Strongly homotopy algebras of a Kähler manifold

S.A. Merkulov

Abstract It is shown that any compact Kähler manifold M gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex. In these algebras the product of two harmonic differential forms is again harmonic.

If M happens to be a Calabi-Yau manifold, there exists a third strongly homotopy algebra closely related to the Barannikov-Kontsevich extended moduli space of complex structures.

1 Introduction

Strongly homotopy algebras have been introduced by Stasheff [7] more than 30 years ago in the context of topological H-spaces. Since that time this remarkable structure (and its cousins) has been found at a number of unexpected places, for example in the string field theory, in topological conformal field theories and in Floer’s theory (see, e.g., [8, 9, 3, 4]).

In this paper we show that in the heart of the classical Hodge theory of Kähler manifolds (see, e.g., [2]) there lies a strongly homotopy algebra which “explains” the following imperfect behavior of harmonic differential forms: the wedge product, α ∧ β, of two harmonic forms α and β is not harmonic in general. This, of course, can be easily fixed by defining a new product ∘,

α ∘ β := [α ∧ β]_{Harm},

[ ]_{Harm} being the projection to the harmonic constituent, but at a price — the product ∘ fails to be associative. Among the main results of the paper is an observation that ∘ does satisfy the higher order associativity conditions thereby unveiling the structure of strongly homotopy algebra in the Hodge theory of Kähler manifolds.

In fact there are at least two strongly homotopy algebras associated with a compact Kähler manifold M. The first one is real and is associated with the Hodge theory of the de Rham complex (Ω^{•}, d); the second one is complex (and is not a complexification of the first) and is associated with the Hodge theory of the Dolbeault complex (Ω^{•,•}, ¯∂).

If M happens to be a Calabi-Yau manifold, there exists a third strongly homotopy algebra closely related to the Barannikov-Kontsevich extended moduli space of complex structures [1].

The paper is organized as follows. In Sect. 2 we remind the definition of a strongly homotopy algebra. In Sect. 3 we consider a specific algebraic data which gives rise to a particular strongly homotopy algebra; a remarkable feature of the construction is that it provides explicit formulae for all the higher homotopies. In Sect. 4 we consider a compact Kähler manifold and prove the results mentioned above.

2 Strongly homotopy algebras

A strongly homotopy algebra, or shortly A_{∞}-algebra, is by definition a vector superspace V equipped with linear maps,

\begin{aligned}μ_{k} : & ⊗^{k}V & ⟶ & V \\ & v_{1} ⊗ … ⊗ v_{k} & ⟶ & μ_{k}(v_{1}, …, v_{k}), & k ≥ 1, \end{aligned}

of parity k̃ := k mod 2ℤ satisfying, for any n ≥ 1 and any v_{1}, …, v_{n} ∈ V, the following higher order associativity conditions,

∑_{k+l=n+1} ∑_{j=0}^{k−1}(−1)^{r}μ_{k} (v_{1}, …, v_{j}, μ_{l}(v_{j+1}, …, v_{j+l}), v_{j+l+1}, …, v_{n}) = 0,  (1)

where r = l̃(ṽ_{1} + … + ṽ_{j}) + j̃(l̃− 1) + (k̃ − 1)l̃ and ṽ denotes the parity of v ∈ V.

Denoting dv_{1} := μ_{1}(v_{1}) and v_{1} ∘ v_{2} := μ_{2}(v_{1}, v_{2}), one may depict explicitly the first four floors of the infinite tower of higher order associativity conditions as follows

n = 1: d^{2} = 0,

n = 2: d(v_{1} ∘ v_{2}) = (dv_{1}) ∘ v_{2} + (−1)ṽ_{1}v_{1} ∘ (dv_{2}),

n = 3: v_{1} ∘(v_{2} ∘v_{3})−(v_{1} ∘v_{2})∘v_{3} = dμ_{3}(v_{1}, v_{2}, v_{3})+μ_{3}(dv_{1}, v_{2}, v_{3})+(−1)ṽ_{1}μ_{3}(v_{1}, dv_{2}, v_{3})+ (−1)^{ṽ_{1}+ṽ_{2}}μ_{3}(v_{1}, v_{2}, dv_{3}),

n = 4: μ_{3}(v_{1}, v_{2}, v_{3})∘v_{4}−μ_{3}(v_{1}∘v_{2}, v_{3}, v_{4})+μ_{3}(v_{1}, v_{2}∘v_{3}, v_{4})−μ_{3}(v_{1}, v_{2}, v_{3}∘v_{4})+(−1)ṽ_{1}v_{1}∘ μ_{3}(v_{2}, v_{3}, v_{4}) = dμ_{4}(v_{1}, v_{2}, v_{3}, v_{4}) − μ_{4}(dv_{1}, v_{2}, v_{3}, v_{4}) − (−1)ṽ_{1}μ_{4}(v_{1}, dv_{2}, v_{3}, v_{4}) −

