we shall use capital Z and Xµ to denote the Poincare patch
coordinates. LetusrecallthatthePoincarecoordinatesonlycoverpartoftheMinkowskiAdS space,i.e. there
5
is a choice of the patch they cover. The one used here is appropriate for the solution and limiting procedure
described below but other simply related options are possible too.
6
˜ ˜
Thus (Z,X ) satisfy the same set of equations (2.10),(2.11) corresponding to the dual AdS
µ 5
metric ds2 = 1 (dZ˜2 +dX˜ dX˜µ). Note that this T-duality is done along non-compact target
Z˜2 µ
space directions that leads to subtleties when, as here, the world sheet theory is defined on a
cylinder (see Appendix C).
The T-duality relations in (2.12) may be written also as
∂aX˜ = −(cid:15)abj , (2.13)
µ b,−µ
j ≡ X ∂ X −X ∂ X , ∂aj = 0 , (2.14)
a,MN M a N N a M a,MN
where j is the SO(4,2) Noether current associated to the AdS space symmetry which is
a,MN 5
conserved on the equations of motion (2.10) with the corresponding charge being 9
(cid:90)
J = dσ j . (2.15)
MN τ,MN
Thus the particular SO(4,2) angular momentum components J get the interpretation of the
−µ
˜ (cid:82) ˜
“winding numbers” in the dual X directions, dσ∂ X = J .
µ σ µ −µ
Let us give some simple examples of T-duality applied to string solutions in AdS. In general,
T-duality formally relates a classical string solution to another solution and thus maps the
corresponding sets of conserved charges. For example, the world line of a point-like string or
massless geodesic in AdS is T-dual to a straight-line Wilson loop surface (in Minkowski world-
sheet signature). In more detail, the T-duality counterparts for the three simple solutions in
AdS ⊂ AdS are (after T-duality it is convenient to interchange σ and τ):
2 5
• massless geodesic in AdS (t = t(τ), ρ = ρ(τ), sinhρ = κτ, tant = κτ, cost = √ 1 )
2 1+κ2τ2
1
˜ ˜
Z = T = , Z = κσ , T = κτ . (2.16)
κτ
˜
The dual surface reaches the boundary at σ = 0 where it is a straight line along T ∼ τ. The
˜
reason we interchanges σ and τ after T-duality is precisely to have the new time coordinate T
proportional to τ (instead of σ).
• massive geodesic (t = κτ, ρ = 0) or a massless BMN geodesic in AdS ×S1 (with S1 angle
2
ϕ = κτ not involved in the T-duality transformation)
1
˜ ˜
Z = , T = cotκτ , Z = sinκσ , T = κτ . (2.17)
sinκτ
˜
The dual surface ends at the boundary at σ = 0 where it is a straight line along T and
then again at κσ = π. Notice that the dual surface has two boundaries corresponding to the
2
intersection of the initial surface with the past and future Poincare Horizons.
• wrapped string (t = κτ, ϕ = κσ where ϕ is an angle of a maximal circle of S5) 10
1
˜ ˜
Z = , T = cotκτ , Z = sinκσ , T = κτ, ϕ = κτ . (2.18)
sinκτ
9Note that while in general T-duality interchanges Noether charges with hidden conserved charges, the AdS
case is special in that the symmetries and thus the Noether charges of the original and the dual backgrounds
are isomorphic (see [19, 20, 21]).
10The wrapped string solutions was considered, e.g., in [34] (case w =0 in section 2.1 of that paper).
7
Here we again interchanged τ and σ after T-duality.
• infinite straight static string in AdS (t = κτ, ρ = ρ(σ), tanh ρ = tan κσ, coshρ = 1 )11
2 2 2 cosκσ
cosκσ sinκσ
˜ ˜
Z = , T = cotκτ , Z = , T = tanκτ . (2.19)
sinκτ cosκτ
Here the original solution reaches the boundary (ρ = ∞) at κτ = π where it ends on a line
2
along T . The dual solution ends at the boundary for κσ = πn on a line along T˜.
