Open string field theory with stubs

There are various reasons why adding stubs to the vertices of open string field theory (OSFT) is interesting: Not only the stubs can tame certain singularities and make the theory more well-behaved, but also the new theory shares a lot of similarities with closed string field theory, which helps to improve our understanding of its structure and possible solutions. In this paper we explore two natural ways of implementing stubs into the framework of OSFT, resulting in an A-infinity-algebra giving rise to infinitely many vertices. We find two distinct consistent actions, both generated by a field redefinition, interestingly sharing the same equations of motion. In the last section we illustrate their relationship and physical meaning by applying our construction to nearly marginal solutions.


Introduction and motivation
It is a well-known fact that the algebraic and geometric structures of open and closed string field theory are fundamentally different.The OSFT action (Witten action) consists of a standard kinetic term and a single cubic interaction and reads [1] The algebraic ingredients are a nilpotent differential given by the BRST-operator, the star product and a symplectic form ω, which all together form a cyclic differential graded algebra.In the following, we will use the coalgebra notation of [2] with m 2 (Ψ 1 , Ψ 2 ) := (−) d(Ψ1) Ψ 1 * Ψ 2 and the shifted degree given by d (Ψ) = gh (Ψ) + 1.Now the defining algebraic properties are In contrast, the classical closed string field theory (CSFT) action [3] contains infinitely many vertices and reads where the vertices are given by integrating certain differential forms over vertex regions V 0,n in the moduli space of n-punctured Riemann surfaces.At quantum level there are also vertices of higher genus to consider, i. e. intrinsic loops already present within the elementary vertices.However, they will not be of interest here for now.
The vertices also can be decomposed into a symplectic form ω ′1 and higher products l n : In contrast to the open string star product, the l n are all totally symmetric.The objects Q =: l 1 , all higher l n and ω give rise to a cyclic L ∞ -algebra: The multiplication of the multilinear products in the first line is defined as l! (k − 1)! l k l l Ψ σ(1) , ..., Ψ σ(l) , Ψ σ(l+1) , ..., Ψ σ(k+l−1) , where the sum runs over all permutations of k + l − 1 elements and the factor (−1) ǫ(σ) is just the sign picked up when permuting the Ψs.
The different algebraic structures have a geometric origin: For a unitary theory it is necessary that the Feynman string diagrams computed from the action cover the full moduli space of n-punctured Riemann surfaces exactly once.For the open string where one needs to consider surfaces with boundaries it has been shown by Zwiebach [4] that this is indeed the case for the action (1) with only one cubic interaction.In contrast, for the closed string, i. e. surfaces without boundaries, such a simple cubic representation of the action seems not to be possible, see [5].
The simplicity of the OSFT action allowed for the discovery of analytic classical solutions, most importantly the open string tachyon vacuum [6].In classical CSFT, no such analytic solutions have been found yet [7,8].To make some effort into this direction, we want to propose the following strategy: If CSFT cannot be simplified easily, maybe we can make OSFT "more complicated" in the sense that its algebraic and geometric structure resembles that of CSFT?There are two motivations for that: First, it could help to make the very abstract language of CSFT more intuitive and tractable.Second, if we can translate the known analytic solutions to the new, deformed OSFT, it could give an idea how CSFT solutions might look like.In total there are three steps to do: 1. Find a consistent deformation of the Witten OSFT such that its mathematical structure resembles CSFT 2. Find analytic solutions of this deformed theory 3. Make an educated guess for analytic solutions of CSFT The first two tasks will be worked out in this paper while the third one is left for the future.
The deformation we will consider has many additional interesting aspects: First, OSFT exhibits some issues with singularities or ill-defined quantities: For example, there exist "identity-based" solutions [9] (for a recent discussion see [10]) which solve the equations of motion but do not have a well defined action.There are hints that these solutions could be better behaved in the deformed theory.Moreover, the deformed products M 2 , M 3 , . . .defined in the next section could allow for a natural definition of a Banach type A ∞ -algebra over the space of string fields.
The paper is organized as follows: In the second section the deformation using stubs [11] is discussed in detail, resulting in an explicit definition of the higher products of the A ∞ -algebra.The necessary mathematical ingredients, namely homological perturbation theory and homotopy transfer, are introduced as well.As a side result, we give a consistent method of applying homotopy transfer to general homotopy equivalences which are not deformation retracts.Section three deals with analytical solutions and the action of the stubbed theory.Surprisingly, we find two consistent actions with the same equations of motion, both generated by a field redefinition.The two field redefinitions, one coming from homological perturbation theory and one motivated by closed string field theory, are derived explicitly.The last section is devoted to the physical interpretation of our results.We apply the whole construction to a suitable class of solutions and compute the action up to first non-trivial order in the field expansion.

