Abstract
For some classical solutions \(\Psi _\textrm{sol}\) in Witten’s bosonic string field theory, it was proven that energy of the solution is proportional to the Ellwood invariant \(\textrm{Tr}(\mathcal {V}\Psi _\textrm{sol})\) with \(\mathcal {V}=c\bar{c}\partial X^0\bar{\partial }X^0\). We examine the relation for solutions involving \(X^0\) variables. As a result, we obtain that the relation may not hold for such solutions. Namely, there is a possibility that the energy is not proportional to the Ellwood invariant.
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1 Introduction
String field theory has been actively studied as a candidate for a non-perturbative formulation of string theory. One of open bosonic string field theories is Witten’s bosonic string field theory and the action is given by [1]
where g is the coupling constant of the string field theory. In this theory, many classical solutions including tachyon vacuum solution were constructed, e.g. [2,3,4,5,6].
To understand the physical interpretation of these solutions, it is important to compute physical observables. In the Witten’s bosonic string field theory, two important observables exist. One is the energy of the classical solution. Because the action evaluated on a static solution is equal to minus the energy of the solution times the volume of the time coordinate, the energy of any static solution \(\Psi _\textrm{sol}\) is given by
Another is
where \(\mathcal {V}\) is a BRS invariant closed string state at the midpoint [7, 8]. This is called Ellwood invariant. It is believed to be equal to the shift in the closed string tadpole amplitude between BCFTs described by the classical solution and the perturbative vacuum solution.
In [9], they proved that the energy is proportional to the Ellwood invariant with
However, it was shown for only some static classical solutions that do not involve \(X^0\). Even if the solution involves \(X^0\), the similar relation should hold as long as the solution is invariant under the shift of \(X^0\) and it depends effectively only on derivatives of \(X^0\). There is a possibility that solutions exist for which these conditions do not hold. In this paper, we examine the relation between the energy and the Ellwood invariant for static solutions that are constructed by K, B, c and matter operators involving \(X^0\). As a result, we obtain that there is a possibility that the energy is not proportional to the Ellwood invariant for such solutions.
This paper is organized as follows. In Sect. 2, we review the discussion in [9] and confirm that the Ellwood invariant is proportional to the energy for regular solutions using only K, B, c. This includes not only Okawa type solution [10] but also ghost brane solution [11] and so on. In Sect. 3, we examine the relation for regular solutions which are constructed by K, B, c and matter operators,Footnote 1 and we obtain that there is a possibility that the energy is not proportional to the Ellwood invariant for such solutions. Additionally, we show the difference between the energy and the Ellwood invariant. In Sect. 4, we present the summary. Appendix A gives formulas for correlation functions of the \(X^\mu \) operators in sliver frame. In Appendices B and C, we examine relations that are needed to show that the energy is proportional to the Ellwood invariant.
2 Review on Ellwood invariant and energy for KBc solution
Many solutions are constructed by using string fields K, B, c. In this section, we consider string fields that are constructed only by K, B, c, and we call such solutions KBc solutions.
The KBc solutions can be written by
where \(F_{1i},F_{2i}\) and \(H_i\) are functions of K. As a concrete example, Okawa type solution [10] is given by
and ghost brane solution [11] is given by
We represent \(\sqrt{F_{ji}},1/H_i\) by a Laplace transform respectively:
The KBc solutions are also represented by the Laplace transform:
where
Let us consider a test state \(\Phi \)
where the string field \(\phi \) is an infinitely thin strip with a boundary insertion of an operator \(\phi (0)\). Similarly to (2.1), \(\Phi \) can be also represented as
Then the trace of \(\Phi \Psi \) is given by the correlation function on the infinite cylinder
where \(C_{L+1}\) is the infinite cylinder with circumference \(L+1\) and the map \(f_2\) is defined (A.1).
Let us consider \(\mathcal {G}\) such that
where \(\mathcal {G}(L,\Lambda ,\delta )\) is defined by
and the contour \(P_{L,\Lambda ,\delta }\) is in Fig. 1 and the contour \(\bar{P}_{L,\Lambda ,\delta }\) is given in a similar way.
