NS-NS Sector of Closed Superstring Field Theory

We give a construction for a general class of vertices in superstring field theory which include integration over bosonic moduli as well as the required picture changing insertions. We apply this procedure to find a covariant action for the NS-NS sector of Type II closed superstring field theory.


Introduction
Though bosonic string field theory has been well-understood since the mid 90's [1,2,3,4], superstring field theory remains largely mysterious. In some cases it is possible to find elegant formulations utilizing the large Hilbert space [5,6,7,8,9], but it seems difficult to push beyond tree level [10,11,12,13] and the presumed geometrical underpinning of the theory in terms of the supermoduli space remains obscure. A somewhat old-fashioned alternative [14] is to formulate superstring field theory using fields in the small Hilbert space. A well known complication, however, is that one needs a prescription for inserting picture changing operators into the action. This requires an apparently endless sequence of choices, and while limited work in this direction exists [15,16,17], it has not produced a compelling and fully explicit action.
Recent progress on this problem for the open superstring was reported in [18], inspired by studies of gauge fixing in Berkovits' open superstring field theory [19]. The basic insight of [18] is that the multi-string products of open superstring field theory can be constructed by passing to the large Hilbert space and constructing a particular finite gauge transformation through the space of A ∞ structures. The result is an explicit action for open superstring field theory which automatically satisfies the classical BV master equation. In this paper we generalize these results to define classical actions for the NS sectors of all open and closed superstring field theories. Of particular interest is the NS-NS sector of Type II closed superstring field theory, for which a construction in the large Hilbert space appears difficult [20]. 4 The main technical obstacle for us will be learning how to accommodate vertices which include integration over bosonic moduli, and for the NS-NS superstring, how to insert additional picture changing operators for the rightmoving sector. These results lay the groundwork for serious consideration of the Ramond sector and quantization of superstring field theory. This is of particular interest in the context of recent efforts to obtain a more complete understanding of superstring perturbation theory [22,23,24,25,26].
This paper is organized as follows. In section 2 we review the algebraic formulation of open and closed string field theory in terms of A ∞ and L ∞ algebras, with an emphasis on the coalgebra description. This mathematical language gives a compact and convenient notation for expressing various multi-string products and their interrelation. In section 3 we revisit Witten's open superstring field theory in the −1 picture [14], but generalizing [18], we allow vertices which include integration over bosonic moduli as well as the required picture changing insertions. We find that the multi-string products can be derived from a recursion involving a two-dimensional array of products of intermediate picture number. The recursion emerges from the solution to a pair of differential equations which follow uniquely from two assumptions: that the products are derived by gauge transformation through the space of A ∞ structures, and that the gauge transformation is defined in the large Hilbert space. In section 4, we explain how this construction generalizes (with little effort) to the NS sector of heterotic string field theory. In section 5 we consider the NS-NS sector of Type II closed superstring field theory. We give one construction which defines the products by applying the open string recursion of section 3 twice, first to get the correct picture in the leftmoving sector and and again to get the correct picture in the rightmoving sector. This construction however treats the left and rightmoving sectors asymmetrically. Therefore we provide a second construction based on the idea of describing closed string multiplication as a tensor product of two "open string" multiplications, where the two open strings represent the left and rightmoving degrees of freedom of the closed string. This allows us to apply the construction of section 3, with slight alterations, simultaneously in the left and rightmoving sectors, ensuring a solution of the L ∞ relations respecting left/right symmetry. We end with some conclusions.

A ∞ and L ∞ Algebras
Here we review the algebraic formulation of open and closed string field theory in the language on A ∞ and L ∞ algebras. For the A ∞ case the discussion basically repeats section 4 of [18]. For more mathematical discussion see [27] for A ∞ and [28,29] for L ∞ .