(−1)^{ṽ_{1}+ṽ_{2}}μ_{4}(v_{1}, v_{2}, dv_{3}, v_{4}) − (−1)^{ṽ_{1}+ṽ_{2}+ṽ_{3}}μ_{4}(v_{1}, v_{2}, v_{2}, v_{3}, dv_{4}).

Therefore A_{∞}-algebras with λ_{k} = 0 for k ≥ 3 are nothing but the differential associative superalgebras with the differential μ_{1} and the associative multiplication defined by μ_{2}. If, furthermore, μ_{1} = 0, one recovers the usual associative superalgebras.

The notion of A_{∞}-algebra is a very natural extension of the usual concept of associative superalgebra. The following well-known fact [6] may serve as a confirmation of this statement: the moduli space of infinitesimal deformations of an associative superalgebra A within the class of associative superalgebras is isomorphic to the second Hochschild cohomology group Hoch^{2}(A, A), while the moduli space of infinitesimal deformations of the same A within the class of A_{∞}-algebras can be identified with the full Hochschild cohomology Hoch^{∗}(A, A). In this sense A_{∞}-algebra is an “utter” concept.

3 An explicit construction of an A_{∞}-algebra

Let (V, d) be a differential associative superalgebra, with d denoting the differential, i.e. an odd linear map d : V ⟶ V satisfying the Leibnitz identity

d(v_{1} · v_{2}) = (dv_{1}) · v_{2} + (−1)ṽ_{1}v_{1} · (dv_{2}),

for any v_{1}, v_{2} ∈ V.

Let W be a linear subspace of V invariant under d. Note that we do not assume that W is a subalgebra of V. Instead we make the following

Assumption 3.1 There exists an odd operator

Q : V ⟶ V

such that for any v ∈ V the element (1 − [d, Q])v lies in the subspace W, where [ , ] is the supercommutator.

Put another way we assume that the operator P := Id−[d, Q] defines a (not necessary surjective) map from V to W. It may not be an identity when restricted to W ⊂ V.

Since [d, P] = 0, the product

v_{1} ∘ v_{2} := (1 − [d, Q])(v_{1} · v_{2})

makes (W, d, ∘) into a differential superalgebra which is not associative in general. We shall show, however, that ∘ does satisfy the higher associativity conditions; moreover, we will be able to compute all the higher homotopies μ_{k} explicitly in terms of d and Q only.

First we define a series of linear maps

λ_{n} : ⊗V^{n} ⟶ V, n ≥ 2,

starting with

λ_{2}(v_{1}, v_{2}) := v_{1} · v_{2}

and then recursively, for n ≥ 3,

λ_{n}(v_{1}, …, v_{n}) := (−1)^{n−1}[Qλ_{n−1}(v_{1}, …, v_{n−1})] · v_{n} − (−1)^{nṽ_{1}}v_{1} · [Qλ_{n−1}(v_{2}, …, v_{n})]

− ∑_{{k+l=n+1 \atop k,l≥2}}(−1)^{k+(l−1)(ṽ_{1}+...+ṽ_{k})}[Qλ_{k}(v_{1}, …, v_{k})] · [Qλ_{l}(v_{k+1}, …, v_{n})].

