Ramond equations of motion in superstring field theory

We extend the recently constructed NS superstring field theories in the small Hilbert space to give classical field equations for all superstring theories, including Ramond sectors. We also comment on the realization of supersymmetry in this framework.


Introduction
The recent construction of NS actions [1,2] has raised the prospect of obtaining a secondquantized, field-theoretic description of all superstring theories. The next step in this programme is to include the Ramond sectors. As is well-known, formulating kinetic terms for the Ramond sector is complicated by the fact that the string field must carry a definite picture [3], which for a holomorphic Ramond state is naturally chosen to be −1/2. This is not the right picture to form the usual string field theory kinetic term, since in the small Hilbert space the BPZ inner product must act on states whose picture adds up to −2. While there are some proposals for circumventing this problem [4][5][6][7], at this time it is not clear what is the most promising way forward. Therefore it is worth considering a simpler problem first: namely, constructing classical field equations for all superstring theories, including Ramond sectors. This is the goal of the present paper. With the classical equations of motion, we can • compute of tree level amplitudes including Ramond asymptotic states around the perturbative vacuum or any classical solution;

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• investigate the broken and unbroken supersymmetries of classical solutions representing distinct string backgrounds; • construct classical solutions in type II closed superstring field theory representing nontrivial Ramond-Ramond backgrounds.
The last point is interesting, since Ramond-Ramond backgrounds are quite difficult to describe in the first quantized RNS formalism. While solving the equations of motion of closed string field theory is a tremendously difficult task, it does not appear to be more difficult for Ramond-Ramond backgrounds than other types of background. The essential idea behind our construction of the equations of motion is already contained in [1]. The main new ingredient will be incorporating additional labels associated with multiplication of Ramond states. Like [1,2], our approach is based on A ∞ and L ∞ algebras, and makes extensive use of associated concepts and notation. We will review the needed apparatus as we go, but for more dedicated discussion see several recent works [1,2,8,9]. A different formulation of the equations of motion using the large Hilbert space has already been provided for the open superstring in [4] and recently the heterotic string in [10,11]. Our approach has the advantage of describing type II closed superstrings as well, and, once suitable Ramond kinetic terms are formulated, might be generalized to give a classical Batalin-Vilkovisky action.

Ramond sector of open superstring
In this section we construct the Neveu-Schwarz and Ramond equations of motion for open superstring field theory using Witten's associative star product [12]. We will discuss the more general construction based on a non-associative product in the next section. The equations of motion involve two dynamical fields for the NS and R sectors: where H N and H R are the NS and R open string state spaces, respectively. Both Φ N and Ψ R are Grassmann odd and carry ghost number 1; the NS field Φ N carries picture −1 while the Ramond field Ψ R carries picture −1/2. For clarity, let us explain why the Ramond string field is a Grassmann odd object. Any state in the Ramond sector can be built by acting oscillators on the Ramond ground state c Θ s e −φ/2 (0)|0 , (2.2) where Θ s denotes the spin field [

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We take the Ramond ground state to be Grassmann even if it is a positive chirality spinor in the GSO(+) sector. Then all GSO(+) projected states built from acting oscillators on (2.2) will be Grassmann even as well. A Ramond string field is created by taking linear combinations of Ramond states with coefficients which are anticommuting spacetime fields, as is appropriate for fermions. The anticommuting fields anticommute with each other, and in addition we will assume that they anticommute with Grassmann odd worldsheet operators. 2 Therefore, in total a Ramond string field must be Grassmann odd, since it is built from Grassmann even states with Grassmann odd coefficients. Let us quickly review some notation and conventions which will be essential for our discussion. When discussing A ∞ algebras in open string field theory, it is very useful to use a shifted grading on the open string state space called degree. The degree of a string field A, denoted deg(A), is defined to be its Grassmann parity plus one (mod Z 2 ). The dynamical string fields Φ N and Ψ R are Grassmann odd, but degree even. Consider a product of m string fields: b m (A 1 , ..., A m ) . (2.5) The degree of the product b m , denoted deg(b m ), is defined to be the degree of its output minus the sum of the degrees of its inputs (mod Z 2 ). It is useful to think of the product b m as a linear map from the m-fold tensor product of the state space into the state space: is the identity operator on H ⊗n . The commutator satisfies the Jacobi identity, and [b m , c n ] defines an m + n − 1 string product. Tensor products of operators act in the natural way on tensor products of states. Given two linear maps b k,ℓ and c m,n from H ⊗ℓ → H ⊗k and H ⊗n → H ⊗m , respectively, the tensor product map b k,ℓ ⊗ c m,n satisfies b k,ℓ ⊗ c m,n (A 1 ⊗ A 2 ⊗ ... ⊗ A ℓ+n ) = (−1) deg(cm,n)(deg(A 1 )+...+deg(A ℓ )) × b k,ℓ (A 1 ⊗ ... ⊗ A ℓ ) ⊗ c m,n (A ℓ+1 ⊗ ... ⊗ A ℓ+n ). (2.10) 2 Another convention would assume that anticommuting spacetime fields commute with anticommuting worldsheet operators. This is closer in spirit to the sign rules of [14], and in this context the A∞ and L∞ algebras we will construct would be equivalently described as super A∞ or L∞ algebras. T.E. thanks U. Schreiber for explaining this convention. 3 Commutators of multi-string products are always graded with respect to degree.

