hep-th/0701248

Comments on Marginal Deformations

in Open String Field Theory

Martin Schnabl

Institute for Advanced Study, Princeton, NJ 08540 USA

E-mail: schnabl at ias.edu

Abstract

In this short letter we present a class of remarkably simple solutions to Witten’s open string field theory that describe marginal deformations of the underlying boundary conformal field theory. The solutions we consider correspond to dimension-one matter primary operators that have non-singular operator products with themselves. We briefly discuss application to rolling tachyons.

## 1 Introduction

One of the important features of string field theory is that it allows us to describe physics of
different string theory backgrounds using data of only a single reference conformal field theory.
This has been recently successfully applied in the context of Witten’s open bosonic string field
theory [1]. It has been shown, in accordance with Sen’s conjectures [2],
that the theory formulated on an arbitrary D-brane describes another vacuum [3] with
no D-branes, and hence no conventional open string degrees of freedom [4].^{1}^{1}1For
related recent development see [5, 6, 7, 8, 9, 10]. Nice reviews of string field theory include [11, 12].

In this letter we shall give a construction of string field theory solutions that correspond to less dramatic changes of the conformal field theory. Our exact solutions will describe conformal field theories deformed by exactly marginal operators. We shall construct such solutions perturbatively in a parameter , which to the first order can be identified with the coupling constant of a given exactly marginal operator [13, 14]. Following [3], we shall use the cylinder conformal frame parameterized by a coordinate . The solution itself will be given by a series expansion in , each term will be given by a cylinder with simple insertions of the ghost, the exactly marginal operator called , and vertical line insertions of the ghost. The mutual distances between insertion points will be parameters that will be integrated over. Unfortunately, for the generic perturbation with singular operator product expansion our solution becomes ill-defined, so we shall restrict ourselves mostly to cases in which is finite when approaches .

One of the more interesting examples of this kind is the time-dependent rolling tachyon solution which is generated by exactly marginal operator studied in [15]. We will look at the time-dependent behavior of the tachyon coefficient to get some clues on the tachyon matter problem [16]. Another example, in fact a simpler one, to which our results apply, are deformations generated by . Physically they correspond to turning on light-like Wilson lines, or in the T-dual picture, where the branes become localized both in space and time, they describe their separation in the light-like direction. We shall not however expand on this solution further.

In Section 4 we propose another type of solution, in what might be called a pseudo gauge. This seems easier to apply to situations with non-trivial self-contractions because of the absence of certain singularities.

Marginal deformation solutions in open string field theory have been studied previously in [17, 18, 19, 20, 21, 22], whereas [23] initiated their study in closed string field theory. In the course of this work we have learned that similar results to ours have been obtained independently by Kiermaier, Okawa, Rastelli and Zwiebach [24] and should appear in preprint at the same time as our work.

## 2 Marginal deformations in SFT

We shall start by solving the string field theory equation of motion

(2.1) |

perturbatively in a parameter . Let us denote the coefficient of order , so that

(2.2) |

At order we find

(2.3) |

To obtain a non-trivial solution we shall take to be a non-trivial element of the
cohomology.^{2}^{2}2Note that similar construction can be used to construct the tachyon vacuum in
the wedge state basis [3]. Therein one takes . The
solution appears to be pure gauge for all , but becomes non-trivial at .
It is well known that for each cohomology class there is a representative of the form

(2.4) |

where is a purely matter operator of conformal dimension one, so that indeed . The solution to the equation of motion (2.1) can be determined recursively order by order using

(2.5) |

The right hand side is manifestly closed, as one can convince themselves by induction, but it is a-priori not clear whether it is also exact. It turns out, that for operators which are exactly marginal in the conformal field theory, the right hand side is always exact.

To invert on an -exact state we have to fix a gauge. Popular option, which works well in
the level truncation, is the Siegel gauge [17]; however, for analytic computations it is
more convenient to use the gauge introduced in [3], or some of its variants.
In principle one could try using gauge^{3}^{3}3The gauge with
was found very useful in computing scattering amplitudes [25]. with different at
each order of , but in this section we shall stick to the simplest gauge. We
remind the reader that is the zero mode of the -ghost in the cylinder coordinate.

Let us work out in detail the order of the solution. Easy computation using the formalism of [3] gives

(2.7) |

where in the second line stands for . The operator , in a notation borrowed from [25], is defined as

(2.8) |

The operator , in turn, is defined as the scaling operator in the cylinder
coordinate. The star denotes a BPZ conjugate and stands for . For more
properties the reader is referred to [3] as well as to older works [26, 27].
Note, that in the last two expressions, we have formally integrated over wedge states with . Such wedge states are ill-defined, have no meaning on their own, but in the present case
they cause no problem. In fact, we use them only for notational convenience to denote a well
defined operation of deleting part of the empty surface from the states . The real problem
can only arise in the limit, where the two insertions of from the two ’s are
approaching each other, squeezing in between a line integral. For generic matter operators
there would be a singularity.^{4}^{4}4The easiest way to deal with the singularity is to introduce
a lower cut-off on the integral and define by the minimal subtraction.
This amounts to defining , which is the right definition for inverting
on weight state . Unfortunately, it turns out that in the gauge there is
also an additional singularity associated with a state on which
cannot be inverted within the gauge. The singularity also appears for
non-exactly marginal operators. For simplicity we shall restrict our discussion in this section to
operators with finite products at coincident points.

