Supersymmetry in Open Superstring Field Theory

We realize the 16 unbroken supersymmetries on a BPS D-brane as invariances of the action of the corresponding open superstring field theory. We work in the small Hilbert space approach, where a symmetry of the action translates into a symmetry of the associated cyclic $A_\infty$ structure. We compute the supersymmetry algebra, being careful to disentangle the components which produce a translation, a gauge transformation, and a symmetry transformation which vanishes on-shell. Via the minimal model theorem, we illustrate how supersymmetry of the action implies supersymmetry of the tree level open string scattering amplitudes.


Introduction
Following the recent constructions of open superstring field theory [1,2,3], an important issue to understand is the realization of supersymmetry. Since the string field does not match fermion and boson degrees of freedom off-shell, supersymmetry is not manifest. It is described by a nonlinear transformation of the form δ susy Ψ = S 1 Ψ + S 2 (Ψ, Ψ) + S 3 (Ψ, Ψ, Ψ) + higher orders, (1.1) where S 1 , S 2 , S 3 , ... are a specific sequence of multi-string products. The goal of this paper is to construct the products of the supersymmetry transformation using the zero mode of the fermion vertex, |z|=1 dz 2πi Θ a e −φ/2 (z), (1.2) picture changing operators, and Witten's associative string star product. We focus on the small Hilbert space formulation of open superstring field theory [2,3], since the supersymmetry transformation in this framework takes a fairly canonical form. In the large Hilbert space formulation [1] there are ambiguities in the choice of supersymmetry transformation related to the enlarged gauge symmetry of the theory, and we postpone discussion to later work [4]. 1 Since classical open superstring field theory does not contain gravity, supersymmetry can only be described as a global symmetry. Therefore our analysis is somewhat different in spirit than other recent discussions of supersymmetry in superstring perturbation theory [6,7], which utilize the fact that closed superstring field theory incorporates supersymmetry automatically as part of the the local gauge symmetry. Finally, we should emphasize that we only consider unbroken supersymmetries. Describing broken supersymmetries is closely related to the issue of background independence in string field theory, and should be important for understanding the appearance of D-brane charges in the supersymmetry algebra. Further progress in this direction may be possible following [8,9,10].
This paper is organized as follows. In section 2 we review the small Hilbert space formulation of open superstring field theory [2,3], mostly to simplify notation and to introduce the concept of cyclic Ramond number which will be convenient for understanding issues related to cyclicity. In section 3 we describe the construction of the supersymmetry transformation and prove that it leaves the action invariant. In section 4 we compute the supersymmetry algebra, explicitly describing the gauge transformation and the on-shell trivial symmetry which appear in addition to the momentum operator when computing the commutator of supersymmetry transformations. Finally, in section 5 we use the minimal model to illustrate how supersymmetry of the action implies supersymmetry of the S-matrix.

Superstring Field Theory in the Small Hilbert space
In this section we review the small Hilbert space formulation open superstring field theory, based on an action realizing a cyclic A ∞ structure [2,3]. This theory is based on the RNS formulation of the superstring worldsheet, with a c = 15 matter superconformal field theory tensored with and a c = −15 ghost boundary superconformal field theory b, c, β, γ. The βγ ghosts will be bosonized to the ξ, η, e φ system [11]. The string field is an element of the state space H of this boundary superconformal field theory. Generally, we consider H to include both Neveu-Schwarz (NS) and Ramond (R) sector states, as well as states in the small and the large Hilbert space. The small Hilbert space consists of states A satisfying ηA = 0, where η is the zero mode of the eta ghost, and the large Hilbert space includes states which do not satisfy ηA = 0. So that we can describe fermions and spacetime ghosts, we assume that states in H can appear in linear combinations with commuting or anticommuting coefficients. In this paper we are interested in supersymmetry, so we require that all states in H are GSO(+) projected.
The following discussion assumes familiarity with the coalgebra representation of A ∞ algebras, in particular as reviewed in [12]. In this formalism it is necessary to use a shifted even/odd grading on the open string state space called degree. The degree of a open string field A, denoted deg(A), is defined to be its Grassmann parity plus one.

Action
The action can be expressed There are three main ingredients: A dynamical string field Ψ, which includes an NS sector component and a Ramond sector component; a symplectic form Ω, which maps two string fields into a number; and multi-string products M n+1 , which multiply n + 1 string fields to produce a string field. The 1-string product M 1 is equal to the BRST operator Q, and the higher products are built from Witten's open string star product with insertions of picture changing operators. Importantly, the products satisfy the relations of a cyclic A ∞ algebra, where the notion of cyclicity is provided by the symplectic form Ω. As it happens, the action is purely quadratic in the Ramond string field. Therefore products M n+1 are taken to vanish when multiplying three or more Ramond states. Let us describe the ingredients of the action in more detail. The dynamical string field Ψ has an NS component and an R component: The NS dynamical field Ψ NS is a degree even NS state in the small Hilbert space at ghost number 1 and picture −1. The Ramond dynamical field Ψ R is a degree even Ramond state in the small Hilbert space at ghost number 1 and picture −1/2. In addition, the Ramond string field satisfies the condition [1] where the operators X and Y are defined

