Modular operads and the quantum open-closed homotopy algebra

We verify that certain algebras appearing in string field theory are algebras over Feynman transform of modular operads which we describe explicitly. Equivalent description in terms of solutions of generalized BV master equations are explained from the operadic point of view.


I. INTRODUCTION
This article is, in particular, an extension of the work [6] by Markl. He showed that algebras considered by Zwiebach in [9] in the context of closed string theory are algebras over the Feynman transform of the modular envelope of the cyclic operad Com. This packed the complicated axioms into standard constructions over Com, an easily understood algebraic object. Moreover, this opens up the way for applications of operad homotopy theory to study of these algebras. The key point is that the Feynman transform is a modular operad which is always cofibrant in a suitable model structure 1 . There are now standard approaches for minimal models and transfer theorems for algebras over cofibrant operads. This has some physical relevance, e.g. gauge fixing [8], [14].
The above mentioned Zwiebach's work deals with closed string theory. In this article, we interpret the analogous algebras in open and open-closed string theory from the operadic point of view. For open strings 2 , the role of the modular envelope of Com is played by the operad QO (Quantum Open), which is easily understood in terms of 2-dimensional surfaces with boundary and some extra structure. We emphasize that only the homeomorphism classes of surfaces are considered, thus the operad has an easy combinatorial description. An operad closely related to QO was considered by Barannikov [1] to describe generalized ribbon graphs. We consider the Feynman transform FQO, an analogue of bar construction in the realm of modular operads, of QO and describe the axioms of algebras over FQO explicitly. A particular case recovered are the axioms of quantum A ∞ algebras of Herbst [5].
In [1], Barannikov explained how algebras over the Feynman transform are equivalently described by solutions of a "master equation" in certain generalized BV algebra. Applying this theory in the closed case, we obtain the BV algebra of Zwiebach's [9]. Application in the open case yields an improvement of the Herbst's [5].
Finally, we introduce a 2-coloured modular operad QOC (Quantum Open Closed) describing the algebraic structures in the open-closed case. We make the generalized BV algebra explicit, thus obtaining a briefly mentioned result of [13] by Kajiura and Stasheff, which is deeply based on the open-closed string-field theory description by Zwiebach [10].
The original motivation for writing this article was understanding the work [7] by Münster and Sachs. Unfortunately, at the moment were not able to prove directly the equivalence of our approach to theirs. We hope to come back to this question in the future. The indirect relation is the following: In [7], the quantum open-closed homotopy algebra structure is obtained as a consequence of Zwiebach's open-closed quantum master equation, in our approach the open-closed homotopy algebra structure is, via Barannikov's approach [1], equivalent to a solution (of a non-commutative version) of the same.
Although the above ideas are simple, the technical execution is not. We had to rethink the basic definitions for modular operads. The standard sources [4] and [2] seem to lack few details relevant for our discussion. We hope to present the basics in a way that explicit calculations are clear (thought not effortless at first). For this purpose, the notation is a bit overloaded and pace is slow.
We kept the operadic prerequisites at minimum -the definitions are stated and the only possibly technical part is the Feynman transform and it is wrapped in Theorem 10 giving practical description of algebras over it.
We found it more convenient to work with operads indexed by sets rather than arities. This makes many definitions easier to understand (see [11] for a similar approach) but the disadvantages are also obvious, e.g. the whole Section III G reexplaining the Barannikov's theory of master equations.
We finish the introduction by noticing an interesting pattern appearing. Let's return back to closed strings to make things more precise. The cyclic operad Com in fact consist of (linear span of) 2-dimensional surfaces of genus zero and boundary components, called closed (string) ends. These surfaces relate to the vertices in the classical limit (genus zero) of the quantum (all genera) closed string field theory. Let the operad QC consists of 2dimensional surfaces of arbitrary genus with closed ends. These surfaces relate to vertices in the Feynman diagrams in the quantum closed string field theory. It is easy to see that QC is in fact the above mentioned modular envelope Mod (Com).
Thus the passage from classical to quantum vertices corresponds to taking the modular envelope of the corresponding operad.
The  3 . We conjecture that QO is the modular envelope Mod (Ass).
Remark 3. If we consider only the first two collections satisfying only Axiom 2., the resulting structure is called Σ-module. Obviously, by forgetting structure, a modular operad gives rise to its underlying Σ-module. If we consider only the first three collections satisfying only Axioms 1., 2., 3., 8., the resulting structure is called cyclic operad. By restricting to G = 0 and forgetting structure, a modular operad gives rise to its underlying cyclic operad.
All these notions are equivalent to their usual counterparts in e.g. [4]. For example, Axiom 2. stands for the Σ-action, 3., 4. express the equivariance and 5. − 8. express the associativity of the structure maps.