A ∞
Let's start with the A ∞ case. Here the basic objects are multi-products b n on a Z 2 -graded vector space H b n (Ψ 1 , ..., then b n acts on such a state as We will find it useful to define the tensor product map: Applying this to a state of the form (2.3) gives There may be a sign from commuting B past the first k states. We are particularly interested in tensor products of b n with the identity map on H, which we denote I.
With these preparations, we can define a natural action of the n-string product b n on the tensor algebra of H: The tensor algebra has a natural coalgebra structure, on which b n acts as a coderivation (see, for example, [27]). We will usually indicate the coderivation corresponding to an n-string product with boldface. The coderivation b n can be defined by its action on each H ⊗N component of the tensor algebra. If it acts on H ⊗N ≥n , we have This means that multi-string products in open string field theory, packaged in the form of coderivations, naturally define a graded Lie algebra. This fact is very useful for simplifying the expression of the A ∞ relations. Open string field theory is defined by a sequence of multi-string products of odd degree satisfying the relations of a cyclic A ∞ algebra. We denote these products (2.12) where Q is the BRST operator and The A ∞ relations imply that the BRST variation of the nth product M n is related to sums of compositions of lower products M k<n . This is most conveniently expressed using coderivations: 14) The first and last terms represent the BRST variation of M n . For example, the fact that Q is a derivation of the 2-product is expressed by the equation, Commutators in this paper are always graded with respect to degree.
Using (2.11), this implies and acting on a pair of states Ψ 1 ⊗ Ψ 2 gives which is the familiar expression of the fact that Q is a derivation (recalling that M 2 has odd degree.) To write the action, we need one more ingredient: a symplectic form 19) and is graded antisymmetric: Gauge invariance requires that n-string products are BPZ odd: ω|I ⊗ M n = − ω|M n ⊗ I, (2.21) so that they give rise to cyclic vertices (in this case the products define a so-called cyclic A ∞ algebra). Then we can write a gauge invariant action which are graded symmetric upon interchange of the arguments. For us, H is the closed string state space, and the Z 2 grading, called degree, is identical to Grassmann parity (unlike for the open string, where degree is identified with Grassmann parity plus one.).
Since the products are (graded) symmetric upon interchange of inputs, they naturally act on a symmetrized tensor algebra. We will denote the symmetrized tensor product with a wedge ∧. It satisfies The wedge product is related to the tensor product through the formula The sum is over all distinct permutations σ of 1, ..., n, and the sign (−1) ǫ(σ) is the obvious sign obtained by moving Φ 1 , Φ 2 , ..., Φ n past each other into the order prescribed by σ. Note that if some of the factors in the wedge product are the identical, some permutations in the sum may produce an identical term, which effectively produces a k! for k degree even identical factors (degree odd identical factors vanish when taking the wedge product). With these definitions, the closed string product b n can be seen as a linear map from the n-fold wedge product of H into H: Acting on a state of the form (2.25), where the right hand side is the n string product as denoted in (2.23). Since the states (2.25) form a basis, this defines the action of b n on all states in H ∧n . We can define the wedge product between linear maps in a similar way as between states: We replace wedge products with tensor products and sum over permutations, as in (2.25). Therefore, the wedge product of linear maps is implicitly defined by the tensor product of linear maps, via (2.7). While this seems natural, expanding multiple wedge products out into tensor products is usually cumbersome. However, the net result is simple. Suppose we have two linear maps between symmetrized tensor products of H: (2.28) Their wedge product defines a map On states of the form (2.25), A ∧ B acts as (2.30) where σ is a permutation of 1, ..., k +m, and Σ ′ means that we sum only over permutations which change the inputs of A and B. (Permutations which only move around inputs of A and B produce the same terms, and are only counted once). The sign ǫ(σ) is the sign obtained from moving the Φ i s past each other and past B to obtain the ordering required by σ. For example, let's consider wedge products of the identity map, where potentially confusing symmetry factors arise. Act I ∧ I on a pair of states using (2.30): (2.31) Here we find a factor of two because there are two permutations of Φ 1 , Φ 2 which switch entries between the first and second maps. Alternatively, we can compute this by expanding in tensor products: Here the factor of two comes because there are two ways to arrange the first and second identity map (which happen to be identical). In this way, it is easy to see that the identity operator on H ∧n is given by The inverse factor of n! is needed to cancel the n! over-counting of identical permutations of I. With these preparations, we can lift the closed string product b n to a coderivation on the symmetrized tensor algebra: 7 b n : SH → SH, On the H ∧N ≥n component of the symmetrized tensor algebra, b n acts as This means that, when described as coderivations on the symmetrized tensor algebra, the products of closed string field theory naturally define a graded Lie algebra. Closed string field theory is defined by a sequence of multi-string products of odd degree satisfying the relations of a cyclic L ∞ algebra. We denote these products The L ∞ relations imply that the BRST variation of the nth closed string product L n is related to sums of compositions of lower products L k<n . In fact, expressed using coderivations, the L ∞ relations have the same formal structure as the A ∞ relations: What makes these relations different is the L n s act on the symmetrized tensor algebra, rather than the tensor algebra as for the open string. Consider for example the third L ∞ relation, which should characterize the failure of the Jacobi identity for L 2 in terms of the BRST variation of L 3 . To write this identity directly in terms of the products, use (2.37): Acting on a wedge product of three states, according to (2.30) we must sum over distinct permutations of the states on the inputs. With (2.27), this gives a somewhat lengthy expression: The first four terms represent the BRST variation of L 3 , and the last three terms represent the Jacobiator computed from L 2 .
To write the action, we need a symplectic form for closed strings: Note that ω| acts on a tensor product of two closed string states (rather than the wedge product, which would vanish by symmetry). Writing ω|Φ 1 ⊗ Φ 2 = ω(Φ 1 , Φ 2 ), the symplectic form is related to the closed string inner product through where c − 0 ≡ c 0 − c 0 . 8 Closed string fields are assumed to satisfy the constraints With these conventions the symplectic form is graded antisymmetric: 9 Gauge invariance requires that n-string products are BPZ odd: This implies that the vertices are symmetric under permutations of the inputs. (This is called a cyclic L ∞ algebra, though the vertices have full permutation symmetry). With these ingredients, we can write a gauge invariant closed string action,

