HU-EP-10/35

Vacuum type space-like string surfaces in

Harald Dorn, George Jorjadze, Chrysostomos Kalousios,

[2mm] Luka Megrelidze,
Sebastian Wuttke
^{2}^{2}2

[15mm]

Institut für Physik der Humboldt-Universität zu Berlin,

Newtonstraße 15, D-12489 Berlin, Germany

[3mm] Razmadze Mathematical Institute,

M. Aleksidze 1, 0193, Tbilisi, Georgia

[3mm] Ilia State University,

K. Cholokashvili Ave 3/5, 0162, Tbilisi, Georgia

Abstract

We construct and classify all space-like minimal surfaces in which globally admit coordinates with constant induced metric on both factors. Up to transformations all these surfaces, except one class, are parameterized by four real parameters. The classes of surfaces correspond to different regions in this parameter space and show quite different boundary behavior. Our analysis uses a direct construction of the string coordinates via a group theoretical treatment based on the map of to . This is complemented by a cross check via standard Pohlmeyer reduction. After embedding in we calculate the regularized area for solutions with a boundary spanned by a four point scattering -channel momenta configuration.

###### Contents

## 1 Introduction

In a series of papers [1, 2] a remarkable correspondence between minimal surfaces in approaching a null polygonal boundary at conformal infinity and gluon scattering amplitudes in super Yang-Mills theory has been established. For generic null polygons a lot of structural insight concerning the dependence of the area on the boundary data has been achieved. However, explicit formulæ for the surfaces are available in the tetragon case only [3, 1]. The tetragon case is very special, since it turned out to be the only flat space-like minimal surface in [4].

The string dual to SYM lives in . There is an extensive literature on dynamical strings, i.e. time-like surfaces, in (see for example [5]
and references therein).
For the correspondence to scattering amplitudes one is interested in space-like surfaces extended up to infinity. Therefore, a more
complete treatment should include the study of minimal surfaces in this ten-dimensional
spacetime with the same null polygonal boundaries for their projection on ,
but with a non-trivial extension in . We started a corresponding analysis
for in [6]. For an example in see [7]. The situation now exhibits two
crucial new aspects. While at first the surface in the total product space of course
has to be minimal, its projections to the factors can be non-minimal. Secondly,
surfaces which are space-like with respect to the induced metric of the full product
space can have projections to AdS of both Euclidean and Lorentzian signature or even with a degenerate induced metric.
In [6]
we concentrated on the case of a space-like AdS-projection and made only
some sketchy remarks on the time-like case. Allowing non-space-like AdS-projections opens
a relatively broad set of possibilities. The present paper is devoted to a full
constructive classification within the set of surfaces which admit coordinates where
the metrics induced from the product space as well as from the individual factors are constant^{1}^{1}1In physical terms these are vacuum solutions, similar to the four cusp solution in pure . A characterization by invariant geometrical quantities is: intrinsically flat with constant mean curvatures
..

The by now standard procedure for generating minimal surfaces as the solution of string equations of motion is Pohlmeyer reduction [8, 4]. The reduced model inherits possible integrable structures of the original string sigma model. After solving the reduced model a first order linear problem has still to be solved to get the coordinates of the wanted surfaces. Based on the bijective maps of and to the group manifolds and , respectively, we perform an analysis which yields the embedding coordinates directly. The analysis of string equations in terms of group variables is usually very helpful for the dynamics with WZ term (see [9, 10] and references therein). It appears that the vacuum configurations of string also have a certain factorized structure, which provides explicit integration of corresponding string equations. For completeness and a cross check we also analyze the problem in Pohlmeyer reduction.