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An A ∞ algebra is defined by a sequence of degree odd multi-string products d n , n = 1, 2, 3, ... which satisfy a hierarchy of identities called A ∞ relations: With these preparations we are ready to discuss the equations of motion. Witten's original proposal for open superstring field theory gives the equations of motion [15] where Q ≡ Q B is the BRST operator, X(z) = Q · ξ(z) is a picture changing operator, and * is the open string star product. As is well known, the Witten theory is singular because of collisions of picture changing operators at the midpoint [16]. We can resolve this problem by spreading the picture changing operators away from the midpoint [8], which following [1] we accomplish by postulating that the equations of motion take the form 0 = QΦ N + M 2 (Φ N , Φ N ) + m 2 (Ψ R , Ψ R ) + higher orders, (2.14) 0 = QΨ R + M 2 (Ψ R , Φ N ) + M 2 (Φ N , Ψ R ) + higher orders, (2.15) with higher order terms that we will construct in a moment. The degree odd product m 2 is Witten's open string star product with a sign needed to shift the grading from Grassmann parity to degree: The degree odd product M 2 must carry picture +1 and takes the form where X is a BPZ even charge of the picture changing operator: The function f (z) is holomorphic in the vicinity of the unit circle and is defined so that ξ is BPZ even and anticommutes with the eta zero mode η ≡ η 0 to give 1: [η, ξ] = 1. (2.19) An important observation of [1] is that M 2 is BRST exact in the large Hilbert space, and therefore may be formally obtained by an improper field redefinition from a free theory [9]. Here µ 2 is called the gauge 2-product 4

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and is degree even. It also satisfies The choice of M 2 in (2.17) was dictated in [1] by cyclicity, i.e. by the assumption that M 2 can be derived by varying the cubic vertex of an action. In the current context we are not attempting to construct an action, so cyclicity is not a meaningful requirement. This means that we can in principle make another choice of M 2 . In fact, there is no reason why the three appearances of M 2 in (2.14) and (2.15) cannot all be chosen to be different products. 5 However, our goal is not necessarily to provide the most general possible form of the equations of motion. We will try, as far as possible, to mimic the construction of the NS sector, which includes some choices which in that context were motivated by cyclicity.

Cubic order
The higher order terms in the equations of motion will be defined by a sequence of degree odd multi-string products,M which satisfy the relations of an A ∞ algebra. We use the tilde over the products to denote a composite object which appropriately multiplies both NS and R sector states. For example, if N 1 , N 2 are NS sector string fields and R 1 , R 2 are R sector string fields, the composite 2-productM 2 is defined to satisfỹ Introducing a composite string field Projecting on the NS output (or picture −1) gives the equation of motion (2.14) and projecting on the R output (or picture −1/2) gives the equation of motion (2.15). Up to cubic order the A ∞ relations are Following the recent suggestion of Sen [7], one can try to construct a tree-level action with two Ramond fields supplemented by a constraint, in a similar spirit as [6]. In this approach the 2-product of NS states can be chosen to be M2 while the 2-product of an NS and R state should be chosen to be Xm2. We did not consider this approach, though it could lead to an interesting refinement of our equations of motion. See also comments in the conclusion.

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Before we get to this, however, it is interesting to consider the product of four Ramond states:M 4 (R 1 , R 2 , R 3 , R 4 ). (2.45) Since this product would contribute to the NS part of the equations of motion (2.14), its ghost number must be −2 and its picture number must be +1. In fact, this is the first product where the ghost number is more negative than the picture number is positive. It is easy to see that any product built from composing Q, m 2 and ξ must satisfy ghost number ≥ −picture number. (2.46) This inequality must be violated for products of four or more Ramond states. Therefore such products potentially present an obstruction to our solution of the A ∞ relations. To see how this problem is avoided, consider the fourth A ∞ relation, acting on four Ramond states: Keeping careful track of the NS and R inputs ofM 3 , this can be further expanded Therefore we can simply chooseM More generally, we claim that all products with four or more Ramond states can be set to zero. Therefore the equations of motion will be cubic in the Ramond string field. At first this seems somewhat strange. If the equations of motion have terms which are cubic in the Ramond string field, cyclicity would naturally imply that they should have terms that are quartic in the Ramond string field as well. This is a clear indication that the equations of motion cannot be derived from an action. While this was expected, one JHEP11(2015)199 might still worry that quartic Ramond terms in the equations of motion are needed to get the correct physics. For example, the quadratic Ramond term m 2 (Ψ R , Ψ R ) is not implied by A ∞ relations or gauge invariance, but is required to incorporate the backreaction of the R field on the NS field. The difference at quartic order is that there is no 4-product of Ramond states at the relevant ghost and picture number which is nontrivial in the small Hilbert space BRST cohomology. Therefore, any quartic term in the Ramond string field can be removed by field redefinition. As a cross check on our equations of motion, it will be shown in [17] that they imply the correct tree level amplitudes.

All orders
A key ingredient in constructing the equations of motion at higher order is to realize that multi-string products can be characterized according to their Ramond number. The Ramond number of a product is defined to be the number of Ramond inputs minus the number of Ramond outputs required for the product to be nonzero: Ramond number = (number of Ramond inputs)−(number of Ramond outputs). (2.51) Generally, products do not have well-defined Ramond number. A product of Ramond number N has the specific property that it will be nonzero only when multiplying N or (N + 1) Ramond states (together with possibly other NS states), in which case it will respectively produce an NS or R state. When Ramond number is defined, we will indicate it by a vertical slash followed by an extra index attached to the product: If the product is nonzero, its Ramond number must be restricted to the range since the number of Ramond inputs cannot exceed the total number of inputs and the number of Ramond outputs cannot exceed one. While generically multi-string products do not possess well-defined Ramond number, they can always be decomposed into a sum of products which do. To see this, consider the projector which selects elements ofH ⊗n which have N Ramond factors. Given an n-string product b n , we can define the component at Ramond number N : Using the resolution of the identity, (2.57) A comment about notation: generally, we use b n | N to denote an n-string product of Ramond number N , but this does not necessarily mean that b n | N is derived from a product b n after projection to Ramond number N . When we do mean this, it should be clear from context. Consider a 1-string product R 1 which acts as the identity on a Ramond state and as zero on an NS state. A product has definite Ramond number N if and only if it satisfies Using the Jacobi identity, this implies that Ramond number is additive when taking commutators of products: Finally, let us mention that the products in the equations of motion always carry even Ramond number (odd Ramond number components vanish), since picture changing operators do not mix NS and R sector states. Products of odd Ramond number will play a role once we consider supersymmetry in section 6. Now let us revisit the results of the previous subsection. The BRST operator has Ramond number zero: (2.60) The composite 2-productM 2 can be written as the sum of products at Ramond number zero and two. Comparing with equations (2.24)-(2.27), we can apparently writẽ with the indicated Ramond projection of M 2 and m 2 . Note that M 2 | 0 can be derived from the Ramond number zero projection of the gauge 2-product µ 2 : Also, The composite 3-productM 3 can likewise be written as the sum of products at Ramond number zero and two:M The Ramond number zero piece corresponds to equations (2.33)-(2.36). The Ramond number two piece m ′ 3 | 2 is seemingly more complicated, as it must produce four distinct expressions (2.41)-(2.44) depending on how it multiplies two or three Ramond states. To

