aa r X i v : . [ h e p - t h ] S e p Comments on Lumps from RG Flows
Theodore Erler Carlo Maccaferri Institute of Physics of the ASCR, v.v.i.Na Slovance 2, 182 21 Prague 8, Czech Republic
Abstract
In this note we investigate the proposal of Ellwood[1] and one of the authors et al [2] toconstruct a string field theory solution describing the endpoint of an RG flow from areference BCFT to a target BCFT ∗ . We show that the proposed class of solutions suffersfrom an anomaly in the equations of motion. Nevertheless, the gauge invariant actionexactly reproduces the expected shift in energy. Email: [email protected] Email: [email protected] ontents
One of the major outstanding problems in open string field theory has been to find ananalytic solution describing a lower dimensional brane (a tachyon “lump”) from the per-spective of a higher dimensional brane. Following Sen’s conjectures[3, 4], such a solutionwas constructed numerically in the Siegel gauge level expansion by Moeller, Sen, andZwiebach[5]. An exact description of the tachyon lump, however, has remained elusive.One of the few concrete proposals for constructing the lump was suggested by Ellwood[1]in his analysis of the gauge structure of open string field theory around the tachyon vac-uum. Later Bonora, Tolla, and one of the authors (henceforth BMT)[2], interpretedEllwood’s proposal as a prescription which, given a boundary RG flow which interpolatesfrom a reference boundary conformal field theory BCFT to a target boundary conformal1eld theory BCFT ∗ [6], produces a formal solution in BCFT describing a configurationof branes corresponding to BCFT ∗ . The construction is fairly general, and includes for-mal solutions for a single lump as a special case. BMT gave a simple example of such asolution, and proved that it has the correct coupling to closed string states[2].In this paper we show that the Ellwood and BMT solutions, as currently understood ,do not satisfy the equations of motion. Carefully evaluating the equations of motionproduces an anomalous, nonzero state, proportional to a projector of the star algebra.The anomaly can be interpreted as saying that the equations of motion are solved onlywith respect to the states of the IR boundary conformal field theory. Surprisingly, wefind that, nevertheless, the action evaluated on the BMT and Ellwood solutions exactlyaccounts for the total shift in the energy between the UV and IR boundary conformalfield theories. This is independent of the particular relevant boundary interaction anddepends only on universal properties of the RG flows we consider.This paper is organized as follows. In section 2 we review the algebraic setup andnecessary assumptions about the relevant boundary interaction. In section 3 we intro-duce the Ellwood and BMT solutions, and explain in general terms why they are singularand why they are not expected to satisfy the equations of motion. In section 4 we studythe BMT solution in a regularization which expresses it as a sum of a tachyon vacuumsolution plus a “phantom term” which builds the IR boundary conformal field theoryon top of the tachyon vacuum. We show that the BMT solution does not satisfy theequations of motion, either when contracted with the solution or with Fock space states,but reproduces the correct difference in energy between the perturbative vacuum and theboundary conformal field theory in the infrared. In section 5 we extend these results tothe (more general) Ellwood solutions, where a few new complications arise. In section 6we investigate whether there is a sense in which the BMT solution supports the correctcohomology of open string states. We argue that the equations of motion are satisfied withrespect to a class of projector-like states which can be put into one-to-one correspondencewith the states of the IR boundary conformal field theory. Within this subclass of states,the kinetic operator around the BMT solution is nilpotent and its cohomology preciselycorresponds to on-shell states in the infrared. We end with some concluding remarks. As we will see, the Ellwood and BMT solutions are singular, and it is possible that a nontrivialregularization exists which solves the equations of motion. We will refer to the Ellwood and BMTsolutions as “solutions,” though, as currently understood, they do not satisfy the equations of motion. ote Added : While this paper was in preparation we were notified of the work of [26]which contains some overlap with our results. The papers should appear concurrently. In this section we review the basic ingredients needed to understand the Ellwood andBMT solutions. We use the same setup as [2], but add a few clarifications.
The construction begins with a relevant matter boundary operator, φ ( s ), which triggers anRG flow from a reference boundary conformal field theory, BCFT , to a target boundaryconformal field theory, BCFT ∗ . For the string field theory manipulations we need toperform, we need to assume that φ ( s ) satisfies three properties: The T - φ OPE is no more singular than a double pole.
This means T ( s ) φ (0) ∼ s ( φ (0) − φ ′ (0)) + 1 s ∂φ (0) , (2.1)where φ ′ ( s ) is some matter boundary operator. The operator φ ′ ( s ) quantifies thefailure of φ ( s ) to be a marginal operator, or, equivalently, the failure of φ ( s ) togenerate a conformal boundary interaction. φ generates a finite boundary interaction without renormalization. This means thatthe operator exp (cid:20) − Z ba ds φ ( s ) (cid:21) , (2.2)is finite without renormalization. We assume that (2.2) can be defined perturba-tively in powers of φ . Finiteness of (2.2) implies that the φ - φ OPE is less singularthan a simple pole: lim s → sφ ( s ) φ (0) = 0 . (2.3)We also assume that the φ - φ ′ and φ ′ - φ ′ OPEs are less singular than a simple pole. φ triggers an RG flow from the reference conformal field theory, BCFT , to a tar-get boundary conformal field theory, BCFT ∗ . For string field theory purposes,this means that a φ boundary interaction in correlation functions on a very large3ylinder[7, 8, 9] imposes BCFT ∗ boundary conditions, while on a very small cylinderit imposes BCFT boundary conditions. Explicitly,lim L →∞ (cid:28) exp (cid:20) − Z L ds φ ( s ) (cid:21) L ◦ O (cid:29) BCFT C L = lim L →∞ (cid:10) L ◦ O (cid:11) BCFT ∗ C L , and lim L → (cid:28) exp (cid:20) − Z L ds φ ( s ) (cid:21) L ◦ O (cid:29) BCFT C L = lim L → (cid:10) L ◦ O (cid:11) BCFT C L . (2.4)where h·i BCFT C L is a correlator on a cylinder of circumference L in the correspondingBCFT, and L ◦ O is a scale transformation of an arbitrary bulk operator O under z → Lz . Scaling (2.4) to a canonical cylinder of circumference 1, these conditionscan be equivalently stated:lim u →∞ (cid:28) exp (cid:20) − Z ds φ u ( s ) (cid:21) O (cid:29) BCFT C = (cid:10) O (cid:11) BCFT ∗ C , and lim u → (cid:28) exp (cid:20) − Z ds φ u ( s ) (cid:21) O (cid:29) BCFT C = (cid:10) O (cid:11) BCFT C , (2.5)where we introduce the operator φ u ( s ) = u ( u − ◦ φ ( us )) . (2.6)The parameter u (equivalently L ) can be interpreted as the RG coupling (or time).Note that equations (2.4) and (2.5) imply that the φ boundary interaction becomestrivial in the UV, and therefore represents a relevant deformation of the referenceBCFT . In general, φ ( s ) will be a sum of different matter operators. To triggera flow to BCFT ∗ as described in (2.4), the coupling constants multiplying eachcomponent matter operator must be precisely chosen. In the language of [2], fixingthese couplings corresponds to tuning the operator φ ( s ).Conditions and are mainly technical assumptions, and, in light of our results, it ispossible that a correct solution for the lump will not need them. It is possible to relax without too many further complications, but relaxing condition would require afundamentally different approach from the one we take here, perhaps something analogousto the construction of marginal solutions with singular OPEs[10, 11].4eferences [1, 2] provide two examples of φ ( s ) satisfying the above criteria. The firstis the cosine relevant deformation[12, 13] describing a codimension one brane on a circleof radius R greater than √ φ u ( s ) = u (cid:20) − u /R : cos (cid:18) X ( s ) R (cid:19) : + A ( R ) (cid:21) , (2.7)where A ( R ) is a constant determined in [2]. The restriction R > √ φ u ( s ) = u (cid:20)
12 : X ( s ) : + γ − πu ) (cid:21) . (2.8)where γ is the Euler-Mascheroni constant[1]. Unlike the cosine deformation, the Wittendeformation leads to a Gaussian worldsheet theory, and Green’s functions can be com-puted exactly[14]. A drawback, however, is that the perturbative vacuum carries infiniteenergy relative to the lump, and this divergence must be treated with care. The Ellwood and BMT solutions are constructed by taking star products of the stringfields
K, B, and c, (2.9)and φ and φ ′ , (2.10)where K, B, c are defined as in [15, 2], and φ and φ ′ correspond to insertions of φ ( s )and φ ′ ( s ) on the open string boundary of correlation functions on the cylinder. Theseobjects satisfy a number of identities summarized in table 1. Since φ ( s ) and φ ′ ( s ) mayhave singular OPEs, we should be careful to avoid contact divergences when taking openstring star products. Explicitly, φ and φ ′ are defined[16] φ = f − S ◦ φ ( ) | I i , φ ′ = f − S ◦ φ ′ ( ) | I i . (2.11)where | I i is the identity string field and f − S ( z ) = tan πz is the inverse of the sliver coordinate map. Weuse the left handed convention for the star product[15]. Qφ = ∂ ( cφ ) − φ ′ ∂c, Q ( cφ ) = φ ′ c∂c, variations QB = K, QK = 0 , Qc = c∂c. algebraic ( cφ ) = 0 ( cφ )( φ ′ c ) = 0identities Bc + cB = 1 , [ K, B ] = 0 , B = c = 0Table 1: A partial list of identities satisfied by K, B, c and φ and φ ′ . Other relations(for example [ c, φ ] = 0) follow easily from the definitions. The notation ∂ O indicates theworldsheet derivative of the corresponding boundary operator O in correlation functionson the cylinder. If O is inserted in a segment of the boundary with BCFT boundaryconditions, ∂ O = [ K, O ]. If O is inserted in a segment with a φ boundary interaction, ∂ O = [ K + φ, O ].A wedge state Ω L = e − LK is a star algebra power of the SL (2 , R ) vacuum Ω = | i ,and, inside correlation functions on the cylinder, corresponds to a strip of worldsheet ofwidth L >
0. A deformed wedge state ˜Ω L = e − L ( K + φ ) . (2.12)also corresponds to strip of worldsheet of width L > φ ( s ) boundary interaction[2, 17]. Probingwith a Fock state, h ˜Ω L , χ i = * exp " − Z L +1 / / ds φ ( s ) f S ◦ χ (0) + C L +1 . (2.13)Note that the L th star algebra power of ˜Ω is directly related to the circumference of thecylinder, and therefore to RG time.The different conformal/noncomformal boundary conditions inside (2.13) make thecorrelator difficult to compute. For the Witten deformation, we can compute (2.13) andsimilar correlators from knowledge of the Green’s function, which can be obtained asthe solution of the appropriate boundary value problem for Laplace’s equation on a unitdisk. We have derived this Green’s function, expressing it as a Fourier expansion whosecoefficients are products and inverses of certain infinite dimensional matrices constructedfrom the Fourier modes of the boundary coupling u . In practice, we must resort tonumerics to compute the required matrix inverses. With these results we can calculatethe overlap of a deformed wedge state ˜Ω L with a plane wave : e ikX :. Figure 2.1 shows that6 Figure 2.1: Fourier transform of the contraction h ˜Ω L , : e ikX : i for the Witten deformationat u = 1, plotted for L = 1 , ...,