Deforming the Witten action using stubs
We have seen that the algebraic structure of CSFT is that of an L ∞ -algebra, so naively we should try to find an L ∞ -based deformation of OSFT.However, this would imply a commutative product of open strings, which would hence be fundamentally different from the Witten product.We still want the deformation be equivalent to Witten theory, so the best we can do is to aim for an A ∞ -algebra with products M n obeying and the multiplication given by From now on we will use the tensor coalgebra notation of [2] where the A ∞ -relations take the simple form Geometrically, to mimic the situation of CSFT, we would like to have a three-vertex that does not give rise to a full moduli space cover, such that higher vertices are necessary.One modification that achieves both of these requirements is to attach stubs on the three vertex, which means, small pieces of propagating strings on each input.In this way, there will appear some regions of moduli space that are not covered by Feynman diagrams and hence have to be taken over by elementary vertices.The higher products which will give rise to those vertices will indeed form an A ∞ -algebra.The Hamiltonian of our CFT which generates time evolution is L 0 , so the operator which inserts a strip of length λ into a string diagram is e −λL0 .The natural modification of the star product which attaches stubs symmetrically on all three inputs is then given by [12]: M 2 is cyclic with respect to the simplectic form ω and Q-invariant: Those relations follow from [Q, L 0 ] = 0 and the fact that L 0 is BPZ-even.It is easy to see that M 2 is not associative as expected for an A ∞ -algebra: The task is now to explicitly determine all the higher products and prove that they satisfy all relevant properties.This will be done in two ways, first in a formal algebraic way using homological perturbation theory and second in a more heuristic way using tree diagrams.

Elements of homological perturbation theory
The starting point of homological perturbation theory (HPT) is a chain homotopy equivalence.Let V , W be two chain complexes, i. e. graded vector spaces with a nilpotent differential of degree one denoted by d V (d W ). Furthermore, we are given two chain maps p : V → W and i : W → V of degree zero as well as a homotopy map h : V → V of degree minus one obeying the relations This ensures that i and p as well as the combination ip leave cohomology classes invariant.The most common case in literature is that of a special deformation retract (SDR) where W is isomorphic to a subspace of V , see e.g.[13].The maps i and p are then given by the canonical inclusion and projection, respectively.It follows trivially that pi = 1 and additionally the following annihilation conditions are demanded: It can be shown that any deformation retract with pi = 1 can be made an SDR by a redefinition of the maps [14].
Lets assume that the differential d V is perturbed by some δ of degree one such that (d V + δ)2 = 0 again.The homological perturbation lemma (HPL) now states that one can construct a new SDR using the perturbed differential with the other maps given by The expression (1 − hδ) −1 requires some explanation: Usually it should be understood as a geometric series We demand for the validity of the HPL that this series either converges or terminates, which will be the case in our examples.The proof of the lemma including all relevant relations is straight forward but tedious and can be found in [14,2].

