Open-closed homotopy algebra in superstring field theory

We construct open-closed superstring interactions based on the open-closed homotopy algebra structure. It provides a classical open superstring field theory on general closed-superstring-field backgrounds described by classical solutions of the nonlinear equation of motion of the closed superstring field theory. We also give the corresponding WZW-like action through the map connecting the homotopy-based and WZW-like formulations.

It is known that several homotopy algebras are naturally realized as algebraic structures in string field theories and play a significant role. This was first recognized in closed bosonic string field theory [1,2], where the L ∞ structure determines the (classical) gauge-invariant action. Open bosonic string field theory was first formulated as a cubic theory using the (Witten) associative product [3] but can be extended to that with an A ∞ structure more generally [4,5]. This is also deformed to the theory on general closed string backgrounds [6][7][8] based on the open-closed homotopy algebra (OCHA) structure [9][10][11]. In the superstring field theories, the homotopy algebra structure is more important. Since it seems inevitable to avoid associativity anomaly [12], the A ∞ structure becomes essential to determine the gauge-invariant action in the open superstring field theory [13][14][15]. The L ∞ structure again plays the role of guiding principle to determine the action with appropriate picture numbers in the heterotic and type II superstring field theories [16][17][18][19][20].
On the other hand, the current understanding is that there is no essential difference between the theory of open string/closed string mixed system and the theory of purely closed string. It merely describes the perturbation on the different backgrounds, those with and without a D-brane [21,22]. They should be derived from non-perturbatively formulated fundamental theory such as string field theory, but it is not a priori clear which one should be considered more fundamental. The closed string field theory is simpler, but the open-closed string field theory has a larger symmetry structure, the OCHA structure 1 . The purpose of this paper is to construct an open-closed string field theory realizing the OCHA structure. The action obtained explains the classical open string field theory on general closed-string backgrounds.
The paper is organized as follows. In section 2, we briefly review the open superstring field theory with general A ∞ structure. After introducing some conventions and fundamental ingredients, we show how we construct the open superstring field theory based on the A ∞ structure. The superstring products with appropriate picture numbers satisfying the A ∞ relations can be obtained by recursively solving the differential equations. We similarly review the closed superstring field theory with the L ∞ structure in section 3. We define the string products multiplying both open and closed string field in section 4 and show the relations they must satisfy to form the OCHA. We also give the differential equations that the products with OCHA structure should follow. They provide an action of the open superstring field theory on the general closed superstring backgrounds. In section 5, we obtain, as a byproduct, the corresponding WZW-like action through the map connecting the homotopy-based and WZW-like formulations, which is a generalization considered in [24]. Section 6 is devoted to the summary and discussion. In order to help the construction of concrete string interactions, some explicit procedure for solving the differential equations is given in Appendix A. Appendix B is added to make the paper self-contained. We introduce two composite string fields, the pure-gauge open string field and the associated open string field, which is nontrivial in the theory with general A ∞ structure.
2 Open superstring field theory with A ∞ -structure We summarize in this section how the open superstring field theory is constructed based on the A ∞ algebra structure.

Open superstring field
The first-quantized Hilbert space of open superstring is composed of two sectors: Correspondingly, the open superstring field Ψ has two components: Ψ = Ψ N S + Ψ R , both of which are Grassmann odd and have ghost number 1. The component Ψ N S (Ψ R ) has picture number −1 (−1/2) and represents space-time bosons (fermions). We impose on it a constraint where π 0 and π 1 are the projection operators onto the NS and R components, respectively: π 0 Ψ = Ψ N S and π 1 Ψ = Ψ R . The picture changing operator (PCO) of open superstring X o and its inverse Y o are defined by The PCO X o is BRST exact in the large Hilbert space: where deg(Ψ) = 1 or 0 if Ψ is Grassmann even or odd, respectively. We also use a natural symplectic form ω o l in the large Hilbert space H o l , which is similarly defined using the BPZ inner product in H o l , and related to ω o s as ω o

