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.

This article is, in particular, an extension of the work [13] by Markl. He showed that algebras considered by Zwiebach in [16] 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. For example, minimal models and transfer theorems have some physical relevance for gauge fixing [2,15]. The key point of the applications of the homotopy theory 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, which hopefully carry over to our setting (e.g. [2], where the homotopy transfer is discussed). 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. 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 in terms of generating operations and relations. A particular case recovered are the axioms of quantum A ∞ algebras of Herbst [6].
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 [16]. Application in the open case yields an improvement of the Herbst's [6].
The approaches to FQO mentioned in the two above paragraphs are dual to each other: the approach via generators and relations (section 5.3.1) in the spirit of the usual A ∞ algebras manifestly uses the dual of the composition in QO, while the approach via master equation (section 5.3.3) manifestly uses the composition directly.
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 [8] by Kajiura and Stasheff, which is deeply based on the open-closed string-field theory description by Zwiebach [17].
The original motivation for writing this article was understanding the work [14] 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 [14], the quantum open-closed homotopy algebra structure is obtained as a consequence of Zwiebach's open-closed quantum master 1 We need a model structure on the category of twisted modular operads (see section 3.3). This has not been systematically studied so far, and we don't attempt to do so in this article. 2 At least for the topological string.

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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. There is 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 2-dimensional 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 However, in the full open-closed case, the pattern seems to broke. The operad for classical vertices, appearing in the work [7] by Kajiura and Stasheff, is not even cyclic, so Mod (P) doesn't make sense. This is discussed in section 6.5.
We finish the introduction by discussing some technical aspects of the paper. The closed case discussed in [13] is very simple and can be dealt without paying too much attention to formal details. In the open case, things get more complicated and one should be more careful.
To start with, we need a definition of (twisted) modular operad which is easy to verify in practice. The standard definition in terms of triples (e.g. [4,12]) is inconvenient for this purpose. Likewise, the biased definition in terms of collections {P(n, G)} indexed by arities n (and genus G) and composition i • j and contraction ξ ij (which is usual for ordinary operads) involves axioms which are too complicated. So we choose an intermediate approach -the collections P(C, G) indexed by finite sets C. This way, the axioms can be stated succinctly and their geometric motivation is obvious (sections 3.1 and 3.3). For our inherently geometric examples of operads QC, QO and QOC, this definition is always easily verified. The passage between collections indexed by integers and collections indexed by sets is discussed in some detail in section 3.4.
We also don't treat cyclic operads as ordinary operads with extra structure, but rather as objects on their own. This emphasizes the geometric nature of the axioms. The same approach has been adopted e.g. in [11].
To make the Feynman transform work, we need twisted operads. The axioms stay succinct: the operadic compositions a • b and contractions ξ ab become degree 1 morphisms, and JHEP12(2015)158 a minus sign is introduced into the associativity axioms (section 3.3). Thus calculations with twisted operads are syntactically similar to those with untwisted operads. The difference is analogous to computations in an algebra A with degree 0 multiplication satisfying the standard associativity relation (ab)c = a(bc), and calculations in its suspension ↓A, where the multiplication has degree 1 and satisfies (ab)c = −(−1) |a| a(bc). We emphasize that although A and ↓A in the above example are in a sense equivalent, the suspension trick doesn't work for modular operad and thus the use of twisted modular operads probably can't be avoided. In this paper, similarly to [9], we try to promote the use of twisted structures by showing that clear and explicit calculations can be performed with them. This is best seen in the proof of Theorem 20.
As the above approach is slightly nonstandard, we felt obliged to provide details. Thus the notation is a bit overloaded and pace is slow. We also kept the operadic prerequisites at minimum -all the basic definitions are stated in full. The only possibly technical part is the Feynman transform and it is wrapped in Theorem 16 giving a practical description of algebras over it.

Conventions and notation
1. N is the set of positive integers, N 0 := N ∪ {0}.
2. k is a field k of characteristics 0. The multiplication in k will be denoted · or omitted.
All (dg) vector spaces are considered over k.
3. Dg vectors spaces have differential of degree +1. Morphisms of dg vector spaces are degree 0 linear maps commuting with differentials.
4. is disjoint union. Whenever A B appears, A, B are automatically assumed disjoint.