Transferring algebraic structure: SDR-case
Our main purpose for using the HPL will be transferring some algebraic structure from one side of the homotopy equivalence to the other.Lets consider an SDR with an associative multiplication defined on V .It is a natural question to ask if there exists some "induced product" defined on W : For example, for a, b ∈ W the product ia • ib in V need not be in W anymore.The only thing that could be done is to project it to W , i. e. define a • b | W = p (ia • ib), but this product will in general not be associative anymore.What happens is that an associative algebra on one side of the equivalence becomes an A ∞ -algebra after transferring to the other side.A formal way to construct all the higher products is given by the so-called tensor trick: One defines a new SDR over the tensor coalgebra 2 of V (W ) with the tensorial versions of the maps given by is the standard tensor coderivation associated to d V (W ) , see (138), i and p are cohomomorphisms while h is defined somewhat asymmetrically which will turn out to be crucial for the formalism to work.One can check directly that those definitions indeed give rise to an SDR again.The product on V , denoted by m 2 , can now be treated as a perturbation δ of the coderivative d V , where m 2 is the coderivation associated to m 2 .D V 2 = 0 will be fulfilled as a consequence of m 2 being associative and d V obeying the Leibniz rule.According to the HPL we will be given a new, perturbed complex with the maps where D W squares to zero and therefore defines an A ∞ -algebra on W .The higher products can be obtained by expanding D W , i. e. projecting onto n inputs and one output: For example we get as already anticipated above.

Transferring algebraic structure: Non-SDR-case
We want to apply those concepts now to the problem of attaching stubs in OSFT.The space V should be given by the space of string fields H BCF T with its grading d (Ψ) = gh (Ψ) + 1 and the differential Q.By comparing ( 16) with (40) it seems natural to define However, we see immediately that this definition does not give rise to an SDR: i and p are not an inclusion and projection anymore and pi = 1.Still, in principle the HPL holds for arbitrary homotopy equivalences, so there is a chance to succeed anyway.Our choice is so far completely symmetric, so lets define The simplest (although not unique) solution for h is It is important to stress that h is well-behaved also for L 0 = 0 and does not have any pole.
One could now try to proceed with the tensor trick as above and eventually compute the map D W , but this runs into problems: Although D W 2 = 0 still, as guaranteed by the HPL, D W is not a coderivation anymore!This means, it does not obey the co-Leibniz rule (136).As explained in the Appendix of [15], the condition pi = 1 as well as the annihilation relations are necessary (and sufficient) for the tensor trick to work.So there has to be some modification to account for that and it will actually turn out to be surprisingly simple: We can just take the expression we get for D W and while expanding to calculate the higher products, pretend that all SDR-relations (23) are satisfied.More precisely, we define an operator P SDR acting on the space of maps from T H → T H which projects on maps in which all SDR-relations hold.This means, every time that pi occurs, it will be replaced by 1 and every time hh, hi or ph occurs, the term will be discarded: P SDR (....hh....) = 0, P SDR (....hi....) = 0, P SDR (....ph....) = 0, P SDR (....pi....) = P SDR (....1....) Now our new higher products will be given as Their associated coderivations can be added together to form a total map called M, It is a coderivation by construction and moreover it squares to zero because in the proof of the HPL, where it is shown that D 2 W = 0 (Eq.1.2.16 of [2]), the SDR-relations are never used.Since we know that D W 2 = 0 independently of the SDR-relations, also M 2 = 0 and M defines the desired A ∞ -algebra.
As an example, lets explicitly calculate M 3 : The second and third term in the second line contained an hi and were deleted whereas in the last term ipi was replaced by i.This whole procedure including the proof of the statement may seem quite handwavy, however, there exists a more formal and precise way of arriving at the same result using operad theory [16].A combinatorial proof using tree diagrams will be given in the next section.

