In section 3.1, 3.1.2 we encountered the $(n-1)$-dimensional moduli spaces such as $\mathcal{M}(b,\ldots,b,\alpha,b^{\prime},\ldots,\beta)$ and $\mathcal{M}(b,\ldots,b,\gamma)$. The special case of holmorphic strips was already mentioned in Example 6.2.

We now consider the moduli $\mathcal{M}(p_{1},\ldots p_{k})$ of polygons with at least 3 corners $p_{1},\ldots p_{k}$, all regarded as inputs, arranged in clockwise order on $\partial\Sigma$, each carrying the local system factors $\Hom(E_{+},E_{-})|_{p_{i}}$ and the orientation factors $|o_{p_{i}}|$. The clockwise composition of the local system hom factors and the parallel transport along $\partial\Sigma$, produces a holonomy factor around $\partial\Sigma$, which is a number in $\mathbb{Q},\mathbb{R},\mathbb{Z}$ depending on the coefficient ring choice. Using (69) and Remark 6.9, as well as the clockwise orientation convention on the Stasheff associahedron, we acquire a (naïve) orientation on $T\mathcal{M}(p_{1},\ldots p_{k})$. To assign orientation and weighting factors to $\mathcal{M}(p_{1},\ldots p_{k})$, we take the product of the holonomy factor, the naïve orientation on $T\mathcal{M}(p_{1},\ldots p_{k})$, and another universal sign factor

$$(-1)^{\deg p_{1}+2\deg p_{2}+\ldots+k\deg p_{k}}(-1)^{\deg p_{k}}.$$

The appearance of this universal sign adjustment is a familiar convention in the open-closed map, cf. [2, eqn 5.24]. The notation $\mathcal{M}$ is a shorthand for the weighted sum of all the $(n-1)$-dimensional moduli spaces involved in the construction of the bordism current.

We equip the domain $\Sigma$ with the complex orientation, and together with an extra minus sign, the orientation on $\mathcal{M}$ induces the orientation on $\mathcal{C}$. This minus sign arises for the same reason as in Example 6.2, namely the discrepancy between our clockwise convention on $\partial\Sigma$, with the standard complex orientation on $\Sigma$.