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derive the 3-string products, consider the third A ∞ relation: This is equivalent to two independent equations at Ramond number 0 and 2: The first equation can be solved following [1]. To review, the solution is where µ 3 | 0 is the gauge 3-product, and m 3 | 0 is the bare 3-product, With these definitions one can show that M 3 | 0 is in the small Hilbert space: The only difference between M 3 | 0 and the 3-product of [1] is the explicit restriction to Ramond number zero. To multiply more than one Ramond state we need m ′ 3 | 2 . By inspection of the Ramond number two component of the A ∞ relation, we can instantly guess the solution Happily, this simple formula reproduces all four equations (2.41)-(2.44) for 3-products of two or more Ramond states. To check, for example, we can compute the NRR product:

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With some experience from [1], it is not difficult to guess the general form of the products to all orders. Let us give the answer first, and then we can prove it. The composite (n + 2)-string productM n+2 can be decomposed As anticipated before, products with four or more Ramond states can be set to zero. In addition we will need to introduce supplemental bare products and gauge products. In total we have four kinds of product: gauge products µ n+2 | 0 : degree even, picture# = n + 1, Ramond#= 0, bare products m n+2 | 0 : degree odd, picture# = n, Ramond#= 0, which are determined recursively by the equations: The recursive procedure for constructing the products, gauge products, and bare products is illustrated in figure 1. Note that these equations are nearly the same as those from [1] determining the NS open superstring field theory. The only major difference is the appearance of a new set of products m ′ n+2 | 2 for multiplying 2 or 3 Ramond states. To prove these formulas it is helpful to work with the coalgebra representation of A ∞ algebras. See also [1,2,8,9]. Consider the tensor algebra generated by taking formal sums of tensor products of string fields: We promote an m-string product into a linear operator on the tensor algebra called a coderivation, which we denote with the same symbol in boldface: The main utility of introducing coderivations is that they can be added to each other. For example, given the products of an A ∞ algebra we can define a coderivation This single object encapsulates all products of the A ∞ algebra. To get the product d n , we act d on the n-string component of the tensor algebra and look at the result in the 1-string component of the tensor algebra. Moreover, the A ∞ relations (2.11) are equivalent to the statement that d is nilpotent: To recover the A ∞ relations (2.11), we act this equation on n-string component of the tensor algebra and look at the result in the 1-string component of the tensor algebra. Let us promote the products, gauge products and bare products to coderivations:

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and define generating functions: Note that Substituting the generating functions and expanding in powers of t, it is straightforward to show that equations (2.75)-(2.78) are equivalent to: Here the operation ξ• is defined by its action on an n-string product This operation defines a homotopy operator for the eta zero mode, in the sense that where b is an arbitrary coderivation, and η is the coderivation representing the η zero mode.

Ramond sector of open superstring with stubs
In preparation for studying the closed superstring, in this section we provide a more general construction of the open superstring equations of motion which does not require the associativity of Witten's open string star product. Specifically, we build the equations of motion by inserting picture changing operators a set of elementary products at picture zero:

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We assume that these products have odd degree, live in the small Hilbert space, and satisfy A ∞ relations. For example, we could define the elementary 2-string product M (0) 2 by attaching "stubs" to Witten's open string star product [2,18] (hence the title of this section). In the following we will need to introduce a multitude of products with different picture and Ramond numbers. We denote the number of inputs, the picture number and the Ramond number of a product as follows: The product M The goal is to construct the NS+R equations of motion, are degree odd composite products which appropriately multiply NS and R states. We require that the composite products live in the small Hilbert space and satisfy A ∞ relations. The composite products can be decomposed into a sum of products of definite Ramond and picture number, with the sum terminating when the Ramond number exceeds the number of inputs. The picture number is correlated with the Ramond number so that the NS component of the equations of motion will have picture −1 and the R component will have picture −1/2. The composite 2-product is a sum of two terms: The term with Ramond number 2 is the elementary 2-product M (0) 2 acting on two Ramond states. The term with Ramond number zero will be defined analogously to (2.17): We also have where the gauge product µ

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So far, the equations of motion are precisely the same as in the previous section with the replacement of Witten's associative star product with M (0) 2 . At higher orders the non-associativity of M (0) 2 begins to play a significant role. The higher order products at zero Ramond number were already described in [2], so let us focus on products with nonzero Ramond number. The composite 3-product is the sum of two terms:M (3.10) The A ∞ relations imply that M Pulling a Q out of this equation we have Here we introduce a new gauge 3-product µ 3 | 2 is in the small Hilbert space. In the previous section we could consistently set this product to zero -indeed, in the previous section all gauge products had vanishing Ramond number. If we postulate that η, µ where M 3 | 2 will be in the small Hilbert space: as follows from the A ∞ relations for the elementary products. In summary, the product M 3 | 2 can be constructed by climbing a "ladder" of products and gauge products, starting from M where the second step inverts (3.14). The Ramond number zero piece of the composite 3-product can be found by climbing a similar ladder, as described in [2].

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Let us proceed to quartic order. The composite 4-product can be written as the sum of three terms:M (3. 19) In the previous section the Ramond number 4 contribution could be set to zero. Now it will not vanish. Projecting the fourth A ∞ relation, onto Ramond number 4 implies Plugging in the expression for M 2 | 2 would be a 3-product at Ramond number four, which must vanish identically. We therefore conclude that where the gauge 4-product µ Therefore, the product M 4 | 4 does not vanish and can be constructed by climbing a "ladder" of products and gauge products, starting from M (3.28)