10. The profiles are Gaussians e − x / ∆ which becomelocalized up to a minimum width ∆ ≈ .
46 for large L . In this plot we have normalizedthe height of the Gaussians to 1. Calculating the absolute normalization in our approachrequires evaluating the trace of infinite dimensional matrices which are only known numer-ically, and convergence is unfortunately too slow to obtain acceptable accuracy, especiallyfor large L . However, our numerics indicates that the normalization is finite and nonzerofor L → ∞ .the overlap has a Gaussian profile in position space, which becomes more localized, up tosome minimum uncertainty, as L becomes large. As L approaches infinity, ˜Ω L approachesa constant nonvanishing state, the deformed sliver , ˜Ω ∞ .Since deformed wedge states are surface states with a nontrivial boundary condition,one might worry that their star products in BCFT would produce divergences analogousto the collision of boundary condition changing operators. The question is whether thefollowing equation holds: lim x → ˜Ω L Ω x ˜Ω L = ˜Ω L + L . (2.14)Since the φ boundary interaction is finite without renormalization, we expect this equationto be true in general. For the Witten deformation we have checked it explicitly, and thelimit x → x ln x . Ellwood’s original proposal for the lump solution[1] is given by writing the Schnabl gaugemarginal solution[18, 19] as a formal gauge transformation of the tachyon vacuum, andthen relaxing the assumption of marginality of the matter operator inside the gaugeparameter. Extending this construction to dressed Schnabl gauges[15] yields a class of7olutions of the form Φ f,g = f (cid:18) cφ − K + φ φ ′ ∂c (cid:19)
11 + − fgK φ Bc g. (3.1)We will call these Ellwood solutions . They are conjectured to be solutions in a string fieldtheory formulated around BCFT which describe the endpoint of an RG flow triggered by φ ( s ), BCFT ∗ . The string fields f and g (subject to a few conditions ) can be any elementsof the algebra of wedge states. Ellwood’s original solution corresponds to choosing f and g to be the square root of the SL (2 , R ) vacuum. The motivation behind this proposal is along story, discussed in depth in [1]; essentially it amounts to a formal argument that thesolution should support the correct cohomology of open string states. Further evidence infavor of (3.1) was provided by BMT, who showed (in the special case f = g = 1) that ithas the correct coupling to closed string states[2]. Note that (3.1) assumes the existenceof an inverse for K + φ . This factor contains most of the physics of the solution, and,paradoxically, also the source of its difficulties.In a few special cases (3.1) reduces to known solutions. If φ is a marginal operator(with nonsingular OPE), φ ′ vanishes and (3.1) reduces to the dressed Schnabl gaugemarginal solution[23]. Choosing φ = φ ′ = 1 gives the dressed Schnabl gauge tachyonvacuum solution[9].The BMT solution[2] is a special case of (3.1) with f = g = 1:Φ = cφ − B K + φ φ ′ c∂c. (3.2)This is by far the simplest solution in the class (3.1). However, the presence of the identity-like term cφ means the BMT solution is more singular than other Ellwood solutions.However, it is not more singular in a sense which is important for the calculation ofobservables. For most of this paper (except section 5) we will study the BMT solutionas a prototypical example. The general Ellwood solution can be obtained from the BMTsolution by a transformation discovered by Zeze Z f,g [Φ] = f Φ 11 + A Φ g, (3.3) Ellwood also suggests a further modification of the gauge parameter to impose the realitycondition[20]. The resulting solution is complicated and we will not consider it. We assume that f ( K ) and g ( K ) is a continuous function of K and f (0) g (0) = 1. To avoid contactdivergences between φ s, we must also assume f ( ∞ ) g ( ∞ ) is finite. There may be other conditions; see[21, 22] for recent discussion of the algebra of wedge states. This transformation first appears in a paper of Kishimoto and Michishita[24], who attribute it to S.Zeze. A = − fgK B . The Zeze map is a gauge transformation only on-shell.The crucial ingredient in the solution is the definition of K + φ . We will follow BMTand define it (via the Schwinger parameterization) as an integral over all deformed wedgestates: 1 K + φ = Z ∞ dt ˜Ω t . (3.4)This integral converges only if the deformed sliver state ˜Ω ∞ vanishes. However, keeping inmind condition , we would not expect the deformed sliver to vanish unless the solutiondescribes the tachyon vacuum. Therefore, in interesting examples, the integral (3.4) isdivergent. We can regulate this divergence, but there is another, more serious problem:Except for tachyon vacuum solutions, the integral (3.4) does not invert K + φ ,( K + φ ) (cid:18)Z ∞ dt ˜Ω t (cid:19) = − Z ∞ dt ddt ˜Ω t = 1 − ˜Ω ∞ , (3.5)so the definition (3.4) does not accomplish its intended purpose. This means that if wesubstitute (3.4) in place of K + φ , the BMT and Ellwood solutions will not satisfy theequations of motion.We can ask whether it is possible to regulate the solution so as to restore the equationsof motion. One way to do this is to replace φ with φ + ǫ for ǫ > φ ′ with φ ′ + ǫ ), which automatically regulates the Schwinger integral by exponentially suppressingthe t = ∞ boundary of moduli space. The expressionΨ tv ( ǫ ) = c ( φ + ǫ ) − B K + φ + ǫ ( φ ′ + ǫ ) c∂c (3.6)is a solution to the equations of motion for all ǫ >
0, since φ + ǫ is simply another choiceof φ . Unfortunately, this regularization drastically alters the physical interpretation ofthe solution: For all ǫ >
0, this is a (non universal) solution for the tachyon vacuum.To see this, note that the boundary interaction of φ ( s ) + ǫ on a very large cylindervanishes (assuming φ ( s ) is already tuned), thanks to the divergent integration of ǫ alongthe boundary.Paradoxically, the unwanted sliver term in (3.5) which threatens the equations ofmotion is physically necessary. If it were possible to invert K + φ , we could trivialize thecohomology around the BMT solution with the homotopy operator[2] BK + φ . (3.7)9urthermore, the same mechanism which produces the unwanted boundary term alsoaccounts for the correct IR coupling to closed string states[2]. Therefore in this setupthere is a basic tension between the equations of motion and the desire to reproduce thephysics of a nontrivial boundary conformal field theory in the infrared. This observationis already enough to exclude the BMT and Ellwood solutions as viable descriptions of thelump. Undoubtedly, another class of solutions, perhaps even closely related, can resolvethis problem, but our goal in this paper is more narrow: We will show that, in spite of thefailure of the equations of motion, the BMT and Ellwood solutions reproduce essentiallyall of the expected physics of the lump, when properly interpreted. The significance ofthis observation is currently unclear to us, but it seems potentially important.Given that K + φ is divergent in the Fock space, one might worry that the BMT andEllwood solutions are also divergent. Actually, this is not the case, because in (3.1) K + φ always appears multiplied by B , which annihilates the linear divergence proportional tothe deformed sliver state ( B ˜Ω t vanishes as 1 /t for large t ). As we will see, this cancellationis not enough to restore the equations of motion. In