Witten's Theory with Stubs
In this section we revisit the construction of Witten's open superstring field theory. Unlike [18], where the higher vertices were built from Witten's open string star product, here we consider a more general set of vertices which may include integration over bosonic moduli. Such vertices are at any rate necessary for the closed string [30]. Witten's superstring field theory is based on a string field Ψ in the −1 picture. It has even degree (but is Grassmann odd), ghost number 1, and lives in the small Hilbert space. The action is defined by a sequence of multi-string products satisfying the relations of a cyclic A ∞ algebra. Since the vertices must have total picture −2, and the string field has picture −1, the (n + 1)st product M (n) n+1 must carry picture 8 The BPZ inner product is conventionally defined with the conformal map I(z) = 1/z for closed strings. 9 The extra sign in front of the closed string inner product in (2.45) was chosen to ensure graded antisymmetry of the symplectic form. Without the sign, the closed string inner product itself has the symmetry of an odd symplectic form, like the antibracket. This symmetry however is somewhat awkward to describe in the tensor algebra language. Note that, with our choice of symplectic form, permutation symmetry of the vertices produces signs from moving fields through the products L n . n. 10 We keep track of the picture through the upper index of the product. The goal is to construct these products by placing picture changing operators on a set of n-string products defining open bosonic string field theory: where the bosonic string products of course carry zero picture. We can choose M can be chosen to vanish. This is the scenario considered in [18]. Here we will not assume that M The presence of stubs means that the propagators by themselves will not cover the full bosonic moduli space, and the higher products

Cubic and Quartic Vertices
We start with the cubic vertex, defined by a 2-product M The picture changing operator X takes the following form: where f (z) a 1-form which is analytic in some nondegenerate annulus around the unit circle, and satisfies The first relation implies that X is BPZ even, and the second amounts to a choice of the open string coupling constant, which we have set to 1. Since Q and X commute, Q is a derivation of M Together with [Q, Q] = 0, this means that the first two A ∞ relations are satisfied. However, M (1) 2 is not associative, so higher products M 3 , M 4 , ... are needed to have a consistent A ∞ algebra.
To find the higher products, the key observation is that M Here we introduce a degree even product is BRST exact means that it can be generated by a gauge transformation through the space of A ∞ structures [18]. So to find a solution to the A ∞ relations, all we have to do is complete the construction of the gauge transformation so as to ensure that M and an array of higher-point products µ (k) l of even degree. We will call these "gauge products." 12 The first nonlinear correction to the gauge transformation determines the 3-product M where we introduce a gauge 3-product µ with picture number two. Plugging in and using the Jacobi identity, it is easy to see that the 3rd A ∞ relation is identically satisfied: 3 ].
(3.12) 11 Note that the cohomology of Q and η is trivial in the large Hilbert space. 12 The notation and terminology for products used here differs from [18]. The relation between here and there is M However, the term [Q, µ 3 ] in (3.11) does not play a role for this purpose. This term is needed for a different reason: to ensure that M with picture 1, satisfying to be in the small Hilbert space implies where we introduce yet another gauge 3-product µ 3 with picture number 1. In [18] it was consistent to set µ is in the small Hilbert space, as is required by (3.13). We define µ where M 3 is the bosonic 3-product. Then taking η of (3.14) implies This is nothing but the 3rd A ∞ relation for the bosonic products. The upshot is that we can determine M for Witten's superstring field theory by climbing a "ladder" of products and gauge products starting from M (0) 3 as follows: 3 ] + [M 2 , µ  interspersed with n gauge products adding picture number one step at a time. Thus we will have a recursive solution to the A ∞ relations, expressed in terms of a "triangle" of products, as shown in figure 3.1.