4)
Y ≡ −c 0 δ ′ (γ 0 ). (2.5) The operators satisfy XYX = X, YXY = Y, [Q, X] = 0, (2.6) and are BPZ even. Note that X is singular when acting on states annihilated by β 0 , and Y is singular when acting on states annihilated by γ 0 . To avoid these singularities, we require that X only acts on Ramond states in the small Hilbert space at picture −3/2, and that Y only acts on Ramond states in the small Hilbert space at picture −1/2 [3]. To describe the subspace of the dynamical string field more efficiently, it is useful to introduce the restricted space: H restricted ⊂ H. (2.7) The restricted space H restricted consists of NS states in the small Hilbert space at picture −1 and Ramond states in the small Hilbert space at picture −1/2 which satisfy the condition XYA = A. The BRST operator preserves this subspace [1]: The dynamical string field Ψ is a degree even state in H restricted at ghost number 1.
The symplectic form Ω operates on a pair of states in the restricted space. Accordingly, we will call Ω the restricted symplectic form. There are actually three symplectic forms which play an important role: All three symplectic forms are graded antisymmetric, and nondegenerate on their respective domains [1]. Moreover, the BRST operator satisfies in all three cases. Generally, an n-string product which satisfies is said to be cyclic with respect to the symplectic form ω. In particular, the BRST operator is cyclic with respect to all three symplectic forms. The eta zero mode is cyclic with respect to the large Hilbert space symplectic form, and Witten's open string star product is cyclic with respect to the small and large Hilbert space symplectic forms. Sometimes it will be useful to write the symplectic form as a "double bra" state ω|, so that (2.14) In this notation, cyclicity of a product b n can be expressed where I is the identity operator on the state space. The notation can be further simplified using in the coalgebra formalism as where π 2 is the projector onto the 2-string component of the tensor algebra and b n is the coderivation corresponding to b n . The symplectic forms ω L , ω S and Ω mentioned above are defined as follows. The large Hilbert space symplectic form is related to the BPZ inner product in the large Hilbert space by a sign: The small Hilbert space symplectic form is be defined by where a, b are states in the small Hilbert space and ηA = a. Note that a product b n which is cyclic with respect to the large Hilbert space symplectic form is also cyclic with respect to the small Hilbert space symplectic form provided η acts as a derivation on b n , i.e. [η, b n ] = 0. Finally, the restricted symplectic form is defined by where a, b are states in H restricted and the operator G −1 is defined as Therefore Ω is given by the small Hilbert space symplectic form together with an insertion of Y between pairs of Ramond states. Finally let us describe the multi-string products M n+1 . We will call these dynamical products. The explicit construction will be reviewed in the following subsections, but here we list the essential properties: (M.a) The dynamical products form an A ∞ algebra. In particular, M n+1 is degree odd, and if M n+1 is the coderivation corresponding to M n+1 , the sum is a nilpotent coderivation in the tensor algebra: where η is the coderivation corresponding to η. This implies that the dynamical products multiply consistently inside the small Hilbert space.
(M.d) The dynamical products preserve the Ramond constraint XY = 1 when acting on states in H restricted .
(M.e) The dynamical products are cyclic with respect to the restricted symplectic form: We can summarize these conditions as requiring that the dynamical products M n+1 define a cyclic A ∞ algebra on H restricted . In particular, conditions (M.b), (M.c) and (M.d) imply that the restricted space is closed under multiplication with M n+1 . Note that conditions (M.a), (M.b) and (M.c) are sufficient for constructing gauge invariant equations of motion, as described in [13]. We additionally require conditions (M.d) and (M.e) to have a gauge invariant action.

Counting Ramond States
To construct the dynamical products it is necessary to introduce some notation for keeping track of the number of Ramond states that are multiplied with a product. We start by considering the tensor algebra generated from the open string state space H: We introduce a projection operator π m , π m : T H → T H, π m π n = δ mn π m , (2.28) which projects onto the m-string subspace of the tensor algebra. We also consider a projection operator π r , π r : T H → T H, π r π s = δ rs π r , (2.29) which selects multi-string states in the tensor algebra containing r Ramond factors (but an undetermined number of NS factors). Multiplying π m and π r defines a projection operator π r m ≡ π m π r = π r π m , (2.30) which selects m-string states containing r Ramond factors. We introduce a coderivation 1 which satisfies The eigenvalue of 1 counts the total number of states. We also introduce a coderivation R which satisfies The eigenvalue of R counts the number of Ramond states. Consider an operator on the tensor algebra O n+1 which has well defined eigenvalue under commutation with 1: The subscript n + 1 denotes the integer eigenvalue n. The operator commutes through the projector π m as We refer to the integer eigenvalue r as the Ramond number of O| r . The Ramond number of an operator on the tensor algebra will be indicated by a vertical slash followed by a subscript. Such an operator commutes through the projector π r as π s O| r = O| r π r+s , (2.36) which means that O| r removes r Ramond states from the tensor algebra. An important case is when the operators are coderivations. A coderivation b n+1 which satisfies is characterized by a corresponding (n + 1)-product b n+1 : The coderivation property uniquely determines a coderivation b n+1 once the product b n+1 has been defined (see e.g. [12]). We may also consider a coderivation b| r which carries definite Ramond number: b| r , R = rb| r . (2.39) Since R and 1 commute, we can have a coderivation b n+1 | r which simultaneously has well defined eigenvalue under commutation with 1 and R. Such a coderivation satisfies π s m b n+1 | r = b n+1 | r π r+s m+n , (2.40) and is uniquely defined by an (n + 1)-string product, which we write b n+1 | r . We say that the product b n+1 | r carries Ramond number r. A product of Ramond number r can be nonzero only when the number of Ramond inputs minus the number of Ramond outputs is equal to r. This means that b n+1 | r must satisfy b n+1 | r r Ramond states = NS state, A generic product b m can be written as a sum of products of definite Ramond number The Ramond number is bounded between −1 and m since b m can have at most m Ramond inputs and 1 Ramond output. The BRST operator carries Ramond number zero Note that Ramond number adds when composing products. Therefore Ramond number defines a grading on the space of products and coderivations, which is of central importance in obtaining a solution of A ∞ relations. However, the concept of Ramond number is less useful when it comes to questions of cyclicity. To see why, let ω • b n+1 denote the cyclic permutation of a product b n+1 , defined through the relation [12]  Sometimes it is useful to view the "vertical slash" as an operation which selects the component of a product with the indicated Ramond number and/or cyclic Ramond number. Thus if we are given a product b n+1 , we can apply the operation | s r to arrive at the product b n+1 | s r . This operation may be defined in terms of the projection operators π r m : We also define The action of | s r on products naturally defines an action of | s r on coderivations. Note that b n+1 | s r does not always derive from operating | s r on a product b n+1 defined for generic Ramond numbers. When this is the case, it should be clear from context.