Before moving to higher orders in , let us check that indeed

(2.9) |

Acting with on given by (2.7) is easy. It annihilates the two factors of , and acting on produces . The integral therefore localizes at the boundary, the contribution gives precisely whereas the contribution vanishes thanks to the two -ghost insertions approaching each other, again assuming absence of singularity from the matter currents. This is such a simple mechanism, that it is rather straightforward to guess the form of the -th order term of the solution

(2.10) |

The proof that this solves (2.5) is easy and is left to the reader. Geometric picture of our solution is given in Fig. 1.

The solution can be viewed as a functional on the Hilbert space of the open string, and as such, it can be represented as a surface with certain punctures. Instead of the conventional upper-half plane, we use a coordinate where the midpoint is sent to infinity so that the surface looks as a cylinder of canonical circumference . But upon taking star product or acting with , the natural circumference of the cylinder changes, and in our case, for the solution (2.10), it is given by . In addition, we get insertions of located at points

(2.11) |

What about the gauge condition, does it remain true for our guess (2.10)? Using the identity

(2.12) |

valid for arbitrary and satisfying , we find successively that , are all in the gauge. To see that, one has to use also the identity .

The tachyon solution was originally found in a similar form [3], but later an elegant closed form was found by Okawa [5]. In the present case, just by simple inspection, we can formally sum up the whole series to obtain

(2.13) |

Here again, the first factor by itself does not make much sense. However, as it acts on , its action is well defined. Actually, it is possible to avoid using the negative wedge states. One can re-write the formula as

(2.14) |

where is a star square-root of the vacuum, or in other words the wedge state , and finally

(2.15) |

is the homotopy operator used in [4] to prove that the cohomology around the tachyon vacuum is trivial.

Let us now ask what are the interesting properties of the marginal solution. For exactly marginal deformation one would expect that the energy of the configuration relative to the original brane is strictly zero. In the time independent setup, the energy is simply given by minus the action. Under the change of the parameter the action changes as

(2.16) |

where we used the fact that is a solution of the equations of motion for all values of . Integrating the equation we find that for all finite values of .

At first sight, there is a little puzzle however. It seems that this proof works not only for exactly marginal deformations, but for all kinds of one-parameter families of solutions continuously connected to zero. One may think of a solution generated by a dimension zero operator

(2.17) |

in a system of two D-branes, where the Chan-Paton factors are given by the Pauli matrices. This corresponds to turning on a constant non-abelian gauge potential along two directions. As is well known, constant non-abelian fields have nonzero potential energy given by ; this is true also in string field theory, as can be shown by integrating out infinite tower of massive fields [28]. It turns out, that for such deformations the recursive procedure for finding the solution breaks down. One has to go back and correct the initial starting point by higher order corrections. Typically what happens is that gets changed to which itself is a nice conformal operator, but its variation with respect to is not. This is the point where our formal proof would break down. The problem does not arise in the fully compact Euclidean case, since there are no operators with continuous spectrum, and so the obstructions in the recursive procedure are unsurmountable. From the field theory perspective these obstructions manifest themselves as impossibility to turn on continuously a flux on a compact manifold.

Another general and interesting question to ask is how the cohomology of the theory changes under
the marginal deformation. Had we worked out in detail, for instance, the solution corresponding to
branes moving apart, we would have to be able to see how does the mass-spectrum of the stretched
strings changes linearly with the brane separation^{5}^{5}5This question was touched upon in the
context of string field theory in [29] and [30].. We do not have the solution
yet, nevertheless, we can address the problem first from a formal viewpoint. Expanding the string field theory around the new vacuum , we get the new BRST-like operator

(2.18) |

and we want to find its cohomology. Formally, this is actually rather easy to determine. Start with a solution to the equation and perturb it in the direction of some operator . The variation of the solution solves

(2.19) |

So the cohomology is given by perturbed solutions. These are very easy to construct. Deform the original theory by an operator , pretending that it is still exactly marginal operator – in reality it is not, of course. The solution will be given by the formula (2.10), but only its first order term in will be relevant for the new cohomology representatives. Concretely the solution is

(2.20) | |||||

where the terms in the first line are obtained by exchanging the position of with the remaining ’s. It is also possible to rewrite the formula in a closed form

(2.21) |

where

(2.22) |

is a formal object, meaningful when sandwiched between two states containing half-strips of size without any insertions on the side adjacent to . Apart of this purely notational formality,the solution (2.21) might be jeopardized when the OPE between and is singular (which is in fact the typical case). As we have been consistently ignoring these issues, we will do so once more. We shall postpone them to a future work. It is perhaps interesting that the straightforward formal proof of (2.21) does not require to be a marginal deformation solution. It can be just any solution to the equations of motion. Of course, we do expect, that in the tachyon vacuum (2.21) will become singular.