2
It is clear from these examples that to interpret the T-dual world surface as that of an open
string stretched inside AdS and ending at the boundary at a point (“quark”) one needs to do
two steps:
(i) Formally “decompactify” the σ direction. Starting with a periodic solution for a small
closed string (defined on a 2d cylinder) and applying the T-duality (in a particular gauge, see
˜
below) we will get a σ-periodic open-string world sheet, apart from the T direction which gets
a term linear in σ. As discussed in Appendix C, the world-sheet theory will still be periodic
˜
since T enters only through its derivatives (assuming there are no vertex operator insertions
˜
depending on T ).
(ii) Interchange the τ and σ coordinates (a familiar operation in the usual flat-space closed–
open string duality relation).12
Let us briefly mention another example – the large spin limit of folded spinning string in
AdS (ds2 = −cosh2ρ dt2 + dρ2 + sinh2ρ dθ2) which is an infinite rotating string stretching
3
all the way to the boundary [22]: t = θ = κτ, ρ = κσ, κ → ∞.13 This solution is known to
be related [3], via a Euclidean world-sheet continuation and an SO(2,4) transformation, to the
null cusp Wilson loop solution [2]. The T-duality provides a different relation to an open-string
world sheet ending at the boundary. Indeed, absorbing κ into σ and τ and thus making their
range infinite we get (X ≡ X )
1
Z = (coshσ sinτ −sinhσ cosτ)−1, T = Zcoshσ cosτ, X = Zsinhσ sinτ , (2.20)
Z˜ = coshσ sinτ −sinhσ cosτ , T˜ = 1(−τ +σ −sinτ cosτ +sinhσ coshσ) ,
2
X˜ = 1(−τ −σ +sinτ cosτ +sinhσ coshσ) ,
2
˜ ˜ ˜ ˜
T +X = −τ +sinhσ coshσ , T −X = σ −sinτ cosτ . (2.21)
˜
The T-dual surface ends at the boundary Z = 0 when tanτ = tanhσ where
1 1
˜ ˜ ˜ ˜
T +X = −arctan(tanhσ)+ sinh2σ , T −X = σ − tanh2σ . (2.22)
2 2
To have an open-string interpretation we are again to replace σ ↔ τ. The boundary trajectory
is a bended curve passing through zero in (T˜,X˜) plane (at large τ one has T˜ +X˜ ∼ e2(T˜−X˜)).
11This solution may be viewed as a special zero-spin limit [22] of a folded spinning string in AdS .
3
12Similar transformation relating oscillating and rigid spinning classes of string solutions in AdS ×S5 was
5
discussed in a different context in [23].
13T-duality applied to finite-spin folded string will be discussed in detail in section 4.
8
Our aim below will be to apply the above T-duality transformation to small (“short”) strings
in the bulk of AdS . To study short strings that effectively probe the near-flat limit of AdS it
5 5
is useful to consider a neighbourhood of linear size (cid:15) → 0 around the point X = 1, X = 0,
0 M(cid:54)=0
namely
X ∼ 1+(cid:15)2 , X ∼ (cid:15) . (2.23)
0 M(cid:54)=0
An example one may have in mind is a small string of size (cid:15) oscillating near ρ = 0. In this case
the components of the Noether current j in (2.14) scale as
a,MN
j ∼ ∂ X ∼ (cid:15), j ∼ (cid:15)2, for M,N (cid:54)= 0 . (2.24)
a,0M a M a,MN
In particular, j ∼ (cid:15) whereas j ∼ (cid:15)2.
a,−0 a,−i
In addition to this scaling limit, an important ingredient of our discussion will be a particular
SO(4,2) transformation – a large boost in the (X ,X ) hyperbolic plane of the embedding 4-
+ −
space in (2.1)
1
X → X , X → (cid:15)X , (cid:15) → 0 . (2.25)
− − + +
(cid:15)
This boost makes a small string near ρ = 0 move fast towards the boundary, following approx-
imately a massless geodesic. This exposes the near-BPS nature of a nearly point-like small
string state.
