Comments on lump solutions in SFT
Abstract
We analyze a recently proposed scheme to construct analytic lump solutions in open SFT. We argue that in order for the scheme to be operative and to guarantee background independence it must be implemented in the same 2D conformal field theory in which SFT is formulated. We outline and discuss two different possible approaches. Next we reconsider an older proposal for analytic lump solutions and implement a few improvements. In the course of the analysis we formulate a distinction between regular and singular gauge transformations and advocate the necessity of defining a topology in the space of string fields.
1 Introduction
After the discovery [1, 2] of the first analytic solution of open SFT à la Witten [3], which links the perturbative vacuum to the tachyon vacuum, there have been a considerable number of papers devoted to related solutions [4, 5] and to marginal deformations thereof [6, 7, 8, 9, 10, 11, 12, 13]. The literature concerning analytic lump solutions, i.e. analytic solutions interpretable as lower dimensional branes (meant to complete the analytic proof of the three conjectures by Sen [14]), is instead poorer. There have been essentially two attempts to find such analytic solutions: the first is the socalled BMT proposal [15, 16, 17], the second is the most recent one formulated in [18], which we will refer to as EM. They are both modeled on the Erler–Schnabl (ES) solution [5], an alternative simpler formulation of the original tachyon vacuum (TV) solution (for recent reviews on the whole subject, see [19, 20, 21, 22]). In [18] the construction is based on previous results on correlators involving boundary condition changing (bcc) operators and on a set of (implicit) prescriptions laid down in order for the solution to satisfy the SFT equation of motion. As we shall see, hidden behind this is the risk of background dependence. More precisely, we wish to clarify first of all if and to what extent these results and prescriptions can be embedded in SFT; second, we would like to discuss their background independence. By the latter we mean that one can derive a variety of backgrounds as solutions of the SFT equation of motion, although such an equation of motion is formulated in a specific background (the D25brane). The only way to clarify these issues is to implement the EM approach in an explicit 2D field theory formulation similar to the one in [5, 12, 15], consistent with the original one in which the SFT is formulated. The first part of our paper is an attempt in this direction. We clarify the problems behind the EM proposal and try to solve them. We do not succeed in carrying out this task, but, nevertheless, we think our scrutiny may be instructive. More explicitly, while we believe the first issue above, although not solved by us in this paper, should anyhow be implementable, the second, i.e. background independence, looks almost impossible to realize.
We then turn to the BMT proposal and, in the second part of the paper, we discuss some aspects of the latter and present a few improvements. In the course of the analysis we propose a definite distinction between singular and regular BRST transformations, which turns out to be instrumental in clarifying some confused issues present in analytic solutions of SFT.
The paper is organized as follows. Sections 2, 3, and 6 are short introductions to the ES solution and to the EM and BMT proposals. Section 4 is devoted to a detailed presentation of the bcc operators used in the EM proposal. Section 5 contains our (failed) attempt to implement the EM approach in an explicit field (oscillator) formulation. Section 7 contains a digression on gauge transformations and identity based solutions, which is needed for the subsequent developments in Sect. 8, where they are applied to the BMT proposal. Section 9 contains some conclusions.
2 Short review of the Erler–Schnabl solution
3 The EM proposal
In the EM ansatz many details are understood, and we must ask: Is it possible to implement this formulation in a concrete field theory formalism, consistent with that of SFT? ^{1}
In the following we would like to bring to light the hidden details and assess their validity. To start with, two BCFT’s are mentioned in [18], BCFT\(_0\) and BCFT\(_*\), and a rule is declared according to which, when writing the solution, any operator that appears to the left of \(\sigma \) or to the right of \(\bar{\sigma }\) belongs to BCFT\(_0\), while operators which appear to the right of \(\sigma \) or to the left of \(\bar{\sigma }\) belong to BCFT\(_*\), but no distinction is made to specify what string fields belong to the former and what to the latter. It may well be that the Q and K that are defined on BCFT\(_0\) are the same that live on BCFT\(_*\) (this is one of the hypotheses we will consider later on). But this is a crucial aspect that must be carefully justified. As we will see more clearly later on, the BCFT\(_0\) is the BCFT with NN boundary conditions for all directions, while BCFT\(_*\) is characterized by DD boundary conditions along one space direction, the 25th, say, and NN along the remaining ones. To keep track of this we will simply label Q and K in these theories with NN and DD, respectively.