## 3 Rolling solutions

The most interesting application of the previous results is to the study of rolling tachyons [15, 16]. Such solutions are generated by a primary field of dimension one (we are using units in which ). For definiteness, we shall take only the plus sign in the exponent – so that the tachyon field is in the perturbative vacuum in the far past. The important property of this vertex operator is that for positive powers its boundary OPE’s are non-singular

(3.1) |

To construct the solution we can simply use the results from the previous section, setting

(3.2) |

The solution itself is given by (2.10); in a form more suitable for level truncation analysis it reads

(3.3) |

where and the are given by formula (2.11). Now let us extract the coefficients . This will tell us the time dependence of the tachyon field. This has been previously studied in various approximation schemes in [15, 16, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. The puzzling feature encountered was that the solution, conjectured to be the tachyon matter, was oscillating with exponentially growing amplitude. The computed pressure was following the same pattern in stark contrast with Sen rolling tachyon conjectures [15, 16]. Logically, there seem to be two possible explanations. Either the solution has a finite radius of convergence in , so that beyond that one has to use proper re-summation formula. An example of such behavior is , which in fact is quite reminiscent of the results from the boundary state analysis. Another possible explanation, perhaps the more likely one, is that the pressure is given by a more complicated formula, containing perhaps some improvement terms that are not given by the Noether procedure. In that case the oscillations in the tachyon field would not have any physical meaning.

The coefficient of the state in the rolling solution is given by

(3.4) |

where , and further

(3.5) |

The matter correlator can be computed using the OPE (3.1)

(3.6) | |||||

To compute the coefficient (3) analytically, it is convenient to pass from to the variables. We were able to compute only the first three coefficients explicitly

(3.7) |

where is rather complicated expression which depends on polygamma function at special points. Equivalently, it can be expressed using the Hurwitz zeta functions for . This is in fact quite natural, since the conformal dimension of is and therefore the transcendentality pattern is similar to the one for the ghost number zero tachyon solution. The value of runs over the values .

Proceeding to higher orders in analytically seems an impossible task, so we have tried to
obtain number of coefficients numerically by Monte Carlo integration^{6}^{6}6Actually we have used
the built-in method QuasiMonteCarlo in Mathematica, so that our approximate values are exactly
reconstructible.. The first few values we got with points are

(3.8) | |||

With less accuracy, points, we went up to values of . The results are plotted in the graph 2. They seem to be fitted remarkably well by a one-parameter fit .

The behavior seems to be consistent with that of Moeller and Zwiebach [31] and Fujita and Hata [35, 37] who also found faster than exponential decay in the coefficients, which means that the sum over powers of has infinite radius of convergence and hence the infinitely growing oscillations will stay. To be completely honest, we have to point out, that our data are not entirely conclusive in this respect. Had the integrand been only moderately peeked around some cube, e.g. of volume for , there would be virtually no chance of detecting this region by the Monte Carlo method. The more reliable numerical or even analytic data for lower order coefficients do not however suggest this scenario. So although we are missing rigorous prove we have enough evidence to believe that decay faster than exponentially, so that our series in has infinite radius of convergence.

## 4 Discussion

We have presented a rather simple solution describing marginal deformation generated by
dimension-one matter primary operator with finite as . In order to be able to
study really interesting examples, such as generic rolling tachyon process, or properties of
unstable systems of branes and (anti)branes one has to understand well the case with singular . Our preliminary computations show that the otherwise successful gauge might not
allow for existence of such solutions. There is one very simple alternative to the construction
presented in section 2. When taking the inverse of an exact object, instead
of demanding that the whole thing be in the gauge, we may as well simply demand that the
argument behind be in the gauge^{7}^{7}7This trick was invented, as far as we
know, by Ian Ellwood in 2002 in the context of the Siegel gauge. He called this a pseudo-Siegel
gauge.. For example for we find

(4.1) |

Working out few more terms, it seems that the pattern is

(4.2) |

where is the characteristic function of a domain specified by the set of inequalities

(4.3) |

for . We leave the proof of this proposal for the future.

## Acknowledgments

I would like to thank Ian Ellwood and Ashoke Sen for useful discussions. I also wish to acknowledge the hospitality of HRI in Allahabad and the organizers of the Indian Strings Meeting in Puri. This research has been supported in part by DOE grant DE-FG02-90ER40542.

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