Interaction with A ∞ -structure
Open superstring interactions are described by the string products M n mapping n open superstring fields to an open superstring field as We identify the one-string product as the open superstring BRST operator: hold by definition. The multi-linear maps M n further satisfy the A ∞ relations , and cyclicity with respect to the symplectic form Ω o , For open superstring field theory, the string interaction M n must be defined for each combination of NS and R inputs so that the picture number must be conserved 3 : where p is the picture number that the the map itself has and 2r is the Ramond number (= number of Ramond inputs − number of Ramond output). The action with A ∞ structure is given by where we introduce a real parameter t ∈ [0, 1] and the group-like element 1 1−Ψ defined by The arbitrary variation of I o is given by using the A ∞ relation in Eq. (2.11) and cyclicity in Eq. (2.12):

Explicit construction of interactions
is defined respecting the cyclic Ramond number (= number of Ramond inputs + number of Ramond output) to make it easier to realize cyclicity 4 . This other A ∞ algebra can be decomposed into two mutually commutative A ∞ algebras (H l , D) and (H l , C) with depending on the picture number deficit of the output. The A ∞ relation [A, A] = 0 can also be decomposed as We consider a generating function for constructing the A ∞ algebra (H o l , ω o l , Q − η + A) and extend the A ∞ relations in Eqs. (2.23) to Note that the cyclic Ramond number has the upper bound p + 2 ≥ r. We consider A (p) p+r+1 | 2r ≡ 0 against the outside of this region. by introducing parameters s and t counting the picture number deficit and the picture number, respectively. Here, in Eqs. (2.27), o 1 (s) = π 0 + sπ 1 , o 2 (t) = tπ 1 and the bracket with subscript [·, ·] O is another simple notation for [·, ·] 1,2 and is defined by inserting the operator O into the intermediate state after taking (graded) commutation relation, At (s, t) = (0, 1), the generating function in Eq. (2.26) and the relations in Eqs. (2.27) reduce to A(0, 1) = A and the A ∞ relations in Eqs. (2.23), respectively. Then, we can show that if A(s, t) satisfies the differential equations   then I(s, t) = J (s, t) = 0. However, the relations in Eqs. (2.33) are nothing less than those satisfied by the geometric string products constructed similarly to those for the bosonic A ∞ algebra, Q + M B (s), without any insertion: We can obtain the cyclic A ∞ algebra, (H o l , ω o l , Q − η + A) by recursively solving the differential equations in Eqs. (2.29), or equivalently, The second equation at p = 0 can be solved for µ (1) (s) as

Closed superstring field theory with L ∞ -structure
Similarly to the open superstring field theory, closed (type II) superstring field theory is constructed based on the L ∞ algebra structure. We next summarize it in this section.

Closed superstring field
The first-quantized Hilbert space, H c , of type II (closed) superstring is composed of four sectors: We impose it closed string constraints and also an extra constraint with where π (0,0) , π (1,0) , π (0,1) , and π (1,1) , are the projection operators onto the NS-NS, R-NS, NS-R, and R-R components respectively: The PCO X c (X c ) and and its inverse Y c (Ȳ c ) are defined by The PCOs X c andX c are BRST exact in the large Hilbert space: where P −3/2 and P −1/2 (P −3/2 andP −1/2 ) are the projectors onto the states with the leftmoving (right-moving) picture numbers −3/2 and −1/2, respectively. We denote the restricted Hilbert space of type II superstring as H res c . Similarly to the relations in Eq.
and thus, G c (G c ) −1 is a projector that is compatible with the BRST cohomology: QP c XY = P c XY QP c XY . The type II superstring field satisfying the constraint in Eq. (3.2) is expanded in the ghost zero-modes as Natural symplectic forms ω c s and Ω c in H c and H res c , respectively, are defined by using the BPZ inner product as Natural symplectic form ω c l in the large Hilbert space H c l is similarly defined by using the BPZ inner product in H c l , and related to ω c s as ω c