Dynamical Products
The dynamical products Taking the Ramond number 0 and 2 components of (2.52) implies The commutator m 2 | 2 , m 2 | 2 automatically vanishes since a 3-string product cannot carry Ramond number 4. The dynamical products have components at Ramond number zero and two: (2.57) In particular, M n+1 vanishes when multiplying four or more Ramond states. In fact, M n+1 will also vanish when multiplying three Ramond states, so the action is quadratic in the Ramond string field. The product M 1 | 0 is identified with the BRST operator Q, and the product m 2 | 2 is identified with the Ramond number 2 component of Witten's open string star product. To construct the dynamical products we introduce auxiliary multi-string products: bare products : m n+2 | 0 degree odd, gauge products : µ n+2 | 0 degree even. (2.58) The bare 2-product m 2 | 0 is the Ramond number zero component of Witten's open string star product. We promote m n+2 | 0 and µ n+2 | 0 to coderivations m n+2 | 0 and µ n+2 | 0 , and define generating functions  Expanding in powers of t, this turns into a recursive system of equations determining the higher order gauge products and bare products in terms of lower order ones: The last equation should be solved to determine µ n+2 | 0 in terms of m n+2 | 0 . This requires a choice of contracting homotopy for η, which determines a configuration of picture changing insertions in the vertices. We will explain how to solve (2.64) in a moment. Once we have solved (2.63) and (2.64), we construct the dynamical products as follows. Define generating functions Expanding in powers of t, this turns into a recursive system of equations: Solving this recursion gives the dynamical products as 1) Ξ is a contracting homotopy for η: [η, Ξ] = I.
3) Ξ and X are defined acting on generic states in H (unlike, in particular, the operator X).
4) X = X when acting on a Ramond state at picture −3/2 in the small Hilbert space.
The definition of Ξ is reviewed in appendix A. We then define the gauge products according to [2,3] Note that µ n+2 | 0 has components at cyclic Ramond number 0 and 2, and these components must be chosen differently. This definition implies that the dynamical products satisfy [3]: The first relation implies that the dynamical products are consistent with the constraint XYA = A in the Ramond sector, as required by condition (M.d), and the second relation implies that the dynamical products vanish when multiplying three or more string fields. It is useful to express the dynamical products in components of definite cyclic Ramond number. Using (2.76) we have To further simplify, it is useful to combine m n+2 | 0 with m n+2 | 2 into a single product and introduce the operator [14] G ≡ I| 0 + X| 2 , (2.79) which acts as the identity on NS states and as X on Ramond states. Then (2.77) can be expressed Therefore the dynamical products have a component at cyclic Ramond number 0 and a component at cyclic Ramond number 2. Using coderivations we may write this as 5 .
where m ≡ m| 0 + m| 2 . To appreciate the structure of (2.81), recall the restricted symplectic form contains the operator G −1 : We have the relation since XY acts as the identity on the restricted space. From this it is clear that the factor of G in (2.81) is required to cancel the factor of G −1 in the restricted symplectic form. Then cyclicity of M n+2 translates to the statement that M n+2 | 0 and m n+2 | 2 are cyclic with respect to the small Hilbert space symplectic form.
It is useful to recall that the construction of M is equivalent to the construction of a field redefinition which relates M to comparatively simple A ∞ structure [12,13]. This can be understood by introducing the cohomomorphism where the path ordering is from left to right in sequence of increasing s. We also defineĜ(t) by replacing the upper limit of the integral in the path ordered exponential with t. From this cohomomorphism the gauge products can be computed using Moreover, any coderivation b(t) which satisfies the differential equation This implies the formulas Therefore, M can be derived by a similarity transformation from two comparatively simple A ∞ structures: Consider the field redefinition where ϕ is a new dynamical string field and denotes the group-like element of a degree even string field A. In [12], the transformation from Ψ to ϕ was called an improper field redefinition, since it does not preserve the small Hilbert space constraint on the string field. The equations of motion and small Hilbert space constraint of Ψ Projecting on to the 1-string component gives Chern-Simons-like equations [16] (Q − η)ϕ + ϕ * ϕ = 0. (2.100) The A ∞ superstring field theory can be viewed as one approach to deriving these equations from an action.