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With sightly more sophisticated use of the Jacobi identity, we can follow this procedure again to construct M 4 | 2 by climbing the "ladder:" The number of steps in the ladder increases with the number of picture changing operators we need to insert in the product. The Ramond number zero piece of the composite 4product was already constructed in [2].
Having provided a few examples, let us describe the general construction. We introduce a list of products and gauge products as follows: where the integers N, p, r take the ranges The restriction on the range of r is a little too generous -many products and gauge products on this list vanish because the Ramond number exceeds the number of inputs. (The first example is M 3 | 4 = 0). For convenience we include them on the list anyway. The restriction on the picture number p comes from the fact that the equations of motion require that products have picture number one less than the number of inputs minus half the Ramond number. Since every step on the ladder to construct a product increases the picture, we do not require products whose picture number exceeds this bound. We can alternatively parameterize the list of products and gauge products by three integers d, p, r: The integer d can be interpreted as the picture number deficit-specifically, the amount of picture that is missing from a product (with a given Ramond number and number of JHEP11(2015)199 Diagram illustrating the construction of products. The products shaded in grey appear in the equations of motion, and the elementary products at picture zero are boxed. At each order in the string field, the products in the equations of motion are distinguished according to their Ramond number, and at each Ramond number the product is derived by climbing a "ladder" of products and gauge products starting from an elementary product at picture zero. The "ladders" are shown above as a stack of products connected by vertical arrows. Each vertical arrow indicates deriving a product with one higher unit of picture from a product of one lower unit of picture by substituting (3.39) followed by (3.40). Climbing each ladder only requires the elementary products and knowledge of products already constructed at lower orders in the string field.
inputs) before it can appear in the equations of motion. The products, gauge products, and bare products introduced in the previous section correspond to a subset of this list: If we assume the associativity of Witten's star product, the remaining products and gauge products can be set to zero. Extrapolating from the first few orders, one can guess that the products and gauge products are determined recursively by a pair of equations, (3.40) The procedure for constructing the products and gauge products from these equations is illustrated in figure 2. Now we need to show that that the composite productsM n+1 which appear in the equations of motion are in the small Hilbert space and satisfy A ∞ relations. We do this JHEP11(2015)199 by lifting the products and gauge products to coderivations on the tensor algebra, and defining generating functions The parameter s counts the picture deficit, t counts the picture number, and u counts the Ramond number. We can express (3.39) and (3.40) in the form of differential equations: These are nearly identical to the differential equations which determine the products of the NS sector [2], the only difference being the label u for Ramond number which just tags along. Next, we claim that Unlike in previous examples, this relation is not directly a statement of the A ∞ relations for the elementary products, since the Ramond projections play a nontrivial role. Expanding in powers of s and u gives the formula To prove this, expand the range of summation on the left hand side to write (3.47) One can check that the additional terms included vanish because the Ramond number of one of the products in the commutator exceeds the number of inputs. Now make the substitution d ′′ = d ′ + r ′ , and relabeling d ′′ → d ′ we obtain which vanishes as a consequence of the A ∞ relations for the elementary products. This proves (3.45). Next note that Next consider the equation

Ramond sector of heterotic string
The generalization of the open superstring with stubs to the heterotic string is straightforward. It requires two steps: first we symmetrize the tensor algebra, which replaces the A ∞ structure of the previous section with an L ∞ structure. Second, we fix the operator ξ

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to be the zero mode ξ 0 for consistency with the level matching and b − 0 constraints. Those familiar with [2] may need no further explanation, but for completeness we spell it out here.
Closed string products are naturally commutative -they are symmetric (up to signs) upon interchange of entries. This means that closed string products are naturally understood as linear maps on symmetrized tensor products of closed string states. We denote the symmetrized tensor product with a wedge ∧. Given string fields Φ 1 , ..., Φ n in the closed string state space H, their symmetrized tensor product is defined where the sum is over all permutations σ of 1, ..., n and the sign (−1) ǫ(σ) is the obvious sign obtained from moving the string fields past each other into the order prescribed by σ. The commuting/anticommuting property of closed string fields is measured by their degree. For a closed string field, the degree is taken to be the same as Grassmann paritynot Grassmann parity plus one, as for open string fields. Symmetrized tensor products of n closed string states form a basis for H ∧n , and a closed string product b n is a linear map: where on the left and side b n is written as a product and on the right hand side b n written as an operator on the symmetrized tensor product of states. Given two linear maps b k,ℓ and c m,n from H ∧ℓ → H ∧k and H ∧n → H ∧m , respectively, we define a wedge product, as follows: where I n is the identity operator on H ∧n . The commutator bracket satisfies the Jacobi identity, and [b m , c n ] is an (m + n − 1)-closed-string product. An L ∞ algebra is defined by a sequence of degree odd closed string products d n , n = 1, 2, 3, ... which satisfy a hierarchy of identities called L ∞ relations:

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Finally, we will find it convenient to consider symmetrized tensor algebra We promote an m-closed-string product to coderivation on the symmetrized tensor algebra, which we denote with the same symbol in boldface It should be emphasized that, as far as our analysis is concerned, the treatment of open string A ∞ algebras and closed string L ∞ algebras is nearly identical. The only difference is that open string products and their commutators act in the appropriate way on tensor products of open string states, while closed string products and their commutators act in the appropriate way on symmetrized tensor products of closed string states. Now let us discuss the NS+R equations of motion of the heterotic string. We need two dynamical fields for the NS and R sectors: 12) where H N and H R are the Neveu-Schwarz and Ramond state spaces of a heterotic string. For the heterotic string, superconformal ghosts and picture only inhabit the left-moving sector. The NS field Φ N should be degree even (and Grassmann even), ghost number 2, and picture number −1. The Ramond field Ψ R should be degree even (and Grassmann even), ghost number 2, and picture number −1/2. With the requisite GSO projection, the Ramond field is a linear combination of Grassmann odd states with Grassmann odd coefficients. We define a composite heterotic string field This must satisfy the usual b − 0 and level matching constraints: where The composite products can be written as a sum of products of definite Ramond and picture numberL where the sum terminates when the Ramond number exceeds the number of inputs. The Ramond number and picture number are correlated so that the NS part of the equations of motion has picture −1 and the R part of the equations of motion has picture −1/2. We want to build these products by inserting picture changing operators on a set of elementary products of odd degree at picture zero: live in the small Hilbert space, and satisfy L ∞ relations. The most natural definition of the elementary products would derive from the polyhedral vertices of Saadi and Zwiebach [19], but for our purposes it will not matter how they are chosen. The procedure for constructing the products is exactly like for the open string with stubs. We introduce a list of products and gauge products: for d, p, r ≥ 0. They are defined recursively following figure 2, using the equations Note that the first equation uses the ξ zero mode Now we discuss the equations of motion for type II closed superstring field theory. For type II closed superstrings, superconformal ghosts appear in both the holomorphic and antiholomorphic sectors. Therefore, string fields and closed string products will have two respective picture numbers, which we call left-or right-moving picture number. We need four dynamical closed string fields in the small Hilbert space, with respective left-/rightmoving pictures: Here H NN is the NS-NS state space, H NR is the NS-R state space, H RN is the R-NS state space, and H RR is the R-R state space. All four string fields are degree even (and Grassmann even), ghost number 2, and satisfy the b − 0 and level-matching constraints. The NS-NS and R-R string fields represent bosons, and are given as a linear combination of Grassmann even states with commuting coefficients. The NS-R and R-NS string fields represent fermions, and are given as a linear combination of Grassmann odd states with anti-commuting (Grassmann odd) coefficients. To be consistent with the Grassmann parity of the string field, we have to appropriately fix the Grassmannality of the Ramond ground states in type IIA and type IIB string theory. Using a bar to denote antiholomorphic or right-moving operators/quantities, we assume c Θ s e −φ/2 = Grassmann even, c Θ s e −φ/2 = Grassmann even, (5.5) where in type IIB both holomorphic and antiholomorphic spinors have positive chirality, and in type IIA the holomorphic spinor has positive chirality and the antiholomorphic spinor has negative chirality. To construct the equations of motion we insert picture changing operators on a set of elementary products of odd degree at left-and right-moving picture zero: L As in the heterotic string, we can take these products to be defined by the polyhedral closed string vertices of Saadi and Zwiebach. The construction of the equations of motion requires the definition of many products at intermediate Ramond and picture numbers.

