A symplectic Calabi-Yau manifold consists of $(M,\omega,J,\psi)$ , where $(M,\omega)$ is a $2n$ -dimensional symplectic manifold, $J$ is an $\omega$ -calibrated almost complex structure on $M$ and $\psi$ is a nowhere vanishing $(n,0)$ -form on $M$ satisfying

$$\nabla\psi=0\,,$$

where $\nabla$ is the Chern connection of $(\omega,J)$ .

Define $dg\mathcal{N}(R)$ to be the category of $R$ -representations in finite-dimensional nilpotent non-negatively graded chain Lie algebras. Let $dg\hat{\mathcal{N}}(R)$ be the category of pro-objects in the Artinian category $dg\mathcal{N}(R)$ .

Here, a chain Lie algebra is a chain complex $\mathfrak{g}=\bigoplus_{i\in\mathbb{N}_{0}}\mathfrak{g}_{i}$ over $k$ , equipped with a bilinear Lie bracket $[,]\colon\thinspace\mathfrak{g}_{i}\times\mathfrak{g}_{j}\rightarrow\mathfrak{g}_{i+j}$ , satisfying:

1. 1.

   $[a,b]+(-1)^{\bar{a}\bar{b}}[b,a]=0$ ,

2. 2.

   $(-1)^{\bar{c}\bar{a}}[a,[b,c]]+(-1)^{\bar{a}\bar{b}}[b,[c,a]]+(-1)^{\bar{b}\bar{c}}[c,[a,b]]=0$ ,

3. 3.

   $d[a,b]=[da,b]+(-1)^{\bar{a}}[a,db]$ ,

where $\bar{a}$ denotes the degree of $a$ , mod $2$ , for $a$ homogeneous.

Define a small extension in $dg\mathcal{N}(R)$ to be a surjective map $\mathfrak{g}\to\mathfrak{h}$ with kernel $I$ , such that $[\mathfrak{g},I]=0$ . Note that the objects of $dg\mathcal{N}(R)$ are cofinite in $dg\hat{\mathcal{N}}(R)$ in the sense of [ Hov ] Definition 2.1.4.

Define $DG\mathrm{Alg}(R)$ to be the category of $R$ -representations in non-negatively graded cochain algebras. Here, a cochain algebra is a cochain complex $A=\bigoplus_{i\in\mathbb{N}_{0}}A^{i}$ over $k$ , equipped with an associative product $A^{i}\times A^{j}\rightarrow A^{i+j}$ , satisfying:

1. 1.

   $ab=(-1)^{\bar{a}\bar{b}}ba$ ,

2. 2.

   $d(ab)=(da)b+(-1)^{\bar{a}}a(db)$ ,

where $\bar{a}$ denotes the degree of $a$ , mod $2$ , for $a$ homogeneous, and a multiplicative identity $1\in A^{0}$ .

For a given HOPS $(\mathcal{A},\varphi)$ we define the corresponding (higher order) free cumulants as a function on ${\mathcal{PS}}(\mathcal{A})$ by

$$\kappa=\varphi*\mu,$$

or more explicitly

$$\kappa({\mathcal{U}},\gamma)[a_{1},\dots,a_{n}]:=\sum_{({\mathcal{V}},\pi),({\mathcal{W}},\sigma)\in{\mathcal{PS}}(n)\atop({\mathcal{V}},\pi)\cdot({\mathcal{W}},\sigma)=({\mathcal{U}},\gamma)}\varphi({\mathcal{V}},\pi)[a_{1},\dots,a_{n}]\cdot\mu({\mathcal{W}},\sigma),$$

for all $n\in{\mathbb{N}}$ , $({\mathcal{U}},\gamma)\in{\mathcal{PS}}(n)$ , $a_{1},\dots,a_{n}\in\mathcal{A}$ .