B. Feynman transform
The Feynman transform of an operad P is a twisted 4 operad denoted FP. Roughly speaking, FP is spanned by graphs with vertices decorated by elements of P # .
We make this more precise in the following example of an element of FP(C, G). Consider a graph G A graph consists of vertices and half-edges. Exactly one end of every half-edge is attached to a vertex. The other end is either unattached (such an half-edge is called a leg) or attached to an end of another half-edge (in that case, these two half-edges form an edge). Every end is attached to at most one vertex/end. The half-edge structure of G is indicated on the picture on the right. We require that every vertex V i is assigned a nonnegative integer G i such that We also require that the legs of G (l 1 , l 2 , l 3 in our case) are in bijection with C. Finally we require for every vertex V i , where |V i | denotes the number of half-edges attached to V i . The graph G is "decorated" by an element where e 1 , e 2 , . . . are all edges of G, ↑ e i 's are formal elements of degree +1, ∧ stands for the graded symmetric tensor product and finally P 1 ∈ P({h 1 , . . . , h 5 }, G 1 ) # and similarly for P 2 and P 3 at vertices with G 2 and G 3 . Then the iso class of G together with (↑ e 1 ∧ · · · ∧ ↑ e 5 ) ⊗ (P 1 ⊗ P 2 ⊗ P 3 ) is an actual element of FP(C, G). The operations ( a • b ) FP and (ξ ab ) FP are defined by grafting of graphs, attaching together two previously unattached ends of two half-edges.
There is a Feynman differential ∂ FP on FP which adds an edge and modifies the decoration using the dual of ( a • b ) P or (ξ ab ) P .
Precise definitions are quite complicated technically (we refer to [4]). Fortunately, we only need a tiny part of the Feynman transform theory, namely Theorem 10 which will come in a moment.
To avoid problems with duals, we assume that the dg vector space P(C, G) is finite dimensional for any (C, G) ∈ Cor whenever FP appears. This is sufficient for our applications, though it can probably be avoided using cooperads. But, to our best knowledge, cooperads have never been investigated in the modular context.
One technical issue we treat in detail here is the notion of twisted operad, since FP is not an operad but a twisted operad.
of degree +1 morphisms of dg vector spaces. These data are required to satisfy the following axioms: for any morphisms ρ, σ in Cor, whenever the expressions make sense.
This notion is equivalent to the modular K-operad of Getzler and Kapranov [2] (also called K-twisted modular operad), where K is the determinant-of-edges coefficient system (also called hyperoperad). However, these explicit axioms have never before appeared in the literature.
To define algebras over the Feynman transform, we need a twisted endomorphism operad.

D. Endomorphism twisted modular operad
Let (A, d) be a dg vector space over a field k of characteristics 0.

Definition 5.
For any set C, we define where ψ's are bijections. The equivalence ∼ is given by with the usual Koszul sign. We denote τ ∈ Σ n and corresponding τ : V ⊗|C| → V ⊗|C| by the same symbol. Let be the natural inclusion i ψ followed by the natural projection. Hence follows from (1). Observe that ι ψ is an iso for any bijection ψ.
This has a differential for any f ∈ E A (C, G). By abuse of notation, we denote it by the same symbol as the differential on A.
We wish to contract indices of elements of E A (C, G) and for this we need a symmetric 2tensor. It arises from a symplectic form: Assume that (A, d) of even dimension n is equipped with a symplectic form ω of degree −1, i.e. ω(u, v) = 0 ⇒ |u| + |v| = 1 for any homogeneous elements of A. Also assume that A is a dg symplectic space, i.e.
where ω ij 's are the components of the matrix inverse of ω ij := ω(a i , a j ). Now the indicescontracting tensor is ω −1 := n i=1 a i ⊗ b i ∈ A ⊗ A, but we prefer to express the contractions using the bases {a i } and {b i }.
Denote µ : k ⊗ k → k the multiplication in k. It will also be denoted · or omitted in the sequel.
where ψ : and ψ 1 , ψ 2 are then defined by ψ 1 (a) := 1, ψ 1 (c 1 ) := ψ(c 1 ) + 1 and where ψ : C The reader can now verify: E A is called endomorphism (twisted modular) operad. In string field theory, A corresponds to a subspace of the space of states of a proper conformal field theory, d to the BRST operator and ω is related to the BPZ product with proper zero mode insertions.
E. Algebra over a twisted operad Definition 8. Let T be a twisted modular operad. An algebra over T on a dg symplectic vector space A is a twisted modular operad morphism of dg vector space morphisms such that (in the sequel, we drop the notation (C, G) at α(C, G) for brevity) Hence every element t ∈ T (C, G) is assigned a linear map α(t) : C A → k. In practice, however, one is rather interested in linear maps A ⊗|C| → k. Of course, A ⊗|C| ∼ = C A, but this is not canonical! We get around this nuisance as follows: Observe that once we know α([n], G) for all corollas with n, G ≥ 0, α is determined on other corollas by Axiom 1. of Definition 8. But for [n], there is a canonical iso A ⊗n ∼ = [n] A, namely ι 1 [n] of Definition 2.
Hence we may replace f : [n] A → k by f • ι 1 [n] : A ⊗n → k. To translate from a generic corolla (C, G) to ([|C|], G), we often invoke the following lemma: Lemma 9. Let Q be any Σ-module 5 , let α : Q → E A be a Σ-module morphism 6 . Then for any (C, G) ∈ Cor, x ∈ Q(C, G) and any ψ : 5 This is defined in Remark 3. 6 It means that α consists of the collection of Definition 8 but satisfies only Axiom 1.

F. Algebra over the Feynman transform
The following theorem is essentially the only thing we need from the theory of Feynman transform. It has already implicitly appeared in e.g. [6] and [1]. Theorem 10. Algebra over the Feynman transform FP of a modular operad P on a dg vector space A is uniquely determined by a collection of degree 0 linear maps (no compatibility with differential on P(C, G) # !) such that where is the dual of a • b on P and ∂ P is the differential on P.