fact, it is in a sense accidental. Thereare other solutions which are in principle equivalent to Ellwood and BMT but actuallydiverge in the Fock space. Consider for example the solutionˆΦ = cφ − B K + φ φ ′ c∂c − φ ′ c∂c K + φ B + 14 B K + φ φ ′ ∂c K + φ φ ′ ∂c K + φ , (3.8)which, unlike (3.1), satisfies the string field reality condition. Regulating K + φ with acutoff, one can easily show that this expression has the correct coupling to closed stringstates. Nevertheless, unless φ triggers an RG flow to the tachyon vacuum, this solutiondiverges in the Fock space since the last term has too many powers of K + φ . At present wedo not know of any solution describing a nontrivial BCFT ∗ which is both real and finite. The BMT and Ellwood solutions (3.2) and (3.1) are not well-defined string fields as theystand (except when they describe the tachyon vacuum), and to give them meaning wemust apply some regularization. Implicitly the computations of [2] regulate by imposinga hard cutoff for the Schwinger integral (3.4). However, a finite cutoff is cumbersome for10ost calculations. Instead, we will consider the BMT solution as the ǫ → ǫ ) = cφ − B K + φ + ǫ φ ′ c∂c, (4.1)where 1 K + φ + ǫ = Z ∞ dt e − ǫt ˜Ω t . (4.2)Here ǫ regulates the Schwinger integral by exponentially suppressing the t = ∞ boundaryof moduli space. Therefore (4.1) is a finite, well-defined string field for all ǫ >
0. We willdiscuss the regularization of Ellwood solutions in section 5.The nice thing about the regularization (4.1) is that Φ( ǫ ) can be neatly separated intotwo terms: the tachyon vacuum solution Ψ tv ( ǫ ) in (3.6), and a term ∆( ǫ ) which “builds”the lump on top of the tachyon vacuum:Φ( ǫ ) = Ψ tv ( ǫ ) + ∆( ǫ ) , (4.3)where ∆( ǫ ) = − ǫc + B ǫK + φ + ǫ c∂c. (4.4)In the ǫ → ǫ ) becomes a sliver-like state, but vanishes in the level expansion .Nevertheless it has a crucial effect on the calculation of observables. In this sense it is akind of “phantom term.” Note the differing roles ǫ plays in (4.3). In Φ( ǫ ) it plays therole of a regulating parameter, but in Ψ tv ( ǫ ) it is a gauge parameter labeling a class ofequivalent solutions for the tachyon vacuum.Equation (4.3) is very useful for calculating observables, since it allows us to cleanlyseparate the “trivial” contribution from the tachyon vacuum from the physically inter-esting contribution of the lump. To illustrate this point, let us present an alternativecomputation of the closed string overlap, which was already computed in [2]. Recall from[25, 2] that the closed string overlap of the BMT solution should satisfylim ǫ → Tr[ V Φ( ǫ )] = −A ( V ) + A ∗ ( V ) , (4.5)where A ( V ) is the disk amplitude in BCFT with one on-shell closed string insertion V = c ˜ c V m , A ∗ ( V ) is the same quantity in BCFT ∗ , and Tr[ V· ] is the 1-string vertex with a The factor ǫK + φ + ǫ approaches the deformed sliver as ǫ →
0. Since B annihilates the deformed sliverwhen contracted with Fock states, ∆( ǫ ) also vanishes. V . Plugging the regularized BMT solution into the right hand sideof (4.5) we find two terms:Tr[ V Φ( ǫ )] = Tr[ V Ψ tv ( ǫ )] + Tr[ V ∆( ǫ )] . (4.6)The first term is the closed string overlap of a tachyon vacuum solution. Since the disktadpole amplitude around the tachyon vacuum vanishes, the first term only contributesminus the disk amplitude around BCFT . Therefore,Tr[ V Φ( ǫ )] = −A ( V ) + Tr[ V ∆( ǫ )] . (4.7)Without any calculation we are already half-way done. The second term from ∆( ǫ )must exclusively account for the nontrivial coupling between closed strings and BCFT ∗ .Modulo ghost factors, we can already see that this is essentially guaranteed since ∆( ǫ ) isproportional to the deformed sliver state in the ǫ → ∗ . Let us see this explicitly:Tr[ V ∆( ǫ )] = ǫ Z ∞ dt e − ǫt Tr[ V ˜Ω t Bc∂c ] . (4.8)Simplifying the ghost correlator and substituting α = ǫt this becomesTr[ V ∆( ǫ )] = Z ∞ dα e − α (cid:16) − ǫα Tr[ V ˜Ω α/ǫ c ] (cid:17) . (4.10)The factor in the integrand can be written as a correlation function on the cylinder: − ǫα Tr[ V ˜Ω α/ǫ c ] = − * exp "Z α/ǫ ds φ ( s ) αǫ (cid:17) ◦ h V ( i ∞ ) c (0) i+ C α/ǫ . (4.11)In the ǫ → . Thus,lim ǫ → (cid:16) − ǫα Tr[ V ˜Ω α/ǫ c ] (cid:17) = −hV ( i ∞ ) c (0) i BCFT ∗ C . (4.12) The − ǫc term does not contribute because of the negative conformal dimension of c [15]. Note˜Ω t Bc∂c = − t ˜Ω t c − t B − ( ˜Ω t c∂c ) , (4.9)where B − is defined in [15]. Using B − invariance of the vertex, the second term does not contribute tothe trace, leading to (4.10). ∗ , as defined in the conventions of [25]. Integrating over α giveslim ǫ → Tr[ V ∆( ǫ )] = A ∗ ( V ) . (4.13)In total lim ǫ → Tr[ V Φ( ǫ )] = −A ( V ) + A ∗ ( V ) , (4.14)as expected. Unlike the closed string overlap, the energy is a nonlinear function of thestring field, and its separation into a tachyon vacuum and lump contribution is less obvi-ous. We will see how this happens in section 4.3. The regularization (4.1) does not produce a solution to the equations of motion, since,despite formal appearances, K + φ + ǫ does not define an inverse for K + φ in the ǫ → K + φ ) 1 K + φ + ǫ = 1 − ǫK + φ + ǫ . (4.15)The second term is not zero because the vanishing of ǫ is compensated by a linear diver-gence in K + φ + ǫ . To see what happens explicitly, plug in the Schwinger integral for K + φ + ǫ and make a substitution α = ǫt : ǫK + φ + ǫ = Z ∞ dα e − α ˜Ω α/ǫ . (4.16)Note that the overall factor of ǫ has disappeared. Taking ǫ →
0, the deformed wedgestate in the integrand becomes the deformed sliver, independent of α . Integration over α then produces a factor of 1 and we findlim ǫ → ( K + φ ) 1 K + φ + ǫ = 1 − ˜Ω ∞ , (4.17)exactly as we found in equation (3.5).It is important to calculate precisely how the equations of motion fail. For this purpose,write the regularized BMT solution in the formΦ( ǫ ) = cφ − A ( ǫ ) φ ′ c∂c, A ( ǫ ) = BK + φ + ǫ , (4.18)where A ( ǫ ) is the homotopy operator which trivializes the cohomology around the tachyonvacuum solution Ψ tv ( ǫ ) in (3.6): Q Ψ tv ( ǫ ) A ( ǫ ) = 1 . (4.19)13ow note from table 1 that the kinetic term of the equations of motion can be written Q Φ( ǫ ) = (1 − QA ( ǫ )) φ ′ c∂c, (4.20)while the quadratic term can be writtenΦ( ǫ ) = − [Φ( ǫ ) , A ( ǫ )] φ ′ c∂c. (4.21)The commutator of the right hand side is there for free, because the product of Φ( ǫ ) with φ ′ c∂c vanishes upon the collision of c s. Adding (4.20) and (4.21) together, we find Q Φ( ǫ ) + Φ( ǫ ) = h − Q Φ( ǫ ) A ( ǫ ) i φ ′ c∂c. (4.22)Therefore, the validity of the the equations of motion is directly related to question ofwhether the BMT solution supports open string states. If it does support open stringstates, then Q Φ( ǫ ) A ( ǫ ) cannot be 1 since otherwise A ( ǫ ) would trivialize the cohomology.But then (4.22) implies the equations of motion cannot be satisfied.Let us complete the computation of (4.22): Q Φ( ǫ ) A ( ǫ ) = Q Ψ tv ( ǫ ) A ( ǫ ) + [∆( ǫ ) , A ( ǫ )] , = 1 + [∆( ǫ ) , A ( ǫ )] . (4.23)The commutator is explicitly,[∆( ǫ ) , A ( ǫ )] = − cB ǫK + φ + ǫ − ǫK + φ + ǫ Bc + ǫK + φ + ǫ ∂c BK + φ + ǫ . (4.24)Now write ∂c = [ K + φ + ǫ, c ]. This allows us to cancel one of the inverse factors of K + φ + ǫ in the third term, and adding up what remains gives simply[∆( ǫ ) , A ( ǫ )] = − ǫK + φ + ǫ . (4.25)Therefore, Q Φ( ǫ ) A ( ǫ ) = 1 − ǫK + φ + ǫ , (4.26)and Q Φ( ǫ ) + Φ( ǫ ) = ǫK + φ + ǫ φ ′ c∂c, ≡ Γ( ǫ ) , (4.27)14 Figure 4.1: Fourier transform of the contraction h Γ(0) , c : e ikX : i assuming the Wittendeformation at u = 1. We were not able to accurately determine the normalization, butwe believe it is finite and nonzero. We are not certain whether the double peak profilehas a useful interpretation, though it is interesting to note that the equations of motionappear to be satisfied far away from the center of the lump. Perhaps not coincidentally,far away from the position of the brane we expect the solution to approximate the tachyonvacuum.where Γ( ǫ ) is the anomaly in the equations of motion.For ǫ >