Thus, in order to be consistent with the EM rules, when passing through \(\sigma \), operators like \(Q_\mathrm{NN}\) and \(K_\mathrm{NN}\), become \(Q_\mathrm{DD}\) and \(K_\mathrm{DD}\), and they switch back to \(Q_\mathrm{NN}\) and \(K_\mathrm{NN}\) when they pass through \(\bar{\sigma }\). We will discuss later on how it is possible to make sense of such switching of operators. In the rest of this section we would like to make it evident that considering only two BCFTs, BCFT\(_0\), and BCFT\(_*\), is not enough.
It is clear that the previous minimalistic cosmetic of the EM rules is too simplistic and only complicates things. A deeper interpretation is necessary. But it is also clear that if the above issues (among others, see below) are not clarified within a concrete field theory formalism, the EM ansatz remains abstract (but see the footnote at the beginning of the section). Our aim in the sequel is to interpret the EM rules in a 2D conformal field theory context, consistent with the formulation of SFT.
4 The bcc operator
One crucial ingredient in the EM ansatz are the bcc operators. We devote this section to a rather detailed description of this subject.
The issue of bcc operators was introduced by Cardy, see [24, 25, 26], and subsequently studied and applied by many authors, see in particular [27, 28, 29, 30]. It is generally believed that the original and twisted theories are characterized by Hilbert spaces that can be related to each other. This is in a sense trivial, because all countable Hilbert spaces are isomorphic as vector spaces. However, a CFT is not simply characterized by a Hilbert space, but also by the central charge, its primary operators, and by the field theory axioms (locality, for one) and its symmetries. Therefore a direct connection between two such theories in the form of an intertwining operator between the two Hilbert spaces (see below) or, even more, an identification of the two, is far from guaranteed and, if it is possible, it is far from trivial to be determined. This is to stress that it is necessary to analyze in depth the concept and application of the bcc operator, a type of analysis the existing literature does not abound with.
In the sequel, for definiteness, we will use an explicit formulation of the bcc operator, following in particular [27].
It is clear that we can repeat word by word the same things for the DN string, whose oscillators are also halfintegralmode. The vacuum in this case will be denoted by \(\bar{\sigma }_*\rangle \).
In conclusion in the \(c=1\) CFT there is room for four different Hilbert spaces \(\mathcal{H}_\mathrm{NN}\), \(\mathcal{H}_\mathrm{DD}\), \(\mathcal{H}_\mathrm{ND}\), and \(\mathcal{H}_\mathrm{DN}\). The last two are the same. The first two are also identifiable except for the zero mode \(\alpha _0\) in the NN case. If the coordinate X is compactified on a circle, the momentum is discrete; dually, in the DD case, we have wrapping modes. In other words the relevant Hilbert spaces are organized in discrete sectors. For the moment let us ignore, for simplicity, such discrete sectors and the momentum in \(\mathcal{H}_\mathrm{NN}\). We see that, as far as the EM proposal is concerned, we have two kinds of Hilbert spaces, one built out of the integralmode matter oscillators \(\alpha _n\) in the 25th direction, the other with halfintegralmode oscillators \(J_{r}\), beside all the other matter and ghost oscillators.
At this point, however, it is worth recalling that OSFT is formulated in terms of NN strings, that is, on the background of the D25brane. The background independence of OSFT therefore does not rely on the original formulation, but on the fact that, starting from it, we can derive all the other possible backgrounds as analytic solutions of its equation of motion. This is the OSFT’s bet. It goes without saying that any background independent solution must be formulated in the original SFT background (the NN string).