G. Barannikov's theory
In [1], Barannikov observed that a twisted modular operad morphism FP → E A is equivalently described as a solution of certain master equation in an algebra succinctly defined in terms of P. In this section, we restate the corresponding theorem in our formalism and then adapt it to our applications. Assume C 1 , C 2 , G 1 , G 2 are given so that C 1 ⊔ C 2 = [n] and G 1 + G 2 = G. Define θ 1 : Thorough this section, the notation θ, θ 1 , θ 2 is fixed for these particular permutations. with P (n, G) being the space of invariants under the diagonal Σ n action on the tensor product and being a formal power series variable. Let P be equipped with operations, given for e ∈ P (n, G), f ∈ P (n + 2, G), g ∈ P (n 1 + 1, G 1 ), h ∈ P (n 2 + 1, G 2 ) by and extended k[[ ]]-linearly. Here τ is the composite exchanging the two middle factors. Recall that θ 1 , θ 2 depend on C 1 , C 2 .
Theorem 12 ([1]). Algebra over the Feynman transform FP on a dg symplectic space A is equivalently given by a degree 0 element, called generating function, Sketch of proof. Consider the iso where {p i } is a k-basis of P(C, G) and {p # i } is its dual basis. Under this iso, (8) becomes the G -component of the master equation of this theorem.
The axioms of noncommutative BV algebra can be verified by a straightforward computation.
By (9), any S ∈ P can be written in the form S = n,G G S n,G , S n, To be able to compare to certain examples in practice, we apply the following iso Ξ: where the RHS is the space of coinvariants with respect to the diagonal Σ n action on the tensor product, I := (i 1 , . . . , i n ) runs over all multiindices in [dim A] ×n , a I := a i 1 ⊗· · ·⊗a in ∈ A ⊗n , {φ i } is the basis dual to {a i } and φ I : Then P ∼ =P and we are interested in an explicit description of the operations after this identification in certain cases that appear in practice: Let's assume that P is a linear span of a modular operad in sets.
That is, we assume that for each ([n], G) ∈ Cor there is a basis {p i } of P([n], G) which is preserved by the Σ n -action and the operations • and ξ. This is obviously satisfied by all the operads QC, QO, QOC considered in this article. Now we can decompose {p i } into Σ n -orbits indexed by r and choose a representative p r for each r. Denote O(p r ) := Σ n /Stab(p r ) and also fix a section Σ n /Stab(p r ) ֒→ Σ n of the natural projection, thus viewing O(p r ) as a subset of Σ n . Hence the orbit of p r in P([n], G) is {P(ρ)p r | ρ ∈ O(p r )} and it has |O(p r )| = n! |Stab(pr)|

elements.
We can get a formula for the operations onP involving p r only. We introduce = 1 n! r ρ∈O(pr) I α((P(ρ)p r ) # )ι 1 [n] (a I ) (P(ρ)p r ⊗ Σn φ I ) = The third equality follows by omitting ρ −1 , which is justified by {ρ −1 (φ I ) | I} = {φ I | I} and {ρ −1 (a I ) | I} = {a I | I}; then the summation over ρ is replaced by the multiplication by |O(p r )|. Thus the generating function S ∈P is Next, we are interested in an explicit description of the operator ∆ onP : That is, we look for the dashed arrow in the commutative diagram P (n + 2, 1)P (n + 2, G) P (n, G + 1)P (n, G + 1) at the top right, we get at the top left. Hence ∆ on (13) is and therefore, at the bottom right, we get In the last expression, we used the notation as in (11). We have
Next, we are interested in an explicit description of the bracket onP : Let {p i }, {q j } be the basis of P(n 1 + 1, G 1 ), P(n 2 + 1, G 2 ) respectively. Starting with at the top right, we get at the top left. Hence {} on (17) is and therefore, at the bottom right, we get As in (11), this equals To rewrite the first line we introduce λ 1 := 1 [n 1 +1] and λ 2 : where the primed numbers are disjoint with unprimed numbers. Using the equivariance of a • b , we obtain we introduce yet one more pairρ 1 ,ρ 2 of relabelling bijections:ρ 1 is increasing on to both tensor factors of (20) to get The left tensor factor looks intimidating, but it is just a natural way to define a " ρ −1 . It doesn't depend on ρ 1 , ρ 2 but only on the values ρ −1 1 (1), ρ −1 2 (1). Analogously, the second line of (19) equals Now it is a straightforward definition unpacking to see that this equals Again, this is completely expected, since the intimidating expression is just the natural way to compose multilinear maps. Now plug (21) and (22) into (19). As in (11), we can omit ρ. Then nothing depends on C 1 , C 2 , hence we replace the summation over C 1 , C 2 by the constant n! n 1 !n 2 ! . We have thus computed that inP , where we use the notation " • " explained below (21). Observe that the expressions depend only on the values ρ −1 1 (1), ρ −1 2 (1) rather than on the permutations ρ 1 , ρ 2 . Finally, we are interested in an explicit description of the differential d onP . To get a simple formula sufficient for our applications, let's assume that P has vanishing differential, ∂ P = 0.
The computation here is even easier than that for ∆, hence we leave it to the reader to check that inP , To summarize,P is now equipped with operations d, ∆ and {} transferred from P along the iso Ξ. The operations are given by formulas (16), (23) and (24) and the generating function by (12).