Supersymmetry Transformation
We are now ready to discuss supersymmetry. We consider open superstring field theory formulated on a maximally supersymmetric D-brane. The goal is to find a transformation of the dynamical string field Ψ realizing all sixteen unbroken supersymmetries. The natural place to start [11] is the zero-mode of the fermion vertex in the −1/2 picture: Let us explain the notation. The index on s 1 indicates that this operator is a 1-string product. The √ 2 factor is included to obtain the canonical normalization of the supersymmetry algebra. The operator Θ a is the spin field: where the scalars H i realize the bosonization of the worldsheet fermions ψ µ through 6 The object ǫ a is a supersymmetry parameter-a constant degree odd spinor. The repeated spinor index a is summed.
To keep notation simple we leave the dependence of s 1 on the supersymmetry parameter implicit. Since we make a GSO(+) projection in both NS and R sectors, the supersymmetry parameter must have positive chirality. Therefore s 1 may represent 16 independent supersymmetries. The massless fermions on the D-brane are described by the vertex operator cΘ a e −φ/2 multiplied by an anticommuting spinor field. Since this should describe a degree even state, 7 the operator Θ a e −φ/2 must be degree odd for positive chirality a. Therefore s 1 is degree even, and carries ghost number 0 and picture −1/2. The operator s 1 has the following algebraic properties: In particular, s 1 commutes with Q and η, and, since it is the zero mode of a weight one primary, is a derivation of the open string star product. Also s 1 is BPZ odd, and is therefore cyclic with respect to the large Hilbert space (and small Hilbert space) symplectic form.

Supersymmetry in the Free Theory
Let's start by considering the supersymmetry transformation in the free theory: We assume that the NS string field transforms as The Ramond string field cannot transform as s 1 Ψ NS , since this carries the wrong picture and is inconsistent with the constraint XYA = A. These problems can be solved simultaneously by multiplying the transformation by X: The pictures match up on both sides, and the constraint is satisfied due to XYX = X. We can package NS and R supersymmetry transformations together in the form for the appropriately defined operator S 1 . The operator S 1 can be decomposed into a piece at Ramond number −1 and a piece at Ramond number 1: It is convenient to replace X with X so that S 1 is defined acting on arbitrary states in H. We can equivalently express S 1 as an operator of cyclic Ramond number 1: using G from (2.79). Here the superscript for cyclic Ramond number is redundant since Let us demonstrate that this is a symmetry of the free action. This relies on two properties: It is easy to see that S 1 is BRST invariant because both X and s 1 are BRST invariant. The fact that S 1 is cyclic with respect to Ω can be shown as follows: where both sides are contracted with states in H restricted . In the second step we used the BPZ even property of G and G −1 , in the third step we used GG −1 = I when operating on H restricted . Finally we used the fact that s 1 is BPZ odd. Therefore we can compute the variation of the action: The free action is supersymmetric.

Supersymmetry in the Nonlinear Theory
In the full string field theory, supersymmetry is realized as a nonlinear transformation of the string field: The degree even products S n+1 will be constructed so that this transformation leaves the action invariant. We will call S n+1 supersymmetry products. Invariance of the action requires the following: (S.e) The supersymmetry products must be cyclic with respect to the restricted symplectic form: These conditions are closely analogous to those defining the dynamical products M n+1 . Conditions (S.b), (S.c) and (S.d) imply that the restricted space is closed under multiplication with the supersymmetry products. Note that conditions (S.a), (S.b) and (S.c) are sufficient to imply supersymmetry at the level of the equations of motion, as described in [13]. We additionally require conditions (S.d) and (S.e) to have a supersymmetric action. Now let us prove that conditions (S.a)-(S.e) imply a symmetry of the action. For this purpose it is helpful to write the action in a form which is closely related to the WZW-like formulation of open superstring field theory [18,19]. We introduce a family of string fields Ψ(t) ∈ H restricted , where t ∈ [0, 1] is an auxiliary parameter, and impose boundary conditions Ψ(0) = 0, Ψ(1) = Ψ, (3.20) where at t = 1 we recover the dynamical string field Ψ. The action can be written is actually the integral of a total derivative, and if we perform the integral we recover the action as expressed in (2.1). In particular, the action only depends on Ψ(t) at t = 1. The supersymmetry transformation (3.15) can be expressed as . .
Note that in computing this we are already assuming conditions (S.b), (S.c) and (S.d), since the supersymmetry variation must be well defined in H restricted . Recall that a coderivation D acts on a group-like element 1 1−A as Then the second term in (3.23) can be simplified to .
Moreover, if a coderivation D is cyclic with respect to ω, we have the relation Noting ω|π 2 D = 0, this can be derived from upon expressing the projector π 2 in the form where △ is the product and △ is the coproduct on the tensor algebra [12]. Since condition (S.e) implies that S is cyclic with respect to Ω, we can simplify the first term in (3.23): . (3.29) Taking (3.25) and (3.29) together, the variation of the action simplifies to