Higher products using tree diagrams
The proposal is that M n is equal to the sum of all distinct, rooted, full binary trees with n leaves such that every leaf represents one input and the root is the output.With every leaf there is one factor of i associated, with every node the product m 2 , with every internal line h and with the root p.In [17] it is argued that this is true for SDRs, since we construct the products in the same way as for an SDR, we conclude that the proposal also holds for our non-SDR case.
In the tree language, the A ∞ -relations can be proven directly: The commutator of Q with an n-leaved tree gives a sum of n − 2 terms, in which one of the n − 2 internal lines h is replaced by 1 − ip.The 1-terms actually cancel away because of the associativity of m 2 : The propagator associated with unity connects two nodes m 2 which leads to an expression of the form m 2 (m 2 (A, B) , C) where A, B, C are three subtrees.In the sum there always exists a second tree with another propagator turned into unity giving rise to the expression m 2 (A, m 2 (B, C)).These two trees cancel away such that only the -ip-factors remain in total.Now the other terms occuring in the relation can be interpreted as follows: If we project on one output, π 1 M k M n+1−k gives a sum of trees where one of the terms in M n+1−k is connected with its root to one of the k leaves of one of the trees in M k and this is done in all possible combinations.The result is a sum of trees with in total n leaves, where one of the internal lines does not contain h but ip; the i from the leaf of the left tree and the p from the root of the right tree.But that is exactly the same sum of terms we have on the l.h.s., indeed it is easy to see that each tree occuring on the r.h.s.must also occur on the l.h.s. and vice versa.An interesting crosscheck of the π 1 -projection of (47) can be done by comparing the total number of trees on both sides: The number of full binary trees with n + 1 leaves is given by the Catalan number This means that on the l.h.s.there are C n−1 trees in M n and n − 2 internal lines that can be changed, hence a total of C n−1 (n − 2) trees.On the r.h.s.we have The Catalan numbers fulfill the following useful recursive relations: Using them one can proceed by induction: Assuming equation ( 49) is valid for some n, then as it should be to complete the induction.This shows that the number of terms in the equation ( 47) is the same on both sides.

Proof of cyclicity to all orders
Using the tree language it is possible to prove that all higher products M n are cyclic with respect to the BPZ-product.We have to show hence we start with a sum of trees on the l.h.s. and use the BPZ-properties of m 2 and h as well as p = i † = i to rewrite it as the sum of trees on the r.h.s..The explicit steps are: 1. Take the p from the root of the tree and write it to the left side of the product where it can be interpreted as i, acting on Ψ 1 .
2. Take the m 2 from the root of the tree and use cyclicity of m 2 to apply it on the first two arguments inside of ω.This gives a sign factor of − (−) d(Ψ1) .In general one will be left with two subtrees then with n + 1 leaves in total.
3. Take the h from the right subtree and use that it is BPZ-even to apply it on the left subtree.This gives an additional sign factor of (−) d(lef t subtree) .
4. Take the m 2 from the root of the right subtree and apply it on the first two arguments inside of ω.It gives a sign factor of − (−) d(lef t subtree)+1 (the +1 comes from the h that was shifted in step 3) which cancels the sign factor of step 3. Again, one is left with two subtrees.5. Repeat steps 3 and 4 until the right subtree only consist of i acting on one input.The total sign factor remains − (−) d(Ψ1) .
6. Remove the i acting on Ψ n+1 and let it act as a p on the left subtree.
Now the left subtree fulfills all requirements to be an element of M n and since we also have the right sign factor, we have obtained a term contained in the r.h.s. of (52).The manipulations are all uniquely invertible so we can conclude that the map between the trees is one-to-one and all terms we need are constructed exactly once.As a result, Eq. ( 52) holds and all higher products are cyclic.Moreover, as already suggested by the name, Eq. ( 52) together with the antisymmetry of ω implies invariance of the vertices under cyclic permutations.

Geometric picture
We have now shown that the higher products fulfill all the algebraic requirements but we do not know anything yet about the geometric picture, if they indeed give rise to a full single cover of the moduli space.
To answer this question, the tree description of the products turns out to be very useful.Lets consider an arbitrary string tree diagram using the stubbed three vertex: As long as the external states are on-shell, the stubs make no difference because the external legs consist of a semiinfinite strip anyway.On the internal lines instead, the stubs make a difference because all internal strips with a length smaller than 2λ do not appear.We can conclude that the additional elemantary vertices we need should consist of all tree diagrams with all internal strips having a length smaller than 2λ.The Siegel-gauge string propagator in the Schwinger parametrization is given by The integral over t can be thought of an integral over strips of propagating strings of all different lengths.Following this logic, the propagators in our new vertices should be given as and hence be equal to minus the homotopy! 3We have constructed the higher products by drawing all binary tree diagrams with Witten vertices, h as propagators and e −λL0 on the leaves and the root.Those are in one-to-one correspondence with all the Feynman tree diagrams that should make up the new elementary vertices.This shows that our higher products M n indeed define a set of vertices which gives rise to a full cover of the moduli space.
3 Analytic solutions and action(s)