A. Loop homotopy algebras
Solutions of Zwiebach's Master Equation [9] for closed string theory are equivalently described by a collection of multilinear maps satisfying certain properties. The resulting algebraic structure is called loop homotopy Lie algebra in [6]. The complicated axioms are succinctly described by operads. In this section, we rephrase the results of M. Markl [6] along these lines in our formalism.
We define the modular operad QC, called Quantum Closed operad, to consist of homeomorphism classes of connected 2-dimensional compact orientable surfaces with labelled boundary components. The homeomorphism class is determined by the genus of the surface, hence, for example, QC( [3], 1) is generated by Definition 13. For a corolla (C, G), let QC(C, G) be just the one dimensional space generated by a symbol C G of degree 0: The structure operations are defined, for any bijection ρ : C ∼ − → D, as follows: Obviously, QC is a modular operad. To prove it, one first makes axioms of algebras over FQC explicit. Second, one uses standard suspension isomorphisms for multilinear maps to translate the axioms to loop homotopy Lie algebras. Here we redo the first step, since it will appear later in a more complicated context.
To lighten the notation, we identify each QC(C, G) # with QC(C, G) by identifying C G with its dual. Applying Theorem 10 and the fact that each QC(C, G) is 1-dimensional, we see that an algebra over FQC is uniquely determined by a collection of degree 0 linear maps satisfying: According to the plan set out in Section III E, we want to express these equations in terms of . Precomposing the above axioms for C G = [n] G with ι 1 [n] , we obtain: (25) is equivalent to f G n : A n → k being completely symmetric: The reader may have expected just G i ≥ 0; the more complicated lower bound is a consequence of the stability condition (1): Here is the bijection used to define (ξ ab ) E A . Now we apply Lemma 9 and the fact that Hence, we consider the unique ψ ∈ Σ n such that its restriction to C 1 is increasing and onto The point is that for ι ψ we can apply the definition of ( a • b ) E A and the above calculation continues: We have thus proved: Theorem 15. An algebra over FQC on a dg symplectic space A is equivalently given by a collection f G n : A ⊗n → k | n, G ≥ 0, 2(G − 1) + n > 0 of degree 0 completely symmetric linear maps satisfying the equations where ψ ∈ Σ n is the (|C 1 |, |C 2 |)-unshuffle uniquely determined by ψ( The operad QC has a nice algebraic interpretation. First observe that restricting to the genus zero part of the operad QC yields the cyclic operad Com. Recall that the modular envelope Mod is the left adjoint of the forgetful functor from modular to cyclic operads. Theorem 16 ( [6], [4]). The modular operad QC is the modular envelope of the cyclic operad Com: Theorem 16 has a physical interpretation: The cyclic operad Com in fact consist of homeomorphism classes of 2-dimensional orientable surfaces with labelled boundary components and genus zero and the composition is gluing of the surfaces along a pair of boundary components. Thus Com is the loopless part of QC and algebras over Com are classical limits of algebras over QC. In other words, the passage from classical to quantum case corresponds to taking modular envelope.
Remark 17. Restricting the Feynman transform for modular operads to cyclic cobar complex for cyclic operads and applying it to Com yields L ∞ algebras after some suspensions. Precise relation between loop homotopy Lie and L ∞ algebras is explained in detail in [6]. We will discuss an analogue for the operad Ass later.

B. Master equation
By applying Barannikov's theory of Section III G, we can get Zwiebach's master equation [9] for closed string theory directly: Since QC(C, G) is the trivial representation Span k C G , which recovers Zwiebach's definition precisely. We can think of S as of a function on a formal supermanifold M with coordinates {φ i }. We have It is now a straightforward calculation to see that this coincides with Zwiebach's definition of the operations in terms of left/right graded derivations: ; finally, · is the extension of the commutative multiplication on S(A # ). The graded derivations act as follows: In particular, the bracket is the Poisson bracket. In this case, it turns out that (P , d, as a straightforward calculation shows. This is the precise result of [9]. Notice that S ′ is obtained from S by allowing the term with f 0 2 := −ω • (d ⊗1), corresponding to an "unstable corolla". ((x 1 , . . . , x n )) = · · · = ((x n−i+1 , . . . , x n , x 1 , . . . , x n−i )) = · · · = ((x 2 , . . . , x n , x 1 )) .
The modular operad QO is closely related to the modular operad S[t] of [1], associated to stable ribbon graphs.

B. Cyclic A ∞ -algebras
In Remark 3, we have already observed that restricting to the G = 0 part of a modular operad P and forgetting ξ ab 's yields a cyclic operad. Similarly, by forgetting the part of the Feynman differential dual to ξ ab 's, the Feynman transform FP becomes the cyclic cobar complex CP (for noncyclic operads, precise analogue is defined in Chapter 3.1 of [4]; in cyclic case, a variant appears in [3]). Expressed in terms of Theorem 10, algebra over CP is given by the collection of α(C, G)'s satisfying the corresponding equations with terms containing ξ ab 's omitted.
Restricting to the G = 0 part of the modular operad QO, we obtain the cyclic operad Ass: Ass(C) = Span k {all |C|-element cycles in C} for |C| ≥ 3 and zero otherwise. Since G = 0 implies g = 0, we omit G and g from the notation.
We make explicit axioms of algebras over CAss to get used to our formalism. Of course, it is well known that we will obtain cyclic A ∞ -algebras [3]. Still, we treat this calculation in detail since standard references avoid it and since it clarifies more complicated calculations in the subsequent parts of the paper.
Invoking the above explained modification of Theorem 10, we see that an algebra over CAss is uniquely determined by a collection As for algebras over FQC, we want to express these equations in terms of α([n])'s. First, to lighten the notation, we identify each Ass(n) # with Ass(n) by identifying each basis vector x ∈ Ass(C) with its dual basis vector x # ∈ Ass(C) # . Thus (32) becomes Hence, we see that it is enough to determine α([n])(x) for a single x from each Σ n -orbit. Obviously, there is only 1 such orbit and we choose x = ((1, . . . , n)). Denote f n := α([n])(((1, . . . , n))) • ι 1 [n] : A ⊗n → k. Now (32) is equivalent to f n having the symmetry f n = f n • τ , where τ is the cyclic permutation as in Definition 18. To understand (33), we must compute ( a • b ) # Ass (((1, . . . , n))). In order to develop some intuition, we think of the cycle ((1, . . . , n)) as n distinct points on a circle (see Figure 2). We immediately see that ( (((1, . . . , n))) vanishes for most of the decompositions C 1 ⊔ C 2 = [n]. In the graphical representation, the only non-vanishing terms are those where we separate [n] into two pieces C 1 , C 2 by cutting the circle exactly twice, see figure 2.
We have thus proved: Theorem 21. An algebra over CAss on a dg symplectic vector space A is equivalently given by a collection {f n : A ⊗n → k | n ≥ 3} of degree 0 linear maps satisfying the equations where τ is the cyclic permutation as in Definition 18 and ψ, depending on s, is as above.