Supersymmetry Products
Now we describe the construction of the supersymmetry transformation. For the time being we will only be interested in implementing conditions (S.a), (S.b) and (S.c), which effectively means that we are constructing a supersymmetry transformation at the level of the equations of motion. In particular, we will not require that the Ramond field satisfies the constraint XYA = A, and we will not assume that the dynamical products M n+1 satisfy (M.d) and (M.e). Later we will account for these conditions and specify the supersymmetry transformation satisfying all conditions (S.a)-(S.e). The supersymmetry product S 1 has components at Ramond number −1 and 1: Both components are degree even and carry ghost number 0, while S 1 | −1 carries picture +1/2 and s 1 | 1 carries picture −1/2. As in subsection 3.1, we assume that s 1 | 1 is the Ramond number 1 component of s 1 . Earlier we chose S 1 | −1 = Xs 1 | −1 , but here we would like to give a more general definition. We postulate that S 1 | −1 can be expressed in the form where σ 1 | −1 is a degree odd operator of ghost number −1 and picture +1/2. The operator σ 1 | −1 is the first example of what we will call a gauge supersymmetry product. We further assume that σ 1 | −1 satisfies where s 1 | −1 is the Ramond number −1 component of s 1 . The operator s 1 | −1 is the first example of what we will call a bare supersymmetry product. It is clear that S 1 | −1 will carry one more unit of picture than s 1 | −1 , which is to say that S 1 | −1 carries picture +1/2. We also have the identities The first follows from (3.32) by construction, and the second follows from (3.32) and (3.33) after noting that s 1 is BRST invariant. Therefore, we have a definition S 1 satisfying conditions (S.a), (S.b) and (S.c), which for the moment is our primary concern. Satisfying conditions (S.d) and (S.e) requires a particular choice of contracting homotopy for η when defining the gauge supersymmetry product from (3.33). To reproduce the formula of subsection 3.1, we must choose It is consistent to assume S 2 has nonvanishing components at Ramond number −1 and 1: Both components are degree even and carry ghost number −1, while S 2 | −1 carries picture +3/2 and S 2 | 1 carries picture +1/2. We can split (3.36) into components of definite Ramond number: The strategy is to solve for S 2 | −1 and S 2 | 1 by pulling a factor of Q, · out of these equations. Noting The objects in the commutator with Q must vanish up Q-exact terms. These terms should be chosen to ensure that S 2 is well defined in the small Hilbert space. In this way we find In the first equation we added a Q-exact term defined by a new gauge supersymmetry product, which we write σ 2 | −1 . In the second equation a Q-exact term is not necessary, since S 2 | 1 is already in the small Hilbert space: Let us introduce a bare supersymmetry product s 2 | −1 satisfying Therefore the bare supersymmetry product s 2 | −1 can be defined up to a Q-exact term. However, such a term is not necessary since this definition already implies that s 2 | −1 is in the small Hilbert space: Now that we have the bare supersymmetry product s 2 | −1 , we may define the gauge supersymmetry product σ 2 | −1 with a choice of contracting homotopy for η. This then determines S 2 consistent with conditions (S.a), (S.b) and (S.c). Let us describe the construction to all orders. We introduce an infinite sequence of bare supersymmetry products and gauge supersymmetry products: bare supersymmetry products s n+1 | −1 : degree even, gauge supersymmetry products σ n+1 | −1 : degree odd. (3.51) We have already described these products when n = 0 and n = 1. At higher order, they can be described by generating functions satisfying the equations Expanding in powers of t gives a recursive definition of these products: Here we introduce a coderivation s| 1 , which represents the t = 1 value of the generating function for a sequence of degree even products s n+1 | 1 . The product s 1 | 1 is the Ramond number 1 component of s 1 . The generating function s| 1 (t) is postulated to satisfy d dt s| 1 (t) = s| 1 (t), µ| 0 (t) . (3.60) Expanding in powers of t gives a recursive formula for s n+1 | 1 : This gives a construction of the supersymmetry transformation satisfying conditions (S.a), (S.b), and (S.c). It is not difficult to verify that S carries the right ghost and picture numbers, so condition (S.b) is satisfied. To verify (S.a) and (S.c), it is useful to express s| 1 and s| −1 in the form The commutators with Q and η drop out since s 1 is BRST invariant and in the small Hilbert space. This leaves which vanishes since s 1 is a derivation of the star product. This proves condition (S.c). Let us explain the relation between the supersymmetry transformation constructed here and the one given in [13]. The supersymmetry transformation of [13] is characterized by a specific choice of gauge supersymmetry products: where σ 1 | −1 is assumed to be the Ramond number −1 component of the operator This is a special case of the supersymmetry transformation we have been describing. To see this, we must verify so that (3.69) and (3.70) represents a choice of contracting homotopy for η in the solution of (3.55). Compute The operator σ 1 in (3.70) is a derivation of the star product, and moreover [η, σ 1 ] = s 1 . Thus we can simplify: which agrees with (3.57). An attractive feature of this supersymmetry transformation is that it corresponds a polynomial transformation of the field ϕ in the Chern-Simons-like equations (2.100). Moreover, the transformation of ϕ requires no picture changing insertions which break conformal invariance, which is convenient for the analysis of analytic solutions. However, the supersymmetry transformation of [13] is not a symmetry of the action since it does not implement conditions (S.d) and (S.e).

