Minimal surfaces in AdS and the eight-gluon scattering amplitude at strong coupling

Abstract

In this note we consider minimal surfaces in three dimensional anti-de Sitter space that end at the AdS boundary on a polygon given by a sequence of null segments. The problem can be reduced to a certain generalized Sinh-Gordon equation and to SU(2) Hitchin equations. The mathematical problem to be solved arises also in the context of the moduli space of certain three dimensional supersymmetric theories. We can use explicit results available in the literature in order to find the explicit answer for the area of a surface that ends on a eight-sided null Wilson loop. Via the gauge/gravity duality this can also be interpreted as a certain eight-gluon scattering amplitude at strong coupling for a special kinematic configuration. Comment: 9 pages

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arXiv:0903.4707v2 [hep-th] 19 Oct 2009
Minimal surfaces in AdS and the
eight-gluon scattering amplitude
at strong coupling
Luis F. Alday and Juan Maldacena
School of Natural Sciences, Institute for Advanced Study
Princeton, NJ 08540, USA
In this note we consider minimal surfaces in three dimensional anti-de Sitter space that end
at the AdS boundary on a polygon given by a sequence of null segments. The problem can
be reduced to a certain generalized Sinh-Gordon equation and to SU(2) Hitchin equations.
The mathematical problem to be solved arises also in the context of the moduli space of
certain three dimensional supersymmetric theories. We can use explicit results available
in the literature in order to find the explicit answer for the area of a surface that ends on
a eight-sided null Wilson loop. Via the gauge/gravity duality this can also be interpreted
as a certain eight-gluon scattering amplitude at strong coupling for a special kinematic
configuration.