7.
A # is the linear dual of A.

Modular operads, Feynman transform and master equation
3.1 Modular operad Definition 1. Denote Cor the category of stable corollas: the objects are pairs (C, G) with C a finite set and G a nonnegative integer such that the stability condition is satisfied:

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Definition 2. Modular operad P consists of a collection of dg vector spaces and three collections of degree 0 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.
Remark 3. If we consider only the first two collections satisfying only Axiom 2., the resulting structure is called Cor-module (more familiar name would be Σ-module, but we reserve this name for slightly different structure, see section 3.4 below), which is simply a functor from Cor to dg vector spaces. Obviously, by forgetting structure, a modular operad gives rise to its underlying Cor-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. [12]. For example, Axiom 2. stands for the Σ-action, 3., 4. express the equivariance and 5. − 8. express the associativity of the structure maps.

Feynman transform
The Feynman transform of a modular operad P is a twisted 4 modular operad denoted FP. Roughly speaking, FP is spanned by graphs with vertices decorated by elements of P # .

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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 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 [12]). Fortunately, we only need a tiny part of the Feynman transform theory, namely Theorem 16 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 [4] (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.

Σ-modules
So now we wish to define algebras over the Feynman transform, and for that, we need a twisted endomorphism operad E A . Informally speaking, E A consists of covariant tensors and the operadic composition and contraction is given by contraction of the tensors using a symplectic form of degree −1. There is the following inconvenience: in E A (C, G), the set C should index inputs of a |C|-times covariant tensor. But while there is no canonical order on C, the inputs of the tensor are ordered by definition. This makes the definition of E A in terms of a Cor-module clumsy and unintuitive.
Thus it is helpful to restrict the category Cor of corollas to a smaller one where a canonical order is available:

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Definition 5. Σ is the skeleton of Cor consisting of all stable corollas of the form ([n], G), n ∈ N 0 . Σ-module is a functor from Σ to dg vector spaces.
Given a (twisted) modular operad P, we can restrict its underlying Cor-module to Σ-module. Then we construct an analogue of operadic composition and contraction on the restricted module as follows: we first need some fixed auxiliary permutations: Given n ∈ N 0 , define a bijection by requiring it to be increasing. Given n 1 , n 2 ∈ N 0 , define bijections by requiring them to be increasing.
Obviously, P with i • j 's and ξ ij 's forgotten is a Σ-module. One easily verifies that the definitions of i • j and ξ ij are independent of the choice of λ, λ 1 , λ 2 .

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The operations i • j and ξ ij satisfy certain properties, call them (P), analogous to those of Definitions 2 and 4. Denote Op Cor the category of (twisted) modular operads. It would be natural to define a new category Op Σ of Σ-modules with i • j 's and ξ ij 's satisfying (P), then notice that the construction P → P of Definition 7 induces an equivalence between Op Cor and Op Σ , and finally use whichever of the two categories is convenient in a given situation.
Unfortunately, it turns out that the formulas corresponding to (P) are too complicated for any practical purposes. Still, we use Op Σ in several places of the paper. For example, the endomorphism operad E A has a particularly easy definition in Op Σ (see Definition 9 below).
So we adopt the following point of view: if we work with and object of Op Σ , we always assume it is of the form P for some P ∈ Op Cor . This is the way we are going to rigorously define the endomorphism operad: we define E A ∈ Op Cor first, then construct E A ∈ Op Σ , and finally observe that E A is nice and simple. This should justify our exposition in the next section.
where ω ij 's are the components of the matrix inverse of ω ij := ω(a i , a j ). Now we can contract indices of tensors Hom k (A ⊗n , k) using 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 }.
From now on, dg symplectic vector space will refer to a structure such as above including the bases {a i } and {b i }.
Let's define a candidate for E A . This is the nice and simple result we wish for: for any permutation ρ.

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Now we need to verify that this candidate is indeed of the form E A for some E A ∈ Op Cor . We need a preliminary on unordered tensor product: Definition 10. 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 (3.2). Observe that ι ψ is an iso for any bijection ψ.
for any (C, G) ∈ Cor. This graded vector space has a differential given by
For f ∈ E A (C {a, b}, G), we define The reader can now verify: Theorem 12. E A of Definition 11 is a twisted modular operad. Taking its E A (in the sense of Definition 7) yields a result described in Definition 9, if we identify The above identification will be done implicitly in the sequel. It is formally justified by first writing an equation in terms of maps from the unordered tensor products and then composing both sides with ι 1 [n] .
Convention 13. We will write just P for both P ∈ Op Cor and P ∈ Op Σ . The former object consists of collections Remark 14. 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.