In [9], they proven that \(\mathcal {G}\) satisfies conditions
for some static solutions \(\Psi _\textrm{sol}\), where \(\chi \) is defied by
Then the action evaluated on the solution can be written by
Here if the following holds
the energy is proportional to the Ellwood invariant.
In this section, we consider
where
and we confirm
Here in appendix C, we check that (2.2) and (2.3) holds.
2.1 Evaluation of \(\mathcal {A}_1\)
Because of
with
\(\mathcal {A}_1\) is given byFootnote 2
First, we evaluate \(\mathcal {T}_\textrm{I}\). Because \(\psi (L)\) does not involve \(X^0\), \(\mathcal {T}_\textrm{I}\) is factorized as
Here since for \(y\rightarrow \infty \) with \(z=x+iy\),
the horizontal part of the contours \(P_{L,\Lambda ,\delta },\bar{P}_{L,\Lambda ,\delta }\) does not contribute \(\mathcal {T}_\textrm{I}\) in the limit \(\Lambda \rightarrow \infty \). Thus we obtain
Next, we evaluate \(\mathcal {T}_{\textrm{IIA}}\).
Here we assume the boundary condition for \(X^\mu \).
Because of the boundary condition, the right-hand side of the above equation vanishes in the limit \(\delta \rightarrow 0\). Thus we obtain
Finally, we evaluate \(\mathcal {T}_{\textrm{IIB}}\). We need to consider the anticommutator between B and \(\kappa \)
because the contours \(P_{L,\Lambda ,\delta }\) and \(\bar{P}_{L,\Lambda ,\delta }\) cross B. Using this, anticommutator between \(\psi \) and \(\kappa \) is given by
where \(L_{2i}+L_{3i}>a>L_{3i}\). Using (A.2), we obtain
where \(\alpha \) is defined by
Hence we obtain
Since for the boundary condition of c
in the limit \(\delta \rightarrow 0\), the below equations hold.
In addition, using (A.2), we can show
Thus \(\mathcal {T}_{\textrm{IIB}}\) is
From our computation, we obtain
2.2 Evaluation of \(\mathcal {A}_2\)
We evaluate \(\mathcal {A}_2\). Here we note
See Appendix B of [9] for the derivation. Using it and \(\mathcal {G}(0,\Lambda ,\delta )=0\), we obtain the below.
Since this can be factorized as
\(\mathcal {A}_2\) can be written by
With the help of (A.3), we can derive the following.
Using it and the assumption \(\alpha (\infty )=0\), we obtain
Therefore we can confirm that (2.4) holds for the KBc solutions which satisfy the assumption \(\alpha (\infty )=0\)
Because we expect \(\alpha (\infty )=0\) for regular solutions, the energy of the regular KBc solutions are proportional to the Ellwood invariant.
3 Ellwood invariant and energy for the solutions including \(X^0\)
Various solutions are constructed by not only K, B, c but also string fields involving matter operators [12, 14,15,16,17]. Especially we focus on
where \(G_{1i},G_{2i}\) are functions of string fields which are an infinitely thin strip with a boundary insertion of matter operators. In this case also, \(F_{ji}\) and \(H_i\) are represented by Laplace transform respectively. Then the solutions can be written by
As a concrete example, simple intertwining solution [6, 17] is given by
where \(\sigma ,\bar{\sigma }\) are defined as an infinitesimally thin strip with the respectively operators insertion by
and \(\sigma _*,\bar{\sigma }_*\) are boundary condition changing operators and both of them are primaries of weight h.Footnote 3
In this section, we study
for the solution (3.1). In Appendix C, it is given that (2.2) and (2.3) are not problematic for this case also but we check that (2.6) does not hold in Appendix B.
3.1 Evaluation of \(\mathcal {A}_1\)
We evaluate \(\mathcal {T}_\textrm{I}\).
In this case, because not only \(\mathcal {C}_\textrm{I}\) but also \(\psi \) involves \(X^0\), the correlation function cannot be factorized as (2.7). However we can derive
in the limit \(y\rightarrow \infty \). Thus no matter what the matter operators which are involved in \(\psi \), the horizontal part of the contours \(P_{L,\Lambda ,\delta },\bar{P}_{L,\Lambda ,\delta }\) does not contribute \(\mathcal {T}_\textrm{I}\) in the limit \(\Lambda \rightarrow \infty \). This leads to the same result as the one obtained in the previous section.