The paper is organized as follows. In section 2 we review the map to a group manifold and then classify and construct the solutions of vacuum type corresponding to space-like surfaces in . We show that three left and three right components of Noether currents related to the isometries of are constants both in and sectors. That also justifies the name vacuum type solutions. Section 3 is devoted to a parallel treatment of the problem within Pohlmeyer reduction. In addition it contains remarks on the relation of our solutions to the complex sin(h)-Gordon type equations, which for the light-like projection degenerate to a linear equation. Then we continue in section 4 with an elaboration of the boundary behavior of our surfaces, pointing out the qualitative differences between the various classes and give a compact listing of their characteristic properties. In section 5 we calculate the regularized area for the solution which is of potential use for a map to 4-point scattering amplitudes in - or -channel configuration. Section 6 contains a summary and some conclusions. Appendices A and B contain some technical details related to the main text.

## 2 Space-like strings in

In this section we describe the string equations in terms of group variables and integrate these equations in the vacuum sector. We use conformal worldsheet coordinates and gauge fixing conditions, based on a holomorphic structure of space-like string surfaces.

### 2.1 and as group manifolds

The and spaces can be realized as the group manifolds and via

(2.1) |

Here are coordinates of the embedding space and the equation for the hyperboloid

(2.2) |

which defines the space, is equivalent to . Similarly, the equation for embedded in

(2.3) |

is equivalent to .

Let us introduce the following basis in

(2.4) |

These three matrices satisfy the relations

(2.5) |

where and is the Levi-Civita tensor with . The inner product defined by

As a basis in we use the anti-hermitian matrices , where are the Pauli matrices ( , ). Here one has the algebra

(2.6) |

and the inner product is introduced by a similarly normalized trace, but with the negative sign

Using (2.4), eq. (2.1) can be written as . The inverse group elements respectively become . With the help of (2.5) and (2.6) one then finds the correspondence between the metric tensors

(2.7) |

We use these relations in the next subsection to write the string equations in terms of the group variables.

### 2.2 String description in terms of group variables

We consider space-like surfaces in . They can be parameterized by conformal complex worldsheet coordinates

(2.8) |

The string action in this gauge corresponds to the sigma model on

(2.9) |

where is a coupling constant. The variation of (2.9) leads to the equations of motion

(2.10) |

From these equations follow the holomorphicity conditions

(2.11) |

for the diagonal components of the induced metric tensors on and on separately. These conditions, together with (2.8), allow to use the gauge

(2.12) |

The remaining freedom of conformal transformations in this gauge is given by translations and the reflection . One can also consider , which corresponds to the reflection .

In the real worldsheet coordinates the equations of motion (2.10) read

(2.13) |

and the gauge fixing conditions (2.12) are equivalent to

(2.14) | ||||||

The isometry transformations of the group manifolds are given by the left-right multiplications

(2.15) |

with constant matrices and They leave the equations of motion (2.13) and the gauge fixing conditions (2.14) invariant. The system (2.13)-(2.14) is also invariant under the discrete transformations and , which correspond to the reflections of , , and , , respectively. Hence, the composition of form the complete group of isometry transformations of The complete isometry group of is obtained as a composition of and , where the latter corresponds to the reflection of and . , and

Using the covariant notation:
, ,
the induced metric tensors on the and
projections can be written as^{2}^{2}2In this paper
(as in [6]) the
index is used for some variables of the spherical part to distinguish
them from similar variables of the AdS part.

(2.16) |

In the next two subsections we construct solutions corresponding to constant and . First we consider the projection, since it is easier to treat. The same method then we apply to the part.

### 2.3 Vacuum solutions in

Let us introduce valued fields related to the right derivatives of

(2.17) |

These fields obey the zero curvature condition

(2.18) |

and the second equation in (2.13) is equivalent to

(2.19) |

The norms and the scalar product of and define the induced metric tensor on the projection. We denote the norm of by , and write the part of the gauge fixing conditions (2.14) in the form

(2.20) |

Our aim is to describe solutions of (2.17)-(2.20)
for constant .
The vectors , and form
an orthogonal basis in .
Expanding the first derivatives of and
in this basis,
from (2.18)-(2.20) one finds that they have vanishing
projections on and . Therefore, the first
derivatives can be written as^{3}^{3}3The coefficients on the r.h.s
are interpreted as the matrix elements of the second fundamental form.