As we said above, based on the operatorstate correspondence, we can assume that in the \(c=1\) CFT a primary \(\sigma _*\) exists of weight \(\frac{1}{16}\) (but nonlocal in the conventional sense, [24, 25, 26]), such that \(\sigma _*(0)0\rangle = \sigma _*\rangle \), where \(0\rangle \) is the vacuum for the (integral) \(\alpha _n\) oscillators. However, this is not yet enough to justify the EM rules. There is no guarantee that the field \(\sigma _*\) admits a free field representation in terms of the free field X or, what is the same, in terms of the free integral oscillators \(\alpha _n\). If it is true, it must be proven. But if this is not the case we have to give a sense to the operations of applying the SFT operator Q or string field K (which we recall are expressed in terms of free (integralmode) matter and ghost oscillators) to \(\sigma _*,\bar{\sigma }_*\).
To summarize we therefore face two alternatives. In the best option we can express the fields \(\sigma _*, \bar{\sigma }_*\) in terms of the \(\alpha _n\) oscillators. In the worst, if this is not possible, we must find a way to deal with different Hilbert spaces. The viability of such options is far from obvious and in any case it has not yet been proved.
5 Two alternatives
The aim of this section is to find the correct \(K,B,c,\sigma _*, \bar{\sigma }_*\) algebra to justify the EM ansatz. As said above, we have two kinds of Hilbert spaces, one built out of the integralmode matter oscillators \(\alpha _n\) in the 25th direction, the other with halfintegralmode oscillators \(J_{r}\), beside all the other matter and ghost oscillators. Let us call them (or, better, their extensions) \(\mathcal{H}\) and \(\mathcal{H}^*\), respectively. The vacuum of the first is the usual string vacuum, made of the tensor product of the various types of oscillators, in particular of the \(0\rangle \) vacuum for the integral oscillators \(\alpha _n\), with the wellknown star product of SFT. The vacuum of the second is obtained by replacing the first vacuum \(0\rangle \) with the state \(\sigma _*\rangle \) (or \(\bar{\sigma }_*\rangle \)) on which the halfinteger oscillators act. In the first (extended) Hilbert space we have the usual K, B, c algebra and the BRST operator \(Q=Q_\mathrm{ii}\) acting on it. In the second K is replaced by \(K^*\equiv K_\mathrm{ND}\equiv K_\mathrm{DN}\) and Q by \(Q^*\equiv Q_\mathrm{ND}\equiv Q_\mathrm{DN}\) and the star product is also modified accordingly.

What is the star product between string states belonging to different Hilbert spaces?

The SFT BRST operator \(Q=Q_\mathrm{NN}\) acts as a derivation on any expression like (57): why is it that it may change by taking the form appropriate to different Hilbert spaces?
The idea of considering two different Fock spaces \(\mathcal{H}\) and \(\mathcal{H}^*\) with an intertwining operator between them, if viable, would solve the problem. However, such a construction would make explicit use of DD, ND, and DN oscillators, beside the NN ones. The former appear in strings attached to a D24brane. In fact they define the D24brane (the dynamics of a brane is defined by the strings attached to it). This means that the information we want the solution to contain, i.e. the description of the D24brane, is already contained in the initial data. This implies that the solution is background dependent. There is nothing wrong, of course, in trying to describe Dbranes in a background dependent way. Background dependence is standard in ordinary string theory approaches, for instance in [29]. But here we are in SFT and, as explained above, the ambition of this theory is background independence, which means that we can derive all the other possible backgrounds as analytic solutions of its equation of motion. It goes without saying that any such solution must be formulated in the original SFT background (the NN string).
This is a good point to recall that the nonanalytic or approximate lump solutions one finds in the literature do not make use of bcc operators or halfinteger modes. This is the case for numerical solutions based on level truncations, [42, 43, 44, 45], but also for the lump solutions in vacuum SFT (see, for instance [46]) and for those in boundary SFT (see, for instance [47]). In analogy we expect that an analytic lump solution should not contain any builtin information about Dbranes, but rather a Dbrane description should emerge from the physical content of the solution. A comparison with the BMT proposal below may help to understand the difference.