Projection cohomomorphism from the HPL
Using the definition the equations of motion of the original Witten theory can be written in coalgebra language as i. e. solutions are Maurer-Cartan elements of the A 2 -algebra defining the theory.In the same spirit, to find solutions of the deformed theory, we have to solve the equation In fact, the homological perturbation lemma already gave us an operator which maps solutions of the Witten theory to solutions of the stubbed theory.This can be seen as follows: The perturbed projection P is a chain map and hence obeys Pm = MP.
Now lets assume Ψ * is a solution of the Witten theory, then where Eq. ( 146) was used.It remains to determine the cohomorphism P for the case where the homotopy equivalence is not an SDR.Again, as for the higher products above, the simplest way is to take the expression from the HPL and apply the operator P SDR on its components to get The resulting maps can then be packaged into a cohomorphism P again.More explicitly, we get for the first few orders ... .
One can now check the equation π 1 Pm = π 1 MP order by order: For order three the calculation is already very tedious but it also turns out to work.The important point is that in the manipulations that are necessary, the SDR-relations were never used.This implies, since we know that (63) works for SDRs, it also works in our case and (63) is a valid definition.Now we have a cohomorphism by construction that obeys the chain map relation (58) such that we can construct analytic solutions of the deformed theory. 4

The action
The on-shell action is one of the most important observables in OSFT, for example for the tachyon vacuum its value is equal to minus the energy of the D-brane which has decayed.Since the stubbed theory should be physically equivalent to the original Witten theory, we expect that the values for the on-shell action we get in the two theories coincide.The Witten action can be written in coalgebra notation [2] as where Ψ (t) is any smooth interpolation between Ψ (0) = 0 and Ψ (1) = Ψ and ∂ t the coderivation associated to ∂ t .Similarly, the stubbed action reads We would expect now a relation with Ψ * a MC-element of m.However, this relation turns out to be non-obvious: Instead where the last line differs from S ′ (Ψ ′ ) by the insertions of P −1 on both inputs of ω.
The invertibility of P is actually a delicate question: In general, a cohomomorphism is invertible iff its linear component, i. e. P 1 = p is invertible.Now p = e −λL0 inserts a strip of length λ, so one would expect the inverse e λL0 to remove a strip of length λ from the world sheet, which is not always possible.On the other hand, e λL0 makes sense on a string field expanded in eigenstates of L 0 as long as the eigenvalues are finite.From now on, we shall assume that this is the case and e λL0 is well-defined on all string fields in question.
Explicitly, the inversion of P works as follows: PP −1 = P −1 P should be equal to the identity cohomomorphism, which is identity in its linear component and zero in all higher components.One can now solve the components P −1 n = π 1 P −1 π n order by order: One can now write out in more detail: It seems that the cohomomorphism P does not define the field redefinition we were looking for, instead it relates the Witten theory to a theory defined by S (Ψ).The equations of motion derived from S are the same as for S ′ , namely M 1 1−Ψ = 0, but it is not at all obvious that the two actions agree even on-shell.The reason is that the HPL does not know anything about the symplectic form ω: To get an invariant action with S ′ (Ψ ′ ) = S (Ψ), not only m has to transform accordingly, but also ω would have to go to otherwise P would fail to be cyclic.As it can be seen from ( 68), S (Ψ) is just a fancy rewriting of the Witten action and therefore defines an equivalent theory.But since we would like to keep the original ω, the field redefinition induced by P from the HPL and giving rise to S (Ψ) is not exactly what we want.However, since it shares the same equations of motion as S ′ (Ψ), it might provide a new family of gauge-invariant observables for solutions of S ′ (Ψ), parametrized by λ.In the last section we will check this explicitly on a special class of solutions.For now the next task is to derive the originally desired field redefinition which relates the actions S (Ψ) and S ′ (Ψ).5

