Taming boundary condition changing operator anomalies with the tachyon vacuum

Motivated by the appearance of associativity anomalies in the context of superstring field theory, we give a generalized solution built from boundary condition changing operators which can be associated to a generic tachyon vacuum in the KBc subalgebra of the Okawa form. We articulate sufficient conditions on the choice of tachyon vacuum to ensure that ambiguous products do not appear in the equations of motion.


Introduction
In recent years a remarkably simple solution to the equation of motion of open bosonic string field theory has been found [1], which can relate any pair of time independent open string backgrounds sharing the same closed string bulk 4 .The solution is formally defined by a choice of tachyon vacuum Ψ tv and a pair of string fields (Σ, Σ), which change the worldsheet boundary conditions from the starting background BCFT 0 , where we initially define OSFT, to the target background BCFT * we wish to describe.The solution takes the form and the equations of motion are satisfied provided Σ Σ = 1. (1. 3) The objects (Σ, Σ) can be expressed in terms of string fields (σ, σ) representing insertions of weight zero primary boundary condition changing operators in correlation functions on the cylinder [1].They multiply as ) where g is the disk partition function of the corresponding BCFT The first relation σσ = 1 is the one which is needed to realize (1.3), but the second creates potential problems with associativity and renders the triple products σσσ undefined if, as is typically the case, g 0 = g * , i.e. if the initial and final D-branes systems have a different mass.This ambiguity does not cause immediate difficulties with the solution presented in [1], but the generalization to superstring field theory is problematic.The analogous solution to the Chern-Simons-like equations of motion at picture zero contains a term 1 where the undesirable product σσ = g * g0 explicitly appears, leading to an anomaly in the equations of motion.
The origin of the difficulty is that [1] assumed that the state Ψ tv representing the tachyon vacuum is given by the "simple solution" of [6]: The factors 1+K are too similar to the identity string field, and do not provide sufficient separation between the boundary condition changing operators inside Σ and Σ to avoid the undesireable product σσ.On the other hand, if the state Ψ tv had been Schnabl's solution [7], the factors √ Ω would ensure that the boundary condition changing operators are always separated by a finite region of worldsheet, and ambiguous products cannot appear.In this note we investigate solutions characterized by a general choice of factorized tachyon vacuum in the K, B, c, γ 2 subalgebra, with the aim of getting a more precise understanding of boundary condition changing operator anomalies.
Interestingly, we find that solutions based on different choice of tachyon vacuum can be related by automorphisms reflecting reparameterization symmetries of the spectrum of K.

Solution
We consider the solution to the Chern-Simons-like equations of motion for the superstring; the solution for the bosonic string can be obtained by setting γ ghosts to zero.The solution lives in a subalgebra of states given by multiplying the string fields which represent corresponding operator insertions on the identity string field.Our conventions for K, B, c, γ 2 follow [8], to which we refer the reader for explicit definitions and algebraic relations (see also section 4).The fields σ and σ are defined as in [1], but with the additional specification (for the superstring) that they represent insertions of matter superconformal primaries of dimension 0. This implies that their BRST variations are given by where represents the worldsheet supersymmetry variation.For the superstring, Qσ and Qσ cannot be expressed using K, B, c, γ 2 , σ, σ and are independent generators of the subalgebra.
We consider a class of factorized tachyon vacuum solutions [9] 5 where F = F (K) is an element of the wedge algebra (a function of K) satisfying the conditions We introduce H = H(K) which is related to F through The simple tachyon vacuum of [6] is defined by equating This choice of tachyon vacuum was assumed in [1].Presently we are concerned with more general choices of F .
Once we have Ψ tv we can build the solution where Σ and Σ are given by One may confirm that ) so that (2.9) formally satisfies the equations of motion.For later analysis it will be useful to substitute the definitions and expand the solution explicitly.We give the solution in two forms.In the first form, we leave Qσ and Qσ as they are, without substituting (2.2) and (2.3).We then have and where "conj" denotes the conjugate of the previous term.In the second form of the solution, we allow ourselves to substitute (2.2) and (2.3): The solution becomes: These expressions substantially simplify if we choose H = F , as for the simple tachyon vacuum, but this choice is not required.