Algebra over a twisted operad
Definition 15. Let T be a twisted modular operad. An algebra over T on a dg symplectic vector space A is a twisted modular operad morphism i.e. it is a collection of dg vector space morphisms such that (in the sequel, we drop the notation (C, G) at α(C, G) for brevity) JHEP12(2015)158 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 15. But for [n], there is a canonical iso A ⊗n ∼ = [n] A, namely ι 1 [n] of Definition 3.3.
Hence we may replace f : [n] A → k by f • ι 1 [n] : A ⊗n → k (compare to Theorem 12).

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. [13] and [1]. Theorem 16. Algebra over the Feynman transform FP of a modular operad P on a dg vector space A is equivalently determined by a collection of degree 0 linear maps (no compatibility with differential on P(C, G) # !) such that is the dual of a • b on P and ∂ P is the differential on P.
To make this compatible with [12] and [4], we have to interpret the sum as follows: if C 1 = C 2 = ∅ and G 1 = G 2 , then the corresponding term appears twice in the sum by definition.
To make algebras over Feynman transform explicit as easily as possible, we modify this theorem. The first idea is that, in our applications, the description of P is easier using Cor-modules, while description of E A is easier using Σ-modules. The second idea is that it is enough to determine α([n], G) for all n, G. Moreover, α([n], G) is determined by its values on orbit representatives. The κ's below are intended to transform a generic element to such an orbit representative; a choice of the representatives and κ will be adapted to a concrete application.

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. For ordinary operads, there is a systematic approach to a similar problem via the convolution operad and its associated Maurer-Cartan equation, e.g. section 6.4.2 of [10]. This theory has been extended in [9] to include Barannikov's example.
In this section, we restate the corresponding theorem in our formalism, reprove it and then adapt it to our applications.
Definition 18. Let P be a modular operad. Recall that we assume with P (n, G) being the space of invariants under the diagonal Σ n action on the tensor product. For e ∈ P (n, G), let For f ∈ P (n + 2, G + 1), let
It is easily seen that these operations take values in P and don't depend on the choice of θ, θ 1 , θ 2 . The following formulas are also easy to verify: where i, j are arbitrary and can even be chosen differently for each ρ in the second formula.

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Notice that the generalized BV algebra is a suspension of Lie algebra with two commuting differentials satisfying the Leibniz relation w.r.t. the bracket.
Proof. Consider the iso where the l.h.s. is the space of all linear Σ C -equivariant maps, {p i } is a k-basis of P(C, G) and {p # i } is its dual basis. Assume α as in Theorem 16 determines an algebra over FP. Let S n,G := Now we apply Z to (3.8) and get 0 = Z(β) + Z(γ) + Z(δ). We will evaluate Z(δ), the other terms are easier and we leave them to the reader. To simplify the notation, we write α for α(C, G). Since |δ| = 1, we get Let {x k } be a basis of P(C 1 a, G 1 ) and {y l } be a basis of P(C 2 b, G 2 ). Then let
Now summing this over n ≥ 0 and G ≥ 0 yields the master equation. Conversely, if S is a degree 0 solution of the master equation, then it gives rises to α satisfying conditions of the Theorem 16. Indeed, it suffices to reverse the above reasoning.
Finally, the axioms of generalized BV algebra can be verified by a straightforward computation. The axioms involving d are easy to prove and we leave them to the reader. In the following computation, we write briefly σx instead of P(σ)x or E A (σ)x; it will always be clear from the context which operad is considered.
As a warmup, we prove ∆ 2 = 0. It is easy to verify that Since θ is arbitrary, we precompose it with σ. Thus (3.13) equals By equivariance of ξ, this equals By associativity, this equals Thus ∆ 2 = 0. Next, we prove (3.10). A tedious but straightforward calculation yields where the bijections θ 1 : For the middle term, recall that the expression doesn't depend on the choice of θ's. We can thus precompose θ 1 with a permutation σ 1 exchanging a and c and leaving everything else in place. Similarly we precompose θ 2 with an exchange σ 2 of b and d. Thus (3.15) equals where the first equality is justified by equivariance of • and ξ, the second one by associativity. Thus the middle term of (3.14) vanishes. As for (3.14), a straightforward calculation yields Equivariance justifies the relabeling occurring below from the first to the second line: cancels with the first term of (3.14). Likewise, {}(1 ⊗ ∆) cancels with the third term.
To prove the Jacobi identity (3.9), let f ∈ P (n 1 , G 1 ), g ∈ P (n 2 , G 2 ), c ∈ P (n 3 , G 3 ). Then {{f, g}, h} can be expressed as a sum of terms of two types: Proceeding formally, a tedious but straightforward calculation shows that where the summations run over all partitions {b} are arbitrary and ψ is the monoidal symmetry Denote F 1 resp. S 1 the terms in the first resp. second sum in (3.17), i.e.
Let σ = 1 2 3 2 3 1 ∈ Σ 3 be the cyclic permutation and similarly let and similarly for g and h. By axioms of operad, The indices a, b, c, d along which we contract can be arbitrarily renamed as in (3.16). This shows that F 1 = −S σ . Similarly, F σ = −S σ 2 and F σ 2 = −S 1 .

