We construct a class of classical solutions in the Berkovits’ open superstring field theory. The resulting solutions correspond to marginal boundary deformations in conformal field theory. The vacuum energy vanishes exactly for the solutions. Investigating the theory expanded around one of the solutions, we find that it reflects the effect of background Wilson lines. The solution has a well-defined Fock space expression and it is invariant under space-time supersymmetry transformation.

## 1 Introduction

String field theory is established as a framework for exploring nonperturbative structures in string theory. Motivated by Sen’s conjecture [1, 2], many people studied classical solutions extensively and intriguing results were provided in bosonic open string field theory [3, 4]. In the supersymmetric case, the most promising theory is formulated in terms of the Wess-Zumino-Witten (WZW) like action proposed by Berkovits [5, 6], which has no problem with contact term divergences [7]. This open superstring field theory is a sufficient framework to elucidate nonperturbative phenomena of the Neveu-Schwarz (NS) sector. Indeed, the tachyon vacuum and kink solutions were found in the superstring field theory by using the level truncation scheme [6, 8, 9, 10, 11]. On the analytical side, there are some attempts to construct exact solutions in terms of a half string formulation [12], a pregeometrical formulation [13, 14], a conjecture of vacuum superstring field theory [15, 16, 17, 18, 19] and an analogy with integrable systems [20].

In the present paper we construct analytic classical solutions in the open superstring field theory using techniques developed in bosonic open string field theory [21, 22, 23]. The resulting solution consists of the identity string field, ghost fields and an operator associated with a current. Taking as the current, we find that the action expanded around the solution can be transformed to the original action by a string field redefinition. In the redefined theory, however, the momentum is shifted in the string field and then the classical solution can be related to a background Wilson line. Generically, we anticipate that our solutions correspond to marginal boundary deformed backgrounds as in the bosonic case.

The analytic solutions are useful for studying gauge structure in string field theory. In bosonic string field theory, the analytic solution corresponding to Wilson lines can be represented as a “locally” pure gauge, and then we find that a “locally” pure gauge configuration in string field theory corresponds to a marginal deformation in conformal field theory [21, 22, 23]. This correspondence is a natural generalization of that of low energy effective theories. Later we will see that the solution in the superstring field theory shares this feature of the bosonic theory.

Marginal deformations in string field theory were often studied using the level truncation scheme. We see that the effective potential for a marginal field becomes flatter as the truncation level is increased [24, 25, 26, 27, 28, 29], and then the vacuum energy of the analytic solution must vanish. Unfortunately, we encounter a difficulty in calculating the vacuum energy in the bosonic theory. Though the vacuum energy formally vanishes, it is given as a kind of indefinite quantities if we calculate it by oscillator representation [21, 22, 23]. However, we will see that the vacuum energy is to be exactly zero in the superstring field theory. This result is a characteristic feature of the supersymmetric case.

In string field theory, the gauge symmetry includes global symmetries generated by [30, 31]. It is a typical symmetry in string field theory because the symmetry mixes various component fields and it has a non-local structure. Based on an analytical approach, we find that the Wilson line parameter in the solution is invariant under the global transformation.

Although it is hard to include the Ramond (R) sector into the action, we have the equations of motion for both of the NS and R sectors [32]. The equations of motion possess a fermionic symmetry which transforms the NS boson (R fermion) to the R fermion (NS boson). Then we expect that the superstring field theory has space-time supersymmetry. Actually, we find that global space-time supersymmetry is realized on-shell as a part of the fermionic symmetry. We show that the solution corresponding to a Wilson line is a supersymmetric solution, namely the solution is invariant under the global space-time supersymmetry transformation.

This paper is organized as follows. In section 2 we construct an analytic classical solution in the open superstring field theory. We find that the solution can be written by a well-defined Fock space expression. After discussing the vacuum energy and the theory expanded around the solution, we relate the solution to background Wilson lines. Michishita gives a covariant action of the R sector by imposing a constraint equation [33]. We discuss the effect of the solution on the R sector in terms of the action proposed by Michishita. Moreover, we show how the solution is transformed under the global symmetry and space-time supersymmetry. In section 3 we extend the Wilson line solution to those which correspond to general marginal boundary deformations generated by supercurrents. We find that generalized solutions also have a favorite feature that the vacuum energy vanishes. We offer some comments related to our results and discuss open questions in section 4. In addition we include four appendices. We represent the identity string field by explicit oscillator expression in the large Hilbert space in appendix A. In appendix B we give a different derivation of the action expanded around a general solution which is originally given in refs. [16, 12]. We use an alternative expression of the action given in ref. [34] to derive the expanded action. In appendix C we show that the fermionic symmetry contains global space-time supersymmetry, and give some comments on supersymmetry in the cubic superstring field theory [30] and its modified theory [35, 36]. In appendix D we construct the analytic solution in bosonic string field theory which corresponds to a general marginal boundary deformations.