Taming Anomalies
We wish to determine sufficient conditions on the choice of tachyon vacuum, or equivalently F (K), such that the solution suffers no difficulties from collisions of boundary condition changing operators.
There can be problems if F (K) is too similar to the identity string field, so that there is not "enough worldsheet" to prevent σ from approaching σ.The degree of similarity to the identity string field can be quantified by the rate of decay of F (K) as K → ∞ [11].For definiteness we assume that it decays as a power: where ν is a real number less than zero.This implies The more quickly F (K) vanishes as K → ∞, the less "identity-like" the tachyon vacuum becomes, and the more regular the solution should appear from the point of view of collisions of boundary condition changing operators.For the simple tachyon vacuum, ν = −1, which for the superstring already poses difficulties.Therefore ν should be bounded from above by −1: The question is whether this bound is sufficient, or needs to be further strengthened.
Associativity anomalies are related to collisions of three boundary condition changing operators, but there can already be problems from the collision of two.For σ and σ themselves the OPE is regular, but there can be singularities in OPEs with ∂σ and ∂σ: suffers from no OPE divergence provided that its leading level in the dual L − level expansion [11] is less than or equal to 0 if the state is GSO even, and less than or equal to 1/2 if the state is GSO odd.
This is a technical way of saying that if we expand G(K) as an integral over wedge states any singularity which appears in the integrand towards t = 0 must be integrable.Now consider the solution expressed in terms of Qσ and Qσ as written in (2.15).The solution contains the terms Computing the BRST variations and ignoring ghosts, which in this case are unimportant, the matter sector component of these states contains the respective factors: By claim 1, these states do not suffer OPE divergence if The remaining terms in the solution only require a weaker bound on ν.This implies that the solution as expressed in (2.15) will generally suffer from OPE divergences for −2 < ν ≤ −1, which includes, in particular, the solution built from the simple tachyon vacuum.On the other hand, the solution as originally presented in [1] is finite.This is because, in the second form of the solution (2.18), OPE divergences are absent all the way up to ν = −1.Therefore in the interval −2 < ν ≤ −1 the solution as expressed in (2.18) will be valid, but generally (2.15) will not be.Now let's turn to issues which concern three boundary condition changing operators.These do not effect the solution Ψ by itself as a state, since it only contains two boundary condition changing operators.However, they concern the validity of the equations of motion since Ψ 2 contains four boundary condition changing operators.We make the following claim:6 Claim 2. Let O 1 , O 2 and O 3 represent three primary operators, and consider the state Simultanous collision of all three operators do not render this state undefined provided that its leading level in the dual L − level expansion is less than h, where h is the lowest dimension of a primary operator which has nonvanishing contraction with the state.
To understand this claim, we contract with a test state where O is a primary operator.Since the singularity which interests us concerns short distance behavior when O 1 , O 2 , O 3 collide, the precise form of the test state is not crucial; we choose (3.16) for convenience.We then have the overlap where g 1 and g 2 are the inverse Laplace transforms of G 1 and G 2 .We change the integration variables, and apply a conformal transformation to the 4-point function so that O 1 is inserted at 1, O 3 is inserted at 0, and O is inserted at infinity.This gives where I(z) = −1/z is the BPZ conformal map.We are interested in the behavior of the integrand towards L = 0, which is when O 1 , O 2 and O 3 collide.For small L we have where ν 1 , ν 2 are the leading levels of the dual L − expansion of G 1 and G 2 .For small L, the integrand of (3.19) is then approximately The integration over θ will be finite assuming that the OPE between O 2 and O 1 , and between O 2 and O 3 , is sufficiently regular; whether this is the case is equivalent to the question of whether the states separately finite, which is not our present concern.Our interest is the convergence of the integration over L towards L = 0.This will be unproblematic if This is precisely a bound on the leading level of the state As a cross check on this argument, consider the state σσσ. (3.23) For matter sector operators, generally the lowest dimension of a probe state will be h = 0. Since the leading (and only) level of σσσ is zero, claim 2 would imply that the state is ill-defined.The state is generally ambiguous due to the associativity anomaly, but the argument given below claim 2 does not immediately seem to apply, since (3.20) assumes that the operators are separated by elements of the wedge algebra whose leading L − level is negative.This can be dealt with by writing σσσ in the form Now the argument below claim 2 applies to all terms; only the first term can be problematic, since the three boundary condition changing operators form a state whose leading level is zero.Indeed, one finds a divergence from integrating 1/L towards L = 0.However, the state σσσ is not necessarily divergent, only ambiguous.In fact one can check that the integration over θ actually vanishes towards L = 0; therefore the first term effectively contains 0 × ∞, where the ambiguity of σσσ is hidden.Now let's understand the implications of this for the solution.We want to be certain that Expanding out the commutator and looking at individual terms, we find that σ and σ are separated To avoid anomalous collisions from such terms, it would appear we need ν < −2.On the other hand, since these contributions appear from expanding out a commutator, there may be cancellation of ambiguities.This can be seen more precisely as follows.Let f /h(t) represent the inverse Laplace transform of F/H, and write In particular, in (3.32) the boundary condition changing operators are separated as which is significantly more mild than (3.33).In this way, terms involving commutators of c with F/H do not require a stronger bound than ν < −1 for the superstring, and ν < 0 for the bosonic string.
Let us make an important caveat to the above discussion.In [1] the bosonic string solution was written in two forms: The second form of the solution was useful for computing coefficients in the Fock basis.However, in the second form the computation of Ψ 2 will be ambiguous due to associativity anomalies.The origin of the problem is that the first form of the solution, where Ψ 2 is well-defined, has been reexpressed as a sum of terms whose star products are individually ambiguous.The fact that this is possible does not reflect poorly on the solution; it is always possible to render a well-defined expression ambiguous by adding and subtracting singular terms.But this leaves open the possibility that something similar might happen for the superstring.We leave this question to future work.