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Remark 21. In physical literature it is customary to introduce a formal parameter whose powers bookkeep the genus. E.g. we might redefine P := n≥0 G≥0 G P (n, G) and extend all BV operations k[[ ]]-linearly; this forces us to replace ∆ by ∆. We won't use this convention except in the special case of section 4.4. Now we will rewrite P and the BV operations into a different form. We note that we are using the convention of Theorem 12, which means that E A (n, G) is Hom k (A ⊗n , k) rather than Hom k ( [n] A, k). There is an iso and, since we are in characteristics 0, there is an iso between invariants and coinvariants Composing these two, we obtain where I := I 1 · · · I n := (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 := φ I 1 ⊗ · · · ⊗ φ In ∈ A # ⊗n . In the bottom left, we have also denoted by φ I the map A ⊗n → k given by We will use this notation in the sequel; it will always be clear from the context which of the two meanings we have in mind. Denotẽ Then P ∼ =P as vector spaces and we transfer the BV operations toP : let Y be the iso (3.18), then d and ∆ are transferred along P (n + 2, G)P (n + 2, G) P (n, G + 1)P (n, G + 1) The formulas can be conveniently expressed in terms of "positional derivations": let Proof. We do the computation for ∆ and leave the other formulas for the reader.
where |I| = n + 2, |J| = n and σ : [n + 2] → [n] {a, b} is an arbitrary bijection. By definitions, The last equality is justified as follows: κ −1 a J = ±a κ −1 J and κ −1 φ J = ±φ κ −1 J with the same sign; thus we can shift the summation index J to κ −1 J. By definitions, where J := J 1 · · · J i−1 dJ i · · · J j−1 eJ j · · · J n is obtained from J by inserting d, e into positions i, j; and (−1) is the Koszul sign of transforming a deJ to a J . Substituting this into (3.20) yields We can rewrite this expression in terms of the positional derivations: where we wish to derivate at positions i and j of φ I , so the position index of the l.h.s. derivation is i := i if i < j and i := i − 1 otherwise. The sign (−1) is indeed the one from (3.21). It it easy to verify, using the fact that ω has degree 1, that the sign coincides in both cases i < j and i > j. We have obtained In this formula, the only dependence of the summands on σ is through i = (θσ) −1 (a) and j = (θσ) −1 (a). Thus we replace σ∈Σ n+2 by n! 1≤i,j≤n+2 i =j to obtain: Solutions of Zwiebach's Master Equation [16] 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 [13]. The complicated axioms are succinctly described by operads. In this section, we rephrase the results of M. Markl [13] along these lines in our formalism.

The modular operad QC
We define the modular operad QC, called Quantum Closed operad, to consist of homeomorphism classes of connected 2-dimensional compact orientable surfaces with labeled boundary components. The homeomorphism class is determined by the genus of the surface, hence, for example, QC( [3], 1) is generated by Definition 23. 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. For the definition of loop homotopy algebras, we refer the reader to loc. cit. To prove the theorem, one first makes axioms of algebras over FQC explicit in terms of operations V ⊗n → k. Second, one passes to operations V ⊗n−1 → V and uses standard suspension isomorphisms for multilinear maps to translate the axioms. Here we redo the first step, since it will appear later in a more complicated context.
To lighten the notation, we identify each QC([n], G) # with QC([n], G) by identifying [n] G with its dual. Applying Lemma 17 and the fact that each QC([n], G) is 1-dimensional, we see that an algebra over FQC is uniquely determined by a collection of degree 0 linear maps satisfying: where we have chosen κ to be increasing on C and κ(a) = n + 1, κ(b) = n + 2; and we have chosen κ 1 to be increasing on C 1 and κ 1 (a) = n 1 + 1 and similarly for κ 2 . Observe that (ρ n 1 +1 1 κ 1 | C 1 ρ n 2 +1 2 κ 2 | C 2 ) −1 is a (|C 1 |, |C 2 |)-shuffle, denote it σ. Now we want to express the r.h.s. of (4.1) in terms of f G n 's. We start by calculating the dual of the structure morphisms of QC: Thus (4.1) becomes We have thus proved:

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Theorem 25. An algebra over FQC on a dg symplectic vector 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 for every x 1 , . . . , x n ∈ A, where ± is the Koszul sign of the evaluation

Relation between Mod(Com) and QC
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 26 ( [12,13]). The modular operad QC is the modular envelope of the cyclic operad Com: Theorem 26 has a physical interpretation: the cyclic operad Com in fact consist of homeomorphism classes of 2-dimensional orientable surfaces with labeled 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 27. 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 [13]. We will discuss an analogue for the operad Ass later.

Master equation
By applying Barannikov's theory of section 3.8, we can get Zwiebach's master equation [16] for closed string theory directly: since QC(C, G) is the trivial representation Span k C G , andP ∼ = S(A # ) := n,g S n (A # ). Let's write G φ I rather than [n] G ⊗ Σn φ I . Then φ's posses a symmetry as if they were graded polynomial variables. The BV differential is -linear (in the obvious sense) and is determined by To use the formulas of Lemma 22 for ∆ and {}, we need to make ξ ij and i • j on QC explicit: Then ∆ is -linear and where the sign consists of (−1) |φ I |+|φ I i | and the Koszul sign of permutation taking φ I 1 ···In to φ I i I j I 1 ··· I i ··· I j ···In ; and {} is -linear and where the sign consists of (−1) |φ I i |(|φ I |+1)+|φ J |+|φ I ||φ J |+1 and the Koszul sign of permutation taking φ I 1 ···IpJ 1 ···Jq to φ I i I 1 ··· I i ···IpJ j J 1 ··· J j ···Jq . The solutions S ∈ S(A # ) of degree 0 of the master equation We wish to compare the above formulas for BV operations to those of Zwiebach in [16]. We introduce left and right derivations with respect to the standard -linear commutative multiplication in S(A # ). Observe that

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It is now natural to ask what is the compatibility between the commutative multiplication · and ∆, {}. One easily verifies Thus in order to get a classical BV algebra, we redefine the operations as follows: The primed operations on S(A # ) indeed form a BV algebra, for example Solutions S of the corresponding master equation The term d (S ) is customary absorbed into S by declaring S := S + 1 4 i,j ω(da i , a j )φ i φ j . Then the master equation reads just N d=1 Notice that S is obtained from S by allowing the term f 0 2 proportional to 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 )) .

Definition 29.
QO(C, G) := where {c 1 , . . . , c b } g is a symbol of degree 0, formally being a pair consisting of g ∈ N 0 and a set of cycles in C with the above properties. We remind the reader that means
The modular operad QO is closely related to the modular operad S[t] of [1], associated to stable ribbon graphs.

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 [12]; in cyclic case, a variant appears in [5]). Expressed in terms of Theorem 16, algebra over CP is given by the collection of α(C, G)'s satisfying the corresponding equations with terms containing ξ ab 's omitted.  ((1, . . . , n)) and ((1, . . . , n)) cut into two pieces C 1 ,C 2 .
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 [5]. 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.
To lighten the notation, we identify Ass( We see that an algebra over CAss is uniquely determined by a collection {f n := α(c n ) ∈ E A (n) | n ≥ 3} of degree 0 linear maps such that f n = f n • τ for all cyclic permutations τ ∈ Σ n , n ≥ 3, and d(f n ) = 1 2 To choose convenient κ 1 and κ 2 , we need to understand the dual of composition in Ass.
where τ is the cyclic permutation as in Definition 28. These maps are required to satisfy for every x 1 , . . . , x n ∈ A and n ≥ 4, where ± is the Koszul sign of evaluating .7)).