(2.21) | ||||||

The consistency conditions of this system are given by the equations

(2.22) |

From the first two equations follows that the coefficients and are harmonic functions , , with . Then, the third equation of (2.22) leads to , which means that and are -independent. Due to the algebraic relation (2.22), the coefficients and are parameterized by one angle variable

(2.23) |

As a result, the following linear combination

(2.24) |

is constant.

Now we introduce valued fields with the left derivatives

(2.25) |

which are related to the right fields in a standard way

(2.26) |

The differentiations of (2.26) lead to a system similar to (2.21)

(2.27) | ||||||

and one finds that the linear combination

(2.28) |

is also -independent.

From (2.24) and (2.28) follows that the field satisfies the equations

(2.29) |

where the new coordinates are related to by the rotation

(2.30) |

The integration of the system (2.29) is straightforward and yields

(2.31) |

with an integration constant . The isometry transformation brings the solution to the form . Denoting by and by , we obtain with

(2.32) |

This field indeed solves equation (2.19) for any pair of constant vectors and .

To verify the orthonormality conditions (2.20) we calculate the induced metric tensor in the coordinates . From (2.32) follows its matrix form

(2.33) |

On the other hand, the map (2.30) defines the same tensor as

(2.34) |

where is the rotation matrix in (2.30). Comparing (2.33) and (2.34), one finds

(2.35) |

The obtained norms of and are consistent with (2.28) and (2.24). A new result from this calculation is the scalar product , which defines the angle between these vectors.

(a) | (b) |

Now we describe how to factorize the left-right symmetry (2.15) and parameterize the orbits of solutions by the pairs .

Since the general case (2.31) reduces to (2.32) by right (or left) multiplications, it is enough to classify the fields (2.32). This factorized form of solutions is invariant under the similarity transformations and by the adjoint representation: , which rotate the vectors

(2.36) |

It corresponds to

(2.37) |

and, in addition, the vectors and are oriented in some symmetric way in the plane. Namely, the two rectangles formed by the components of the vectors and have the same area (see fig. 1(a)). The choice (2.36) is motivated by its relevance for the generalization to the case. It is also helpful to establish contact with Pohlmeyer reduction.

The calculation of the exponents and corresponding to (2.36) is straightforward. However, in general, the solution (2.32) has a rather complicated matrix form. One can simplify it by an isometry transformation. For this purpose we rewrite (2.36) as

(2.38) |

where and are the polar coordinates in the plane for the vectors and , respectively (see fig. 1(a)). Using then (A.4) and multiplying the solution (2.32) by from the l.h.s and by from the r.h.s., we obtain the new solution

(2.39) |

Here and are the rescaled worldsheet coordinates

(2.40) |

and corresponds to the angle between and

(2.41) |

By (2.35) one has

(2.42) |

The embedding coordinates for the solution (2.39) are given by

(2.43) |

The pairs (, ) and (, ) here are on the circles

(2.44) |

Therefore, eq. (2.43) describes a torus in with the radii and .

The solution (2.43) was obtained in [6] via Pohlmeyer reduction. It was shown that the mean curvature of the embedding in is equal to

(2.45) |

From (2.32) follows the equation

(2.46) |

which defines a surface in in a form independent of worldsheet coordinates. Since and are linear functions of the embedding coordinates, the l.h.s. of (2.46) provides a quadric in these variables. The corresponding surface expressed in embedding coordinates is given as an intersection of the and

(2.47) |

Taking and from (2.38), after simple transformations one can rewrite (2.46) in the form

(2.48) |

which reproduces (2.44). Though this surface in SU(2) is described only by , this parameter is not a complete characteristic of the SU(2) part for our system in . Since we are in a fixed gauge, also has a certain gauge invariant meaning. Namely, defines the area measure on the surface induced from .