## 2 Classical solutions and background Wilson lines

The open superstring field theory action [5, 6] is given by

(2.1) |

where denotes a string field of the GSO(+) NS
sector which corresponds to a Grassmann even vertex operator of ghost
number 0 and picture number 0 in the conformal field theory. CFT
correlators
are
defined in the large Hilbert space and
.^{1}^{1}1For details of the definition, see
for instance ref. [8].
The action is invariant under the infinitesimal gauge
transformation,

(2.2) |

where and are infinitesimal
parameters.
Integrating this infinitesimal form, we can obtain the finite gauge
transformation as^{2}^{2}2The gauge transformation can be
expressed as where , since
each of the operators, and , has trivial cohomology in the
large Hilbert
space. Here, and are group elements in the “stringy gauge
group” in superstring field theory.

(2.3) |

where and are finite parameters. Variating the action (2.1), we can derive the equation of motion to be

(2.4) |

For simplicity, we mainly consider superstring field theory describing the dynamics of a D brane without Chan-Paton degrees of freedom. We single out a direction on the world volume of the brane, writing the string coordinate as and its supersymmetric partner as . Our later analysis can be easily extended to include Chan-Paton indices.

### 2.1 classical solutions in open superstring field theory

In this subsection, we will show that one of the classical solutions is given by

(2.5) |

where is the identity string field
and the operator is
defined as^{3}^{3}3We note that ( : odd) is a fermionic
operator.
More precisely, we need a cocycle factor to represent statistical
property of the operator.

(2.6) |

Here, denotes a counter-clockwise path along a half
of the unit circle, i.e., for .
is a function on the unit circle satisfying
[22, 37].^{4}^{4}4Under this condition,
cannot be a non-zero constant.
We have to impose an additional constraint on due to
the reality of the string field as explained in the next subsection (see
also appendix A).

First, we introduce half string operators similar to :

(2.7) | |||||

(2.8) |

where is the ghost field and is defined as

(2.9) |

By definition, the commutation relation holds. We also define the operators, , and by replacing the integration path by which rotates counter-clockwise along . For these half string operators, we can derive their (anti-)commutation relations from similar procedures in refs. [22, 37]. The operator product expansions (OPEs) among local operators in the integrand are easily calculated as

(2.10) | |||||

(2.11) |

Using these OPEs, we obtain (anti-)commutation relations of these local operators on the unit circle, :

(2.12) | |||||

(2.13) |

where the delta function is defined as
.^{5}^{5}5The delta function satisfies

We integrate (2.12) and (2.13) to derive (anti-)commutation relations of half string operators:

(2.17) | |||||

(2.18) |

where we have used in the latter equation. The similar
relations hold for the right-half string operators, and
other (anti-)commutation relations become zero.
Here, we emphasize that these equations
hold for any functions and
defined on the unit circle, because we have only to use the
equal-time (anti-)commutation relations to derive the equations.^{6}^{6}6
To derive (2.17)
and (2.18), it is sufficient for and to be square
integrable. The condition is unnecessary for
these (anti-)commutation relations.
Namely, the
functions are not necessarily to be holomorphic, though we express them
as functions of a complex variable.

Next, we consider some properties of the half string operators associated with the star product and the identity string field. Suppose that two string fields and are defined as and , where and are conformal fields on the unit discs and , respectively. The star product is defined in terms of the gluing Riemann surface by the identification [38, 39]. Accordingly, it follows that

(2.19) |

where is a primary field with dimension , and
denotes the statistic index defined to be if is a bosonic
(fermionic) operator.
Multiplying a function which satisfies
^{7}^{7}7 is the same as the function
in ref. [22].
We note that our analysis is easily extended to the case of
in ref. [22]. to both sides of
(2.19) and
integrating it along the path ,
we can find the generic formula [22, 37]

(2.20) |

where the operator is defined as

(2.21) |

Similarly, we can obtain a generic formula associated with the identity string field :

(2.22) |

If we choose , and as the primary
field,^{8}^{8}8The dimensions of , and are
, and , respectively. we can derive the following equations
from the generic formulae:

(2.23) | |||||

(2.24) | |||||

(2.25) | |||||

(2.26) | |||||

(2.27) | |||||

(2.28) |

where the function satisfies .
Again, these equations hold if
is defined on the unit circle . As in the case of
eqs. (2.17) and (2.18), the function
does not need holomorphicity.
Here, the function
in eq. (2.26) should behave like
near in order that the state
has a well-defined Fock space expression.
Because the ghost field has a single pole at on the
identity state as seen in the next subsection
[21, 22, 40]. This condition is assured
by imposing if the function is expandable in a Taylor
series.^{9}^{9}9Actually, we can expand the function as
if holds.