Automorphisms
It is known that the string field K can meaningfully be understood to have a spectrum consisting of nonnegative real numbers. 7It is interesting to consider the connected component of the diffeomorphism group on the spectrum8 Diff 0 (R ≥0 ).( This automatically defines a group of automorphisms of the algebra of wedge states, defined through More surprisingly, it is possible to generalize this into an automorphism group of the KBc subalgebra [13].Several interesting applications of this symmetry have been found in [11][12][13][14][15][16][17][18].Here we show that the automorphisms can be further extended to act on boundary condition changing operators. Applying such automorphisms to the solution (1.1) turns out to be equivalent to changing the choice of tachyon vacuum.Therefore the solutions discussed in this work can be interpreted as representing different parameterizations of the spectrum of K.
The solution lives in a subalgebra given by multiplying generators This is a graded differential associative * -algebra.An automorphism of such an algebra should satisfy gh(φ where "gh" denotes ghost number and ‡ denotes reality conjugation of the string field.An important part of realizing the automorphism group is understanding what relations the generators of the algebra should satisfy.This is not completely trivial, since the relevant collection of identities is actually a proper subset of those satisfied by K, B, c, γ 2 , σ, σ, Qσ, Qσ as defined in the conventional way by operator insertions on the identity string field, as assumed in section 2. We postulate the following algebraic relations, the following differential relations, and the following properties under conjugation: The conventional understanding of the generators as operator insertions on the identity string field results in an infinite number of additional (and less important) relations, refered to as "auxiliary identities" in [11].A notable example of such a relation is This identity is absent from (4.8), and henceforth we assume that it does not hold.
Given φ ∈ Diff 0 (R ≥0 ) we have a corresponding automorphism of the above subalgebra defined by ) ) Bc. (4.17) One can check that the transformed generators (denoted with hat) satisfy all relations (4.8)-(4.10).
However, they do not satisfy auxiliary identities, such as (4.11).The automorphism is characterized by two states in the wedge algebra These states look potentially singular at K = 0. To ensure that they are well-defined, we work with the connected component of the diffeomorphism group, and further assume that the diffeomorphisms are (at least) once differentiable at K 0. This implies φ(0) = 0, φ ′ (0) > 0, (4.19) which is sufficient to ensure that the automorphism is regular at K = 0.One particularly simple type of diffeomorphism is a scale transformation of the spectrum of K φ(K) = αK, α > 0. (4.20)This leads to