Second alternative. A radically different and more appealing alternative could be inspired by the example of marginal deformation to the TV solutions. To this end let us recall the KOS solutions [12].
5.1 The KOS bcc operator
The KOS solutions are based essentially on Eq. (65). Returning to the EM ansatz, the first difference we notice is that in the latter the field analogous to V does not exist (in the relevant case it is singular [18]). Thus we have to proceed in another way. In [27] the halfintegralmode sector is called ‘Ramond’. This is an unconventional terminology, but it is reminiscent of the Ramond sector in superstring theories and suggests a parallel with that situation. We recall that in open superstring theory we have two vacua, the NS one, analogous to the usual vacuum in bosonic string theory, and the Ramond vacuum (which is made of spinor states forming a representation of the gamma matrix algebra in 10D). It is, however, possible to define a primary operator that, applied to the usual vacuum, creates the Ramond vacuum under the stateoperator correspondence. This primary is constructed using (matter and ghost) fields in the theory (with fermion and superghost fields in bosonized form).
Applying this analogy to the EM proposal we may ask whether the \(\sigma _*\rangle \) (and \(\bar{\sigma }_*\rangle \)) vacuum can be created in an analogous way starting from the ordinary vacuum and applying a primary operator formed with the fields in the theory. In such a case all Q’s and K’s would collapse to the same operator and the previous difficulties would disappear, because there would be no need to distinguish between string states belonging to the four different Hilbert spaces. This scheme is certainly most appealing, in particular it would guarantee background independence. But it is not easy to implement. It cannot be done straight away, because we have at our disposal only the bosonic NN field X (which has no charge at infinity, and thus does not offer any chance to use the Coulomb gas method^{3}). At page 5 of [28] we find the peremptory sentence: “In open strings there is no such thing as a twisted state”, where ‘twisted state’ refers to \(\sigma _*\rangle \). Perhaps this is too strong a statement, but, certainly, there seems to be no straightforward way to implement this scheme.
In conclusion we have not been able to implement the EM prescriptions in a concrete 2D field theory formalism, consistent with 2D CFT on which SFT is defined, while avoiding background dependence. But, since ours is not a mathematical theorem, we cannot completely exclude that it is possible.
The parallel with the KOS solution suggests one more consideration. When one tries to translate the KOS solution into the EM scheme one comes across a singularity (in the analog of V). This may be an indication that lump solutions are inevitably, in some sense, singular. A similar peculiarity is met in numerous classical field theory solutions and, in itself, is not really important. What is important in these cases is that the physical quantities related to the solutions can be computed. This is what happens also for the BMT solution.
6 The BMT proposal
In [15] a general method has been proposed to obtain new exact analytic solutions in open string field theory, and in particular solutions that describe inhomogeneous tachyon condensation. The method consists in translating an exact renormalization group (RG) flow generated in a twodimensional worldsheet theory by a relevant operator, into the language of OSFT. The soconstructed solution is a deformation of the ES solution. It has been shown in [15] that, if the operator has suitable properties, the solution will describe tachyon condensation only in specific space directions, thus representing the condensation of a lower dimensional brane. In the following, after describing the general method, we will focus on a particular solution, generated by an exact RG flow first analyzed by Witten [48]. On the basis of the analysis carried out in the framework of 2D CFT in [47], we expect it to describe a D24brane, with the correct ratio of tension with respect to the starting D25brane.
 1.
\(\frac{1}{K+\phi }\) is in some sense singular, but
 2.
the RHS of (70) is regular and
 3.
\(\frac{1}{K+\phi }(K+\phi )=1\).
The \(\phi _u\) just introduced satisfies all the requested properties. According to [47], the corresponding RG flow in BCFT reproduces the correct ratio of tension between D25 and D24branes. Consequently \(\psi _u\equiv \psi _{\phi _u}\) is expected to represent a D24brane solution.
In SFT the most important gauge invariant quantity is of course the energy. Therefore in order to make sure that \(\psi _u\equiv \psi _{\phi _u}\) is the expected solution we must prove that its energy equals a D24brane energy.
