We revisit the identity-based solutions for tachyon condensation in open bosonic string field theory (SFT) from the viewpoint of the sine-square deformation (SSD). The string Hamiltonian derived from the simplest solution includes the sine-square factor, which is the same as that of an open system with SSD in the context of condensed matter physics. We show that the open string system with SSD or its generalization exhibits decoupling of the left and right moving modes and so it behaves like a system with a periodic boundary condition. With a method developed by Ishibashi and Tada [N. Ishibashi and T. Tada, Int. J. Mod. Phys. A

Open bosonic string field theories on a D-brane have a stable vacuum where a tachyon field is condensed and the D-brane disappears. At the tachyon vacuum, it is proved that we have no open strings as physical degrees of freedom and there is a belief that closed strings living in a bulk space-time are found. Since the on-shell closed string spectrum is obtained in a scattering amplitude in the perturbative vacuum of open string field theories, it is reasonable that there are some observables made by open string fields to clarify the existence of closed strings even on the tachyon vacuum. If it was realized, a pure closed string theory would be formulated in terms of open string fields.

In string theories, it has long been considered that closed string dynamics is described by open string degrees of freedom. It is naively intuitive that a closed string is given by a bound state of an open string and so an open string is regarded as a fundamental degree of freedom. This idea is realized in matrix theories, which are comprehensive or constructive theories including closed strings and D-branes, where matrices are considered as collections of zero modes of open strings. The anti-de Sitter/conformal field theory (AdS/CFT) correspondence also provides a sophisticated version of this thought.

About a decade ago, Gendiar, Krcmar, and Nishino made an important discovery concerning discretized open systems in condensed matter physics [

After their seminal papers [

Interestingly, it has been pointed out in Ref. [

If we change the variable to

The identity-based solution is constructed by an integration of some local operators along a half string and the identity string field. As a result, the theory expanded around the solution includes the modified BRST operator expressed as the integration of local operators on a worldsheet, which is a useful tool for investigating the corresponding vacuum. Actually, we can find the perturbative vacuum in the theory expanded around the identity-based tachyon vacuum solution with high precision by a level truncation scheme [

These facts about SSD and string field theories (SFT) indicate that we are able to formulate a pure closed string theory in terms of open string fields and the identity-based tachyon vacuum solution. In this paper, motivated by this possibility, we will try to find closed string symmetries in the open string field theory at the identity-based tachyon vacuum. To do so, we will mainly use the technique developed in dipolar quantization of 2D conformal field theories [

This paper is organized as follows. In Sect.

The Hamiltonian of an open string is given by

Here, we consider the deformation of Eq. (

As mentioned in the introduction, if we change the variable to

Since

By using this expansion form and the Virasoro algebra, we can obtain a commutation relation for

By using Eq. (

It should be noted that the zeros of

Equal-time contours generated by

According to Ref. [

The contour at the time

The upper half plane corresponds to the worldsheet swept by the left moving sector of a string. Equal-time contours can be depicted by using Eqs. (

Equal-time contours on the

Similarly, we can illustrate equal-time contours for propagation of the right moving sector of a string by using Eqs. (

These contours have a remarkable feature that the string boundaries are fixed at

The equal-time contours generated by

Hence, the equal-time contours by

Now that we have obtained two decoupled Hamiltonians for the left and right moving sectors, we can construct two independent Virasoro operators according to [

For a constant time

By setting

We illustrate this point in the simplest case of Eq. (

We notice that, unlike the radial quantization, it is impossible to deform the integration paths of ^{1}

Regularization for the essential singularities.

Then, what to be proved is that the integration along

The right moving sector of the Virasoro operator

Thus, we have found two independent Virasoro algebras in a deformed open string system, which can be regarded as Virasoro algebras for closed strings, i.e., the holomorphic and antiholomorphic parts.

The continuous Virasoro algebra (

The term

We consider this shift of

We notice that this term is included in Eq. (

This algebra is the same as that in Ref. [

Similarly, we can find antiholomorphic Virasoro algebra by introducing a complex coordinate

Here, it should be noted that the definitions (

Thus, the continuous Virasoro algebras (

We can also derive the continuous Virasoro algebra in the natural frame directly from the definitions (

We consider cubic open bosonic string field theory. For the string field

Expanding the string field around the solution

The operators

The identity-based solution ((

For the moment, we assume that

First, we decompose the modified BRST operator (

Assuming that

A derivation is explained in Appendix

Now, we consider the identity-based tachyon vacuum solution generated by

It should be noted that this decomposition of the modified BRST operator occurs only for the tachyon vacuum solution. For trivial pure gauge solutions,

This interpretation is applicable to closed string states in the open string field theory:

These are analogous relations to Eq. (

In closed string field theories, a gauge transformation for a closed string field

On the perturbative vacuum, the string Hamiltonian in the Siegel gauge is derived from the anticommutation relation of the BRST operator with the zero mode of the antighost. Analogously, we can construct the Hamiltonian

Let us consider finding the continuous Virasoro algebra at the tachyon vacuum. It might be helpful to use the ghost twisted energy–momentum tensor

From the operator product expansions (OPEs) of

Here, it should be noted that ^{2}

By using

Thus, we have found the continuous Virasoro algebra in an open string field theory at the tachyon vacuum.