Next, we evaluate \(\mathcal {T}_{\textrm{IIA}}\). Because of the discussion in the previous section, we derive
In the limit \(\delta \rightarrow 0\), to avoid collision between \(X^0\) and matter operators involved \(\psi \), we regularize \(\sqrt{F_{ji}}\) by
Owing to the regularization, using the boundary condition (2.8), we obtain
Finally, we evaluate \(\mathcal {T}_{\textrm{IIB}}\). In the same way as in the previous section, we need to consider only the anticommutator between B and \(\kappa \) because the contours \(P_{L,\Lambda ,\delta }\) and \(\bar{P}_{L,\Lambda ,\delta }\) do not cross the matter operators. Using (2.9), we can derive
where \(\alpha \) is defined by
Thus as in the previous section, it is enough to consider
Here using (A.5) we can derive
Hence even if \(\psi \) involves matter operators, we can use
Thus we obtain
This is the same result as in the previous section.
Therefore we obtain
3.2 Evaluation of \(\mathcal {A}_2\)
We evaluate \(\mathcal {A}_2\). In a similar way as in the previous section, we use
Because \(\psi \) involves \(X^0\), this cannot be factorized as (2.10). If we focus on the case that \(G_{1i}\) and \(G_{2i}\) are constructed only by plane wave vertex operators, the right-hand side can be written by
where \(\Delta \) id defined by (A.6). The first term can be written by
in the same way as in the previous section. However the second term presents an obstruction. Because of the second term, we obtain
Here the trace in the last line leads to
On the other hand, we were unable to evaluate \(\Delta \). Hence it is not clear whether (2.4) does not hold for the solution. If \(\Delta \) does not vanishes, the difference between the energy and the Ellwood invariant is given by
4 Summary
We examine condition (2.4) for the solution which is constructed by K, B, c and matter operators involving \(X^0\). As a result, we obtain that \(X^0\) presents an obstruction. If the solution involves \(X^0\), we need to calculate \(\Delta \). Hence (2.4) may not hold for the solution. Because we confirm that (2.2) and (2.3) are not problematic in Appendix C, if we can evaluate \(\Delta \), it will be clear whether the energy is proportional to the Ellwood invariant. Unfortunately, \(\Delta \) depends on \(X^0\) included in the solution and we were unable to evaluate \(\Delta \). Thus at present, it is not clear whether the energy is proportional to the Ellwood invariant. However, according to the numerical result in Appendix A, \(\Delta \) does not vanish (Fig. 2). Therefore the energy may be not proportional to the Ellwood invariant.
If one would like to clarify whether the energy is proportional to the Ellwood invariant for such solutions, it may solve the problem to modify \(\mathcal {G}\). It is required that \(\mathcal {G}\) satisfies
but such \(\mathcal {G}\) is not unique. In [19, 20], they found operator sets that satisfy the algebraic relation of the KBc algebra. Using such operator sets even if a solution involves \(X^0\), it looks like the KBc solution. They may be helpful to modify \(\mathcal {G}\).
In this paper, we focused on regular solutions and did not consider solutions in which regularization is necessary e.g. [21,22,23,24]. In [9], it is already examined for Murata-Schnabl solution but it may be interesting to examine also for other solutions. Especially the solution which is constructed in [25] involves \(X^0\) and regularization is necessary. It would be intriguing to examine the relation between the energy and the Ellwood invariant for the solution.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical study and no experimental data has been listed.]
Code Availability Statement
This manuscript has no associated code/software. [Authors’ comment: The Fig. 2 and 3 were generated using a Mathematica program which is available from the authors upon request.]
Notes
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Acknowledgements
The authors would like to thank Nobuyuki Ishibashi for reading a preliminary draft and helpful comments. Additionally, the authors would like to thank a referee for proposing a numerical approach to (A.7) in Appendix A. The work of Y.A. was supported by JST, the establishment of university fellowships towards the creation of science technology innovation, Grant Number JPMJFS2106.