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We will indicate the number of inputs, the left-/right-moving picture number, and the left-/right-moving Ramond number of products through indices as follows: The factors of 2 in the last three equations arise from the fact that the fields can appear in either order in the 2-product. Note that the picture numbers of the products are fixed consistently with the picture of the linear terms. The products L can be taken from [2] and are defined: 14) where and ξ 0 and ξ 0 are the holomorphic and antiholomorphic ξ zero modes: These products are the natural closed superstring counterparts of (3.6).

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Let us introduce the composite string field We can write the equations of motion in the form 0 = QΦ +L 2 Φ ,Φ +L 3 Φ ,Φ,Φ + higher orders, (5.19) whereL n+1 are composite products of odd degree which appropriately multiply the four sectors of states. We require that the composite products satisfy L ∞ relations and are in the small Hilbert space. The composite products can be decomposed into a sum of products with definite left-and right-moving Ramond and picture numbers: The products ofL 2 were described in the previous paragraph. Nevertheless, let us explain how to "construct" them in a way that generalizes to higher orders. The product L The products denoted with λ and λ will be called left and right gauge products, respectively. Following [2], L (1,1) 2 | 0,0 can be constructed as follows: 6 Note that the procedure splits into two independent threads. This is the first indication that the recursion defining the products is effectively 2-dimensional, reflecting the need to insert picture in both the left-and right-moving sectors. We illustrate the construction of 2-string products diagrammatically in figure 3. In particular, the above construction of L (1,1) 2 | 0,0 corresponds to filling a "diamond" of products of intermediate left-and rightmoving picture number.
Proceeding to the next order, the construction of 3-string products is illustrated in figure 4. This requires filling two "diamonds" to construct L (2,2) 3 | 0,0 and L (1,1) 3 | 2,2 , and two "prisms" to construct L (2,1) 3 | 0,2 and L (1,2) 3 | 2,0 . To emphasize the pattern, we also illustrate the construction of 4-string products in figure 5. The explicit formulas that go along with these diagrams are easiest to motivate by proceeding directly to the differential equations for the generating functions. We start with a list of all required products and left/right gauge products:    The integers N, p, r, p, r take the respective ranges L (p,p) N +1 | 2r,2r :

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The allowed range of r and r is generous, as some products and left/right gauge products listed above vanish because the left-or right-moving Ramond number exceeds the number of inputs. Ignoring products which vanish, one can double check that this list maps onto the diagrams of figures 3-5 (note that left and right arrows represent left and right gauge products, respectively). In defining the generating functions, at first one might guess that we need six parameters s, s, t, t and u, u, counting left/right picture deficit, left/right pic-JHEP11(2015)199 ture, and left/right Ramond number, respectively. This is almost correct, but would be over-counting since the products only have five integer labels. This is a consequence of the fact that the left/right picture deficits, picture and Ramond numbers are not all independent: they are constrained by the fact that the rank of multiplication in the holomorphic and antiholomorphic sectors is the same. The correct way to deal with this is to identify t = t, so the generating functions depend on only five parameters s, s, t, u, u. Promoting the products and left/right gauge products to coderivations on the symmetrized tensor algebra, we therefore define generating functions We have inserted Kronecker deltas into the sums to enforce the appropriate relation between left/right picture deficit, picture and Ramond number and the number of inputs in the product.

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Based on our results for the open string, it is natural to guess that the generating functions should satisfy the same set of differential equations which determine the products of the NS-NS sector [2] with extra parameters u, u that go along for the ride. In particular we should have ∂ ∂t L(s, s, t, u, u) = L(s, s, t, u, u), λ(s, s, t, u, u) + λ(s, s, t, u, u) , Here the operation ξ 0 • is defined by its action on an n-closed string product This defines a homotopy operator for the coderivation η: The same equations hold for ξ 0 • with ξ 0 and η replaced with ξ 0 and the coderivation for the right moving eta zero mode η. Next, we claim that at t = 0 the generating function for products is nilpotent: where the sets S n and S n defining the summations over r ′ and r ′ are characterized by the conditions To prove (5.44), note that we can expand the range of summation on the left hand side N n=0 r ′ ∈Sn r ′ ∈Sn since the additional terms all vanish on account of the fact that the left-or right-moving Ramond number of one of the products in the commutator exceeds the number of inputs.