All Vertices
We now explain how to determine the vertices to to all orders. We start by collecting superstring products into a generating function so that the (n + 1)st superstring product can be extracted by looking at the coefficient of t n . Here we place an upper index on the generating function (in square brackets) to indicate the "deficit" in picture number of the products relative to what is needed for the superstring. In this case, of course, the deficit is zero. The superstring products must satisfy two properties. First, they must be in the small Hilbert space, and second, they must satisfy the A ∞ relations: Expanding the second equation in powers of t gives the A ∞ relations as written in (2.14).
To solve the A ∞ relations, we postulate the differential equation (1) bosonic string products superstring products where is a generating function for "deficit-free" gauge products. Expanding (3.30) in powers of t gives previous formulas (3.8) and (3.13) for the 2-product and the 3-product. Note that this differential equation implies  [18]. In this context, the differential equation (3.30) says that changes of the coupling constant are implemented by a gauge transformation through the space of A ∞ structures, and µ [0] (t) is the infinitesimal gauge parameter. The statement that the coupling constant is "pure gauge" normally means that the cubic and higher order vertices can be removed by field redefinition, and the scattering amplitudes vanish [31]. This does not happen here because µ [0] (t) is in the large Hilbert space, and therefore does not define an "admissible" gauge parameter. But then the nontrivial condition is that the superstring products are in the small Hilbert space despite the fact that the gauge transformation defining them is not. To see what this condition implies, take η of the differential equation (3.30) to find where is the generating function for products with a single picture deficit. Now we can solve (3.33) by postulating a new differential equation is a generating function for gauge products with a single picture deficit. Now we are beginning to see the outlines of a recursion. Taking η of (3.35) implies a constraint on the generating function for products with two picture deficits M [2] (t), which can be solved by postulating yet another differential equation, and so on. The full recursion is most compactly expressed by packaging the generating functions and the lower order products The lower order products are either again determined by (3.46), or they are products of the bosonic string, which we assume are given. So now we must find the gauge products µ m+n+2 . The solution is not unique. However there is a natural ansatz preserving cyclicity: or, more compactly, we can write µ  The final step in this ladder is the n + 1-string product of Witten's open superstring field theory. Incidentally, note that the nature of this construction guarantees that the superstring products will define cyclic vertices if the bosonic products do (see appendix B of [18]).

NS Heterotic String
Our analysis of the open superstring almost immediately generalizes to a construction of heterotic string field theory in the NS sector. An alternative formulation of this theory, using the large Hilbert space, is described in [7,8]. The closed string field is a degree (and Grassmann) even NS state Φ in the superconformal field theory of a heterotic string. Note that the βγ ghosts and picture only reside in the leftmoving sector. The string field has ghost number 2 and picture number −1, and satisfies the b − 0 and level matching constraints (2.47). An on-shell state in Siegel gauge takes the form where O m is a matter primary operator with left/rightmoving dimension ( 1 2 , 1). The symplectic form (2.45) is nonvanishing only on states whose ghost number adds up to five and whose picture number adds up to −2.
The action is defined by a sequence of degree odd closed string products which, or course, have vanishing picture. The explicit definition of the closed bosonic string products is an intricate story [2,32,33,34,35], but for our purposes all we need to know is: 1) they satisfy the relations of a cyclic L ∞ algebra, 2) they are in the small Hilbert space, 3) they carry vanishing picture number, and 4) they are consistent with the b − 0 and L − 0 constraints. The problem we need to solve appears completely analogous to the open superstring. Aside from replacing tensor products with wedge products, there is one minor difference. Since the products of the heterotic string must respect the b − 0 and L − 0 constraints, the picture changing operator X in the 2-product must be identified with the zero mode X 0 . This way, we can pull b − 0 and L − 0 past X 0 to act on L (0,0) 2 , which vanishes. More generally, we must construct closed superstring products using the ξ zero mode rather than a more general charge which would be consistent for the open string. Following the discussion of the open superstring, we introduce a "triangle" of products and gauge products, λ of intermediate picture indicated in the upper index. We build the (n + 1)-heterotic string product L (n) n+1 by climbing a "ladder" of products The only differences from the open superstring are that the coderivations act on the symmetrized tensor algebra, and ξ has been replaced by ξ 0 .