Equivalent descriptions
Here we sketch other equivalent descriptions of cyclic A ∞ algebras. Denote T A := n≥0 A ⊗n the tensor algebra on A and let i n : A ⊗n → T A be the canonical inclusion. There are the usual isos, the first term being the space of all coderivations T A → T A: Hom((↑A) ⊗n , ↑A) A description of the isos is implicit below. The usual commutator of coderivations gives rise to brackets on the other spaces.
1. For F, G ∈ n≥0 Hom(A ⊗n , k), we get where ψ ∈ Σ n exchanges the last two blocks of i 2 and i 3 elements. This is a degree +1 symmetric bracket. Setting F n := G n := f n and f • i n = f n , (34) becomes, because of the cyclic symmetry of f n , Compare this to the description 4. below. 3. Defining degree 2 − n maps m ′ n : (↑A) ⊗n → ↑A by m ′ n := ↑ m n ↓ ⊗n and m ′ 1 := d (the differential on A), (34) becomes which is the usual cyclic A ∞ -algebra relation. The symmetry of m ′ n is determined by is graded symmetric and f ′ n := f n ↓ ⊗n is graded "anticyclic".

Finally, we apply the Barannikov's theory of Section III G to get a description of cyclic
A ∞ -algebras in terms of solutions of the master equation. We proceed as in Section IV B: A permutation cycle in Σ n of order n is called a cyclic permutation (to distinguish it from cycles in sets of Definition 18). There are n cyclic permutations in Σ n . Fix a cycle c n in [n], say c n := ((1, · · · , n)). Let the one and only Σ n -orbit in the set of all cycles be represented by c n . Then O(c n ) can be chosen to be {ρ ∈ Σ n | ρ(1) = 1}.
For every cycle c in [n] there is ρ c ∈ Σ n such that ρ c (c n ) = c. Cyclic permutations fix any cycle. Thus there is an iso Again, we will write φ I in place of c n ⊗ φ I , keeping the "cyclic symmetry" of φ's implicit. Thus the generating function S ∈P is where we denoted f n := f cn = α(c # n ) • ι 1 [n] . The bracket is (1) for any ρ 1 ∈ O(c n 1 +1 ), ρ 2 ∈ O(c n 2 +1 ) and

Again, it suffices to focus on C = [n]. Recall that a basis of QO([n], G) consists of elements
where b k is the number of cycles of length k.
Of course, (b k ) is eventually zero, thus the last two sums contain only finitely many nonzero terms. Obviously, two elements of the above form belong to the same Σ n -orbit in QO([n], G) iff their b-sequences coincide. Conversely, any sequence satisfying (38) and G = 2g + b − 1 determines an orbit 9 in QO([n], G). Hence given such a (b k ), we choose a representative of the corresponding orbit as follows: where each c i = c 1 i , . . . , c |c i | i and these satisfy b i=1 c i = [n] and c k i < c l j whenever i < j or i = j and k < l. Hom(A ⊗n , k).
Recall that G = 2g + b − 1, thus given b, g determines G and vice versa. Now we dualize the operad structure maps on QO. The empty cycles appearing in the elements of QO make formulas look complicated. The reason is that while you can always distinguish cycles c i , c j of which at least one has nonzero length, you can't distinguish them if both are empty. This forces us to treat the empty cycles c 1 = · · · = c b 0 = ∅ separately. Also, the stability condition (28) affects the fourth term below.
Let's dualize (ξ ab ) QO (Equations (31) and (30)) and evaluate on (b k ) g = {c 1 , . . . , c b } g : where the upper indices of c i are counted modulo |c i | so that their values belong to [|c i |] and analogously for the upper indices of c j resp. c m ; and δ··· equals 1 if the lower index condition is met, otherwise equals 0. Now we dualize (29). We are in fact interested in the sum over C 1 , C 2 , G 1 , G 2 of duals appearing in (36). A moment's reflection convinces us that where I = {i 1 , . . . , i |I| } and J = {j 1 , . . . , j |J| }, the upper indices are counted mod |c m | as explained above. G i 's and g i 's in the first term are related by G 1 = 2g 1 + (1 + e + |I|) − 1 and m , · · · , c s+l m } ⊔ |I| k=1 c i k and analogously for C 2 . Then is forced by the stability condition (28). Similarly, (43) Now we evaluate (36) on (b k ) g and then compose with ι 1 [n] . There are three contributions: two coming from the first four resp. last two summands in (40) and one coming from the right-hand side of (41).