0, the anomaly is (of course) nonzero, since Φ( ǫ ) is not intended to be asolution for nonzero ǫ . In the limit ǫ → ∞ φ ′ c∂c. (4.28)In [2] it was implicitly assumed that this state vanishes. The intuition behind this expec-tation is that the deformed sliver tends to drive the boundary conditions to the IR, where φ ′ vanishes by conformal invariance. Nevertheless, (4.28) is a nonvanishing state. Thereason is because φ ′ appears at the outer edge of the deformed sliver, where test states donot see the boundary conditions as conformal. More formally, since φ ′ is on the edge itcan have contractions with nearby operators which do not vanish in the sliver limit, eventhough the 1-point function of φ ′ does vanish.For the Witten deformation, we have calculated the overlap of Γ(0) with a plane wave,giving the position space profile shown in figure 4.1. However, it is possible to see thatthe anomaly is nonvanishing with a simpler calculation. For the Witten deformation,consider a test state χ = ˜Ω / (: X : ∂ c ) ˜Ω / , (4.29)where the strips on either side of the insertion are deformed. Contracting this withΓ(0) gives a correlator whose entire boundary shares the same nonconformal boundary15 .1 0.2 0.3 0.4 - - - - Figure 4.2: Anomaly in the equations of motion contracted with the state (4.29), plottedas a function of the boundary coupling u of the Witten deformation (2.8) for 0 < u < . h Γ(0) , χ i = − u h cos(2 πu )ci(2 πu ) + sin(2 πu )si(2 πu ) i , (4.30)where u is the boundary coupling of the Witten deformation and ci and si are the sineand cosine integralsci( x ) = − Z ∞ x ds cos ss si( x ) = − Z ∞ x ds sin ss . (4.31)We plot this as a function u in figure 4.2. Note that the overlap vanishes for very large u : h Γ(0) , χ i ∼ − π u (large u ) . (4.32)This suggests that there is a sense in which the equations of motion are satisfied whenformulating the solution directly at the infrared fixed point of the RG flow. We will saymore about this when discussing the cohomology. Leaving the equations of motion to the side for a moment, let us proceed to calculatethe energy of the BMT solution. For static backgrounds, we can calculate the energy byevaluating the action, and the result should be[2] E = − S = 12 π ( − g (0) + g ( ∞ )) , (4.33)where g (0) is the norm of the SL (2 , R ) vacuum in BCFT and g ( ∞ ) is the norm of the SL (2 , R ) vacuum in BCFT ∗ . Note that g (0) and g ( ∞ ) are related via RG flow of the disk16artition function g ( L ) = Tr[ ˜Ω L ] m , (4.34)where Tr[ · ] m is the trace in the matter component of BCFT . Since the equations of motion are not satisfied, there is no reason to expect to find thecorrect energy by computing the kinetic or cubic terms alone. Therefore, we will calculatethe full gauge invariant action S = Tr (cid:20) −
12 Φ( ǫ ) Q Φ( ǫ ) −
13 Φ( ǫ ) (cid:21) . (4.35)in the limit ǫ → S = Tr (cid:20)
16 Φ( ǫ ) −
12 Φ( ǫ )Γ( ǫ ) (cid:21) . (4.36)Our strategy is to use the decompositionΦ( ǫ ) = Ψ tv ( ǫ ) + ∆( ǫ ) (4.37)to separate the action into a contribution from the tachyon vacuum and a contributionfrom the lump. We start by substituting this decomposition into the Φ( ǫ ) term: S = Tr (cid:20)
16 (Ψ tv ( ǫ ) + ∆( ǫ )) −
12 Φ( ǫ )Γ( ǫ ) (cid:21) , = Tr (cid:20)
16 Ψ tv ( ǫ ) + 12 ∆( ǫ )Ψ tv ( ǫ ) + 12 ∆( ǫ ) Ψ tv ( ǫ ) + 16 ∆( ǫ ) −
12 Φ( ǫ )Γ( ǫ ) (cid:21) . (4.38)Now write the last four terms back in terms of Φ( ǫ ), keeping the Ψ tv ( ǫ ) term: S = Tr (cid:20)
16 Ψ tv ( ǫ ) (cid:21) + Tr (cid:20)
12 ∆( ǫ )Φ( ǫ ) −
12 ∆( ǫ ) Φ( ǫ ) + 16 ∆( ǫ ) −
12 Φ( ǫ )Γ( ǫ ) (cid:21) . (4.39)The first term is the gauge invariant action evaluated on a solution for the tachyon vacuum.Without any calculation, we know what this quantity is: it is the vacuum energy of thereference BCFT : 16 Tr[Ψ tv ( ǫ ) ] = 12 π g (0) . (4.40)We will take this equation as given. We do not have a general proof except to notethat Ψ tv ( ǫ ) is related to other solutions for the tachyon vacuum, whose energy has been Implicitly, we assume that matter correlators include a trivial ghost factor to ensure vanishing centralcharge, and likewise ghost correlators contain a trivial matter factor. For the Witten deformation we are ableto verify (4.40) numerically, as discussed in appendix B.Assuming (4.40), all we have to do is show that the four remaining terms in (4.39)conspire to give the energy of the lump. To simplify, we use the identity∆( ǫ )Φ( ǫ ) = Γ( ǫ ) . (4.41)This equality is easy to verify using the relations of table 1, though we don’t have aninterpretation of why it holds. There does not seem to be an analogous relation forEllwood solutions. At any rate, plugging (4.41) into (4.39), the term with the anomalycontracted with the solution cancels, and we are left with S = 12 π g (0) + Tr (cid:20) −
12 ∆( ǫ )Γ( ǫ ) + 16 ∆( ǫ ) (cid:21) . (4.42)On the right hand side, both terms in the trace are sliver-like, and so they drive theboundary conditions to the IR. The term with the anomaly however has an insertionof φ ′ , which tends to kill the correlator as the boundary conditions become conformal.Therefore we can anticipate that only the ∆( ǫ ) term contributes to the energy of thelump. This is exactly as we would expect if ∆( ǫ ) were a genuine solution of the equationsof motion for the lump expanded around the tachyon vacuum. However, as we know fromprevious discussion, it is not.We now compute the right hand side of (4.42) explicitly. Start with the termTr[∆( ǫ )Γ( ǫ )] = Tr (cid:20) ǫK + φ + ǫ Bc∂c ǫK + φ + ǫ φ ′ c∂c (cid:21) . (4.43)Expanding out the Schwinger integrals, making a change of variables, and separating thetrace into matter and ghost components gives the expressionTr[∆( ǫ )Γ( ǫ )] = 1 ǫ Z ∞ dα α e − α Tr[ ˜Ω α/ǫ φ ′ ] m Z dq Tr[Ω − q Bc∂c Ω q c∂c ] gh . (4.44) For example, the “simple” tachyon vacuum of [15] can be expressed as a limit lim ǫ →∞ ǫ L − Ψ tv ( ǫ ),which is a combination of a reparameterization and a shift in the gauge parameter ǫ . Note that the gaugetransformation needed to relate these two solutions is not unique. Note a curious thing: The anomaly Γ(0) is a nonzero state, even though ∆( ǫ ) vanishes in the levelexpansion when ǫ →