(2)

For example,

λ_{3}(v_{1}, v_{2}, v_{3}) = [Qλ_{2}(v_{1}, v_{2})] · v_{3} − (−1)ṽ_{1}v_{1} · [Qλ_{2}(v_{2}, v_{3})],
λ_{4}(v_{1}, v_{2}, v_{3}, v_{4}) = −[Qλ_{3}(v_{1}, v_{2}, v_{3})] · v_{4} − (−1)^{ṽ_{1}+ṽ_{2}}[Qλ_{2}(v_{1}, v_{2})] · [Qλ_{2}(v_{3}, v_{4})]
−v_{1} · [Qλ_{3}(v_{2}, v_{3}, v_{4})]
λ_{5}(v_{1}, v_{2}, v_{3}, v_{4}, v_{5}) = [Qλ_{4}(v_{1}, v_{2}, v_{3}, v_{4})] · v_{5} + (−1)^{ṽ_{1}+ṽ_{2}+ṽ_{3}}[Qλ(v_{1}, v_{2}, v_{3})] · [Qλ_{2}(v_{4}, v_{5})]
−[Qλ_{2}(v_{1}, v_{2})] · [Qλ_{3}(v_{3}, v_{4}, v_{5})] − (−1)ṽ_{1}v_{1} · [Qλ_{4}(v_{2}, v_{3}, v_{4}, v_{5})].

Setting formally

λ_{1} := −Q^{−1}

(which makes sense because in all our formulae below λ_{1} enters in the combination Qλ_{1} = −Id), one can rewrite the recursion (2) as follows

λ_{n}(v_{1}, …, v_{n}) = − ∑_{{k+l=n+1 \atop k,l≥1}}(−1)^{k+(l−1)(ṽ_{1}+...+ṽ_{k})}[Qλ_{k}(v_{1}, …, v_{k})] · [Qλ_{l}(v_{k+1}, …, v_{n})], (3)

where now n ≥ 2.

Lemma 3.2 The tensors λ_{k}, k ≥ 2, satisfy the identities

Φ_{n}(v_{1}, …, v_{n}) ≡ ∑_{{k+l=n+1 \atop k,l≥2}} ∑_{j=0}^{k−1}(−1)^{r}λ_{k} (v_{1}, …, v_{j}, λ_{l}(v_{j+1}, …, v_{j+l}), v_{j+l+1}, …, v_{n}) = 0,

r = l(ṽ_{1} + … ṽ_{j}) + j(l − 1) + (k − 1)l,

for any n ≥ 3 and any v_{1}, …, v_{n} ∈ V.

Proof. First we split Φ_{n},

Φ_{n}(v_{1}, …, v_{n}) = ∑_{{k+l=n+1 \atop k,l≥2}}(−1)^{(k−1)l}λ_{k} (λ_{l}(v_{1}, …, v_{l}), v_{l+1}, …, v_{n}) +

∑_{{k+l=n+1 \atop k,l≥2}}(−1)^{l(ṽ_{1}+...+ṽ_{k−1})+k−1}λ_{k} (v_{1}, …, v_{k−1}, λ_{l}(v_{k}, …, v_{n})) +

∑_{{k+l=n+1 \atop k,l≥2}} ∑_{j=1}^{k−2}(−1)^{r}λ_{k} (v_{1}, …, v_{j}, λ_{l}(v_{j+1}, …, v_{j+l}), v_{j+l+1}, …, v_{n}) = 0.

(4)

It is an easy calculation using (3) to check that

− ∑_{{k+l=n+1 \atop k,l≥2}}(−1)^{l+k(ṽ_{1}+...+ṽ_{l})}λ_{l}(v_{1}, …, v_{l}) · [Qλ_{k−1}(v_{l+1}, …, v_{n})]

+ ∑_{{k+l=n+1 \atop k+l≥2}}(−1)^{l(ṽ_{1}+...+ṽ_{k−1})}[Qλ_{k−1}(v_{1}, …, v_{k−1})] · λ_{l}(v_{k}, …, v_{n}) = 0.

Then the first two sums in (4) reduce, again with the help of (3), to the following expression

− ∑_{{k+l=n+1 \atop k,l≥2}} ∑_{{s+t=k \atop s≥2} ; t≥1} (−1)^{p} [Qλ_{s} (λ_{l}(v_{1}, …, v_{l}), v_{l+1}, …, v_{s+l−1}))] · [Qλ_{t}(v_{l+s}, …, v_{n})]

− ∑_{{k+l=n+1 \atop k,l≥2}} ∑_{{s+t=k \atop s≥1} ; t≥2} (−1)^{q} [Qλ_{s} (v_{1}, …, v_{s})] · [Qλ_{t} (v_{s+1}, …, v_{k−1}, λ_{l}(v_{k}, …, v_{n}))] ,

where p = k(l − 1) + s + (t − 1)(ṽ_{1} + … + ṽ_{s+l−1} + l) and q = l(ṽ_{1} + … + ṽ_{k−1}) + k − 1 + s + (t − 1)(ṽ_{1} + … + ṽ_{s}). The third sum in (4) splits into the following two sums