The integral in (77) is well defined in the IR (s very large, setting \(s=2uT\)) but has a UV (\(s\approx 0\)) singularity, which must be subtracted away.^{4} Once this is done, Eq. (77) can be numerically computed, the result being \(\approx \)0.069. This is not the expected result, but this is not surprising, for the result depends on the UV subtraction. Therefore one cannot assign to it any physical significance. To get a meaningful result we must return to the very meaning of Sen’s third conjecture, which says that the lump solution is a solution of the theory on the tachyon condensation vacuum. Therefore we must measure the energy of our solution with respect to the tachyon condensation vacuum. Simultaneously the resulting energy must be a subtractionindependent quantity because only to such a quantity can a physical meaning be assigned. Both requirements have been satisfied in [16] in the following way.
Before we pass to a closer scrutiny of this solution and its properties, we need a discussion of the relations between various solutions of the SFT EoM provided by gauge transformations. What we would like to stress is that gauge equivalent solutions may take very different (even singular) forms. To this end we make a detour about identity based (IB) solutions.
7 A detour: TV solution as gauge transformation of IB solutions
The energy for these class of solutions in (87) is independent of both \(\alpha \) and \(\lambda \) and it is equal to the energy of the ES TV solution. The same is true for the closed string overlap. Since the original Schnabl’s TV solution is also gauge equivalent to the ES solution, we see that all the known TV solutions are regular gauge transforms of the IB solutions.
7.1 BMT lump solution as gauge transformation of IB solutions
8 Discussion of BMT solution
As was repeatedly noticed in [16, 17, 54], this is analogous to the procedure of removing singularities, or defining distributions, in ordinary function theory. In [54] a first attempt was made to do the same for \(\frac{1}{K+\phi }\). A space \({\varvec{\mathcal {F}}}\) of test string states or regular states with a definite topology was introduced. The dual of it, \({\varvec{\mathcal {F}}'}\), with the appropriate topology was defined. The state \(\frac{1}{K+\phi }\) belongs to the latter, i.e. it is a distribution. As such the condition 2 and 3 of Sect. 6 are satisfied and the SFT equation of motion is verified (on the other hand, in a commonsense approach, a simple continuity argument would be enough to prove the latter; see [16]). Moreover, as was shown above, the procedure to compute the energy is well defined.
As for point 1 of Sect. 6 it is selfevident, but a comment is in order. The string state \(\frac{1}{K+\phi }\) must be singular so that the wouldbe homotopy operator \(\frac{B}{K+\phi }\) (understood as the starmultiplication operator by the string state \(\frac{B}{K+\phi }\)) is not well defined, otherwise the perturbative spectrum on the brane would be trivial. A true homotopy operator is supposed to map normalized states (annihilated by the BRST operator) into normalized states. The state \(\frac{1}{K+\phi }\) is singular and it has to be regularized. As was shown in [54] this can be done in a weak sense (in physical language, for correlators), while the previous requirement would require an operator topology argument, which does not seem to exist. For this reason we conclude that \(\frac{B}{K+\phi }\) does not exist as a homotopy operator.
The previous conclusions have important consequences for the themes discussed in this paper. For instance, it is debated what the allowed gauge transformations are in SFT. Based on the above we propose the following distinction: a gauge transformation is allowed if it is regular in the above sense, that is, if it does not need to be regularized; otherwise it is not allowed. In other words, an allowed gauge transformation cannot be a true distribution. Once this is established, we can return to the end of the previous section: the state \(\frac{1}{K+\phi +\varepsilon }\) for \(\varepsilon \ne 0\) is not a distribution (it does not need a regularization), it is a regular state. So the states X and \(X^{1}\), which lead from the ES solution to the \(\Psi _{\phi ,\varepsilon }\) solution (98), (99), are regular. Therefore the corresponding gauge transformation is a genuine one, and the \(\Psi _{\phi ,\varepsilon }\) solution is equivalent to the ES one. This implies in particular that they have the same energy.