First contribution
First four summands, called S 1 , of (40) contribute to where the first equality follows from Lemma 9; ψ : [n] ⊔ {a, b} ∼ − → [n + 2] is given by ψ(a) := 1, ψ(b) := 2, ψ(i) = i + 2 for 1 ≤ i ≤ n; we have denotedc j i := c j i + 2; and we have abused the notation by writing all the summands as one -notice that the above element of QO is correct even if c i or c j is ∅. Now we can insert QO(ρ ′−1 ) • QO(ρ ′ ) between α([n + 2], G − 1) and {· · · } g for arbitrary ρ ′ ∈ Σ n+2 . Let (b ′ k ) be the b-sequence for cycles 1c p+1 i · · ·c where the first equality follows from (3) is invariant under exchanges of cycles of the same length. Now fix i 0 and j 0 and observe that the terms with i = i 0 < j = j 0 and i = j 0 > j = i 0 coincide, since corresponding ρ ′ 's differ only by an exchange of values at 1, 2. Thus (46) equals Notice that the coefficient 2 doesn't appear at the last summand.

Third contribution
For S 3 coming from (41), we have where we had to first use ι 1 [n] = ι ψ • ψ to be able to use the definition of ( a • b ) E A ; ψ ∈ Σ n and ψ 1 , ψ 2 are as in (5); we have chosen ψ explicitly as follows: Write the cycles ac s+1 m · · · c s+l m , c i 1 , . . . , c i |I| in a sequence 11 with nondecreasing cycle lengths such that the cycle containing a is written on the leftmost possible position. Do the same for bc s+l+1 m · · · c s+m m , c j 1 , . . . , c j |J | and append the latter sequence to right of the former one. Then omit the elements a, b and replace each cycle by an increasing sequence of its elements. Thus we obtain a sequence s 1 , . . . , s n of distinct numbers. Then ψ(s i ) = i for 1 ≤ i ≤ n. ψ obviously satisfies the required property ψ( Doing a calculation analogous to that in (44) and (46) (ambiguity of ψ, and hence of ψ 1 , ψ 2 , is handled in the same way), we obtain that (50) equals where (b 1 k ) is the b-sequence for cycles ac s+1 m · · · c s+l m , c 1 , . . . , c e , c i 1 , . . . , c i |I| and ρ 1 ∈ Σ |C 1 |+1 is any permutation such that QO(ρ 1 ψ 1 )({ ac s+1 m · · · c s+l m , c i 1 , . . . , Explicitly, we can choose ρ 1 as follows: Let Then, for 1 ≤ k ≤ |C 1 | + 1, and ρ 2 are defined analogously. Theorem 22. Let (A, d, ω) be a dg symplectic space with bases {a d }, {b d } as in (4). Then an algebra over FQO on the space A is equivalently given by a collection of degree 0 linear maps f (b k ),g : A ⊗n → k indexed by all eventually zero sequences (b k ) ∞ k=0 of nonnegative integers, where we have denoted n := ∞ k=0 kb k ; these maps are required to satisfy: The equation holds.
Remark 23. If b = 0 is also included in the definition of QO, the only new thing we get is {} g ∈ QO(∅, 2g −1) for g ≥ 2. These elements can't be composed using a • b nor ξ ab and even don't affect the dual of the structure maps. Moreover α({} g ) : k → k and so d E A α({} g ) = 0. We easily see is tautological. Hence, for algebras over FQO extended in this way, we are only getting a collection of scalars which don't interact in any way with the rest of the operations f (b k ),g .