Elements from closed string field theory
In [18], Zwiebach and Hata have shown how to relate slightly different, consistent sets of vertices in CSFT via an infinitesimal field redefinition.Our strategy is now to apply their method to our problem in OSFT and integrate the result to the finite case.This will not only provide us the field redefinition we are looking for, but also give some insight into the rather abstract formalism of CSFT.First, it is useful to collect some basic information about the structure of CSFT.As already explained in the introduction, the vertices are given by integrating basic differential forms Ω g,n Ψ1Ψ2...Ψn defined by They are living in the tangent space of the fibre bundle Pg,n over the moduli space M g,n , with the fiber being the space of local coordinates around the punctures modulo phase rotations.The dimension of this bundle is infinite, but the degree of Ω g,n Ψ1Ψ2...Ψn is just the real dimension of the base space M g,n .It takes as arguments tangent vectors Vi ∈ T Pg,n , which represent deformations of the world sheet Riemann surface Σ, either by changing the moduli or the local coordinates.The v i are Schiffer vectors on Σ, supported around the punctures, which generate those deformations.This means that the local coordinate around the nth puncture transforms as for some small ǫ.The b-ghost insertions are then defined as We will be only interested in the classical action without the loop vertices, so the genus g shall be zero from now on.The basic forms can now be integrated over sections of P0,n defining the vertices V 0,n .The quantization procedure makes use of the Batalin-Vilkovisky formalism; although we are not interested in quantum effects, the BV-antibracket is used in constructing the symmetry generator.It is defined as where the Ψ * i are antifields of opposite parity associated to each basis element of H.The BV-master action takes the same form as (7) with the only difference that the Ψ are not restricted in ghost number and run over fields as well as antifields.

Constructing the field redefinition
We are looking for a non-linear field redefinition of the form that relates the Witten action to the stubbed A ∞ -action in the form (66). To be consistent with the results of Zwiebach and Hata [18] we demand Since the kinetic term is identical, we immediately find In [18] it is shown that under an infinitesimal field redefinition of the form the classical action transforms as where {} denotes the BV-antibracket.In the paper it is now argued that for any small change of vertices, the change of the action indeed takes this form and the generator e is constructed explicitly: Here we assume that there exists some family of consistent vertex sets V 0,n (u) parametrized by some real number u and everything is evaluated at the point u 0 .The vector u is some Schiffer vector which generates a deformation of the V 0,n (u 0 ) in the direction of u, i. e. it generates diffeomorphisms which push V 0,n (u 0 ) into V 0,n (u 0 + δu).For the case of varying the stub length, this Schiffer vector takes a particular simple form: First, lets notice that the stub length λ for closed strings is defined as the geodesic distance from the location | z |= 1 of the local coordinate to the begin of the semiinfinite cylinder associated with the puncture.This implies that λ can be changed by just rescaling the coordinate: Sending By comparing with Eq. ( 73) we read off The b-ghost insertion is then given by We want to use the above expression for e in the context of changing the stub length for open strings, hence a few modifications and simplifications are necessary: First, the combinatorial factor n! originates from total symmetrization of the vertices and is not necessary for open strings.Second, the insertions of b + 0 should get replaced simply by b 0 since there is no antiholomorphic sector.Moreover, the string coupling κ will be set to one.Now the generator simplifies to If we make the ansatz as the infinitesimal version of (76), then the f n are determined as where δλ plays the role of t in (79).
To find f 2 (Ψ, Ψ) we need to consider e (λ) for n = 3: The vertex V (λ) 0,3 is zero dimensional, so there is no integral and the surface state Σ| is just the Witten vertex with stubs of length λ, Inserting into (84) yields The BV-bracket with Ψ can be straightforwardly evaluated; after carefully checking the signs the result is We expect now the relation to hold up to order 3 in Ψ; by directly inserting we can compute explicitly The last and most interesting term yields where the last three terms in the second line cancel after applying the Leibniz rule and cyclicity.On the other hand, which is the same expression up to order Ψ ⊗3 (Again, cyclicity was used in the last line.).The explicit form of f 2 (Ψ, Ψ) suggests the following general structure: We can guess the ansatz where b 0 again denotes the coderivation associated to b 0 .At first sight, Eq.( 94) looks a bit strange now from the coalgebra perspective because it is a commutator of two odd objects, so one would more naturally expect an anticommutator.However, the first term in (94) stems from the application of b 0 on the first argument of the symplectic form ω, hence the sign contains implicit information about ω.From the discussion about P from the HPL we could anticipate that ω has to enter the calculation at some point.The more natural looking expression π 1 [b 0 , M n ] would have been independent of ω.
One can prove now that the ansatz (94) is indeed correct by directly inserting into the action.