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Notice that the cyclic symmetry of f l+1 allows us to move a d to an arbitrary input. Similarly for b d . This is reflected by the ambiguity of the choice of κ 1 , κ 2 .
Next, we sketch two equivalent description of the cyclic A ∞ algebras. First, denote T A := n≥0 A ⊗n the tensor algebra on A. There are isos Let m 1 :=↑ d ↓ be the differential on ↑A. In terms of these maps m n of degree 2 − n, (5.8) becomes which is the usual cyclic A ∞ -algebra relation. Next, we apply the Barannikov's theory of section 3.8 to get a description of cyclic A ∞ -algebras in terms of solutions of the master equation. We proceed as in section 4.4: recall that the element c n = ((1, 2, . . . , n)) ∈ Ass(n) is a representative of the single Σ n orbit. Its stabilizer obviously consists of the cyclic permutations, hence To get a formula for the BV bracket, one first easily verifies i • j (c n 1 +1 ⊗ c n 2 +1 ) = = ((1, . . . , i − 1, j + n 1 , . . . , n 2 + n 1 , 1 + n 1 , . . . , j − 1 + n 1 , i, . . . , n 1 )) = π ij c n 1 +n 2 , where π ij ∈ Σ n 1 +n 2 is a permutation of blocks of lengths i − 1, n 1 + 1 − i, j − 1, n 2 + 1 − j swapping the second and fourth block. Symbolically, Then If one thinks of each φ I i in the tensors φ I as sitting on a circle at position i, the summands above have a direct relation to the combinatorics of Ass: for brevity of notation, let's write simply φ I rather than c n ⊗ Σn φ I . Then The sign ± consist of the sign in (5.10), the sign of positional derivatives and the Koszul sign of π ij . For example, if φ I = φ J = 0 and I = (I 1 , . . . , I 5 ), J = (J 1 , . . . , J 4 ), then To conclude, solutions S of the master equation

Quantum A ∞ -algebras
Definition 32. A quantum A ∞ algebra is an algebra over FQO.

Axioms for algebras over FQO
In this section, we make the axioms for these algebras explicit. We

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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 (5.13) and G = 2g + b − 1 determines an orbit 7 in QO([n], G). Hence given such a (b k ), we choose a representative of the corresponding orbit as follows: and c k i < c l j whenever i < j or i = j and k < l. By abuse of notation, we will often write c i meaning a tuple ranging over all b-sequences and integers g ≥ 0. Recall that G = 2g + b − 1, thus given b, g determines G and vice versa. Lemma 17 lists the axioms which these maps are required to satisfy. We make these axioms explicit in the sequel. We first 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 (5.1) affects some terms below.
Let's dualize (ξ ab ) QO (equations (5.4) and (5.3)) and evaluate on (b k ) g = {c 1 , . . . , c b } g : In fact, if all ci's are nonempty, {c1, . . . , c b } can be seen as a decomposition into independent cycles of a permutation on [n]. Then Σn acts by conjugation and the sequence of lengths of cycles is a familiar invariant of the orbits.

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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 δX equals 1 if the lower index condition X is met, otherwise it equals 0. Now we dualize ( a • b ) QO (equation (5.2)). We are in fact interested in the sum over C 1 , C 2 , G 1 , G 2 of the duals of the composition. A moment's reflection convinces us that is forced by the stability condition (5.1). Similarly, (5.18) Now we evaluate the equation of Lemma 17 on (b k ) g . There are three contributions: two coming from the first four resp. last two summands in (5.15) and one coming from the right-hand side of (5.16).
First contribution. First four summands of (5.15) contribute to

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of Lemma 17 by i,j By abuse of notation, we have written all four terms as one. Choose arbitrary κ ∈ Σ n+2 so that Of course, the summands above directly reflect the combinatorics of QO, although this is obscured by the complicated definition of κ and ψ. An example makes that clearer: (1) Notice that you can directly read off the order of arguments from the values of κ −1 . Different choices of κ lead to orders: ( . Of course, this choice is irrelevant due to the symmetry of f (0,0,1,0,0,1,0,...),g . Now we rewrite the first four terms of (5.15) more carefully: In the second (resp. third) sum, In the fourth sum, (b k ) = (b 0 −2, b 1 , b 2 +1, . . .). The positions of a d and b d and ψ (depending on i, j, p, q) are described above and the choices made in determining them don't affect the expression f (b k ),g (· · · ⊗ a d ⊗ · · · ⊗ b d ⊗ · · · ) • ψ because of the symmetries of f (b k ),g . 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. Hence we get Notice that the coefficient 2 doesn't appear at the last summand.
The only other possible order of the arguments of the left factor is (a d , x 2 , x 1 ).