Finally, note that and are related to the Noether integrals of the isometry group and therefore they have also a gauge invariant meaning.

### 2.4 Vacuum solutions in

Based on the isometry between the algebra and 3d Minkowski space, we call a vector space-like if , time-like if and light-like if .

Note that the vector is space-like in
the gauge (2.14).
Similarly to the case, we introduce the notation
and write the induced metric tensor on the projection
in the form^{4}^{4}4This form differs from the one used in
[6]. There the -component of the induced
metric tensor was denoted by for the space-like surfaces
and by for the time-like ones. Note also, that in our convention is time-like in the time-like case.

(2.49) |

The sum with the metric (2.20) gives the total metric induced from and is by construction a multiple of the identity matrix. The metric (2.49) is space-like if it is time-like if and it becomes light-like if . The last case corresponds to a degenerate metric with a light-like vector .

We consider constant and apply the scheme of the previous subsection. The case of light-like is special and we consider it separately. If is space-like or time-like, the vectors and form an orthonormal basis in . Repeating then the same steps as in we come to the factorized type solution (2.32)

(2.50) |

Here is a pair of constant elements of the algebra and ( are worldsheet coordinates obtained from by a rotation on an angle which parameterizes the coefficients of the linear system (see (2.21))

(2.51) |

Now and satisfy again (2.22). For further convenience, we use here a different parameterization

(2.52) |

which is obtained from (2.23) by the replacements . This provides the following coordinates (compare with (2.30))

(2.53) |

If is light-like, one gets the commutator^{5}^{5}5Note that (2.54) is equivalent to .

(2.54) |

Hence, in this case, and do not form a basis in . Completing and in a suitable way to a basis, one can show that in the end the third independent direction is not needed to express the derivatives of and . For details see the parallel discussion of this issue in the framework of Pohlmeyer reduction in section 3.3 below. Altogether the system (2.21) is modified and replaced by the linear system

(2.55) | ||||||

The consistency conditions for this system lead to the equations

(2.56) |

which, in contrast to other cases, do not necessarily require constant and . In section 3.3 we describe also the general solution of the consistency conditions (2.56) and indicate how to integrate the linear system (2.55). Here we identify vacuum configurations with the solutions for constant and . In this case (2.56) reduces again to and one can use the parameterization (2.52). Constant currents are then constructed in a same way and one again obtains solutions in the factorized form (2.50). Thus, (2.50) represents solutions in the vacuum sector for all three cases: and as well.

The induced metric tensor in -coordinates, calculated by (2.49) and (2.53), becomes

(2.57) |

and comparing it with the calculation from (2.50), one finds

(2.58) |

These equations fix the norm of the commutator to

(2.59) |

Thus, is time-like for space-like surfaces , is light-like for light-like surfaces () and is space-like for time-like surfaces .

We use these metric characteristics of the commutator and also the scalars (2.58) to classify the fields by the adjoint orbits similarly to the previous subsection. Since the adjoint representation of is given by the group of proper Lorentz transformations , the orbits can be identified with the hyperbolas and cones in 3d Minkowski space. Choosing canonical pairs from the orbits, one can construct solutions by (2.50) and then simplify them as in (2.39). This procedure also simplifies the form of the quadratic relation between the embedding coordinates, which is now given by

(2.60) |

It is natural to divide the solutions in three classes according to the signature of the induced metric tensor and then continue classification inside the classes.

Before starting classification it is useful to note that the reflection freedom allows to set (see eqs. (2.51)-(2.52) and (2.55)). This corresponds to . Below we assume this condition.

1. Space-like surfaces .

This case corresponds to space-like , and time-like . One can take the commutator proportional to .
However, in contrast to the case there are two possibilities

(2.61) |

which are not on the same adjoint orbit. These two cases are related by the discrete isometry transformation .

Let us consider the negative sign in (2.61) and choose the pair