0. This suggests that the BMT solution is in a sense “infinite,” even though it isfinite in the level expansion. ǫ )Γ( ǫ )] = − π ǫ Z ∞ dα α e − α Tr[ ˜Ω α/ǫ φ ′ ] m . (4.45)To get rid of the φ ′ use the relation Tr[ ˜Ω L φ ′ ] m = − ddL g ( L ) . (4.47)Note that this implies that the 1-point function of φ ′ on a very large cylinder vanishes,since the disk partition function approaches a constant in the infrared (the IR norm ofthe SL (2 , R ) vacuum). This is the sense in which the anomaly was naively expected tovanish. Plugging (4.47) into (4.46) we findTr[∆( ǫ )Γ( ǫ )] = − π Z ∞ dα αe − α (cid:20) L ddL g ( L ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) L = α/ǫ . (4.48)Since g ( L ) approaches a constant its derivative must vanish faster than 1 /L . Thereforelim ǫ → Tr[∆( ǫ )Γ( ǫ )] = 0 . (4.49)From the perspective of the gauge invariant action, the anomaly in the equations of motionvanishes.Finally, compute the term16 Tr[∆( ǫ ) ] = 16 Tr "(cid:18) ǫK + φ + ǫ Bc∂c (cid:19) . (4.50)Expanding the Schwinger integrals, and making a change of variables, and separating intomatter/ghost components,16 Tr[∆( ǫ ) ] = 16 Z ∞ dα α e − α g (cid:0) αǫ (cid:1) Z dq Z q dr Tr[Ω − q Bc∂c Ω q − r Bc∂c Ω r Bc∂c ] gh . (4.51) This can be shown using invariance of the vertex under reparameterizations by L − [15]. Noting that L − is a derivation and that L − φ = φ − φ ′ and L − K = K we can show L − ˜Ω L = Z L ds ˜Ω L − s φ ′ ˜Ω s + L ddL ˜Ω L (4.46)Taking the trace of both sides, L − kills the vertex and we are left with (4.47). Z dq Z q dr Tr[Ω − q Bc∂c Ω q − r Bc∂c Ω r Bc∂c ] gh = − π (4.52)gives 16 Tr[∆( ǫ ) ] = − π Z ∞ dα α e − α g (cid:0) αǫ (cid:1) . (4.53)In the ǫ → SL (2 , R ) vacuumin the infrared, and the integration over α gives a factor of 2!. Thuslim ǫ →
16 Tr[∆( ǫ ) ] = − π g ( ∞ ) . (4.54)and the total energy is E = − S = 12 π ( − g (0) + g ( ∞ )) , (4.55)as expected.We may ask how this result depends on our choice of regularization. It depends tosome degree, since the tachyon vacuum solution (3.6) can be viewed as a regularizationof the BMT solution. That being said, we believe that the lump energy works for alarge class of regularizations of the BMT solution. For example, with an assumption ,the energy works for any regularization which represents K + φ as a limit of states in thedeformed wedge algebra while leaving the rest of the solution unchanged. This includes,for example, regulating the solution with a hard cutoff for the upper limit of the Schwingerintegral (3.4). Nevertheless, the regularization we have used produces highly nontrivialsimplifications which may have a deeper explanation. It is interesting to ask whether we would have obtained the correct energy calculatingonly the cubic or kinetic terms of the action. The answer to this question depends onwhether the anomaly contracted with the solution,lim ǫ → Tr[Φ( ǫ )Γ( ǫ )] , (4.56)vanishes. We see no reason why this quantity should vanish in general, but its explicitcomputation depends on the choice of relevant deformation and, since it is not gauge The needed assumption is that Tr[Φ ] is an absolutely convergent integral over correlation functionson the cylinder. This is true for the Witten deformation. ǫ )Γ( ǫ )] = − Tr (cid:20) BK + φ + ǫ φ ′ c∂c ǫK + φ + ǫ φ ′ c∂c (cid:21) . (4.57)Now on the right hand side we expand the Schwinger integrals and separate them intoan integration over the total width of the cylinder and an integration over the relativepositions of the φ ′ c∂c insertions within a cylinder of fixed width. In the ǫ → φ ′ c∂c insertions:lim ǫ → Tr[Φ( ǫ )Γ( ǫ )] = − lim L →∞ (cid:18) L Z dq Tr[ B ˜Ω Lq φ ′ c∂c ˜Ω L (1 − q ) φ ′ c∂c ] (cid:19) . (4.58)The integration variable q is the ratio between the circumference of the cylinder and theseparation between the φ ′ c∂c insertions. To go further we need to evaluate the mattercorrelator, which requires us to specialize to the Witten deformation. The 2-point functionof φ ′ in the presence of the Witten boundary interaction can be easily computed from(A.14) in appendix A. Dropping the contribution from the square of the 1-point functionof φ ′ , which vanishes as 1 /L , and computing the ghost correlator leaves the integral lim ǫ → Tr[Φ( ǫ )Γ( ǫ )] = − g ( ∞ )2 π lim L →∞ (cid:18) L Z dq h π (1 − q ) sin 2 πq + cos 2 πq − i G ( Lq, L ) (cid:19) , (4.59)where G ( s, L ) is the boundary Green’s function for the Witten deformation of two X insertions separated by a distance s on a cylinder of circumference L : D e − R L dt φ u ( t ) X ( s ) X (0) E BCFT C L = g ( L ) G ( s, L ) . (4.60)Its explicit form is given in (A.10). For generic q the Green’s function in the integrandvanishes for large L , because the boundary conditions on a very large cylinder are effec-tively Dirichlet which forces X = 0. The integral therefore can only receive contributionfrom singular contractions between two X s in the vicinity of q = 0 and q = 1. Further-more, at q = 1 the singular contraction between the X s is suppressed by a vanishingcontraction between the pair of c∂c s. Therefore, in the large L limit the integral only With the appropriate substitution we set the coupling parameter u of the Witten deformation tounity. q = 0. To clearly extract this contribution, wemake a substitution of variables x = 2 πLq :lim ǫ → Tr[Φ( ǫ )Γ( ǫ )] = − g ( ∞ )4 π lim L →∞ (cid:18) L Z πL dx π h π (cid:16) − x πL (cid:17) sin xL + cos xL − i G (cid:16) x π , L (cid:17) (cid:19) . (4.61)Making a series expansion of the trigonometric factor in the integrand, the integral furthersimplifies lim ǫ → Tr[Φ( ǫ )Γ( ǫ )] = − g ( ∞ )4 π Z ∞ dx x lim L →∞ G (cid:16) x π , L (cid:17) . (4.62)Thus we need to know the 2-point function of X s with fixed separation on a very largecylinder. This limit is given in (A.18). Thus the equations of motion contracted with thesolution is lim ǫ → Tr[Φ( ǫ )Γ( ǫ )] = − g ( ∞ ) π w, (4.63)where w is the value of the integral w = Z ∞ dx x (cos( x ) ci( x ) + sin( x )si( x )) = 0 . . (4.64)This means that if we had evaluated the cubic term in the action expecting to find thecorrect energy, instead we would have found − lim ǫ →
16 Tr[Φ( ǫ ) ] = E −
12 lim ǫ → Tr[Φ( ǫ )Γ( ǫ )]= 12 π h − g (0) + (1 + w ) g ( ∞ ) i . (4.65)which is incorrect. So far we have focused our analysis on the BMT solution. In a sense, the BMT solutionis a degenerate example of Ellwood’s more general construction, and it is important tounderstand how much our analysis depends on accidental simplifications or singularitiesof this particular example.Therefore in this section we study Ellwood solutions. The first question which arisesis how the Ellwood solutions should be regularized. One obvious approach is to define22he regulated Ellwood solution as a gauge transformation of the regulated BMT solution(4.1), by some off-shell extension of the Zeze map (3.3). However, this approach leads toregularizations which seem artificial, and it does not give any independent confirmation ofthe physics behind the BMT solution. Furthermore, one of the most important propertiesof Ellwood solutions is that they are (in general) much less singular than the BMT solutionfrom the perspective of the identity string field. However, any gauge transformation of theBMT solution produces a state which is essentially as identity-like as the BMT solution.Therefore we search for a way to regulate Ellwood solutions directly. Compared toBMT, where the physics of the solution appears to be reasonably independent of thechoice of regularization, the regularization of Ellwood solutions is delicate. Many natu-ral proposals turn out to be unphysical. We study one particular regularization whichworks: the ǫ → f,g ( ǫ ) = f (cid:20) cφ − K + φ + ǫ φ ′ ∂c (cid:21) B − fgK ( φ + ǫ ) cg. (5.1)Under reasonable assumptions about f and g , this is a well-defined string field for all ǫ >