Cyclic Ramond Number Decomposition
To realize supersymmetry in the action we must make a specific choice of gauge supersymmetry products. Assuming that the dynamical products M n+1 are given as in section 2, we claim the proper choice is In this subsection our task is to use this form of σ m+1 | −1 to express the supersymmetry products in components of definite cyclic Ramond number. In this process we will see that condition (S.d) is satisfied. The proof of cyclicity of the supersymmetry products in the next subsection will then proceed by demonstrating cyclicity of the cyclic Ramond number components. Assuming the gauge products µ n+2 | 0 are defined as in section 2, the cohomomorphismĜ −1 takes a special form when it produces one Ramond output [3]: From this it follows that the gauge supersymmetry products also take a special form with one Ramond output: In the final step we used Ξ 2 = 0. Similar computations show that [3] π 1 1 m| 0 = π 1 1 m 2 | 0Ĝ , (3.77) The last two relations reexpress (2.76). We now use these formulas to compute the cyclic Ramond number decomposition of S. Since S n+1 has components at Ramond number −1 and 1, potentially it can have components at cyclic Ramond number 1 and 3: (3.80) First consider the cyclic Ramond number 3 component. We know that S n+1 | 3 3 vanishes since S n+1 does not carry Ramond number 3. Cyclicity should then imply that S n+1 | 3 1 also vanishes. However, this fact is nontrivial. We can compute S n+1 | 3 1 using (2.49): Plugging in (3.58) gives where we used (3.79) to drop one term from the commutator. Next we plug in (3.76) for σ| −1 and expand s| 1 using (3.75) to find The first term can be further simplified using m| 2 =Ĝ −1 m 2 | 2Ĝ . The second term drops out since s 1 | 1 produces only an NS output. Therefore In the second step substituted commutators since m 2 | 2 and s 1 | 1 only produce NS outputs. Finally we used that s 1 is a derivation of the star product. Therefore the cyclic Ramond number 3 component vanishes. By process of elimination, this means that the supersymmetry products only carry cyclic Ramond number 1. In particular, there can only be one Ramond state in the input and output of the supersymmetry transformation. When the supersymmetry products produce an NS state, they take the form and when they produce a Ramond state, Using (3.78) and (3.76) this can be further expressed The last term drops out since Q and s 1 commute. Now use (3.75) to insert a factor ofĜ −1 in front of s 1 | −1 in the first term: In the second term we can switch the order of s 1 | −1 and m 2 | 0 since s 1 is a derivation of the star product. Inserting a factor ofĜĜ −1 then gives Taking the NS and R outputs together, we therefore have It is natural to combine s| −1 and s| 1 into a single object: The supersymmetry products can then be expressed (3.92) Note that the Ramond output is proportional to X, and is therefore consistent with the condition XY = 1 on the Ramond string field. Therefore the supersymmetry products realize condition (S.d).

Proof of Cyclicity
We are now ready to prove cyclicity: From (3.92) we see that the factor of G −1 in the restricted symplectic form cancels a factor of G in the supersymmetry products. Therefore cyclicity is equivalent to The argument goes more easily if at intermediate steps we allow ourselves to contract with states in the large Hilbert space. Therefore we will prove the stronger relation which holds contracted with arbitrary states in H. The restriction to cyclic Ramond number 1 can be implemented by operating with the projector π 1 : Next we separate s and m into components of definite Ramond number, and commute the projector π 1 through to operate on π 2 . This produces Our task is to show that the two terms cancel. Let us start with the first term. Since σ| −1 necessarily produces a Ramond output, one piece of the commutator vanishes against π 0 2 , which only accepts NS inputs. Also, it follows from the construction of the NS open superstring field theory [20] that the cohomomorphismĜ −1 is cyclic when it produces only NS outputs [3]. In particular this means Therefore the first term in (3.97) simplifies to where we substituted s| 1 =Ĝ −1 s 1 | 1Ĝ and m| 2 =Ĝ −1 m 2 | 2Ĝ . Evaluating the second term in (3.97) requires more work. We represent the projector π 2 2 in terms of the product and coproduct [12,3] π 2 2 = △ (π 1 1 ⊗ ′ π 1 1 )△, (3.100) and act the coproduct to the right. This gives We can evaluate the action of π 1 1 on the above coderivations using (3.75)-(3.79): Note that the last two terms cancel against the first two terms using so this simplifies to We expand further by writing (3.106) Using the BPZ even property of Ξ we can write the second term The same manipulation applies to the fourth term in opposite order. Therefore (3.108) Using the BPZ even property of Ξ we can write the sixth term (3.109) The same manipulation applies to the eighth term in opposite order. Thus we find There is some cancellation: The structure of these terms is such that we can pull the coproduct back to the left towards the product and replace again with π 2 2 . This leaves Adding the first and second terms in (3.97) then gives which vanishes since both s 1 and m 2 are cyclic with respect to the large Hilbert space symplectic form. This completes the proof of cyclicity. Therefore the supersymmetry products satisfy all conditions (S.a)-(S.e) required to define a symmetry of the action.

Supersymmetry Algebra
Now that we have the supersymmetry transformation, it is interesting to investigate the form of the supersymmetry algebra. On general grounds we expect the supersymmetry algebra to appear as [δ ′ susy , δ susy ]Ψ = −2P 1 Ψ + trivial terms, (4.1) where P 1 is the momentum operator and ǫ a and ǫ ′ a are the parameters defining the supersymmetry transformations δ susy and δ ′ susy , respectively. We will use a prime to denote objects defined with the primed parameter ǫ ′ a . The supersymmetry algebra may in addition contain a gauge transformation or symmetry transformation which vanishes on-shell. These transformations act trivially on physical observables.
It is clear that the commutator of supersymmetry transformations will be a nonlinear function of the string field. The momentum operator only acts on one string field, so the remaining nonlinear terms must be a combination of gauge transformations and symmetries which vanish on-shell. For a generic supersymmetric field theory, determining the explicit form of these transformations may be difficult. In our case, with some motivation from the deformation theory of A ∞ algebras, we can anticipate that they will take a fairly specific form. Consider a deformation of an With this understanding, it is natural to expect that the commutator of supersymmetry transformations may not exactly produce the momentum operator, but a symmetry which is equivalent to the momentum operator in the cohomology of [M, ·]. Expressed in terms of coderivations, this means that the supersymmetry algebra will take the form S, S ′ = −2P 1 + M, T , (4.7) for some degree odd coderivation T. To understand what this implies, compute [δ ′ susy , δ susy ]Ψ = π 1 S, S ′ The second term above represents an infinitesimal gauge transformation of Ψ with a gauge parameter The third term vanishes assuming the equations of motion and therefore represents a symmetry which vanishes on-shell. Therefore our main task is to compute T. For the above interpretation of the supersymmetry algebra to be consistent, we must require the following properties: (T.a) T must satisfy (4.7).
(T.b) The products of T must carry the appropriate ghost and picture number so that (4.9) is an allowed gauge transformation of the dynamical string field.
(T.c) The products of T multiply consistently in the small Hilbert space. In particular, we require η, T = 0. (T.e) T must be cyclic with respect to the restricted symplectic form: Conditions (T.b), (T.c) and (T.d) imply that the restricted space is closed under multiplication with the products of T, so that in particular (4.9) is a well-defined gauge transformation. Condition (T.e) is required so that [M, T] is a cyclic coderivation, and therefore generates a symmetry of the action.