∑_{{k+l=n+1 \atop k,l≥2}} ∑_{j=1}^{k−2} ∑_{{s+t=k \atop 1≤s≤j} ; t≥1} (−1)^{a} [Qλ_{s}(v_{1}, …, v_{s})]·[Qλ_{t} (v_{s+1}, …, v_{j}, λ_{l}(v_{j+1}, …, v_{j+l}), v_{j+l−1}, …, v_{n})]

− ∑_{{k+l=n+1 \atop k,l≥2}} ∑_{j=1}^{k−2} ∑_{{s+t=k \atop s≥j+1} ; t≥1} (−1)^{b} [Qλ_{s} (v_{1}, …, v_{j}, λ_{l}(v_{j+1}, …, v_{j+l}), v_{j+l+1}, …, v_{s+l−1})]·[Qλ_{t}(v_{s+l}, …, v_{n})] ,

where a = r + s − 1 + (t − 1)(ṽ_{1} + … + ṽ_{s}) and b = r + s + (t − 1)(ṽ_{1} + … + ṽ_{s+l−1} + l).

Substituting these two expressions back into (4) one easily gets the following recursive formula

Φ_{n}(v_{1}, …, v_{n}) = ∑_{{k+l=n \atop k≥3} ; l≥1} (−1)^{(l−1)(ṽ_{1}+...+ṽ_{k})+k} [QΦ_{k}(v_{1}, …, v_{k})] · [Qλ_{l}(v_{k+1}, …, v_{n})]

− ∑_{{k+l=n \atop k≥1} ; l≥3} (−1)^{l(ṽ_{1}+...+ṽ_{k})} [Qλ_{k}(v_{1}, …, v_{k})] · [QΦ_{l}(v_{k+1}, …, v_{n})] ,

where n ≥ 4.

Finally we compute

Φ_{3}(v_{1}, v_{2}, v_{3}) = (v_{1} · v_{2}) · v_{3} − v_{1} · (v_{2} · v_{3}) = 0.

Thus Φ_{n} = 0 for all n ≥ 3. □

Lemma 3.3 The tensors λ_{k}, k ≥ 2, satisfy, for any n ≥ 2 and any v_{1}, …, v_{n} ∈ V, the identities

Θ_{n}(v_{1}, …, v_{n}) ≡ dλ_{n}(v_{1}, …, v_{n}) + ∑_{j=0}^{n−1}(−1)^{n−1+ṽ_{1}+...+ṽ_{j}}λ_{n}(v_{1}, …, v_{j}, dv_{j+1}, v_{j+2}, …, v_{n})

− ∑_{{k+l=n+1 \atop k,l≥2}} ∑_{j=0}^{k−1}(−1)^{r}λ_{k} (v_{1}, …, v_{j}, [d, Q]λ_{l}(v_{j+1}, …, v_{j+l}), v_{j+l+1}, …, v_{n})

= 0

where r = l(ṽ_{1} + … ṽ_{j}) + j(l − 1) + (k − 1)l.

A scetch of the proof. Following the same scenario as the one used in the proof of Lemma 3.2 (i.e. studing separately the group of terms with extreme values of the index j, where some cancellations occur, and applying (3) throughout), one gets the following recursion formula

Θ_{n}(v_{1}, …, v_{n}) = ∑_{{k+l=n \atop k≥2} ; l≥1} (−1)^{(l−1)(ṽ_{1}+...+ṽ_{k})+k} [QΘ_{k}(v_{1}, …, v_{k})] · [Qλ_{l}(v_{k+1}, …, v_{n})]

− ∑_{{k+l=n \atop k≥1} ; l≥2} (−1)^{l(ṽ_{1}+...+ṽ_{k})} [Qλ_{k}(v_{1}, …, v_{k})] · [QΘ_{l}(v_{k+1}, …, v_{n})] ,

where n ≥ 3.

Next we compute

Θ_{2}(v_{1}, v_{2}) = d(v_{1} · v_{2}) − (dv_{1}) · v_{2} − (−1)ṽ_{1}v_{1} · (dv_{2}) = 0.