Interaction with L ∞ -structure
Type II superstring interactions are descried by the string products L n that map n closed superstring fields to a closed superstring field as where Φ 1 ∧ · · · ∧ Φ n is the symmetrized tensor product defined by We identify the one-string product as the closed superstring BRST operator: L 1 = Q c . By definition, these products must satisfy and cyclicity The string interaction L n of the type II superstring field theory must be defined for each combination of NS-NS, R-NS, NS-R and R-R inputs so that the picture numbers of left-and right-moving sectors must be conserved separately: where we used the diagonal matrix representation L n,m = δ n,m L n . The superscript p (p) is the left-moving (right-moving) picture number that the map itself has, and the subscript 2r (2r) is the left-moving (right-moving) Ramond number. Introducing a real parameter t ∈ [0, 1], the action with L ∞ structure is given by with the group-like element where I SHc is the identity in SH c that satisfies The arbitrary variation of I c is given by δI c = Ω c (δΦ, π 1 L(e ∧Φ )).

Explicit construction of interactions
The cyclic L ∞ algebra (H res c , Ω c , L) is constructed in two steps. We consider first an introducing the degree odd coderivation This L ∞ algebra is equivalent to three mutually commutative L ∞ algebras (H l , D), (H l , C), and (H l ,C) with decomposed according to the picture number deficit. Then, the L ∞ relations are written as Here to the cyclic L ∞ algebra (H res c , Ω c , L) and two (trivial) L ∞ algebras (H c l , η) and (H c l ,η) as is not cyclic with respect to ω c l unlike the open superstring case. However, we can show, in a similar way given in the Appendix C of Ref. [18], that the L in Eq. (3.33) is cyclic with respect to Ω c if B is cyclic with respect to ω c l . In the next step, we consider a generating function and extend the L ∞ relations in Eq. (3.31) to for constructing the L ∞ algebra (H c l , O). The parameters s,s, and t counting the left-moving picture number deficit, right-moving picture number deficit, and the total picture number, respectively. The bracket with subscript is defined by inserting 4 Open-closed superstring field theory with OCHA-structure Now, we are ready to discuss OCHA, the main subject of this paper. In this section, we first see what OCHA is and how it is realized in the superstring field theory and then give a prescription to construct them explicitly.

Interaction with OCHA structure
We define classical interactions among the open and closed (type II) superstrings mixed system by the vertices described by the following two kinds of surfaces: • A sphere with n (≥ 3) closed-superstring punctures; • A disk with n (≥ 0) closed-superstring punctures on the bulk and l + 1 (≥ 1) opensuperstring punctures on the boundary with n + l ≥ 1.
We can identify the former as the linear maps {L n } given in the previous section, which form the cyclic L ∞ algebra (H res c , Ω c , {L n }). The latter includes both the open-superstring interactions (n = 0) and interactions between open and closed superstrings (n > 0) and is described by the string products N n,l that maps n closed-superstring fields and l open-superstring fields to an open-superstring field: with the identification N 0,l = M l . By definition, the condition holds. The linear maps {L n , N n,l } satisfying the OCHA relation Ψ 1 , · · · , Ψ i , N n−m,j (Φ σ(m+1) , · · · , Φ σ(n) ; Ψ i+1 , · · · , Ψ i+j ), Ψ i+j+1 , · · · , Ψ l ), (4.3) and the cyclicity condition Here, the sign factor µ m,i (σ) in Eq. (4.3) is given by Note that the OCHA relation in Eq. where 2r is the total Ramond number defined by total Ramond number =

# of R-NS inputs + # of NS-R input + 2(# of R-R inputs) + # of open R inputs
− # of open R output.
The action with OCHA structure is given by which describes the open superstring field theory on the closed-superstring background 6 . The open-superstring field Ψ is dynamical and the closed-superstring field Φ is the background field satisfying the equation of motion π 1 L(e ∧Φ ) = 0. (4.13) The arbitrary variation of I oc is given by (4.14) We can show that the action in Eq. (4.12) is invariant under the gauge transformation using the relation in Eq. (4.9): The open-closed superstring interaction N deforms by the background closed superstring field Φ gives a weak A ∞ algebra (H res o , M (Φ)) with Here, I is the identity map in T H res o and π o is the projector onto T H res o . 6 We omitted here the terms corresponding to a disk with closed strings in the bulk and no open strings on the boundary, which are included in the action proposed in Ref. [7]. These terms give a constant determined by a closed string background but do not relevant to symmetry structure of the theory [10].