Relation to Herbst's quantum A ∞ algebras
For algebras over FQO on A satisfying f (b k ),g = 0 whenever b 0 > 0 a slightly simpler description is possible. In this Section, we recover the results of Herbst in [5].
In physics, so called string vertices are used rather than the collection {f (b k ),g }. Roughly speaking, the string vertex is an f (b k ),g evaluated at some fixed vectors of A. However, the string vertex depends on the order of cycles in (b k ) g , contrary to f (b k ),g . This dependence is purely formal, but the precise relation between the string vertices and f (b k ),g 's might be confusing, so we treat this in detail. Let OP (n, b) be the set of all Ordered Partitions of n into b positive integers. In other words, such a partition is a b-tuple (l 1 , . . . , l b ) ∈ N ×b such that l 1 + · · · + l b = n. Consider A ⊗n ⊗ Span k OP (n, b).
Notice that if l i = l j and i = j, then the corresponding stabilizer is nontrivial. There is also a Σ b action on A ⊗n ⊗ Span k OP (n, b) given by The ± is the Koszul sign and β (l 1 ,...,l b ) ∈ Σ n is the block permutation permuting blocks 1, . . . , l 1 and l 1 + 1, . . . , l 1 + l 2 and so on, according to β.
The coefficient −1/2 is purely conventional. The intuition behind the formula (54) is roughly this: the blocks in the subscript of F , separated by "|", correspond to nonempty cycles of {c 1 , . . . , c b } g . The subscript also determines an order of the cycles. Before evaluating, we reorder the cycles using β so that the lengths of the cycles form a nondecreasing sequence.
Notice that the above requirement doesn't determine β uniquely: if β ′ ∈ Σ b is another permutation such that l β ′−1 (1) ≤ · · · ≤ l β ′−1 (b) , then it is easy to verify that ...,l b ) and the reduced string vertex is independent of this choice.
It is now easy to express the equation (52) in terms of the reduced string vertices. Actually, we will be interested only in the first terms of the three contributions.
Example 27. We express the first term in (47) evaluated on v 1 ⊗ · · · ⊗ v n ∈ A ⊗n . With the forthcoming Theorem 30 in mind, we assume b 0 = 0. To lighten the notation, we will write k instead of v k in the subscript of F . Similarly, we write a instead of a d and b instead of b d . Further, given (b k ) g = {c 1 , . . . , c b } g as in (39), we write c k instead of (c 1 k , . . . , c |c k | k ) in the subscript of F . Finally, if an empty cycle appears in the subscript of F , then we omit it.
It is then only a matter of recalling the definition of ρ ′ to see that (47) yields (omitting the summations) where ± is the Koszul sign The first term in (49) is handled analogously.
Proof. The first part follows immediately from the definition of the reduced string vertex.
Now we can state the precise comparison to Herbst's work: Theorem 30. Let (A, d = 0, ω) be a symplectic dg vector space with zero differential. If A carries a structure of algebra over FQO satisfying where δ 1 is as in (42) with e = b 0 = 0 and the signs are Koszul signs produced by permuting a d ⊗ b d ⊗ v 1···n into the orders indicated by the subscripts of F 's. Notice that we are using the abbreviations introduced in Example 27. The above equation is precisely the Herbst's minimal quantum A ∞ relation of Theorem 1 of [5].
Proof. Assume the algebra over FQO is given and let's rewrite Equation (52) in terms of the reduced string vertices. The first sum was worked out in detail including the sign in Example 27. The second sum is completely analogous. The third sum was worked out in Example 28. This yields (55).
At this point, our notation is almost the same as Herbst's in Equation (24) of [5]. We just need to adjust the summation in the last term: First, we consider a pairing similar to that used to get (49): Let the numbers m and those of the tuple (I, J, g 1 , g 2 , s, l) have the meaning as in the summations (51) or the RHS of (55). Given m, pair (I 0 , J 0 , g 0 1 , g 0 2 , s 0 , l 0 ) ↔ (J 0 , I 0 , g 0 2 , g 0 1 , s 0 + l 0 , |c m | − l 0 ). The corresponding ±F ′ s coincide. Thus the RHS of (55) becomes This is surjective but not injective. Fortunately, it is easy to see that each preimage has |I|!|J|! elements. Using Lemma 29, we see that (56) equals |I|,|J|≥0 |I|+|J|=b−1 .
Finally, the stability condition, represented by the δ 1 above, corresponds to the Herbst's notion of minimality.
Remark 31. A priori, the obvious converse of Theorem 30 doesn't hold. In fact, algebras over FQO with f (b k ),g = 0 for b 0 > 0 satisfy not only equations (55), but also equations (52) with b 0 = 0, whose LHS vanishes but there can be nonzero terms on the RHS.

Master equation
In this section, we apply the Barannikov's theory of Section III G to get a master equation describing quantum A ∞ -algebras.
Recall that each Σ n -orbit of QO([n], G) has the canonical representative (b k ) g satisfying k kb k = n and 1 2 (G − b + 1) =: g ∈ N 0 , where b := k b k . The number of orbits is the number of the corresponding b-sequences (b k ). Each element of the orbit containing (b k ) g has the stabilizer consisting of k≥1 b k !k b k permutations. Thus the orbit contains can be seen as a tensor product of φ's with symmetries given by the stabilizer of (b k ) g . Therefore the generating function S ∈P is where (b k ) g runs through the set consisting of the representatives of each orbit (in other words: through all b-sequences with given n, G) and We make the noncommutative BV algebra operations explicit though the final expressions are no more illuminating than the general case, because there is no way to choose O(p r )'s so that ρ −1 (1) = 1 for all ρ ∈ O(p r ) as we did in the QC and Ass cases. One possible choice The idea is to use the stabilizer of (b k ) g to write the sequence of labels (ρ(1), . . . , ρ(n)) in a canonical form described above. The first set of conditions can be fulfilled using the cyclic symmetry of each cycle and the second set of conditions by permuting the blocks corresponding to cycles of the same length. Then it is easy to verify that {ρ −1 (1) | ρ ∈ O((b k ) g )} = {1, b 1 + 1, b 1 + 2b 2 + 1, b 1 + 2b 2 + 3b 3 + 1, . . .}. This is the set of the initial positions of blocks of cycles of the same length.
where the subscript numbers are positions in the tensor product.
Computing the stabilizer of (l 1 , . . . , l b ) g , we get an analogue of Lemma 29: This justifies the notation. Using it in the last line of (58), we get (abbreviating ρ := ρ (l 1 ,...,l b ) ) b,g l 1 ,...,l b Now we replace f (b k ),g (a I ) by the reduced string vertex and then use Lemma 29 to rearrange its subscript to i ρ −1 (1) · · · i ρ −1 (l 1 ) |i ρ −1 (l 1 +1) · · · i ρ −1 (l 1 +l 2 ) | · · · (writing k instead a k here). The Koszul sign is easily seen to coincide with the ± above. Thus we get Finally, substituting this to (57) and omitting ρ after summing over I, we obtain i.e. b 0 = 0, this is the Herbst's formula (42) in [5]. However, Herbst's φ's are probably graded symmetric, which is stronger than our requirement (59). Consequently, his generating function (satisfying a master equation) doesn't determine quantum A ∞ algebra. Let us note that the reduced string vertices are graded symmetric only with respect of permutations of non-empty cycles. In order to be later, when discussing the quantum openclosed string field theory, compatible with the physics notation, we introduce the string vertices as follows: where on the left hand side we allow for l i ≥ 0 and on the right hand side the sequence of l ′ i 's is obtained from the sequence l 1 , l 2 , . . . by omitting zeroes while keeping the order of nonzeroes. In terms of so defined string vertices we have where ′ is the product of nonzero l k 's.