Thus Φ_{n} = 0 for all n ≥ 1. □

Theorem 3.4 Let (V, d) be a differential associative superalgebra and let (W, d) ⊂ (V, d) be a subcomplex satisfying the Assumption 3.1. Then the linear maps

μ_{k} : ⊗^{k}W ⟶ W, k ≥ 1,

defined by

μ_{1} := d,
μ_{k} := (1 − [d, Q])λ_{k}, for k ≥ 2,  (5)

with λ_{k} being given by (3), satisfy the higher order associativity identitites (1) for all n ≥ 1 and thus define on W the structure of A_{∞}-algebra.

Proof. Denote the l.h.s. of the equation (1) by Ψ_{n}. Then Ψ_{1} and Ψ_{2} vanish because (W.d) is a subcomplex of (V, d). For n ≥ 3 one has

Ψ_{n} = (1 − [d, Q]) (Φ_{n} + Θ_{n}) .

By Lemmas 3.2 and 3.3, the tensors Φ_{n} and Θ_{n} vanish for all n ≥ 3. This completes the proof. □

Note that the higher homotopies μ_{n}, n ≥ 2, can be written as follows

μ_{n}(v_{1}, …, v_{n}) = − ∑_{{k+l=n+1 \atop k,l≥1}}(−1)^{k+(l−1)(ṽ_{1}+...+ṽ_{k})}[Qλ_{k}(v_{1}, …, v_{k})] ∘ [Qλ_{l}(v_{k+1}, …, v_{n})].

4 A_{∞}-algebras of a Kähler manifold

4.1. Hodge theory: a brief overview. Let M be a compact Kähler manifold, Ω^{•}M the algebra of real smooth differential forms on M, Ω^{•,•}M the algebra of complex differential forms with Hodge gradation, and

∂ : Ω^{•,•}M ⟶ Ω^{•+1,•}M,  ¯∂: Ω^{•,•}M ⟶ Ω^{•,•+1}M,

the standard derivations. We also consider two real derivations

d = ∂ + ¯∂: Ω^{•}M ⟶ Ω^{•+1}M, d_{c} = i(∂ − ¯∂) : Ω^{•}M ⟶ Ω^{•+1}M.

The Kähler metric g on M gives rise to a Hermitian metric on Ω^{•,•} defined, say for (p, q)-forms

α = ∑ α_{i_{1}...i_{p}j̄_{1}...j̄_{q}}dz^{i_{1}} ∧ … ∧ dz^{i_{p}}dz̄j̄_{1} ∧ … ∧ dz̄j̄_{q},

and

β = ∑ β_{s_{1}...s_{p}t̄_{1}...t̄_{q}}dz^{s_{1}} ∧ … ∧ dz^{s_{p}}dz̄t̄_{1} ∧ … ∧ dz̄t̄_{q},

by the integral

(α, β) := ∫_{M} (∑ g^{i_{1}s̄_{1}} … g^{i_{p}s̄_{p}}g^{t_{1}j̄_{1}} … g^{t_{q}j̄_{q}} α_{i_{1}...i_{p}j̄_{1}...j̄_{q}} \overline{β_{s_{1}...s_{p}t̄_{1}...t̄_{q}}}) vol_{g},

where g_{ij̄} are components of g in a local holomorphic coordinate chart {z^{i}}. The scalar product (α, β) of differential forms with different Hodge type is set to be zero.

This scalar product is used to define conjugate operators

∂^{∗} : Ω^{•,•}M ⟶ Ω^{•−1,•}M, ¯∂^{∗} : Ω^{•,•}M ⟶ Ω^{•,•−1}M, Λ : Ω^{•,•}M ⟶ Ω^{•−1,•−1}M,

by the formulae

(∂α, β) = (α, ∂^{∗}α), (¯∂α, β) = (α, ¯∂^{∗}β), (α ∧ ω, β) = (α, Λβ),

where ω denotes the Kähler form on M and α, β ∈ Ω^{•,•}M are arbitrary. We shall also need the following real operators

d^{∗} = ∂^{∗} + ¯∂^{∗}, d_{c}^{∗} = −i(∂^{∗} − ¯∂^{∗}).

There are four Laplacians,

Δ_{∂} = ∂∂^{∗} + ∂^{∗}∂, Δ ¯∂ = ¯∂¯∂^{∗} + ¯∂^{∗} ¯∂,
Δ_{d} = dd^{∗} + d^{∗}d, Δ_{d_{c}} = d_{c}d_{c}^{∗} + d_{c}^{∗}d_{c}.