Proof of the ansatz for the infinitesimal field redefinition
If we focus solely on terms of order n + 1 in Ψ we get (95) The last term is the contribution from the original S (Ψ), so the first two terms denoted by δS (n+1) should yield the infinitesimal variation δS ′(n+1) ?
= δλ d dλ Inserting the ansatz and performing some straight forward manipulations gives The last two terms of the second line actually cancel each other: Because of cyclicity of the (n + 1)-vertex the last two lines contain the same terms, just differing by a sign which comes from commuting Q with b 0 .Therefore they add to zero.The second term in the second line of (97) can be further manipulated using the A ∞ -relations: Now again, the terms in the last line cancel after using cyclicity: All terms except for the last one can be further manipulated using the antisymmetry of ω and cyclicity of M k .For example, so in all of the terms the M n+1−k can be moved to the outermost right.After all, the terms can be summed up as which is after the summation over k identical to the first term in the second line of (99), just with opposite sign.So we arrive at the expression which should now be compared to the result of formula (96).
The derivative with respect to λ can act on the stubs as well as on the homotopy h.The action on e −λL0 inserts a factor of −L 0 on every input string field of the (n + 1)-vertex.Since the vertices are cyclically symmetric, we get n + 1 identical terms, which cancels the prefactor 1 n+1 .The result is d dλ which is equal to the first term of (103).To compute the action on h, the tree representation turns out to be useful again: First of all, hence we get a sum of all possible tree diagrams with one propagator replaced by −2b 0 e −2λL0 .We can cut through the diagram along this replaced propagator and think of the factor e −2λL0 as arising from two e −λL0 -stubs from the leaf and the root of the two subtrees.Both subtrees are now part of a higher product M k for some k, 2 ≤ k ≤ n − 1.So the whole expression can be written as a combination of two higher products M k , M n+1−k with a factor −2b 0 inserted: Because of cyclicity of the k + 1-vertex, the different lines contain the same terms so we have The last bracket can be further manipulated: In the last step cyclicity of the (n + 1 − k)-vertex was used again.We see that in the sum of (107) the kth term and the (n + 1 − k)th term are identical so the sum can be rewritten as which is precisely the second term in (103).This completes the proof that the ansatz indeed yields the correct infinitesimal field redefinition.

Finite field redefinition
So far we have only been concerned with infinitesimal variations of λ, now we want to generalize the results to finite changes.We know where we have written the superscript λ to indicate that the f n also depend on λ explicitly.This equation can be integrated to Inserting Ψ λ back we get a perturbative expansion in the original solution Ψ 0 : Ψ λ =Ψ 0 + ˆλ 0 dtf t 2 (Ψ 0 , Ψ 0 ) + ˆλ 0 dtf t 3 (Ψ 0 , Ψ 0 , Ψ 0 ) + This formula provides an algorithm to find the associated A ∞ -solution to each known solution of the Witten OSFT.

Physical interpretation
To summarize, we found two distinct field redefinitions However, both actions share the same equations of motion, namely the Maurer-Cartan equation of the stubbed A ∞ -algebra for A ∞ -or L ∞ -algebras, in particular about the solutions of closed string field theory.One way to proceed would be to transform the whole construction to the sliver frame, where many analytic solutions of OSFT are formulated.
Another possible future direction is to examine wether the stubbed theory is "more well-behaved" in the sense that some typical singularities and ambiguities, for example connected to identity-like solutions, are ameliorated.(142) Again, the individual products can be recovered from f as Of special importance are elements of T V of the form for some v ∈ V.They fulfill the following useful properties: for any cohomomorphism f .
A bilinear map ω|: T V × T V → C is called a symplectic form if it satisfies Given two symplectic forms ω|, ω ′ |, a cohomomorphism f is cyclic if which generate two different actions S and S ′ via S Ψ = S (Ψ) , S ′ (Ψ) = S (Ψ ′ ) .