As seen in Eq. (

These are also frame-dependent operators as

We give a ghost number current at the tachyon vacuum by using

Since it should be defined by the normal-ordering prescription

This is also a frame-dependent operator due to

Now that a ghost number current is given at the tachyon vacuum, we can define an operator counting the ghost number with respect to

These satisfy the following commutation relations with the BRST operators (Appendix

According to the correspondence between

Here, it is interesting to consider the relation of

So, this might suggest that the ghost number for open strings has a linear relation to the ghost number for closed strings. However, the integration on the right-hand side is ill defined due to poles on the unit circle, which arise from the relation

So far, we have found

A transformation of this type was first used to prove vanishing physical cohomology at the tachyon vacuum [

These directly lead to the same OPEs and commutation relations as those of the perturbative vacuum. Moreover, the similarity transformation is given as a Bogoliubov-type transformation and the inner product of the vacua related by the transformation is ill defined because

We can apply the notion of frame-dependent operators to the BRST current. By the similarity transformation generated by

As well as other operators,

We have seen that the identity-based tachyon vacuum solution leads to closed string symmetries in open SFT by using results of SSLD, provided that

First, we consider the solution for

Here, we consider the difference between these two cases. For example, we consider the following function with fourth-order zeros at

As discussed in Sect.

The equal-time contour for the holomorphic sector is depicted in Fig.

The equal-time contours of a string for: (a) the weighting function (

As seen in Fig.

If

From this asymptotics, we find that

Secondly, let us consider the solution generated by

The case of

other than

As a characteristic example, we consider the weighting function for

This function has zeros at the fourth roots of

Before considering the

String pictures before and after SSLD. The solid and dashed lines correspond to holomorphic and antiholomorphic parts of a string. (a) The case of the weighting function (

We apply this picture of SSLD to the case of

This inconsistency is resolved by considering physical observables in open SFT. A gauge-invariant observable in open SFT is given by

Returning to the

In this paper, we have studied SSLD in open string systems and clarified that the left and right moving modes in the SSLD system are decoupled and uncorrelated by zeros of the weighting function of the Hamiltonian. We have shown that, as a result of the decoupling, the SSLD system is equivalent to a closed string system, in the sense that we find holomorphic and antiholomorphic Virasoro algebra in the SSLD system.

Next, we considered open SFT expanded around the identity-based tachyon vacuum solution. We have found that the modified BRST operator is decomposed to holomorphic and antiholomorphic parts, which are anticommutative and nilpotent. These operators are analogous to closed string BRST operators and so the gauge symmetry of the theory is regarded as that of closed string field theories. By using these BRST operators, we have constructed the local operators in the tachyon vacuum, including the energy–momentum tensor, ghost and antighost fields, the ghost number current, and the BRST current. It is a remarkable feature that these operators depend on the frame of the worldsheet, which is chosen by the identity-based tachyon vacuum solution. From these operators, we have found the continuous Virasoro algebra and the ghost number operator. The important point is that these have holomorphic and antiholomorphic parts, which are realized by the SSLD mechanism.

The theory at the identity-based tachyon vacuum solution possesses a gauge symmetry generated by the holomorphic and antiholomorphic BRST operators, which is identified with a gauge symmetry of closed string field theories. Since gauge symmetry is an essential ingredient in SFT, we conjecture that the theory at the tachyon vacuum provides a kind of closed string field theory. That is, it is expected that the theory includes the dynamics of closed strings, although the field in the theory is an open string field. Actually, observables of pure closed strings might be calculable in terms of gauge invariants in this theory. It is known that a similar situation has occurred in matrix theories, where closed string amplitudes are obtained in terms of matrices replacing an open string field [

We should comment on the cohomology of the modified BRST operator at the tachyon vacuum. As proved in Refs. [

While closed string states are related to the state (

Finally, we should comment on the level-matching condition, which is imposed on a conventional closed string field to assure the

The authors are grateful to Nobuyuki Ishibashi and Tsukasa Tada for valuable discussions and comments. We would like to thank Mayuri Kasahara for her help at the beginning of this work. T.T. would also like to thank the organizers of the workshop on “Sine square deformation and related topics” at RIKEN, where this work was initiated, for their hospitality. T.K. would also like to thank the Yukawa Institute for Theoretical Physics at Kyoto University for its hospitality during the workshop “Strings and Fields 2018”. This work was supported in part by a Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (C) (JP15K05056).

Open Access funding: SCOAP

For the energy–momentum tensor on the

With this algebra, we can derive Eq. (

(

Actually, the commutation relation (

Here, for the transformation

The integration range is changed to a contour

For the delta function, we find the following relations:

For the left and right moving modes of the Hamiltonian,

We illustrate the evaluation of the integral along

Following the definition of

Suppose that the operator

Here we have used

By taking the limit

As a result, we can add the integral along

Here, we show that the essential singularity of

On the

Expanding the BRST current

(

Therefore, the nilpotency of

Here, we show Eq. (

From the relation

In particular, we find that

For the holomorphic and antiholomorphic parts of the Hamiltonian,

Using Eqs. (

The BRST current

For the ghost number current

If we define

Integrating the above, we have
^{3}

From Eq. (

Integrating the above and using Eq. (

Because

These relations imply Eq. (

On the modified BRST current, we note that the difference of Eqs. (

It is possible to provide an alternative derivation of Eq. (

Since ^{4}

Integrating the above, we have

The equal-time contour in Fig.

The function

In the case of Eq. (

We note that the conventional modes satisfy