One of the central results of the Hodge theory says that

Δ_{∂} = Δ ¯∂ = \frac{1}{2}Δ_{d} = \frac{1}{2}Δ_{d_{c}}.

A differential form α ∈ Ω^{•,•} is called harmonic if Δ_{d}α = 0. The vector space Harm of all real (or complex) harmonic forms on M is finite dimensional and is isomorphic to its de Rham (or Dolbeault) cohomology. There is a natural orthogonal projection

[ ]_{Harm} : Ω^{•,•} ⟶ Harm
α ⟶ [α]_{Harm}.

Note that all the Laplacians Δ become isomorphisms when restricted to the orthogonal complement, Harm^{⊥}, of the space of harmonic forms.

There are very important for our purposes Hodge identities

[Λ, ¯∂] = −i∂^{∗}, [Λ, ∂] = i∂^{∗}, [Λ, d] = d_{c}^{∗},  (6)

and the Hodge decompositions, for an arbitrary α ∈ Ω^{•,•},

α = [α]_{Harm} + ∂G_{∂}∂^{∗}(α) + ∂^{∗}G_{∂}∂(α)
α = [α]_{Harm} + ¯∂G ¯∂ ¯∂^{∗}(α) + ¯∂^{∗}G ¯∂ ¯∂(α)
α = [α]_{Harm} + dG_{d}d^{∗}(α) + d^{∗}G_{d}d(α)
α = [α]_{Harm} + d_{c}G_{d_{c}}d_{c}^{∗}(α) + d_{c}^{∗}G_{d_{c}}d_{c}(α),

where G : Ω^{•,•} ⟶ Ω^{•,•} are the Green operators defined by

G_{∂} |_{Harm}= 0, G_{∂} |_{Harm}_{⊥}= Δ_{p}^{−1},

and analogously for all others. A classical fact:

G_{∂} = G ¯∂ = 2G_{d} = 2G_{d_{c}}.

Note that ∂G_{∂} = G_{∂}∂, ∂^{∗}G_{∂} = G_{∂}∂^{∗}, etc.

For further details about Hodge theory on Kähler manifolds we refer to [2].

4.2. Real A_{∞}-algebra of a Kähler manifold. Consider the subcomplex (W := Kerd_{c}^{∗}, d) in the de Rham differential superalgebra (Ω^{•}M, d). It does satisfy the Assumption 3.1 with

Q = d_{c}G_{d}Λ.

Indeed,

1 − [d, Q] = 1 − d_{c}G_{d}[Λ, d] = 1 − d_{c}G_{d}d_{c}^{∗}

which, according to the Hodge decomposition, is precisely the projector from Ω^{•}M to its subspace Kerd_{c}^{∗}.

By Theorem 3.4, the space Kerd_{c}^{∗} has canonically the structure of an A_{∞}-algebra with the higher homotopeis given explicitly by the formulae (5) and (2). For example,

μ_{1} = d,
μ_{2}(v_{1}, v_{1}) = (1 − d_{c}G_{d}d_{c}^{∗})(v_{1} ∧ v_{2}) ≡ v_{1} ∘ v_{2},
μ_{3}(v_{1}, v_{2}, v_{3}) = [d_{c}G_{d}Λ(v_{1} ∧ v_{2})] ∘ v_{3} − (−1)ṽ_{1}v_{1} ∘ [d_{c}G_{d}Λ(v_{2} ∧ v_{3})] ,

etc.

In this algebra the product of two harmonic forms is obviously harmonic as well. Another interesting observation is that the complex (Kerd_{c}^{∗}, d) is precisely the one which gives rise to a Frobenius manifold structure on the de Rham cohomology of M [5]. The same comments will apply to our next example.

4.3. Complex A_{∞}-algebra of a Kähler manifold. Consider the subcomplex (W := Ker∂^{∗}, ¯∂) in the Dolbeault differential superalgebra (Ω^{•,•}M, ¯∂). It satisfies the Assumption 3.1 with

Q = i∂G_{∂}Λ.

Indeed,

1 − [d, Q] = 1 − i∂G_{∂}[Λ, ∂] = 1 − ∂G∂^{∗}

which, according to the Hodge decomposition, is precisely the projector from Ω^{•,•} to the subspace Ker∂^{∗}.

Therefore, by Theorem 3.4, the space Ker∂^{∗} has canonically the structure of an A_{∞}- algebra and one may explicitly write all the higher homotopies μ_{k} in terms of operators ∂, ∂^{∗}, G_{∂} and Λ.