NS-NS Closed Superstring
We are now ready to discuss the NS-NS sector of Type II closed superstring field theory. A recent proposal for defining this theory in the large Hilbert space appears in [21]. The closed string field is a degree even (and Grassmann even) NS-NS state Φ in the superconformal field theory of a type II superstring. Now βγ ghosts and picture occupy both the leftmoving and rightmoving sectors. The string field has ghost number 2, satisfies the b − 0 and L − 0 constraints (2.47), and has left/rightmoving picture number (−1, −1). Onshell states in Siegel gauge take the form where O m is a superconformal matter primary of weight ( 1 2 , 1 2 ). The symplectic form (2.45) is nonvanishing on states of ghost number 5 and left/right picture (−2, −2).
The theory is defined by a sequence of degree odd closed string products , ... , (5.2) satisfying the relations of a cyclic L ∞ algebra. The (n + 1)st closed string product must have left/right picture (n, n). These products should be constructed from the products of the closed bosonic string, , we surround it once with a leftmoving picture changing operator X 0 , and again a rightmoving picture changing operator X 0 , to produce the expression Note that since X 0 and X 0 commute it does not matter which order we apply them to the bosonic product.

First Solution: Composing Open Strings
The easiest solution for the closed superstring is to apply the open string construction twice: The first time to get the correct picture number for leftmovers and a second time to get the correct picture number for the rightmovers. More specifically we proceed as follows. Starting with the bosonic product L again using (4.9) and (4.10), but this time the leftmoving picture is a spectator index, and the right moving zero mode ξ 0 appears in (4.10) rather than the leftmoving one. Thus, for example the 2-product is constructed by climbing two ladders: = given by first ladder, ].

(5.7)
This is the simplest construction we have found the NS-NS superstring, in the sense that it requires the fewest auxiliary products of intermediate picture number in defining the recursion. However, it suffers from a curious asymmetry between left and rightmoving picture changing operators. This asymmetry first appears in L (2,2) 3 , which for example has a term of the form 8) and no corresponding term with left and rightmovers reversed.