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The sum over r ′ , r ′ now reproduces the 2r, 2r projection of the (N + 1)st L ∞ relation for the elementary products, which gives zero: The sums in the last equation are over all values of p ′ , r ′ and p ′ , r ′ such that the products and left/right gauge products in the commutators have admissible left-and right-moving Ramond and picture numbers, as defined by the list (5.35). Explicitly, (p ′ , r ′ ) is in a set JHEP11(2015)199

Supersymmetry
Now that we have equations of motion including all NS and R sectors, it is interesting to understand in what sense they are supersymmetric. Here we will limit our discussion to the open superstring using Witten's associative star product, leaving generalizations to future work. For heterotic and type II closed superstrings, spacetime supersymmetry can be described in a different way as part of the gauge symmetry at the interacting level [7]. For other discussions of supersymmetry in open string field theory, see [4,15,20].
We will work with open string field theory on a BPS D-brane with sixteen supersymmetries. As described in [3], the simplest way to realize a supersymmetry transformation is by integrating the zero-momentum fermion vertex in the −1/2 picture: where the index on s 1 indicates that this operator is a 1-string product. The object ǫ s is the supersymmetry parameter -a constant Grassmann odd spinor. To keep the notation simple, we leave the dependence of s 1 on the supersymmetry parameter implicit. We make a GSO(+) projection in both NS and R sectors, so the supersymmetry parameter must have positive chirality. Then, following the Grassmann assignments in section 2, the supersymmetry operator s 1 will be Grassmann even (and degree even). 7 It is natural to identify the linearized supersymmetry transformation of the NS field as Likewise, the linearized supersymmetry transformation of the R field should be proportional to the NS field. However, to get the pictures to line up we need a supersymmetry operator of picture +1/2. This can be defined by integrating the zero momentum fermion vertex at picture +1/2: where Γ µ s, s ′ are 10 dimensional gamma matrices. This operator is also Grassmann even. The linearized supersymmetry transformation of the Ramond field will therefore be We can write both supersymmetry transformations in a single equation using the composite string fieldΦ = Φ N + Ψ R δΦ =S 1Φ , (6.5) with the indicated Ramond projections of S 1 and s 1 . Now the question is how this supersymmetry generalizes to the nonlinear theory. 7 The factor √ 2 is inserted to obtain the canonical normalization of the supersymmetry algebra.

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Before we get to this, however, let us mention a few essential properties of s 1 and S 1 . Both operators are in the small Hilbert space and BRST invariant It will also be useful to introduce a Grassmann odd supersymmetry operator in the large Hilbert space: and is also a derivation of the star product. Note the relationship between the supersymmetry operators S 1 , σ 1 and s 1 is somewhat analogous to the relation between the products M 2 , µ 2 , and m 2 .

Perturbative construction of supersymmetry transformation
In this subsection we give a perturbative construction of the supersymmetry transformation. It turns out that the final form of the supersymmetry transformation is easiest to understand in a different set of field variables, described in the next subsection, and some readers may wish to skip ahead. (See, in particular, equation (6.90)) The following derivation, however, has the advantage that it likely generalizes to supersymmetry transformations in other forms of superstring field theory. Our task is to find a sequence of degree even products in the small Hilbert space, S 1 ,S 2 ,S 3 , ... , (6.13) so that the supersymmetry transformation can be written whereM n+1 and related products in the equations of motion are taken from section 2. We also introduce a group-like element 8 The equations of motion and the supersymmetry transformation can be expressed in the form:M 1 We claim that the productsS n+1 generate a symmetry if M ,S = 0. (6.19) To see this, supposeΦ is a solution to the equations of motion. Then calculate the equations of motion on the transformed string fieldΦ + δΦ: Therefore the transformation is a symmetry. Note that the condition [M,S] = 0 is somewhat stronger than the statement that the transformation maps solutions into solutions. It places a nontrivial condition on the off-shell form of the supersymmetry transformation. In fact, (6.19) is the nearest we can come to the statement that the transformation is a symmetry of the action -all that is missing is a symplectic structure which would allow us to define an action and impose cyclicity. Equation (6.19) implies that the productsS n+1 satisfy a hierarchy of identities:

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To solve the first equation, we can pull a Q out of either M 2 or S 1 . The solution we prefer is to pull a Q out of M 2 , obtaining We can check that this is in the small Hilbert space: where we used conservation of Ramond number and the fact that S 1 is a derivation of m 2 .
To solve for the Ramond number 1 component ofS 2 , we pull a Q in the natural way out of the two terms in (6.26) It turns out to be convenient to introduce separate symbols for these two terms: Check that this is in the small Hilbert space: where we used conservation of Ramond number and the fact that s 1 is a derivation of m 2 . This completes the definition of the 2-productS 2 in the supersymmetry transformation.
To establish a pattern it is helpful to continue on to the 3-productS 3 , which must satisfy Q,S 3 + M 2 ,S 2 + M 3 ,S 1 = 0. (6.33) Let us look at the component of this equation at Ramond number 3: Substituting (6.29) for s 2 | 1 and (2.72) for m ′ 3 | 2 , where the additional terms either cancel or vanish identically because the Ramond number exceeds the number of inputs. From this we conclude that the Ramond number 3 component ofS 3 can be set to zero. ThereforeS 3 must be a sum of products at Ramond number This suggests we identify Check that this is in the small Hilbert space: Finally, let's look at the Ramond number 1 component of (6.33): Therefore we can identify Thus we see a pattern where the supersymmetry product at Ramond number 1 breaks up into a product denoted with (I) and a product denoted with (II), each determined by independent recursions. Let us check that s 3 | 1 is in the small Hilbert space: This completes the definition of the 3-productS 3 in the supersymmetry transformation. Now we can guess the form of the supersymmetry transformation at higher orders. The (n + 1)st product can be writteñ Components with higher Ramond number can be set to zero. In addition, s n+2 | 1 can be written as a sum The products are determined recursively by the equations, 2 | 1 = m 2 | 2 , σ 1 | −1 . Now let's prove that these products have the required properties. We promote the products to coderivations and define generating functions (6.50)

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Using the generating function of the gauge products (2.91), the recursive equations (6.47)-(6.49) can be reexpressed The generating functions for the products in the equations of motion and supersymmetry transformation take the form (6.54) The differential equations for the generating functions imply a set of equations: which proves that the productsS n+1 generate a symmetry. To prove that the transformation acts in the small Hilbert space, consider the following set of equations:

Supersymmetry in Berkovits' superstring field theory
Following [8,9], our analysis should imply a natural form for the supersymmetry transformation of the NS+R equations of motion in Berkovits' open superstring field theory [4,21,22]. To derive this supersymmetry transformation, as an intermediate step we must perform a field redefinition from our string fieldΦ to a new string fieldφ whose equations of motion and constraint are only polynomial [9]. The field redefinition is defined by the cohomomorphismĜ where the path ordered exponential is defined in sequence of increasing t ′ . In particular, we can express the relation betweenΦ andφ as where G[Φ] is given by +higher orders, (6.75) and π 1 denotes the projection onto the 1-string component of the tensor algebra. Note that this field redefinition is only defined in the large Hilbert space -it is not compatible with the small Hilbert space constraint on the string field. In [9], such a field redefinition was called an improper field redefinition. Once we have transformed fromΦ toφ, we must perform a second transformation to map fromφ into Berkovits' superstring field theory. However, it will be useful to work out the supersymmetry transformation forφ first. Let us find the equations of motion and constraint forφ. The generating functions given in (2.88)-(2.90) can be written The field redefinition implies that we can breakφ into NS and Ramond components where the NS field ϕ N is Grassmann odd and has picture −1 and the R field ψ R is Grassmann odd and has picture −1/2. Then (6.83) is equivalent to four equations: where the commutator of string fields is computed with the star product and is graded with respect to Grassmann parity. The first two equations should be interpreted as equations of motion, since they are the image of the equations of motion forΦ after the field redefinition. The second two equations should be interpreted as equations of constraint, since they are the image of the small Hilbert space constraint onΦ after the field redefinition. Note that the equations of motion forφ are quadratic the the Ramond string field, while the equations of motion forΦ are cubic in the Ramond field. This happens because the field redefinition (6.75) is at most linear in the Ramond field, and, in conjunction with the quadratic term in the Ramond field in the equations of motion forφ, this produces cubic terms in the Ramond field in the equations of motion forΦ. Now let's derive the new form of the supersymmetry transformation. The differential equations (6.51)-(6.53) imply that the supersymmetry transformation can be written in the formS =Ĝ −1sĜ . (6.89) This implies that the supersymmetry transformation ofφ can be expressed Explicitly in terms of the NS and R components, We can check thats is a symmetry of the equations of motion:

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The terms either cancel or vanish because the Ramond number exceeds the number of inputs. We can also check thats preserves the constraints: This vanishes since s 1 , S 1 and σ 1 are derivations of the star product and because the star product is associative. Conjugating byĜ, this provides an alternative proof that the supersymmetry transformationS preserves the equations of motion forΦ and is consistent with the small Hilbert space constraint. Now that we have the supersymmetry transformation forφ, it is straightforward to translate into Berkovits' superstring field theory. Berkovits' superstring field theory uses a Grassmann even NS field Φ in the large Hilbert space and at picture and ghost number 0, and a Grassmann odd R string field Ω in the small Hilbert space at picture −1/2 and at ghost number 1 [4]. These are related to ϕ N and ψ R through 9 The NS gauge parameters u, v are Grassmann even and carry ghost and picture number zero; u is BRST closed while v is in the small Hilbert space. The Ramond gauge parameter Λ is Grassmann odd and in the large Hilbert space, and carries ghost number −1 and picture number +1/2. The supersymmetry transformation is Plugging into (6.96) gives the supersymmetry transformations for the string fieldφ, and further mapping gives the original supersymmetry transformationS for the string fieldΦ. One interesting potential application of these results is to check whether or not classical solutions in open superstring field theory are supersymmetric. However, the supersymmetry transformation we have derived is not necessarily the most convenient for this purpose. Specifically, our supersymmetry transformation is natural for an NS string field at picture JHEP11(2015)199 −1, but usually analytic solutions in Berkovits superstring field theory are closely related to an NS string field at picture 0. 10 For this reason, we will find it convenient to consider a "dual" formulation of open superstring field theory using the string field where ϕ * N is a Grassmann odd NS field at picture 0 and ghost number 1 and ψ * R is a Grassmann odd Ramond field at picture −1/2 and ghost number 1. The "dual" fields satisfy the equations When ψ * R = 0, these are the equations of motion of the NS sector of the modified cubic superstring field theory [24,25]. Note that the new field equations forφ * are identical to the previous field equations forφ after the "duality" map: picture ↔ −(1 + picture). (6.106) We indicate the "dual" fields with a star * . We have Taking the "dual" of the supersymmetry transformation forφ we arrive at a new supersymmetry transformation forφ * δ * 1 1 −φ * =s * 1 1 −φ * , (6.108) Relating this "dual" formulation to Berkovits' superstring field theory requires a slight change of perspective on the Ramond sector. In the previous paragraph we described the Ramond sector of the Berkovits theory using a string field Ω in the small Hilbert space. Now it will be more natural to use a string field Ω * which is not in the small Hilbert space but is BRST closed. The Ramond field Ω * is Grassmann odd and carries ghost number 1 and picture −1/2. On shell, the Ramond fields Ω and Ω * are related by This relation is not meaningful off-shell since it is not consistent with the assumption that Ω * is BRST closed while Ω is in the small Hilbert space. The "dual" fields ϕ * N and ψ * R can be then be related to Berkovits' superstring field theory through

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In this way, (6.102) and (6.103) reduce to identities while (6.104) and (6.105) are equivalent to the equations of motion of Berkovits' superstring field theory expressed in the form In these variables, the "dual" supersymmetry transformation of the Berkovits theory takes the form Plugging into (6.111) produces to the supersymmetry transformation of ϕ * N and ψ * R given in (6.109).
Note that the supersymmetry transformation δ and the "dual" supersymmetry transformation δ * assume different off-shell degrees of freedom, and therefore really apply to different string field theories. On shell, however, they should be equivalent. In fact, one can show that they are equal up to an infinitesimal gauge transformation: (on shell), (6.114) where the Ramond gauge parameter is given by This shows that the two supersymmetry transformations are physically equivalent.
We are now ready to discuss supersymmetry of classical solutions. On a BPS D-brane there are not many classical solutions whose existence is well-established. The only known solutions represent marginal deformations of the reference boundary superconformal field theory. We will consider specifically a transverse displacement of the reference D-brane, for which it is sufficient to consider the solution [26,27] e Φ = 1 + F X 1 in the context of Berkovits' superstring field theory. As is appropriate for a classical background, the Ramond string field is zero. We use the algebraic notation for analytic solutions introduced in [28], following the conventions of [29]. The string field F is a function of K which we can take to be the square root of the SL(2, R) vacuum. The fields X and J are defined where λ is the marginal parameter describing the displacement of the D-brane, and the index + indicates a lightcone direction whose spatial component is transverse to the Dbrane. 11 The Berkovits solution implies an expression for the NS field ϕ N at picture −1