Explicit construction of interactions
Let us construct a cyclic OCHA (H res c ⊕ H res o , Ω c ⊕ Ω o , L + N ). We assume that the cyclic sub-L ∞ -algebra L is already constructed in the way given in the previous subsection. Similarly to the previous cases, we can construct N satisfying the relation in Eq. (4.9) and the cyclicity condition in Eq. (4.4) in the following two steps. First consider a degree odd nilpotent coderivation ) If we find such A, B, and C, the cohomomorphism transforms D and C to the ones we eventually construct as We can construct C similarly to A and B, which we already find. By introducing parameters s and t , we consider a generating functions in Eq.
We can show that if C(s, t) satisfy with degree even coderivation ν(s, t) = ∞ p,n,l,r,m=0 which can constructed similarly to those of the bosonic open-closed string field theory [7]. Therefore, we can obtain C(s, t) satisfying Eq. (4.27) by recursively solving the differential equations in Eqs. (4.28) under the initial condition in Eq. (4.34) to be cyclic with respect to ω o l . In Appendix A, we give a concrete procedure to solve them for some lower orders. The cyclic OCHA (H res , Ω, L + N ) is eventually constructed by transforming using cohomomorphism in Eq. (4.23).
Let us first focus on the NS ⊕ NS-NS sector, which we simply call the NS sector in this section. The map between two formulations, the homotopy-based and WZW-like formulations, is given by the cohomomorphismĝ =ĝ c ⊗ĝ o [14,18,20] witĥ This cohomomorphism maps the string fields (Φ N S-N S , Ψ N S ) to those in the WZW-like formu- is the pure-gauge string fields of type II superstring identically satisfying where V c is a background field satisfying the equation of motion of the closed-superstring Q c G c (V c ) = 0. In order to give the WZW-like action deriving this equation of motion in Eq. (5.8), we define the associated string field as where d = t, δ or Q and ξ d is the coderivation derived from ξ∂ t , ξδ or −ξπ 1 M N S , respectively. We can show that the relations where D η is the nilpotent linear operator defined by acting on an open superstring field ϕ ∈ H N S . The coderivation L η acts as L η c + Lη c on H c and as L η o on H o . Then, the WZW-like action for the NS sector is given by , (5.14) which is invariant under the gauge transformation It is straightforward to extend these results of the NS sector to all the sectors. Sinceĝ c and g o act as the identity operators outside the NS sector, we find that and can identify the components Thus, these components are also annihilated by η c andη c (or η o ) and satisfy the constraint in Eq. (3.2) (or Eq. (2.1)). The WZW-like action of the open superstring field theory on the general closed-string backgrounds is eventually written as , The closed superstring backgrounds (V c , Ψ c ,Ψ c , Σ c ) satisfy π 1L e ∧(Gc(Vc)+Ψc+Ψc+Σc) = 0 (5.21) withL =ĝLĝ −1 . Note that, sinceĝ acts as the identity except on the NS sector,Ñ andL preserve the constraints in Eqs. (2.1) and (3.2), respectively. The WZW-like action in Eq. (5.19) is invariant under the gauge transformation which is also obtained through the mapĝ. Here, Λ and λ are the gauge parameters in the NS and R sectors, respectively, and λ is annihilated by η and satisfies the constraint in Eq. (2.1).