Second Solution: Tensoring Open Strings
To restore symmetry between left and rightmovers we consider a different solution of the L ∞ relations. To motivate the structure, consider the 2-product L ]. (5.9) Now we have introduced two gauge products. The first λ (1,1) 2 will be called a "left" gauge product, and is defined by replacing X 0 in the expression (5.4) for L (1,1) 2 with ξ 0 . The second λ (1,1) 2 will be called a "right" gauge product, and is defined by replacing X 0 in L (1,1) 2 with ξ 0 . Once we act with Q, λ is derived by filling a "diamond" of products and gauge products, as shown in figure 5.1. Also shown is a "diamond" illustrating the derivation of the 3-product, which has four "cells" giving a total of 21 intermediate products. The explicit formulas associated with this diagram are difficult to guess, so we will proceed to motivate the general construction.
To find the closed superstring product L (n,n) n+1 , we need a diamond consisting of (n + 1) 2 products L (p,q) n+1 , 0 ≤ p, q ≤ n, (5.15) n(n + 1) left gauge products 16) and n(n + 1) right gauge products We would like to package the products into three generating functions L(s, s, t), λ(s, s, t), λ(s, s, t), (5.18) which depend on three variables, corresponding to the three indices characterizing the products. The variable t counts the total picture number, s the deficit in leftmoving picture number, and s the deficit in rightmoving picture number. Thus we have The summations in the generating functions may seem awkward. Unlike the open string, there is no simple relation between the picture number, picture number deficits, and the number of entries in the product. This is a consequence of the fact that the closed superstring has left/rightmoving sectors with separate picture numbers, but not separate notions of multiplication. We can solve this problem by formally introducing an extra index to indicate "rightmoving multiplication:" This suggests a formal analogy between the closed string products and the tensor product of two open string products: Note that if closed string products were simply a pair of open string products, we would need another variable t in the generating function to indicate the rank of rightmoving multiplication. Then (5.31) would split into two differential equations with respect to t and with respect to t. But since left/right multiplication is identified, these two equations merge into one. Note that at t = 0 L(s, s, t) reduces to a generating function for the bosonic products: ] . (5.36) The sum over r, s include all values such that the product and gauge product in the commutator have admissible picture numbers. Explicitly, in the commutator with λ, and in the commutator with λ, Similar to (3.46), this formula determines the products recursively given the products of the bosonic string and the left/right gauge products. The left/right gauge products are defined by solving (5.32) and (5.33), and following the argument of section 3.2 we find natural solutions Once we know all products and gauge products with up to n + 1 inputs, we can determine the (n + 2)nd superstring product L (n+1,n+1) n+2 by filling a "diamond" of products of intermediate picture number, starting from the bosonic product L (0,0) n+2 at the bottom. Filling the diamond requires climbing 4(n + 1) levels, 2(n + 1) of those require computing gauge products from products using (5.39) and (5.40), and the other 2(n + 1) require computing products from gauge products using (5.36).
Just to see this work, let's write the necessary formulas to determine the 3-product L In the first level we have two gauge products, In the second level, two products: In the third level, four gauge products: In the fourth level, three products: In the fifth level, four gauge products: In the sixth level, two products: In particular, the (n + 1)st superstring product derived from (5.62) will be related to the (n+1)st superstring product derived from (5.31) by factor of (cc) n , which can be absorbed into a redefinition of the coupling constant. A more substantial generalization would take c and c to be functions of t. This generalization can be understood as follows. If we were describing a tensor product of open strings, the generating functions would depend on an extra parameter t and (5.31) would split into two equations ∂ ∂t L(s, s, t, t) = L(s, s, t, t), λ(s, s, t, t) , ∂ ∂t L(s, s, t, t) = L(s, s, t, t), λ(s, s, t, t) .
(5.64) For the closed string these equations are overdetermined because of the identification between left and right multiplication. However, we can impose a linear combination of these equations tangent to a given curve in the t, t plane, defined by functions t(τ ), t(τ ) of some parameter τ . Thus we have ∂ ∂τ L(s, s, τ ) = L(s, s, τ ), dt(τ ) dτ λ(s, s, τ ) + dt(τ ) dτ λ(s, s, τ ) . (5.65) In this case the parameter τ no longer counts picture number, and L(s, s, τ ) loses the interpretation as a generating function-in particular, the coefficients of a possible expansion in s, s and τ do not represent the products, but are more general coderivations describing superpositions of the products. Extracting the products from (5.65) is therefore somewhat more complicated. One advantage of this setup, however, is that the superstring products described in this section and those described in section 5.1 can be understood in a common language. They correspond to two different choices of curves in the t, t plane: This section : t(τ ) = τ, t(τ ) = τ, In the former case, the products follow by evaluating L at s = s = 0 and τ = T and expanding in powers of T , while in the latter case, they follow from evaluating L at s = s = 0 and τ = 2T and expanding in powers of T . This gives one possible avenue to the proof of gauge equivalence between the products derived here and in section 5.1.

Conclusion
In this paper we have constructed explicit actions for all NS superstring field theories in the small Hilbert space. Closely following the calculations of [18], one can show that they reproduce the correct 4-point amplitudes. Since these actions share the same algebraic structure as bosonic string field theory, relaxing the ghost number of the string field automatically gives a solution to the classical BV master equation. This is a small, but significant step towards the goal of providing an explicit computational and conceptual understanding of quantum superstring field theory. The next steps of this program include • Incorporate the Ramond sector(s) so as to maintain a controlled solution to the classical BV master equation.
• Quantize the theory. Specifically determine the higher genus corrections to the tree-level action needed to ensure a solution to the quantum BV master equation.
• Understand how the vertices and propagators of classical or quantum superstring field theory provide a single cover of the supermoduli space of super-Riemann surfaces.
• Understand how this relates to formulations of superstring field theory in the large Hilbert space, which may ultimately be more fundamental.
Progress on these questions will not only help to assess whether superstring field theory can be a useful too beyond tree level, but may provide valuable insights into the systematics of superstring perturbation theory.