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and the "dual" NS field ϕ * N at picture zero: Note that the picture zero field is simpler, which is why the "dual" supersymmetry transformation δ * is more convenient. The Berkovits solution also implies a solutionΦ of the original equations of motion in the small Hilbert space, which up to second order in the marginal operator takes the form +higher orders. (6.121) Translation of a D-brane does not break any supersymmetries. Therefore we should be able to show that this solution is supersymmetric. The only nontrivial supersymmetry transformation to compute is for the fermion. Using the "dual" supersymmetry transformation we find Thus we need to compute s 1φ * N . This is zero because s 1 is a derivation of the star product and because s 1 K = 0, s 1 B = 0, s 1 J = 0. (6.123) The first equation follows because s 1 is the zero mode of a weight 1 primary, and the last two equations follow because there are no poles in the OPE between the −1/2 picture fermion vertex and either b or J. Therefore the solution is supersymmetric. However, the solution is not identically invariant under the supersymmetry transformation δ which is natural at picture −1. From (6.115) one finds that the supersymmetry transformation of the fermion is δΩ = η QΛ + e −Φ Qe Φ , Λ . (6.124) with Λ = −e −Φ σ 1 e Φ . While σ 1 annihilates K and B, it does not annihilate J, and the Ramond gauge parameter Λ is not zero. Translating back into our original equations of motion, this means that the supersymmetry transformationS constructed in the previous section leaves the solutionΦ in (6.121) invariant up to a gauge transformation. While it is interesting to know the supersymmetry transformation on a BPS D-brane, it is not really enough to give a full account of the role of supersymmetry in open superstring field theory. This is because at least half of the supersymmetries are spontaneously broken by the reference boundary superconformal field theory, and broken supersymmetries can in principle be restored upon expanding around a nontrivial classical solution. The most JHEP11(2015)199 dramatic example of this is tachyon condensation on a non-BPS D-brane [32], where all 32 supersymmetries are restored at the tachyon vacuum. This can be seen as follows. While we do not presently know the explicit form of broken supersymmetry transformations in superstring field theory, in any case they will generate at most an infinitesimal deformation of the tachyon vacuum. But since the kinetic operator around the tachyon vacuum has no cohomology, any infinitesimal deformation can be removed by gauge transformation, which shows that the tachyon vacuum is invariant under all supersymmetries. (In fact, this argument shows that the tachyon vacuum is invariant under all symmetries of the closed string background). For more general BPS solutions the explicit realization of this story will be more nontrivial, but progress along these lines may be possible following [30].

Supersymmetry algebra
Now that we have a supersymmetry transformation, it is interesting to compute the supersymmetry algebra. For this purpose it is easier to work with the polynomial string field ϕ in (6.74) rather than the original string fieldΦ in the small Hilbert space. Since the supersymmetry transformation ofφ is given by the coderivations, we compute [s,s ′ ], (6.125) where the prime indicates that the operator is defined with a second independent supersymmetry parameter ǫ ′ s . Once we find s,s ′ ], we can easily recover [S,S ′ ] for the original string fieldΦ by making a similarity transformation withĜ.

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The particular gauge parameters which appear the supersymmetry algebra are λ N ≡ −2 π(ǫ, ǫ ′ )ϕ N − (σ 1 s ′ 1 − σ ′ 1 s 1 )ϕ N − [σ 1 ϕ N , σ ′ 1 ϕ N ], (6.155) One can check that the gauge parameters satisfy the constraints, which is basically a consequence of the fact thatλ satisfies (6.138). Now let's write down the on-shell trivial terms. For short, let us write the NS and Ramond Euler-Lagrange functions Once we impose the equations of motion, we obtain a supersymmetry algebra of the expected form modulo gauge transformations.

Conclusions
In this paper we have constructed consistent classical field equations for all superstring theories, and for the open superstring given an explicit analysis of supersymmetry. A proof that our field equations imply the correct tree-level amplitudes will be provided in upcoming work [17]. Let us conclude by discussing future directions. Though we don't know how to write a fully satisfactory action for the Ramond sector, it should be possible to formulate a tree-level action which includes two Ramond string fields (typically, at picture −1/2 and −3/2), which are afterwards related by imposing a "selfdual" constraint on classical solution space [6]. See [33] for recent discussion. One version of this idea was recently suggested in [7], and would be particularly natural to implement using the methods of this paper. However, the required products in the equations of motion will be different from those introduced here; in a sense they will be more complicated, since even at a given Ramond number the products will differ depending on the number of Ramond states being multiplied. However, this is probably not an insurmountable complication. It would be particularly nice if an action with constraint could be realized for type II closed superstring field theory, as it would give a potentially interesting gauge invariant observable for Ramond-Ramond backgrounds. However, it remains to be seen whether an action with constraint helps in defining the quantum theory.
One important question is whether recent developments in superstring field theory can help in understanding higher genus amplitudes in superstring perturbation theory. The

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conservative approach to this problem requires first constructing a satisfactory classical action, and then quantizing following the methodology of the Batalin-Vilkovisky formalism [34]. However, given present limitations in the Ramond sector, a more pragmatic approach may be to construct a 1PI effective superstring field theory, as suggested by Sen [7,35]. The main question in this respect is whether the methods developed here and in previous works can be adapted to handle spurious singularities which appear in superconformal ghost correlators at higher genus [36]. It may also be helpful to clarify the relation between our construction of vertices and the method of "vertical integration" introduced in [37] and further developed in [38]. We hope to return to these questions soon.