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To summarize the application of Lemma 17 for P = QO, notice that its main equation takes the following form: on the left-hand side, we have d(f (b k ),g )(x 1 , x 2 , . . .) for some fixed q = (b k ) g and x 1 , . . . , x n ∈ A. To get the right-hand side, we collect all possible basis elements y ∈ QO such that ξ ab (y) = q and all possible pairs z 1 ⊗ z 2 of basis elements such that a • b (z 1 ⊗ z 2 ) = q. To each such y, there is a term on the right-hand side of the form d ±f (b k ),g κ(a d , b d , x 1 , x 2 , . . .). The calculations above clarify how to get (b k ),g andκ ∈ Σ n+2 from y in a very easy way. Similarly, each z 1 ⊗ z 2 contributes the term Theorem 36. An algebra over FQO on a dg symplectic vector space A is equivalently given by a collection of degree 0 linear maps indexed by all eventually zero sequences (b k ) ∞ k=0 of nonnegative integers satisfying the stability condition where n := ∞ k=0 kb k and b := ∞ k=0 b k . These maps are required to satisfy: The equation holds.
Example 37. F := f (0,0,0,2,0,...),g : A ⊗6 → k has symmetries generated by the following three permutations: Remark 38. 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. Call QO this extension of QO.

Relation to Herbst's quantum A ∞ algebras
In this section, we recover the results of Herbst in [6] concerning the quantum A ∞ algebras satisfying f (b k ),g = 0 whenever b 0 > 0. 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.
Define the reduced string vertex where the b-sequence (b k ) and β ∈ Σ b on the r.h.s. are determined as follows: for k ≥ 1, b k is the number of l i 's equal to k; 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 sign ± is the Koszul sign of the action of β (l 1 ,...,l b ) on v's. The coefficient −1/2 is purely conventional. The intuition behind the formula (5.23) 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 (5.22) in terms of the reduced string vertices. With the forthcoming Theorem 42 in mind, we assume b 0 = 0 and thus we will be interested only in the first terms of the three contributions.
Example 40. We express the first term in (5.19) evaluated on v 1 ⊗ · · · ⊗ v n ∈ A ⊗n . 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 (5.14), we

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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.
We easily see that the first term of (5.19) yields (omitting the summations) where ± is the Koszul sign of permuting v ab1···n to v The first term in (5.20) is handled analogously.
Example 41. Using the abbreviations as in the previous example, the first term in (5.21) can be expressed, upon evaluation on v 1 ⊗ · · · ⊗ v n , by bc Finally, let's observe that the reduced string vertices have the expected symmetries:

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The above equation is precisely the Herbst's minimal quantum A ∞ relation of Theorem 1 of [6].
Proof. Assume the algebra over FQO is given and let's rewrite equation (5.22) in terms of the reduced string vertices. The first sum was worked out in detail including the sign in Example 40. The second sum is completely analogous. The third sum was worked out in Example 41. This yields (5.25).
At this point, our notation is almost the same as Herbst's in Equation (24) of [6]. We just need to adjust the summation in the last term: first, we consider a pairing similar to that used to get (5.20): let the numbers m and those of the tuple (I, J, g 1 , g 2 , s, l) have the meaning as in the summations (5.21) or the r.h.s. of (5.25). Given m, pair The corresponding ±F s coincide. Thus the r.h.s. of (5.25) becomes b m=1 I,J I J= bc This is surjective but not injective. Fortunately, it is easy to see that each preimage has |I|!|J|! elements. Using (5.24), we see that (5.26) 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 43. A priori, the obvious converse of Theorem 42 doesn't hold. In fact, algebras over FQO with f (b k ),g = 0 for b 0 > 0 satisfy not only equations (5.25), but also equations (5.22) with b 0 = 0, whose l.h.s. vanishes but there can be nonzero terms on the r.h.s.

Master equation
In this section, we apply the Barannikov's theory of section 3.8 to get a master equation describing quantum A ∞ -algebras. We obtain results dual to those of the preceding section 5.3.2, except that we allow for empty boundaries.
The quantum A ∞ -algebras are degree 0 solutions S of the master equation solved in the spaceP = n,G QO(n, G) ⊗ Σn A # ⊗n .
We introduce a notation similar to that for the reduced string vertices (5.23): for each where (b k ) is the b-sequence corresponding to b cycles of lengths l 1 , . . . , l b and b − b empty cycles, n = l 1 + · · · l b and β (l 1 ,...,l b ) ∈ Σ n is the block permutation described below (5.23) and I k is an l k -tuple (I k 1 · · · I k l k ) for each k; and G = 2g + b − 1. Notice that every φ I 1 |···|I b ;g,b ∈P is restricted by the stability condition 2(2g + b − 2) + b k=1 l k > 0. Obviously, every element ofP can be written in the form for some coefficients C g,b I 1 |···|I b ∈ k. Notice that this expression is not unique (see also section 5.3.4 below).
Using Lemma 22, we easily obtain the following formula for the BV operator ∆: In the first term, the sign consists of (−1) and of the Koszul sign of permuting (this Koszul sign already includes the signs coming from the positional derivatives of Lemma 22). Notice that this permutation brings the order of the superscript indices I 1 · · · I b appearing in the argument of ∆ on the first line to their order in the second line. The sign in the second term is similar.