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Appendices
Appendix A Correlation functions in the sliver frame
The conformal transformation \(f_s\) from the infinite cylinder \(C_s\) with circumference s to upper half plane (UHP) and the inverse transformation \(f_s^{-1}\) are given as
Using them, we obtain the 2-point correlation function of \(\partial X\) on \(C_s\).
Especially, since for
we can evaluate the correlation function of \(g_z\)
and the correlation function of \(\mathcal {G}\)
Similarly, we obtain the correlation function of vertex operators on \(C_s\)
where \(u_i\) is defined by
In general, correlation functions of X variables can be evaluated from above result. For example, the correlation function of \(\partial X^\mu e^{ik\cdot X}\) is given by
In particular, we give important correlation function.
Using them, we obtain
Since for
with \(z_0=x_0+iy_0\) in the limit \(y_0\rightarrow \infty \), we obtain
and
where we define
Since for
with \(z=x+iy\) in the limit \(y\rightarrow \infty \), we obtain
and
Using the above results, we give the correlation function of \(\mathcal {G}\) and plane wave vertex operators
In the limit \(\Lambda \rightarrow \infty \), the horizontal part of the integration vanishes
On the other hand, the vertical part of the integration does not vanish. Thus in the limit \(\Lambda \rightarrow \infty \), we obtain
Unfortunately, we are unable to evaluate this integral and we do not know whether this vanishes.
We numerically evaluate
and in Fig. 2, we plot \(\Re \hat{\Delta }(10,6,z_m,z_o)\) as a function of \(z_m,z_o\) where \(\hat{\delta }_z,\hat{\delta }_{\bar{z}}\) are defined by
According to Fig. 2, It is clear that \(\Delta \) does not vanish in general.
Appendix B Examination of condition (2.6)
Even if \(\Delta \) does not vanish, it does not imply that the energy is not proportion to the Ellwood invariant. This is because there is still a possibility that (2.6) does not hold.
In the this appendix, we show that the above relation for string fields involving \(X^0\) variables does not hold.
First let us consider a string field
The Ellwood invariant for the string field is
On the other hand, the trace of \(\chi Q\Psi _{L,L'}\) is given by
As shown in [9], they are the same.
Next let us consider a string field
where we set \(k_1^\mu =(\sqrt{h},0,\dots ,0)\) and \(k_2^\mu =-k_1^\mu \). The Ellwood invariant for the string field is
where
Since for (A.4), we obtain
Thus the Ellwood invariant is
where
Finally, we evaluate the trace of \(\chi Q\Psi \). \(Q\Psi \) is
We focus on the second term and we consider the correlation function of it and the first term in (2.5). This is given by
Similarly, we obtain
We evaluate also the the trace of \(\chi \) and the remaining term in (B.2). They are given by
and
To satisfy (2.6), the sum of (B.3), (B.4) and (B.5) has to coincide with (B.1) and we examine it order by order in h. Because of
we obtain
Because the first term can be expressed
the higher-order terms in h has to vanish. We focus on the term at \(\mathcal {O}\left( h^2\right) \). It is given by
In Fig. 3, we present numerical result. It is clear that (B.6) is nonzero. Therefore the equation (2.6) does not hold for the string field involving \(X^0\) variables.
Appendix C Examination of condition (2.2) and (2.3)
We would like to examine the conditions (2.2) and (2.3). In this appendix, we consider more general condition
than (2.2) and (2.3). We show that the above equation holds for arbitrary string fields \(\Psi _i\) which are constructed only by K, B, c and matter operators.
If we represent \(\Psi _i\) by a Laplace transform, the left-hand side can be written by
where s is given by
Using the cyclicity of the cylinder, the vertical part of the contours cancel each other (Fig. 4).
Additionally, using (A.5), we obtain
This is what we wanted to show. Therefore it is clear that (2.2) and (2.3) hold for the solutions which are constructed only by K, B, c and matter operators.
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Ando, Y., Suda, T. Energy from Ellwood invariant for solutions involving \(X^0\) variables. Eur. Phys. J. C 84, 578 (2024). https://doi.org/10.1140/epjc/s10052-024-12888-2
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DOI: https://doi.org/10.1140/epjc/s10052-024-12888-2