Computation of Trivial Term in Supersymmetry Algebra
As a first step we will give a definition It is useful to introduce the operators (4.14) ̟ 1 is degree odd, ghost number −1 and picture 0, and p 1 is degree even, ghost number 0, and picture −1. We can think of ̟ 1 as a "momentum gauge product" and p 1 a "momentum bare product." We have the relations which follow from the fact that P 1 , ̟ 1 and p 1 are zero modes of weight 1 primaries. The operator p 1 appears in the "supersymmetry algebra" generated by s 1 : This is not quite a supersymmetry algebra since p 1 is not the standard momentum operator.
To find T we start by using (3.58) to compute the commutator This is almost has the structure of (4.7), but the momentum operator is missing. We therefore add and subtract 2P 1 and attempt to absorb 2P 1 into the commutator with M. This can be achieved as follows. Since the gauge products are independent of the the position coordinate, we have the identity which we can further write as Moreover, since ̟ 1 is a derivation of the star product we have Absorbing the factors ofĜ into the commutator gives where In this way we can absorb 2P 1 into the commutator with M, giving From this we can read off T: In principle we could add an [M, ·]-exact term, but we will show that this is not necessary. Note that T carries Ramond number zero. Now we must confirm that T is in the small Hilbert space. For this we need the identities Thus we find (4.32) The first three terms cancel as follows: where we used (4.19). For the remaining terms, note that since products cannot carry Ramond number −2. Therefore we can rearrange Consider the object in parentheses in the first term: which vanishes because s 1 is a derivation of the star product. The object in parentheses in the second term vanishes for the same reason after interchanging ǫ a and ǫ ′ a . Therefore we have found T satisfying conditions (T.a)-(T.c).

Cyclic Ramond Number Decomposition and Cyclicity
We now demonstrate that T satisfies conditions (T.d) and (T.e) provided the dynamical products and supersymmetry products satisfy (M.d)-(M.e) and (S.d)-(S.e). The first step is to compute the cyclic Ramond number decomposition of T. Since T carries Ramond number zero, it can have components at cyclic Ramond number zero and two: (4.37) At first we might anticipate that T| 2 will vanish, since it must vanish when operating on two Ramond states (since T has Ramond number 0) and by cyclicity it should then vanish when operating on one. The exception to this reasoning is if T| 2 is composed entirely of a 1-string product, which of course cannot be cyclically permuted to a product with two Ramond inputs. Let us see how this occurs. We compute T| 2 using where in the second step we substituted (4.28) and expanded the commutators. Now we substitute the formulas (3.75), (3.76) and (3.79) for the Ramond outputs ofĜ −1 , σ| −1 and m| 2 : The terms in parentheses in the first step correspond sequentially to the terms in (4.38). In the second step we dropped some terms which vanish by Ramond number counting. Continuing, we can insert commutators in some terms as follows: Using (4.19) and the fact that s 1 is a derivation of the star product, we find To go further we will need to know something about how ̟ 1 commutes with Ξ. In appendix A we show Taking the commutator with η implies the identity [Ξ, ̟ 1 ] = Ξp 1 Ξ. (4.43) Applying this to (4.41) gives Canceling theĜs we find As anticipated, T| 2 is composed entirely of a 1-string product. However, it is not obvious that this operator preserves the constraint on the Ramond string field in the restricted space. To address this question it is sufficient to consider the action of ̟ 1 − Ξp 1 on X. In appendix A we will prove the identity where ψ 0 is the zero mode of the worldsheet fermion and show that the operator (4.46) is preserved when acting XY. Therefore T preserves the constraint on the Ramond string field, as required by condition (T.d). In appendix A we will also show that the operator (4.46) is BPZ odd, which will be important in a moment. We now turn to the proof of cyclicity. First we will consider the cyclic Ramond number zero component of T: which holds for arbitrary states in H. The computation is straightforward: (4.50) In the second step we noted that σ| −1 necessarily produces a Ramond output, and therefore vanishes against π 0 2 . For the same reason, in the third step we drop the Ramond number restriction on s and m. In the fourth step, we used thatĜ is cyclic when it produces only NS outputs, and finally we obtain zero since ̟ 1 , s 1 and m 2 are cyclic with respect to the large Hilbert space symplectic form. Next we consider cyclicity of the cyclic Ramond number 2 component of T. For Ramond states A and B in the restricted space we have In the last step we used the fact that ψ 0 b 0 δ(β 0 ) is BPZ odd. Continuing This completes the proof of cyclicity of T. In summary, we have shown that the supersymmetry algebra can be expressed through (4.7), with an explicit T satisfying all required properties (T.a)-(T.e).