Assuming that the (cid:15) parameters in (2.23) and in (2.25) are the same, the resulting scaling of
the components of j after the boost is
a,MN
j ∼ 1, j , j ∼ (cid:15), j , j , j ∼ (cid:15)2, j ∼ (cid:15)3 . (2.26)
a,−0 a,−i a,0i a,+− a,ij a,+0 a,+i
The components entering the T-duality relation in (2.13) are then
j (cid:39) ∂ X ∼ 1, j = X ∂ X −X ∂ X ∼ (cid:15) , (2.27)
a,−0 a − a,−i − a i i a −
implying that for J in (3.23) one has J ∼ 1 and J ∼ (cid:15). This means, in the T-dual
MN −0 −i
˜ ˜
interpretation, that the dual string is extended along X = T with fluctuations of order (cid:15) in the
0
˜ ˜
directions X . Since X ∼ 1 the dual string world sheet extends also along Z = 1/Z = X .
i=1,2,3 − −
It is useful to record the scaling of the conserved charges representing the energy and the
√
spin of a short string we started with (we omit the factor of string tension, i.e. E = λE¯,
2π
etc.)14
(cid:90)
E¯ ≡ J +J = dσ∂ X +O((cid:15)2) , (2.28)
−0 +0 τ −
S¯ ≡ J = O((cid:15)2) . (2.29)
ij ij
Tosummarise, (i)expandingaroundthepointX = 1, X = 0inAdS and(ii)performing
0 M(cid:54)=0 5
a large boost in the X plane with the same parameter (cid:15) as the expansion scale produces a
±
˜ ˜
string configuration which has the T-dual interpretation as a world sheet extended along T , Z
˜
with small fluctuations in the spatial X directions. We shall study this relation in detail
i=1,2,3
in the following section.
14Note also that S¯ ≡ J −J = O((cid:15)) may be interpreted, from the point of view of dual conformal field
4i −i +i
theory on R3,1, as a linear combination of the momentum generator and the generator of special conformal
transformations.
9
3 T-duality relation between short closed strings and
long wavy open strings
Letuselaborateonthesmallstringexpansiondiscussedabove. Startingwithglobalcoordinates
in (2.4) we may set
ρ = (cid:15)ρ¯, t = (cid:15)y , (3.1)
0
so that the metric becomes nearly flat
ds2 = −cosh2ρdt2 +dρ2 +sinh2ρdΩ2
[3]
(cid:0) (cid:1)
(cid:39) (cid:15)2 −dy2 +dy dy , y = ρ¯nˆ , (3.2)
0 r r r r
where nˆ is the unit vector in (2.3).
r
The corresponding small string equations will be the same as in flat space. Let us fix the
conformal gauge and further choose a particular conformal parametrisation (light-cone gauge)
where
y ≡ y −y = τ . (3.3)
− 0 4
Then the solution for the transverse spatial directions y is the usual flat-space one
i
y (σ,τ) = y (σ,τ) ≡ y+(σ +τ)+y−(σ −τ) , i = 1,2,3 (3.4)
i i i i
with an initial value at τ = 0 given by
y (σ,0) = y (σ) = y+(σ)+y−(σ) . (3.5)
i i i i
As we shall explain below, this leading-order small-string configuration is T-dual to an open-
string solution ending at the boundary along the following trajectory (after τ ↔ σ interchange)
(cid:90) τ
X = τ , X = x (τ) = dτ(cid:48)y (τ(cid:48)) . (3.6)
0 i i i
It thus represents a small deformation of the straight-line Wilson loop X = τ.
0
Let us recall how the small-string limit looks in the embedding coordinates (2.23) – as an
expansion around the point X = 1, X = X = 0:
0 ± i
X ∼ 1+(cid:15)2, X ,X ∼ (cid:15), (cid:15) → 0 . (3.7)
0 ± i
TheequationofmotionforX in(2.10)impliesthatΛ ∼ (cid:15)2 andtheconstraintX X −X Xµ =
0 + − µ
1 determines the (cid:15)2 term in X . To the lowest order in (cid:15) in each of the equations in (2.10),(2.11)
0
we find
∂ ∂aXi = 0, ∂ ∂aX = 0, (3.8)
a a ±
∂ X ∂ X +∂ X ∂ X −∂ X ∂ X −∂ X ∂ X = 0 ,
σ + σ − τ + τ − σ i σ i τ i τ i
∂ X ∂ X +∂ X ∂ X −2∂ X ∂ X = 0 . (3.9)
σ + τ − τ + σ − σ i τ i
10