0. Like (4.1), this expression can be separated into a tachyon vacuum solution Ψ tv f,g ( ǫ )plus a term ∆ f,g ( ǫ ) which “builds” the lump on top of the tachyon vacuum:Φ f,g ( ǫ ) = Ψ tv f,g ( ǫ ) + ∆ f,g ( ǫ ) , (5.2)where Ψ tv f,g ( ǫ ) = f (cid:20) c ( φ + ǫ ) − K + φ + ǫ ( φ ′ + ǫ ) ∂c (cid:21) R ( ǫ ) Bcg, ∆ f,g ( ǫ ) = f (cid:20) − ǫc + ǫK + φ + ǫ ∂c (cid:21) R ( ǫ ) Bcg, (5.3)and for short we have defined the factor R ( ǫ ) ≡
11 + − fgK ( φ + ǫ ) . (5.4)Note the extra ǫ in the denominator of the rightmost factor of (5.1). This ǫ plays no rolein regulating the Schwinger integral, and if it was not there (5.1) would still be a well-defined string field for ǫ >
0. Nevertheless this ǫ turns out to be necessary to recover the This regularization can be obtained as a particular extension of the Zeze map: Φ f,g ( ǫ ) = f Φ( ǫ ) A Ψ tv ( ǫ ) g . Other naively equivalent possibilities, such as f Φ( ǫ ) A Φ( ǫ ) g and f tv ( ǫ ) A Φ( ǫ ) g ,appear not to work. K + φ as the BMT solution,so we would not expect it to satisfy the equations of motion. Computing the equationsof motion from (5.1) we find Q Φ f,g ( ǫ ) + Φ f,g ( ǫ ) = f h Γ( ǫ ) − Φ( ǫ )∆( ǫ ) i R ( ǫ ) Bcg + h f Φ( ǫ ) R ( ǫ ) Bcg ih f ∆( ǫ ) R ( ǫ ) Bcg i , (5.5)where Φ( ǫ ) , ∆( ǫ ), and Γ( ǫ ) are defined in section 4. The anomaly is a complicated expres-sion, and we will not attempt to calculate its overlap with test states. We do not expectit to vanish in the ǫ → We now show that the regularization (5.1) correctly captures the physics of the lump, justlike the BMT solution.We start by computing the closed string overlap:Tr[ V Φ f,g ( ǫ )] = Tr[ V Φ( ǫ ) R ( ǫ ) Bc ω ]= Tr[ V cφR ( ǫ ) Bc ω ] − Tr (cid:20) V K + φ + ǫ φ ′ ∂cR ( ǫ ) Bc ω (cid:21) , (5.6)where ω ≡ f g for short. Now in the second term insert a trivial factor of Bc next to the ∂c . This allows us to eliminate the Bc between R ( ǫ ) and ω to findTr[ V Φ f,g ( ǫ )] = Tr[ V cφR ( ǫ ) Bc ω ] − Tr (cid:20) V Bφ ′ c∂c (cid:18) R ( ǫ ) ω K + φ + ǫ (cid:19)(cid:21) . (5.7)The factor in parentheses on the right can be simplified in a useful way. To see how, wewrite ω in a particular form: ω = − − ωK K + 1 , = − − ωK ( K + φ + ǫ ) + 1 + 1 − ωK ( φ + ǫ ) , = R ( ǫ ) − − − ωK ( K + φ + ǫ ) . (5.8)Therefore R ( ǫ ) ω K + φ + ǫ = 1 K + φ + ǫ − R ( ǫ ) 1 − ωK . (5.9)24lugging this into the parentheses of (5.6) the K + φ + ǫ term above combines with theinsertions to give the overlap of the regularized BMT solution. Keeping track of the otherterms givesTr[ V Φ f,g ( ǫ )] = Tr[ V Φ( ǫ )] + Tr h V cφR ( ǫ ) Bc ω i + Tr (cid:20) V Bφ ′ c∂cR ( ǫ ) 1 − ωK (cid:21) . (5.10)The overlap of the BMT solution was already computed in section 4.1 and [2], and weknow it gives the correct shift in the closed string tadpole. All we have to do is show thatthe last two terms cancel in the ǫ → V Φ f,g ( ǫ )] − Tr[ V Φ( ǫ )] = Tr (cid:20) V R ( ǫ ) Bc (cid:18) ωcφ + B − ωK φ ′ c∂c (cid:19)(cid:21) , = − Tr (cid:20) V R ( ǫ ) Bc Q (cid:18) B − ωK cφ (cid:19)(cid:21) . (5.11)Replace the R ( ǫ ) Bc factor in the trace by an equivalent expression dressed up with re-dundant factors of Bc :Tr[ V Φ f,g ( ǫ )] − Tr[ V Φ( ǫ )] = − Tr (cid:20) V
11 + B − ωK c ( φ + ǫ ) Q (cid:18) B − ωK cφ (cid:19)(cid:21) . (5.12)The right hand side is almost the overlap of a pure gauge solution. We just have to fixup the string field inside the BRST variation:Tr[ V Φ f,g ( ǫ )] − Tr[ V Φ( ǫ )] = − Tr (cid:20) V
11 + B − ωK c ( φ + ǫ ) Q (cid:18) B − ωK c ( φ + ǫ ) (cid:19)(cid:21) , + ǫ Tr (cid:20) V R ( ǫ ) Bc Q (cid:18) B − ωK c (cid:19)(cid:21) . (5.13)The first term is the closed string overlap of a pure gauge solution. Assuming R ( ǫ ) admitsa convergent geometric series expansion in powers of − ωK ( φ + ǫ ), this term can be shownto vanish order by order. Therefore only the second term contributes for finite ǫ , givingTr[ V Φ f,g ( ǫ )] = Tr[ V Φ( ǫ )] + ǫ Tr (cid:20) V R ( ǫ ) Bc Q (cid:18) B − ωK c (cid:19)(cid:21) . (5.14)The second term vanishes since the string field in the trace is regular in the ǫ → ǫ > ǫ → ǫ in the rightmost factorof (5.1). In this case the regularized Ellwood solution would beΦ f,g ( ǫ ) = f Φ( ǫ ) R (0) Bcg, (5.15)where R (0) is (5.4) at ǫ = 0. The calculation of the overlap proceeds analogously, but, toextract the overlap of the regularized BMT solution, instead of (5.9) we need the relation R (0) ω K + φ + ǫ = 1 K + φ + ǫ − R (0) 1 − ωK − R (0) 1 − ωK ǫK + φ + ǫ . (5.16)The main difference between this and (5.9) is the third term. While the third termis proportional to ǫ , it is nonvanishing in the ǫ → V Φ f,g ( ǫ )] = Tr[ V Φ( ǫ )] + Tr (cid:20) V Bφ ′ c∂cR (0) 1 − ωK ǫK + φ + ǫ (cid:21) . (5.17)In the ǫ → (cid:20) V Bφ ′ c∂cR (0) 1 − ωK ˜Ω ∞ (cid:21) . (5.18)This does not appear to vanish for generic choice of f and g . Therefore the fact that theclosed string overlap works in (5.14) is not a consequence of a formal gauge equivalence,but is an independent confirmation of the physics behind the construction.We can calculate the energy of the Ellwood solutions (5.1) by analogy with the BMTsolution. The idea is to extract the negative energy from the tachyon vacuum, and toreduce the remaining terms to their BMT counterparts by repeated use of the identity(5.9). The calculation requires keeping track of many terms, and is too lengthy and mostlyroutine to be worth presenting here. Some aspects however deserve mention. The first isthat for the Ellwood solutions (with this regularization) there is no simple relation betweenthe anomaly, solution, and phantom term analogous to (4.41). This relation was crucialfor simplifying the action from (4.39) to (4.42). For Ellwood solutions this simplification26oes not happen automatically, and the terms which would otherwise simplify have tobe expanded and shown to cancel in a nontrivial fashion. The second point is that thecalculation produces many spurious terms which do not cancel identically for ǫ >
0. Mostof these terms are impractical to explicitly compute for ǫ >
0, and they must be arguedto vanish for general reasons in the ǫ → ǫ Tr[ X ( ǫ )] , ǫ Tr (cid:20) X ( ǫ ) ǫK + φ + ǫ (cid:21) , Tr (cid:20) X ( ǫ ) B ǫK + φ + ǫ (cid:21) , (5.19)where X ( ǫ ) is a finite and not sliver-like string field, generally some combination of R ( ǫ ) , ω, − ωK , φ, φ ′ , and ghosts. The first two classes of terms vanish because an overallfactor ǫ multiplies a trace we believe is finite in the ǫ → ǫ → B annihilates the sliver in the Fock space. With this understanding, thecalculation of the energy is straightforward and reproduces the expected answer (4.55). It is interesting to ask whether the BMT solution supports the expected cohomology ofopen string states. Of course, taken literally this question has no meaningful answer, sincethe shifted kinetic operator is not nilpotent: Q ǫ ) = [Γ( ǫ ) , · ] (6.1)So the existence of cohomology is closely related to the equations of motion. In this sectionwe argue that the BMT solution satisfies the equations of motion when contracted withstates of the IR boundary conformal field theory, in a sense described below. Then, theBMT kinetic operator is nilpotent in BCFT ∗ , and defines a cohomology.As a first step, let us explain what it means to contract the equations of motion, whichis a state in BCFT , with states in BCFT ∗ . SupposeΠ ∗ i = Ω / π ∗ i Ω / (6.2)are a basis of Fock states of BCFT ∗ , where π ∗ i are vertex operators and (in this equation)Ω is the SL (2 , R ) vacuum of BCFT ∗ . This basis can be equivalently characterized inBCFT as a singular, projector-like limit of states of the formΠ i ( ǫ ) = ˜Ω / ǫ π i ( ǫ ) ˜Ω / ǫ , (6.3)27hen ǫ →