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For the BV bracket, we obtain where, for each k, l k is the length of the tuple I k ; and similarly m k 's are lengths of J k 's. The sign consists of the factor (−1) of Lemma 22) and of the Koszul sign of permutation taking Again, notice that this permutation corresponds to the change of order of indices from the first to the second line except for the switch of I p i and J q i . Notice that the restrictions due to stability are included already in the allowed "monomials" φ I 1 |···|I b ;g,b and, contrary to sections 5.3.1 and 5.3.2, don't complicate the master equation.

Comparison to Herbst's generating function
Lemma 44. Every element inP (n, G) can be uniquely expressed as where the summation runs through g ≥ 0, b ≥ 1, b ≥ 0 and I k ∈ [dim A] ×l k for each 1 ≤ k ≤ b, such that b ≤ b and l 1 + · · · + l b = n and G = 2g + b − 1; and the coefficients C g,b I 1 |···|I b ∈ k are required to satisfy for each k, where I k = (I k 1 · · · I k l k ). If l 1 ≤ · · · ≤ l b and (b k ) is the corresponding b-sequence, let Then n,G S n,G satisfies the master equation (5.27) iff f (b k ),g 's satisfy the conditions of Theorem 36.
Proof. Let's describe a basis ofP (n, G). Let Stab(b k ) g be the stabilizer of (b k ) g ∈ QO([n], G) inside Σ n . Observe that

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This substitution leads to the Herbst's formula (42) of [6] for the generating function: Let us note that the reduced string vertices are graded symmetric only with respect to permutations of non-empty cycles. In order to be later, when discussing the quantum open-closed string field theory, compatible with the physics notation, we introduce the string vertices as follows: where on the l.h.s., multiindices I i of zero length are allowed and I i on the r.h.s. are obtained by omitting these zero length multiindices while keeping the order of the others. In terms of so defined string vertices we have where is the product of nonzero l k 's.

Relation between Mod (Ass) and QO
Motivated by the relationship between QC and its genus zero part, cyclic Com, in closed string case (Theorem 26), 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). An affirmative answer is given in [3]: Theorem 45. There is a modular operad isomorphism QO ∼ = Mod (Ass) compatible with the obvious morphisms from Ass to QO resp. to Mod (Ass).
6 Quantum open-closed operad and related algebraic structures

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 46. Let Cor 2 be the category of stable 2-coloured corollas: the objects are pairs (O, C, G) with O, C finite sets 9 and G a nonnegative half-integer 10 such that the stability condition is satisfied: 2(G − 1) + |O| + |C| > 0. (6.1) 9 O stands for Open, C for Closed. 10 That is, G is of the form a/2 for some a ∈ N0. Algebra over a 2-coloured twisted modular operad is again defined by replacing Cor by Cor 2 in Definition 15.
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 modular operad QOC
The idea is that the 2-coloured modular operad QOC consists of labeled 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 reader can now verify: Theorem 50. QOC is a 2-coloured modular operad.
This suggests that our notion of stability is unnecessarily strict and should be refined in any more systematic approach to modular operads.

Further questions
Inspired by Theorems 26 and 45, one asks whether QOC is the modular envelope of its genus zero part. Recall that the operadic genus G of a surface {c 1 , c 2 , . . .} g C ∈ QOC(O, C, G) is given by the formula G = 2g + b − 1 + |C|/2. Hence Notice that the nontrivial part is a suboperad in open inputs isomorphic to Ass. Also notice that the surfaces ∅ 0 C for any C (these would correspond to a suboperad in closed inputs isomorphic to Com) are not present (the stability condition removes ∅ 0 C for any |C| ≤ 2). Consequently, the modular envelope of QOC(−, −, 0) is trivial whenever C = ∅ and thus is not QOC.
However, the genus zero part of QOC 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. 11 Thus we are led to consider the Open Closed operad OC 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. [7]. We briefly explain it here. Observe that one can compose open ends arbitrarily in OC, but composition of closed ends

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is possible only if one of the composed surfaces has no open boundary. One easily sees that this restriction is satisfied if we choose a distinguished end of each surface so that closed 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 [7], 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.