D. Relation between Mod (Ass) and QO
Motivated by the relationship between QC and its genus zero part, cyclic Com, in closed string case (Theorem 16), one can ask the same question for open strings: Is QO the modular envelope of its genus zero part, cyclic Ass? This is obvious neither from the topological viewpoint (in terms of 2-dimensional surfaces), nor from the algebraic viewpoint (in terms of adding the results of ξ ab compositions to Ass as freely as possible). But there are evidences, which we won't explain here, in favor of the claim, hence: Conjecture 34. There is a modular operad isomorphism QO ∼ = Mod (Ass) compatible with the obvious morphisms from Ass to QO resp. to Mod (Ass).

VI. QUANTUM OPEN-CLOSED OPERAD AND RELATED ALGEBRAIC STRUCTURES
A.

2-coloured modular operad
Here we define a 2-coloured modular operad with half-integral genus and related notions. This is clearly only a provisional definition -it is coined to express the master equation for open-closed string theory. We expect it to fit into a more general framework for "operads of modular type".
Definition 35. Let Cor 2 be the category of stable 2-coloured corollas: the objects are pairs (O, C, G) with O, C finite sets 12 and G a nonnegative half-integer 13 such that the stability condition is satisfied: 12 O stands for Open, C for Closed. 13 That is, G is of the form a/2 for some a ∈ N 0 . Definition 36. To define a 2-coloured modular operad, replace Cor by Cor 2 everywhere in Definition 2 and also require a • b and ξ ab to be defined only if both a, b are open or both are closed.
In the same way, we obtain the definition of twisted 2-coloured modular operad. Now we define the 2-coloured twisted modular endomorphism operad:  (5) and (6), using the o-indexed (resp. c-indexed) bases if a, b are open (resp. closed).
Algebra over a 2-coloured twisted modular operad is again defined by replacing Cor by Cor 2 in Definition 8.
The notion of Feynman transform of a 2-coloured modular operad is defined using a suitable definition of 2-coloured graphs. We leave it to the reader to fill in the details.
The analogue of Theorem 10 is:

B. The modular operad QOC
The idea is that the 2-coloured modular operad QOC consists of labelled 2-dimensional orientable surfaces with both open and closed ends, as explained for QC and QO. We are only allowed to glue two open ends together or two closed ends together. The operadic composition along closed ends is defined by The operadic contraction along open ends is defined as follows: If there are i < j such that o i = ((a, x 1 , . . . , x m )) and o j = ((b, y 1 , . . . , y n )), then define The reader can now verify: Theorem 39. QOC is a 2-coloured modular operad. Notice that the sphere with 0 or 1 closed ends are excluded since for these G < 0. In physics, it is desirable to include the surface 3. in QOC(∅, [1], 1/2). Call it c. But " 1 • " 1 (c ⊗ c) is the surface 2., hence that closedness under the operadic compositions implies that 2. has to be included too. It is easy to check that no more unstables are produced by gluing 2., 3. and stable surfaces. Thus we obtain an extension QOC ′ of QOC such that QOC ′ (∅, {c}, 1/2) and QOC ′ (∅, ∅, 1) are 1-dimensional. In an algebra α : FQOC ′ → E A , α(c) : A c → k is a cohomology class: 0 = d(α(c)) = α(c) • d A , since one easily verifies ∂ FQOC ′ (c) = 0.
The corollas (∅, {c}, 1/2) and (∅, ∅, 1) are not stable, thus formally QOC ′ is not a modular operad. However, there is no problem: The purpose of stability is to guarantee the finiteness of number of iso classes of graphs appearing in the Feynman transform (see Lemma 2.16 in [2]), i.e. finiteness of dimension of FP(n, G) for each n, G. Allowing the corollas (∅, {c}, 1/2) and (∅, ∅, 1), this property is easily seen to be still true.
This suggests that our notion of stability is unnecessarily strict and should be refined in any more systematic approach to modular operads.

E. Further questions
Inspired by Theorem 16 and Conjecture 34, one might ask whether QOC is the modular envelope of its genus zero part. This is probably again true. However, the genus zero part is not the right operad to consider. For this result to have an interesting physical interpretation, we would like QOC to be a modular envelope of an operad describing vertices of "classical limit" of Feynman diagrams, that is diagrams with no circles. In the classical limit, the vertices are genus zero surfaces with any number of open and closed ends, but with at most one open boundary component 14  The operadic composition should be inherited from QOC via the obvious map OC → QOC. The ξ ab composition is, of course, omitted. The a • b composition is however not well defined, since a • b (o C ⊗ o ′ C ′ ) ∈ OC in general. Hence OC is not a cyclic operad. This problem was partially overcome in the work of Kajiura and Stasheff, e.g. [12]. We briefly explain it here. end is distinguished only if there is no open boundary component on the surface. This way, OC becomes a 2-coloured noncyclic operad. In Appendix of Kajiura and Stasheff's [12], this operad is seen to be the Koszul dual of the 2-coloured operad for a Leibniz pair. This answer is unsatisfying, since in the absence of cyclicity, there seems to be no way to relate OC with the modular operad QOC in terms of some "free" construction, e.g. modular envelope.
However, there are more ways to view OC. E.g. it is an operadic module over the cyclic operad Com and one can consider a variants of modular envelopes for modules. We plan to address these questions in future.