4.4. A_{∞}-algebra of a Calabi-Yau manifold. A compact Kähler manifold M is called Calabi-Yau if c_{1}(M) = 0 or, equivalently, if it admits a nowhere vanishing holomorphic section Ω of the canonical line bundle of M.

Consider a differential associative superalgebra

V := Λ^{•}T M ⊗ Λ^{•}\overline{Ω^{1}M}

with the differential ¯∂ and the multiplication · being given by the ordinary wedge products on Λ^{p}T M and Λ^{q}\overline{Ω^{1}M}. It was first considered by Barannikov and Konstevich [1] in the context of mirror symmetry.

The holomorphic volume form defines an isomorphism

Ω : Λ^{•}T M ⊗ Λ^{•}\overline{Ω^{1}M} ⟶ Λ^{dim M−•}Ω^{1}M ⊗ Λ^{•}\overline{Ω^{1}M}
α ⊗ β  ⟶  (α⌟ Ω) ⊗ β.

Then for any linear operator

s : Ω^{•,•} ⟶ Ω^{•,•}

there is associated a linear operator

ŝ: V ⟶ V

defined as the composition:

ŝ: V ⟶^{Ω} Ω^{•,•} ⟶^{s} Ω^{•,•} ⟶^{Ω^{−1}} V.

In this way one obtains the operators ˆ∂, ˆ∂^{∗}, Ĝ_{∂}, etc. Note that ˆ ¯∂ = ¯∂. Note also that ˆ∂ is not a derivation of (V, ·).

Consider the following subcomplex

(W := Kerˆ∂, ¯∂)

in the Barannikov-Kontsevich differential superalgebra (V, ¯∂). It does satisfy the Assumption 3.1 with

Q = −iΛ̂Ĝ_{∂} ˆ∂.

Indeed,

1 − [d, Q] = 1 − i{ \atop [Λ, ¯∂]}Ĝ_{∂} ˆ∂ = 1 − ˆ∂^{∗}Ĝ_{∂} ˆ∂,

which, according to the Hodge decomposition, is precisely the projector from V to the subspace Kerˆ∂.

Therefore, by Theorem 3.4, the space Kerˆ∂ has canonically the structure of an A_{∞}- algebra and one may explicitly write down all the higher homotopies μ_{k} as explained in Sect.3. It is again a remarkable fact that the same complex (Kerˆ∂, ¯∂) admits not only an A_{∞} but also a very special Gerstenhaber-Batalin-Vilkoviski structure which makes the extended moduli space of complex structures on M into a Frobenius manifold [1].

Acknowledgments. This work has been done during author’s visit to the Max-Planck Institute for Mathematics in Bonn. It is a pleasure to thank Yu.I. Manin and V. Schechtman for valuable discussions.

References

[1] S. Barannikov and M. Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields, Internat. Math. Res. Notices, no.4 (1998), 201-215.

[2] P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons, 1978.

[3] M. Kontsevich, Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, Vol. II (Paris, 1992) (Bürkhäuser, Basel), 1994, pp. 97-121.

[4] M. Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol.1 (Zürich, 1994) (Bürkhäuser, Basel), 1995, pp. 120-139.

[5] S.A. Merkulov, Formality of canonical symplectic complexes and Frobenius manifolds, Internat. Math. Res. Notices, no. 14 (1998), 727-733.

[6] M. Penkava and A. Schwarz, A_{∞} algebras and the cohomology of moduli spaces, Lie Groups and Lie algebras: E.B. Dynkin’s seminar, AMS Providence, 1995, pp. 91-107.

[7] J.D. Stasheff, On the homotopy associativity of H-spaces, II, Trans. Amer. Math. Soc. 108 (1963), 293-312.

[8] J.D. Stasheff, Higher homotopy algebras: string field theory and Drinfeld’s quasi-Hopf algebras, in Proceedings of the XXth International Conference on Differential Geometric methods in Theoretical Physics (New York, 1991)(River Edge, NJ), vol.1, 1992, pp.408-425.

[9] J.D. Stasheff, Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space, Perspectives on Mathematics and Physics (R.C. Penner and S.Y. Yau, eds.), International Press, 1994, hep-th/930461, pp. 265-288.

Department of Mathematics University of Glasgow 15 University Gardens Glasgow G12 8QW, UK