We can now reconsider one of the unsatisfactory aspects of the energy calculation in [16]. As pointed out above, the energy of (96) was calculated only numerically and turned out to be 0. It was remarked in [16] and repeated above that this value is only conventional, because it is obtained via a UV subtraction which introduces an arbitrariness in the absolute result. Consequently it was stressed in [16] that only quantities that are independent of such subtractions can be assigned a physical meaning, and it was precisely in this way that the lump energy was calculated. From what we have just argued (see also the end of Sect. 7.1) we are now able not only to confirm all this, but also that the energy of \(\Psi _{\phi ,\varepsilon }\) is precisely the TV energy, thus confirming the intuition in [16].
8.1 A comment about homotopy operators
This is puzzling and we face two possibilities: (1) either \(\psi _{\alpha =0}\) is truly not equivalent to the ES solution though degenerate in energy with it, in which case it may well be that the perturbative spectrum supported by this solution is not empty; but the physical interpretation would be obscure: what is this vacuum degenerate with the TV supposed to be? or (2) \(\psi _{\alpha =0}\) is truly equivalent in some nonevident way to the ES solution, but in this case a homotopy operator should exist. Then we are again faced with two possibilities. The first possibility is that the homotopy operator for \(\psi _{\alpha =0}\) exists, but cannot be expressed as a function K, B, c. Although unlikely, because the solution \(\psi _{\alpha =0}\) is a simple function of K, B, c, we cannot completely exclude this exotic possibility. A second, more likely, alternative is that the homotopy operator exists as a limit of a sequence of homotopy operators of analogous solutions. This of course requires the existence of a topology in the space of solutions, that is, in the space of string states. Such a topology should not be confused with the Hilbert space norm, because most of the string states that enter this game have infinite Hilbert space norm. An attempt in this direction has been initiated in [54].
9 Conclusions
In this paper we have pointed out the problems connected with the proposal of [18] to construct analytic lump solutions in OSFT. We have remarked that for it to become effective (in a background independent way) one must show that it is possible to implement it in the very same language (2D CFT) in which OSFT is formulated. We have made some attempts in this direction without succeeding. However, we do not completely exclude that such an implementation be possible. We have also made some additional improving remarks on the BMT proposal for analytic lump solutions. This proposal too has a problem of a different nature related to its mathematical background: a formulation of a welldefined distribution theory for string fields that generalizes the approach of [54]. We hope anyhow we have at least brought enough evidence, with this and other examples in Sect. 8.1, that it is time to face the problem of the topology in the space of string fields.
Footnotes
 1.
This is what we deem necessary. Not everybody shares this opinion. The referee of this paper believes that this problem we think to be so important is already considered and implicitly solved in the first ref. [38]; see eqs. (8.13–14) there. We take note of this, but we believe that just because of this it should be possible to formulate the EM proposal in a more explicit form using the oscillator formalism. In the process the issue of background dependence or independence will automatically emerge.
 2.
For the oscillators we use the same symbols in the NN and DD case and in the ND and DN case. The reader is invited to keep in mind the difference.
 3.
Needless to say \(e^{\frac{1}{4} X}\), where X is as in (33), is not what we need.
 4.
Of course this singularity is present also in the ES solution, it is represented by the infinite volume. The difference here is that the infinite volume appears in the form of a zero mode which generates a singularity \({\sim } \frac{1}{\sqrt{u}}\) in g(u).
 5.
The parameter \(\varepsilon \) is originally a gauge parameter, but due to the UV subtraction such a gauge nature is broken and the energy functional depends softly on the value of \(\varepsilon \) [49].
 6.
We do not say anything about the properties of the homotopy operator, for instance if it is continuous. To this end one should introduce a topology in the space of string states. This problem is usually ignored in the existing literature, although it is very likely at the origin of many ambiguities that arise in the search for analytic solutions in SFT.
Notes
Acknowledgments
We would like to thank Carlo Maccaferri for explaining to us his work prior to publication and for other comments, and Ilmar Gaharamanov for collaboration in an early stage of the research. The work of D.D.T. was supported by the Grant MIUR \(2010YJ2NYW_001\).
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