0. The equivalence between Π ∗ i and Π i ( ǫ ) can be explained as follows: Incorrelation functions on the cylinder, Π i ( ǫ ) represents a strip of worldsheet of width 1 /ǫ with deformed boundary conditions and an operator π i ( ǫ ) inserted in the middle. Witha reparameterization, we can squeeze the strip to width 1, whereupon the boundaryconditions flow to BCFT ∗ , and the operator π i ( ǫ ), if appropriately chosen , flows to π ∗ i .In particular, this means that N -string vertices of Π i ( ǫ ), when ǫ →
0, are equal to thecorresponding N -string vertices of Π ∗ i , and so for string field theory purposes the statesare indistinguishable.With this understanding, the BMT solution satisfies the equations of motion in BCFT ∗ in the following sense:lim ǫ → D Π i ( ǫ ) , Q Φ( ǫ ) + Φ( ǫ ) E = lim ǫ → D Π i ( ǫ ) , Γ( ǫ ) E = 0 . (6.4)To see why (6.4) holds, note that, because of the very large width of the test stateΠ i ( ǫ ), contractions between φ ′ and the vertex operator π i ( ǫ ) are suppressed by clusterdecomposition. Therefore (6.4) should be proportional to the one point function of φ ′ ona very large (deformed) cylinder. This vanishes faster than ǫ because the disk partitionfunction is constant in the infrared, by (4.47). The ghost correlator diverges as ǫ , butthis is not enough to cancel the vanishing matter correlator.Let us clarify a possibly confusing point: (6.4) does not imply that the BMT solutionis a well defined state in BCFT ∗ satisfying the equations of motion. In fact it is not:the overlap of Φ( ǫ ) with Π i ( ǫ ) diverges in the ǫ → is a well-defined state in BCFT ∗ , and it is precisely zero. This is allwe need for the cohomology.Acting within BCFT ∗ , the BMT kinetic operator takes the form D Π i ( ǫ ) , Q Φ( ǫ ) Π j ( ǫ ) E = D Π i ( ǫ ) , Q Π j ( ǫ ) + [Φ( ǫ ) , Π j ( ǫ )] E . (6.5)Since the BMT solution is not well-defined in BCFT ∗ , it is not obvious that the BMTkinetic operator should be meaningful either. To see what happens, concentrate first on We will not attempt here to construct the full basis of states Π i ( ǫ ) explicitly for a particular relevantdeformation, though we have studied a few examples. However, a few points are worth mentioning.First, the operators π i ( ǫ ) are not fixed uniquely. For example, for the Witten deformation, both ǫc and − ǫu cφ flow to the zero momentum tachyon c in the infrared. Second, many operators, such as theenergy momentum tensor, experience divergent contractions with the boundary interaction and must beappropriately renormalized. Lastly, the states Π i ( ǫ ) in general diverge in the Fock space of BCFT inthe ǫ → Q . When Q acts on a deformed wedge state, it can benaturally separated into two pieces: Q = − [ cφ, · ] + ˜ Q. (6.6)If φ were a marginal operator, the first part would be the BRST variation of the boundarycondition changing operator, and the second part, ˜ Q , would be the BRST operator of themarginally deformed boundary conformal field theory. Since φ is not marginal, ˜ Q is nota BRST charge, and it is not nilpotent:˜ Q = [ φ ′ c∂c, · ] . (6.7)However, the operator φ ′ c∂c vanishes in BCFT ∗ for the same reason that the anomalyvanishes. Therefore, in the ǫ → Q is nilpotent and can be naturally identified withthe BRST operator of the infrared boundary conformal field theory:lim ǫ → h Π i ( ǫ ) , ˜ Q Π j ( ǫ ) i = h Π ∗ i , Q Π ∗ j i . (6.8)The − [ cφ, · ] term in the BRST operator diverges as ǫ →
0, but thankfully it cancels againstthe corresponding divergence from the BMT solution in (6.5). The remaining piece of theBMT solution, − K + φ + ǫ Bφ ′ c∂c , does not contribute in the ǫ → φ ′ kills the matter correlator. Adding everything up gives the simple result:lim ǫ → D Π i ( ǫ ) , Q Φ( ǫ ) Π j ( ǫ ) E = h Π ∗ i , Q Π ∗ j i . (6.9)Therefore the BMT kinetic operator in the infrared is the same as the BRST operator ofBCFT ∗ , and they share the same cohomology.It is interesting to see how the cohomology disappears for the tachyon vacuum regu-larization of the BMT solution, Ψ tv ( ǫ ) in (3.6). Repeating the above steps, in the tachyonvacuum kinetic operator in the IR takes the formlim ǫ → D Π i ( ǫ ) , Q Ψ tv ( ǫ ) Π j ( ǫ ) E = h Π ∗ i , Q Π ∗ j i − lim ǫ → D Π i ( ǫ ) , [∆( ǫ ) , Π j ( ǫ )] E , (6.10)Performing a scale transformation of the second term, we can replace Π i ( ǫ ) with Π ∗ i and∆( ǫ ) with − c + K +1 Bc∂c , where “ K ” is now the K of the IR boundary conformal fieldtheory. Thereforelim ǫ → D Π i ( ǫ ) , Q Ψ tv ( ǫ ) Π j ( ǫ ) E = (cid:28) Π ∗ i , Q Π ∗ j + (cid:20) K + 1 ( c + Q ( Bc )) , Π ∗ j (cid:21)(cid:29) . (6.11)29he right hand side is precisely the kinetic operator of the “simple” tachyon vacuumsolution described in [15]. Note that becauselim ǫ → D Π i ( ǫ ) , ∆( ǫ ) E = − (cid:28) Π ∗ i , K + 1 ( c + Q ( Bc )) (cid:29) , (6.12)∆( ǫ ) is a well defined state in BCFT ∗ , and is precisely the perturbative vacuum of BCFT ∗ as seen from the tachyon vacuum. This is consistent with the interpretation of ∆( ǫ ) as“building” the lump on top of the tachyon vacuum. In this paper we studied a class of formal solutions which were conjectured to describelower dimensional branes as tachyon lumps in open string field theory. We found thatthe solutions do not satisfy the equations of motion. Nevertheless, they have the correctcoupling to closed string states, and evaluating the action gives the expected energy.The current situation is puzzling since the correct solution remains to be found, yetclearly this construction is capturing the physics of the desired solution in a nontrivialfashion. The question now is how to proceed. We offer a few possibilities: • It is possible that while the specific solutions (3.1) are problematic, other solutionswithin the subset of states generated by multiplying
K, B, c, φ , and φ ′ could describea tachyon lump. It is difficult to analyze the full set of candidate solutions ingenerality, but we have found that the difficulties with Ellwood’s proposal are fairlygeneric. Perhaps a novel mechanism selects a particular subclass of solutions forwhich the equations of motion can be made non-anomalous. • It is possible that the Ellwood and BMT solutions satisfy the equations of mo-tion when correctly defined, but we have not identified the necessary definition ofexpressions such as K + φ when they appear inside the solution. It is worth mention-ing that analogous problems with defining K appear when studying of multibranesolutions[27], and new developments on this front may also have implications forlump solutions. • Finally, it is possible that the current setup is for some reason inadequate to capturenonsingular lump solutions. Perhaps a different approach, for example based onboundary condition changing operators, as suggested in [17], is needed.30e hope that the current work will stimulate further thought on this important problem.
Acknowledgments
We would like to thank the organizers of the conference SFT 2010 in Kyoto where thiscollaboration began, and Micheal Kiermaier , Yuji Okawa, and Martin Schnabl for usefulconversations. We would like to thank L. Bonora for making his numerical computationsavailable to us. We thank Ian Ellwood for comments on the second version of the paper.This research was supported by the EURYI grant GACR EYI/07/E010 from EUROHORCand ESF.