Summary and discussion
In this paper, we constructed interactions for the open-closed superstring field theory based on the OCHA structure. It provides the open-closed superstring field theory on general closedsuperstring backgrounds. We also give a corresponding WZW-like action for open-closed superstring field theory through a field redefinition.
Recently, the open string field theory deformed with a gauge invariant open-closed coupling is studied [24,[32][33][34][35]. The effective open superstring field theory is governed by a weak A ∞ structure which includes non-trivial tadpole term, destabilizing the initial perturbative vacuum. It requires to shift the vacuum to a new equilibrium point. The open-closed superstring field theory, given in this paper, provides a basis for such an analysis on more general closedsuperstring backgrounds described by classical solutions of the nonlinear equation of motion of the closed superstring field theory.
In order to quantize the classical superstring field theory, we must extend the classical action to the quantum master action satisfying the quantum BV equation. Such an open-closed superstring field theory is recently given in Ref. [23] based on the formalism using the extra free field [36,37]. It is interesting to give a quantum master action using the formulation based on the homotopy algebra, which requires to extend the OCHA structure to the quantum OCHA structure [38]. The quantum open-closed superstring field theory is also practically useful to study the string dynamics on the Ramond-Ramond backgrounds [39], the D-brane backgrounds [40][41][42][43][44], and so on, which are difficult in the first-quantized formulation using the RNS formalism. The (quantum) OCHA structure should shed new light on such nonperturbative studies.
We first rewrite the differential equations (3.39) in the form where we expanded c 1 (s,s, t) in the power of t as c 1 (s,s, t) = c 0 1 (s,s) + tc 1 1 (s,s) with c 0 1 (s,s) = π (0,0) + sπ (1,0) +sπ (0,1) + ssπ (1,1) and c 1 1 (s,s) = (sX +sX)π (1,1) . The first one (A.1) determines several B (p,p) (s,s) with the same total picture number simultaneously. We must split them by each left-and right-moving picture number. The explicit decomposition for the NS-NS sector was given in Ref. [16], but we have not yet extend it to the whole sectors in a closed form. Instead, we give an explicit decomposition for some lower picture numbers and show how the equations determine B (p,p) (s,s) for all the higher picture numbers. First, setting p = 0 in Eqs.  Repeating the procedure, we can obtain B (p,p) (s,s) for arbitrary p andp independently. Similar but slightly different analysis was give in Ref. [20]. The differential equations in Eqs. (4.28) for open-closed interactions is also solved recursively with the initial condition in Eq. (4.34). The differential equations in Eqs. (4.28) are rewritten as This is solved as so as to respect the cyclicity, where ξ o 0 • is defined on general coderivation C = ∞ n,l=0 C n+1,l by (A.23) Then, Eq. (A.18) at p = 0 ,  One can obtain any C (p) (s) one wants by repeating the procedure. Finally, it makes sense to mention that if we specify the type and number of inputs, the procedure ends in finite steps. We can explicitly determine any C (p) n+1,l | 2r you want in order from the one with the smallest number of inputs 8 . The one with the smallest number of inputs is the open-closed interaction: 1,0 (s) = C 1,0 (s) = ν It is a little more non-trivial for C (p) 1,1 (s) : 1,1 (s) = C 1,1 (s) = ν

B Composite string fields in open superstring field theory
In this Appendix we show that the pure-gauge string field G o (V ) for the open superstring field theory with general A ∞ structure is obtained in a similar way given in the heterotic string field theory [29]. The pure-gauge string field G o (V o ) is associated with a finite form of the "gauge transformation" with the infinitesimal parameter δV o , and is obtained by integrating along a straight line connecting 0 and V o that we parameterize as τ V o with 0 ≤ τ ≤ 1. Considering that the difference between G o (τ V o + dτ V o ) and G o (τ V o ) is an infinitesimal gauge transformation, we obtain a differential equation where we introduced a coupling constant g for convenience.The pure-gauge string field G o (V o ) corresponds to G o (τ V o ) at τ = 1 1and is obtained by solving this differential equation with the initial condition G o (0) = 0. Expanding G o in the power of g as G o = ∞ n=0 g n G and is solved as Similarly, we can find G o up to any order of g we want: In order to find an explicit form of associated string field B d (V o ) (d = ∂ t , δ or Q), we consider and its τ derivative . (B.10)