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1. Introduction
Recently there has been some interest in Wilson loops that consist of a sequence of
light-like segments. These are interesting for several reasons. First, they are a simple
subclass of Wilson loops which depend on a finite number of parameters, the positions of
the cusps. Second, they are Lorentzian objects with no obvious Euclidean counterpart.
Finally, it was shown that they are connected to scattering amplitudes in gauge theories
[1,2,3,4,5], for a review see [6].
In this note we study these Wilson loops at strong coupling by using the gauge/string
duality. One is then led to compute the area of minimal surfaces in AdS [7][8]. We consider
a special class of null Wilson loops which can be embedded in a two dimensional subspace,
which we can take as an R1,1subspace of the boundary of AdS. For these loops, the string
worldsheet lives in an AdS3subspace of the full AdSdspace, d ≥ 3.
In order to analyze the problem one can use a Pohlmeyer type reduction [9,10,11,12].
This maps the problem of strings moving in AdS3to a problem involving a single field α
which obeys a generalized Sinh-Gordon equation.
The same mathematical problem appears in the study of SU(2) Hitchin equations
[13]. Interestingly, these Hitchin equations also appear in the study of the supersymmetric
vacua of certain gauge theories [14,15]. This connection is specially useful because the
authors of [14,15] have studied this problem, exploiting its integrability, and have proposed
a method for finding the answer. For the simplest case, the metric in moduli space for the
corresponding field theory problem is known [16,17,14,15]. These results can be used to
compute the area for the simplest non-trivial case which is the eight sided Wilson loop.
This note is organized as follows. In section 2 we describe the interplay between
classical strings on AdS3and the generalized Sinh-Gordon model. In section 3 we describe
several features of the solutions at hand and in section 4 we give the full answer for the
case of the null Wilson loop with eight sides.
Note: A much more detailed exposition of the material reported here will be presented
in a forthcoming publication [18].
2. Sinh-Gordon model from strings on AdS3
Classical strings in AdS spaces can be described by a reduced model which depends
only on physical degrees of freedom. In terms of embedding coordinates, Yµin R2,2, with
Y2= −1, the conformal gauge equations of motion and Virasoro constraints are
∂¯∂?Y − (∂?Y .¯∂?Y )?Y = 0 ,∂?Y .∂?Y =¯∂?Y .¯∂?Y = 0(2.1)
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where we parametrize the world-sheet in terms of complex variables z and ¯ z. For the case of
AdS3the above system can be reduced to the generalized sinh-Gordon model [9,10,11,12].
We define
e2α(z,¯ z)=1
2∂?Y .¯∂?Y ,
p = −1
2
Na=e−2α
2
ǫabcdYb∂Yc¯∂Yd,
?N.∂2?Y ,
¯ p =1
2
?N.¯∂2?Y
(2.2)
Then, as a consequence of (2.1) it can be shown that p = p(z) is a holomorphic function
and that α(z, ¯ z) satisfies the generalized Sinh-Gordon equation
∂¯∂α(z, ¯ z) − e2α(z,¯ z)+ |p(z)|2e−2α(z,¯ z)= 0(2.3)
The area of the world-sheet is simply given by
A = 4
?
d2ze2α
(2.4)
For solutions relevant to this note this area is divergent and we will need to regularize it.
Given a solution of the generalized sinh-Gordon model, one can reconstruct a classical
string world-sheet in AdS3by solving two auxiliary linear problems (which we will denote
as left and right)
∂ψL
α,a+ (BL
z)β
αψL
˙β
˙ αψR
β,a= 0,
¯∂ψL
α,a+ (BL
¯∂ψR
¯ z)β
αψL
˙β
˙ αψR
β,a= 0
∂ψR
˙ α,˙ a+ (BR
z)
˙β,˙ a= 0,
˙ α,˙ a+ (BR
z)
˙β,˙ a= 0
(2.5)
where the SL(2) left and right flat connections are given by
BL
z=
?
1
2∂α
−eα
−1
e−αp(z)
1
2∂α
−e−αp(z)
?−1
−eα
2∂α
?
,BL
¯ z=
?−1
?
2¯∂α
−eα
−e−α¯ p(¯ z)
1
2¯∂α
−eα
−1
?
?
BR
z=
2∂α
?
,BR
¯ z=
1
2¯∂α
e−α¯ p(¯ z)
2¯∂α
(2.6)
Internal SL(2)L×SL(2)Rindices α, ˙ α denote rows and columns of these connections, while
the indices a and ˙ a denote the two different linearly independent solutions of each linear
problem (2.5) . The space-time isometry group SO(2,2) = SL(2) × SL(2) acts on these
indices. We require that the two pairs of solutions obey the normalization condition
ψL
a∧ ψL
b≡ ǫβαψL
α,aψL
β,b= ǫab,ψR
˙ a∧ ψR
˙b= ǫ˙ a˙b
(2.7)
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Once a pair of solutions has been found, the explicit form of the space-time coordinates
Ya˙ a(z, ¯ z) is given by a particular bilinear combination of the left and right solutions
Ya˙ a=
?Y−1+ Y2
Y1+ Y0
Y1− Y0
Y−1− Y2
?
a,˙ a
= ψL
α,aMα˙β
1ψR
˙β,˙ a,Mα˙β
1
=
?10
10
?
(2.8)
It turns out that the left connection BLcan be promoted to a family of flat connections
by introducing a spectral parameter
Bz= Az+ Φz→ Bz(ζ) = Az+1
ζΦz,B¯ z= A¯ z+ Φ¯ z→ B¯ z(ζ) = A¯ z+ ζΦ¯ z
(2.9)
where we have decomposed the connection into its diagonal part Azand off diagonal part
Φz. Actually, both, left and right connections can be simply obtained (up to a constant
gauge transformation) from B(ζ) by setting ζ = 1 or ζ = i respectively. The zero curvature
condition for B(ζ) can be rephrased as
D¯ zΦz= DzΦ¯ z= 0,Fz¯ z+ [Φz,Φ¯ z] = 0
DµΦ = ∂µΦ + [Aµ,Φ],Fµν= ∂µAν− ∂νAµ+ [Aµ,Aν]
(2.10)
These are the Hitchin equations, which arise by dimensional reduction of the four dimen-
sional self-duality condition (instanton equations) to two dimensions. A has the interpre-
tation of a gauge connection in two dimensions and Φ is a Higgs field. In our case we have
the Hitchin equations for SU(2).
After we compute the area we can relate these results to results for Wilson loops or
amplitudes in N = 4 super Yang Mills as
?W? ∼ Amplitude ∼ e−
R2
2πα′(Area),
R2
α′=
√
λ =
?
g2N (2.11)
where the area is computed by setting the AdS radius to one and we have embedded
AdS3appropriately in AdS5. For other field theories with gravity duals we should use the
corresponding expression for R2/α′.
3. Finding the surface and computing its area
Since p(z) is a holomorphic function we can simplify the generalized sinh-Gordon
?p(z)dz. In the w−plane the equation takes
the form
∂w¯∂¯ wˆ α − e2ˆ α+ e−2ˆ α= 0 ,
equation by defining a new variable dw =
ˆ α ≡ α −1
4logp¯ p (3.1)
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The expression for the area (2.4) becomes A = 4?d2we2ˆ α. Note that w has branch cuts
where p has zeros. So, we locally simplify the equation but we complicate the space on
which the equation is defined.
Let us first write the solution for the four sided polygon obtained in [19,1]. In this
case p = 1, w = z and α = 0. The connections (2.6) are constant. Up to a constant gauge
transformation the two solutions can be chosen as
ψL
+=
?ew+ ¯ w
0
?
,ψL
−=
?
0
e−(w+ ¯ w)
?
(3.2)
The ψR
±are the same but with w + ¯ w →
|z| → ∞. This implies that the AdS embedding coordinates (2.8) are diverging, which
means that we are going to the AdS boundary. Different components of the Y coordinates
in (2.8) are different combinations of the solutions ψL,R
w− ¯ w
i
. Some of these solutions diverge when
± . The solutions that diverges
determine a point on the AdS boundary. This point depends on whether the solution that
diverges is ψ+or ψ−. Thus, in each of the four quadrants of the w plane we approach a
different point on the AdS boundary. Each of these points is a cusp. When we change
quadrants the solutions change dominance only for the left problem or only for the right
problem. This implies that two consecutive cusps are separated by a null line.
Consider now the case in which p(z) is a generic polynomial of degree n − 2, p ∼
zn−2+ ···. In this case w ∼ zn/2for large z. As we go once around the z plane we
go around n/2 times in the w plane. Since we expect that the solution near each cusp
is similar to the solution for the four sided polygon, we demand that ˆ α → 0 as |z| goes
to infinity. As a result, the solutions for large w become approximately as in (3.2). The
problem displays the Stokes phenomenon. Namely, an exact solution is given in terms of
a linear combination of each of the two solutions (3.2), with coefficients that change as
we move between Stokes sectors. For instance, consider the left problem in the large |w|
region, such as Im(w) < 0. Within that region (3.2) is a good approximate solution. Let
us consider now what happens as we cross the line where w is real and positive, with very
large |w|. The solution that decreases as Re(w) → +∞ is accurately given by (3.2) and
it is the same on both sides of the line. On the other hand, the large solution will have
a jump in its small solution component. More precisely, we choose a basis of solutions
which has the asymptotic form in (3.2) in one Stokes sector. We denote these two exact
solutions as ψbefore
±
. After we cross the Stokes line, we enter into a new Stokes sector. We
can now choose another basis of solutions which has the asymptotic form in (3.2) in this
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