Supersymmetry and the S-matrix
It is interesting to illustrate how the supersymmetry transformation of string field theory is related to the usual on-shell supersymmetry which operates on open string scattering amplitudes. In string field theory we can derive the S-matrix in the standard way by gauge fixing and deriving Feynman rules. However, the theory of A ∞ algebras gives an elegant but equivalent alternative via what is known as the minimal model. The minimal model is defined by a map (an A ∞ -quasi-isomorphism) which takes the A ∞ algebra M into an A ∞ algebra M min which operates on states satisfying the mass shell condition. The multi-string products of M min represent multi-string scattering amplitudes. Let us review the definition of the minimal model. Since the construction is in principle well-known [22,23,24], we will mostly content ourselves with providing the formulas. See especially [24] for recent discussion in the context of superstring field theory, which motivates the construction from the perspective of homological perturbation theory. The first step is to define a subspace of physical states H p where we wish to define the minimal model. We require that the subspace contains all elements of the cohomology of Q, Q + is degree odd, ghost number −1 and picture zero. In addition, we assume that Q + satisfies (Q + ) 2 = 0, (5.5) Q + Π = ΠQ + = 0.
(5.6) For string field theory amplitudes computed in Siegel gauge, the physical subspace H p consists of states satisfying the mass shell condition L 0 = 0. The projector onto this subspace may be formally represented as The contracting homotopy operator Q + is precisely the Siegel gauge propagator: It is clear that the Siegel gauge propagator satisfies (5.4) and (5.5), whereas we assume that (5.6) holds in a formal sense.
Next we promote Π and Q + to natural operations on the tensor algebra. We lift Π to a cohomomorphismΠ which acts on an n-string state simply asΠ π n = Π ⊗ ... ⊗ Π n times π n . (5.9) Also, [Q,Π] = 0. The contracting homotopy Q + is lifted into an operator Q + : Note that Q + is not quite a coderivation because inputs to the right of Q + above are projected by Π. Nevertheless the coproduct acts in a simple way: The rationale for the definition of Q + is the property which can be viewed as a tensor algebra analogue of (5.4). We also have (Q + ) 2 = 0, Q +Π =ΠQ + = 0, (5.14) corresponding to (5.5) and (5.6). The minimal model for an A ∞ algebra M can be expressed in the form M min =PMÎ, (5.15) where the cohomomorphismsP andÎ are called projection and inclusion maps, respectively. The projection map P takes an element of T H restricted into an appropriate element of T H p , while the inclusion mapÎ takes an element of T H p into an appropriate element of T H restricted . They are given by the formulaŝ where δM is the interacting part of the A ∞ algebra M: It is also useful to introduce a nonlinear generalization of Q + : With some computation one can establish the following properties: In particular M min is nilpotent, and the second relation implies that it acts as a coderivation on T H p . Therefore the products of M min define an A ∞ algebra on the subspace of physical states. The n + 1 string product of M min defines the color-ordered n + 2 string scattering amplitude: To see that this identification is plausible, note that after substituting the formulas forP andÎ we may express (5.15) in the form From this we can compute (for example) the 4-point amplitude by extracting the 3-string product. For the Siegel gauge amplitude we obtain M min,3 π 3 = π 1Π Q + δM In the last step we formally assumed that b 0 /L 0 annihilates Π. The first term gives the contribution from the quartic vertex, and the second and third terms give the contributions from a pair of cubic vertices connected by a propagator in the s and t channels. Let us evaluate the amplitude on a pair of Ramond states R 1 , R 2 and a pair of NS states N 1 , N 2 in H p : Here we simplified the amplitude knowing the form of the products with Ramond output and using the Ramond constraint XY = 1. Note that the diagram containing an intermediate Ramond state inherits a factor of X in the propagator, as would be expected from the propagator as derived by gauge fixing the Ramond kinetic term. Now let us discuss supersymmetry. By analogy to (5.15), one might guess that the supersymmetry transformation in the minimal model will be described by S min =PSÎ, (5.29) which acts as a coderivation on T H p . Assuming that [S 1 , Q + ] = [S 1 , Π] = 0, which holds in Siegel gauge, this can be written explicitly as S min =Π S 1 + 1 1 + δMQ + δS Note that the second two terms contain the projector Π between multi-string products. Such terms will only contribute if the external momenta are adjusted so as to produce intermediate states on the mass shell. Therefore for generic external momenta this relation simplifies to 0 =Π S 1 , δM 1 1 − Q + δM − Q, 1 1 + δMQ + δS 1 1 + Q + δM Π , (for generic momenta). (5.37) Moreover, for physical scattering amplitudes the external states will be BRST invariant. Therefore the second term, which is Q exact, will drop out. All that remains is the first term, which can be written [S 1 , M min ] = 0, (for BRST invariant states at generic momenta). (5.38) All nonlinear terms in the supersymmetry transformation have dropped out. With the identification (5.25), we conclude that scattering amplitudes satisfy A(S 1 Φ 1 , Φ 2 , ..., Φ n ) + A(Φ 1 , S 1 Φ 2 , ..., Φ n ) + ... + A(Φ 1 , Φ 2 , ..., S 1 Φ n ) = 0, (5.39) where Φ i are BRST invariant states with generic momenta. This is the expected statement of supersymmetry at the level of the S-matrix.