.21)
In this case the automorphism is equivalent to acting with the operator α where the dependence on F is explicitly shown in the brackets.This immediately implies that the solution itself transforms as If we wish to transform from F whose leading level is ν < 0 in the dual L − expansion, to F ′ whose leading level is ν ′ < 0, the leading level of φ(K) should be The level is positive, so φ(K) necessarily diverges as K → ∞; this is consistent with the assumption that φ is in the connected component of the diffeomorphism group of R ≥0 .Note that if we want to produce a more regular solution, so that ν ′ is more negative than ν, the diffeomorphism necessarily grows more than linearly for large K.In a sense, to produce a more regular solution we need to push the spectrum of K out to infinity.This means that the more regular the desired solution, the more singular the state φ(K) must be from the perspective of the identity string field [11].An extreme example of this is the diffeomorphism relating the simple tachyon vacuum to Schnabl's solution: The inverse wedge state Ω −1 is so singular that it does not even have a well-defined Fock space expansion.For this reason, it may be appropriate to impose conditions on the asymptotic behavior of elements in Diff 0 (R ≥0 ) towards K = ∞, at the cost of limiting the range of solutions that can be related by automorphism; we will not do so here.If a choice of F can be related to 1 1+K through diffeomorphism, the above analysis implies that it is formally possible to express the solution in the same form as the solution based on the simple tachyon vacuum: with the appropriately transformed generators.One must be a little careful however, since this expression inherits many of the problems with boundary condition changing operator collisions which appear for the simple tachyon vacuum.For example, the term contains the undesirable product σσ.If the leading level of F is −2 or lower, we know from the previous section that the problematic collisions here are an artifact of how the solution is being written.
It is natural to ask whether any two choices of F can be related by applying automorphisms.The answer is no, since it is easy to see that a diffeomorphism of R ≥0 will preserve the number of extrema of F and its values at the extrema. 9Therefore the space of F s can be partitioned into equivalence classes, labeled by n-tuples (a 1 , a 2 , ..., a n ), (4.28)where n is the number of extrema and a i is the value of F at the ith extremum, counting towards increasing K. From (2.6) it follows that the a i s should satisfy the inequalities a 2k < 1, a 2k > a 2k−1 , a 2k > a 2k+1 (4.29) with a n < 0 if n is odd and a n > 0 if n is even.This leads to an interesting "topological" classification of tachyon vacuum solutions in the KBc subalgebra, though as far as is known [19,20] all such solutions are physically equivalent.This raises the question as to whether there is broader significance to structures which are preserved by diffeomorphism of the spectrum of K.This remains to be understood.

1 2 L
− .Therefore the automorphisms generalize the well-known scale transformations of correlation functions on the cylinder, generated by the operator L − .Now we can understand how the automorphisms act on the solution (1.1).If the solution is characterized by some F (K), one can show that the tachyon vacuum (2.5) and the boundary condition changing fields (2.14) transform as .23) Therefore the automorphism simply changes the choice of F characterizing the solution, or equivalently, the choice of tachyon vacuum.Note that (4.19) implies that the automorphism preserves the conditions (2.6) on the choice of F .Particularly interesting for the purposes of the present work is understanding how to choose a diffeomorphism which regulates the collision of boundary condition changing operators.
Cross terms which do not involve (3.26) place a strictly weaker bound on ν.Therefore, ν < −1 is sufficient to ensure that (2.18) is safe from anomalies due to collisions of boundary condition changing operators.The stronger bound ν ≤ −2 is sufficient to further guarantee that the formally equivalent expression (2.15) is also safe.Note that the contributions to the solution which give the strongest bound on ν are only present for the superstring.For the bosonic string, anomalous collisions are 25)is a well-defined state.The cross terms in Ψ 2 which provide the strongest bound on ν arise from the following contributions to the solution as given in (2.18):