Now, it can be easily shown that given by (2.5) is a classical solution:

(2.29) | |||||

(2.30) | |||||

(2.31) |

where we have used (2.28) and in the first equality, and eq. (2.17) in the last equality. We should note that the state is well-defined because due to . The zero mode is not contained in both operators and and the identity string field satisfies . As a result, we find that and then is a solution in open superstring field theory.

### 2.2 Fock space expressions

The operator in the solution can be expressed in terms of integration with respect to ( on the contour ) :

(2.32) |

where operators in the integrand are given by oscillator expansions:

(2.33) |

and is expanded in eq. (A.23). Using formulae: eqs. (A.17), (A.18) and (A.19) and computing straightforwardly, they are expressed on the identity state as

(2.34) | |||||

(2.35) | |||||

(2.36) | |||||

(2.37) | |||||

In computing , we have used a relation for the Neumann coefficients:

(2.38) |

where is the conformal map for the identity string field (see appendix A). Using the above expressions and the reality , we find a relation between the BPZ and hermitian conjugations:

(2.39) |

where we take a convention: . As a result, the reality condition for our solution imposes for the coefficient function in the integrand of (2.32), which is expanded as

(2.40) | |||||

Here we have used in the second equality, which follows from the condition for the classical solution. The reality condition implies that should be real and should be pure imaginary.

Putting the above expansions together, we obtain the explicit Fock space expression of the classical solution as follows:

In particular, the integration with respect to gives finite coefficients for each term of the form because both

More concretely, the lowest few terms of the solution are computed as

In the above explicit expression, the first term implies the condensation of the massless vector field because it is expanded as and is the vertex operator for massless vector with zero momentum [41]. This coefficient constant for the lowest level can be rewritten as

### 2.3 background Wilson lines

We found that the classical solution involves the condensation of the massless vector field. This result suggests that our solution is related to a background Wilson line. In this subsection, we will discuss the vacuum energy of the classical solution, the theory expanded around the solution, and other characteristic features of the solution. Accordingly, we will show that the solution corresponds to a background Wilson line.

In order to evaluate the vacuum energy, it is convenient to use an alternative expression for the action:

(2.43) | |||||

(2.44) |

The equivalence of the actions (2.1) and (2.43) is proved in ref. [34]. In general, the state has no zero mode. For the solution (2.5), it is easily seen that

(2.45) |

and then the state also does not
contain the zero mode.
As a result, we find that the integrand in (2.43) becomes
zero
for the classical solution since there is no zero mode in the
correlation function of
the integrand.^{11}^{11}11In the large Hilbert space, correlation
functions are normalized as to be
.
Hence we confirm that the vacuum energy of the solution vanishes due
to the ghost charge non-conservation in the large Hilbert space.

Let us consider the expansion of the string field around the solution (2.5). Generally, if we expand the string field around a classical solution as , the action becomes

(2.46) | |||||

(2.47) | |||||

where corresponds to the vacuum energy and has the same form as the original action (2.1) except the kinetic operator , which is defined as

(2.48) |

A proof is given in appendix B.
For the new BRS charge, nilpotency
holds automatically but is satisfied owing to the
equation of motion, .
For the solution (2.5), we find as evaluated
above. Substituting
(2.5) into (2.48) and using
(2.23), (2.24), (2.26), (2.27) and
(2.31), we can write the new BRS charge
as^{12}^{12}12We can check nilpotency of the new BRS charge
in terms of (2.18).

(2.49) |

Using (2.17), the new BRS charge is rewritten as a similarity transformation from the original BRS charge:

(2.50) |

Here, we introduce the following half string operators,

(2.51) | |||||

(2.52) |

Using a similar procedure in the previous subsection, we can obtain (anti-)commutation relations between these operators in terms of their OPEs:

(2.53) | |||||

(2.54) | |||||

(2.55) |

If satisfies , it follows from (2.20) and (2.22) that

(2.56) | |||

(2.57) |

Precisely speaking, is not a primary field and we can not apply the formula (2.20) for the case that . However, it is directly shown that the equation (2.19) holds for [42, 43, 44] and then we can derive the same formula in which behaves like a primary field with dimension 0 on the string vertex. The same holds for the formula associated with the identity string field.

In the theory expanded around the solution (2.5), we redefine the string field as

(2.58) | |||||

Under this redefinition, the action of is transformed to the exactly same form as the original action, because is transformed to :

(2.59) |

where use has been made of (2.53) and (2.54). This equivalence between the original and expanded actions suggests that is a pure gauge solution. Actually, we can represent the solution as a pure gauge form by using (2.55), (2.56) and (2.57):