A Witten Deformation
In this appendix we give some formulas which allow for explicit computation of correlationfunctions on the cylinder in the presence of the Witten boundary interaction. Most of theformulas follow immediately from [14] with the appropriate transcription.The Witten deformation is generated by inserting the operatorexp (cid:20) − Z L ds φ u ( s ) (cid:21) (A.1)into correlation functions on the cylinder in a reference BCFT which includes a noncom-pact free boson X ( z, ¯ z ) subject to Neumann boundary conditions, where φ u ( s ) = u (cid:20)
12 : X ( s ) : + γ − πu ) (cid:21) (A.2)and φ ′ u ( s ) = u ddu φ u ( s ) = φ u ( s ) + u. (A.3)Here u is a parameter which we are free to choose. Different u s are related by the scaletransformation (2.6). For short, let’s write h ... i uC L = (cid:28) exp (cid:20) − Z L ds φ u ( s ) (cid:21) ... (cid:29) BCFT C L . (A.4)The X part of the worldsheet theory is Gaussian and therefore completely defined by thezero point and bulk 2-point functions h i uC L = g ( L ) (A.5) h X ( z , ¯ z ) X ( z , ¯ z ) i uC L = g ( L ) G ( z , ¯ z ; z , ¯ z ; L ) , (A.6)31here, leaving the u dependence implicit, g ( L ) is the disk partition function and G is thebulk Green’s function: g ( L ) = r uL π (cid:16) euL (cid:17) uL Γ( uL ) (A.7) G ( z , ¯ z ; z , ¯ z ; L ) = − (cid:0) ln | Z − Z | − ln | − Z ¯ Z | (cid:1) − uL + 2Re (cid:2) Φ( Z ¯ Z , , uL ) (cid:3) ( Z = e πiL z ) , (A.8)where Φ( z, s, a ) is the Lerch zeta functionΦ( z, s, a ) = ∞ X k =0 z k ( k + a ) s . (A.9)For most applications we are interested in computing n -point functions of φ u ( s ). Forthis purpose it is helpful to define the boundary Green’s function and the normalized1-point function: G ( s, L ) = − uL + 2Re h Φ (cid:16) e πiL s , , uL (cid:17)i , (A.10) W ( L ) = − u (cid:20) uL + ψ ( uL ) − ln( uL ) (cid:21) , (A.11)where ψ is the digamma function. With these objects can write explicit expressions forthe 0 , , h i uC L = g ( L ) , (A.12) h φ u ( s ) i uC L = g ( L ) W ( L ) , (A.13) h φ u ( s ) φ u ( s ) i uC L = g ( L ) (cid:20) W ( L ) + u G ( s , L ) (cid:21) , (A.14) h φ u ( s ) φ u ( s ) φ u ( s ) i uC L = g ( L ) (cid:20) W ( L ) + u W ( L ) (cid:0) G ( s , L ) + G ( s , L ) + G ( s , L ) (cid:1) + u G ( s , L ) G ( s , L ) G ( s , L ) i . (A.15)We can write similar expressions for higher point functions as well, but we will not needthem.For computations related to the anomaly it is useful to have asymptotic formulas forthe large L behavior of these n -point functions. For this purpose we list some large L g ( L ) = 1 + 112 1 uL + 1288 1( uL ) − uL ) + ... = exp " ∞ X n =1 B n n (2 n −
1) 1( uL ) n − , (A.16) W ( L ) = − u (cid:20) −
112 1( uL ) + 1120 1( uL ) − ... (cid:21) = − u " − ∞ X n =1 B n n uL ) n , (A.17)where B n are the Bernoulli numbers. For the boundary Green’s function we can derivean asymptotic expansion for large L and fixed separation s between the X insertions: G ( s, L ) = − h cos(2 πus )ci(2 πus ) + sin(2 πus )si(2 πus ) i + 2 ∞ X n =1 B n n ( uL ) n cos n (2 πus ) , (A.18)where si and ci are the sine and cosine integrals (4.31) and cos n is a partial sum of thecosine series, cos n ( x ) = n − X k =0 ( − k x k (2 n )! . (A.19)Another useful formula is the large L , fixed q = sL behavior of the boundary Green’sfunction G ( qL, L ) = 1( uL )
12 csc ( πq ) − uL )
14 csc ( πq ) (cid:2) πq ) (cid:3) + ... = 2 ∞ X n =0 ( − n (cid:18) csc( πq )2 uL (cid:19) n +2 2 n X k =0 (cid:28) n + 1 k (cid:29) cos (cid:2) π ( k − n ) q (cid:3) , (A.20)where (cid:10) mn (cid:11) are the Eulerian numbers. Note that, in this expansion, the boundary Green’sfunction vanishes as 1 /L for large L , as we would expect since the boundary conditionson the cylinder become Dirichlet ( X = 0) in the L → ∞ limit. However, the leading 1 /L behavior comes with a coefficient which depends on the normalized separation q betweenthe X insertions which has a double pole when the insertions become coincident. This isnot the usual logarithmic behavior we would expect from the X - X OPE, and the double We start from the asymptotic formula appearing in equation (7) of [28]. This formula correctsequation (D.24) in [2]. L but fixed s limit of G ( s, L ) (A.18) is actuallynonzero despite the fact that the boundary conditions are becoming Dirichlet as L → ∞ .This is essentially the reason why the anomaly in the equations of motion (4.27) does notvanish. B Tachyon Vacuum Energy for Witten Deformation
In this appendix we compute the action of the tachyon vacuum solution (3.6) for theWitten deformation. Our computation serves as an independent check of the effectivelyequivalent numerical computation first appearing in [29]. Thanks to improved analyticcontrol of the Green’s functions, we are able to obtain more precision.Using the results of equations (4.54) and (4.63), we can compute the action of thetachyon vacuum in terms of the cubic vertex evaluated on the BMT solution:Tr (cid:20)
16 Ψ tv ( ǫ ) (cid:21) = lim ǫ → Tr (cid:20)
16 Φ( ǫ ) (cid:21) + g ( ∞ )2 π (1 + w ) . (B.1)Focus on the computation of Tr[Φ( ǫ ) ]. Substituting the solution, expanding out theSchwinger integrals, and evaluating the matter and ghost correlators gives an expressionof the form lim ǫ → Tr (cid:20)
16 Φ( ǫ ) (cid:21) = −
16 lim ℓ → Z ∞ ℓ dL Z dq Z − q dr F ( L, q, r ) K ( q, r ) . (B.2)We have already taken the ǫ → L . However, we introduce a unrelated regularization ℓ → F and K above come from evaluating the appropriate matter/ghost correlators. The ghost factoris given by K ( q, r ) = − π sin πq sin πr sin π ( q + r ) , (B.3)and the matter factor comes from the 3-point function of φ ′ , which can be written as thesum of three terms: F ( L, q, r ) = F ( L ) + F ( L, q, r ) + F ( L, q, r ) , (B.4)34here F ( L ) = L g ( L ) (cid:16) W ( L ) + 1 (cid:17) , (B.5) F ( L, q, r ) = 12 L g ( L ) (cid:16) W ( L ) + 1 (cid:17)(cid:16) G ( Lq, L ) + G ( L ( r + q ) , L ) + G ( Lr, L ) (cid:17) , (B.6) F ( L, q, r ) = L g ( L ) (cid:16) G ( Lq, L ) G ( L ( r + q ) , L ) G ( Lr, L ) (cid:17) , (B.7)The integral (B.2) is independent of u and we are free to choose a canonical value. Weset u = 1.The matter correlator F ( L, q, r ) diverges as √ L for small L , and therefore the integral(B.2) diverges as √ ℓ in the ℓ → SL (2 , R ) vacuum in BCFT , which corresponds to the ℓ → g ( ℓ ) = 1 √ π √ ℓ + O ( √ ℓ ) . (B.8)This suggests that the integral (B.2) can be defined by subtracting the √ ℓ divergence andreplacing it with g (0)2 π . Note that because the subleading terms in (B.8) vanish as ℓ → only the √ ℓ divergence and leave the finite remainder untouched.Therefore we writelim ǫ → Tr (cid:20)
16 Φ( ǫ ) (cid:21) = g (0)2 π − Z ∞ dL Z dq Z − q dr F ∗ ( L, q, r ) K ( q, r ) . (B.9)The function F ∗ ( L, q, r ) is related to F ( L, q, r ) by adding a total derivative term: F ∗ ( L, q, r ) = F ( L, q, r ) + 158 √ π ddL (cid:18) L / e − L (cid:19) . (B.10)The coefficient in front of the total derivative has been fixed so that integration producesa boundary term at L = ℓ which precisely cancels the √ ℓ divergence from F . The finitecontribution from F is unchanged because the subleading contributions from the boundaryterm at L = ℓ vanish as ℓ →
0, and the L = ∞ boundary term vanishes due to the e − L suppression. Actually, in the following we will compute the contribution to the energyfrom the three terms F , F and F separately. Accordingly, we define subtracted functions F ∗ , F ∗ and F ∗ following the above prescription. The numerical factor in front of √ ℓ is irrelevant, up to an overall sign, since it can be absorbed intoa redefinition ℓ → constant × ℓ , and at any rate ℓ is going to zero. F integral iseasily done since only the ghost sector enters into the integration over q and r , which canbe performed analytically. The remaining numerical integral over L gives the result − Z ∞ dL Z dq Z − q dr K ( q, r ) F ∗ ( L ) = − π (0 . . (B.11)To evaluate the contribution from F we observe that because of a symmetry q → − q of the Green’s function, G ( Lq, L ), we can replace F ( L, q, r ) → L (cid:16) W ( L ) + 1 (cid:17) G ( Lq, L ) . (B.12)inside the integral. The integral over r now only involves the ghost factor K ( q, r ), andthe remaining integration over q and L can be done numerically. (To help the computerin the L ∼ L ∈ (0 , − ), so that the q -integration can beperformed exactly in this region. This introduces a small error which, as we checked, isunder control and of order 10 − ). In total − Z ∞ dL Z dq Z − q dr K ( q, r ) F ∗ ( L, q, r ) = − π (0 . . (B.13)To compute the final contribution from F we use a trick to get rid of one integral an-alytically. Note that K ( q, r ) F ( L, q, r ) can written as the product of three copies of asingle function evaluated at q , r and q + r . Inserting the appropriate step functions wecan extend the range of integration over q and r from plus to minus infinity, and insertingan auxiliary integral over s (with a delta function) allows us to write Z dq Z − q K ( q, r ) F ( L, q, r ) = − π L g ( L ) Z R dqdrds δ ( q + r − s ) h ( L, q ) h ( L, r ) h ( L, s ) , (B.14)where h ( L, q ) = sin( πq ) G ( Lq, L ) θ [0 , ( q ) , (B.15)and θ [0 , ( q ) is a unit step function with support on the interval q ∈ [0 , h in momentum space, with a delta function formomentum conservation. With a Fourier transform, the three integrals over the momentaturn into a single integral over the interaction point x : Z dq Z − q K ( q, r ) F ( L, q, r ) = − π L g ( L ) Z R dx ˜ h ( L, x ) (cid:12)(cid:12)(cid:12) ˜ h ( L, x ) (cid:12)(cid:12)(cid:12) , (B.16)36here ˜ h ( L, x ) = Z R dq h ( L, q ) e iqx , (B.17)The function ˜ h ( L, x ) can be computed analytically, although its form is not particularlyinteresting to write it down. Therefore, to compute the F contribution we only need toevaluate a numerical integral over over L and x . This gives the result − Z ∞ dL Z dq Z − q dr K ( q, r ) F ∗ ( L, q, r ) = − π ( − . . (B.18)Adding the contributions from F , F , and F together, we findlim ǫ → Tr (cid:20)
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