We now revisit Prop. 3.4 from the moduli space integral perspective, which we expect is better suited for generalization to compact Calabi-Yau settings. All transversality requirements of moduli spaces will be assumed, and in this sense the calculations below are formal.

In Remark 3.5 we sketched that under the extra assumption $HF^{-1}(L_{0},L_{0})=0$, there is an $(n+2)$-dimensional universal family $\tilde{\mathcal{C}}$ over some $n$-dimensional moduli $\tilde{\mathcal{M}}$, such that the boundary of $\tilde{C}$ has three $(n+1)$-dimensional contributions, corresponding up to sign to the three bordism currents $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ between $L_{0},L_{0}^{\prime},L$, which are in turn the universal families over the $(n-1)$-dimensional moduli spaces $\mathcal{M}_{i}$ for $i=1,2,3$. Here $\mathcal{M}_{i}$ can be viewed as certain boundary strata of the compactification of $\tilde{\mathcal{M}}$. The integrand $\mathcal{I}$ naturally makes sense as an $(n-1)$-form on $\tilde{\mathcal{M}}$, and restricts naturally to $\mathcal{M}_{i}$. The change of reference formula (22) amounts to

$$\int_{\mathcal{M}_{3}}\mathcal{I}=\int_{\mathcal{M}_{1}}\mathcal{I}+\int_{\mathcal{M}_{2}}\mathcal{I}.$$

(40)

Our strategy is to use Stokes formula on the moduli spaces. As usual, the holonomy weighting factors will be suppressed in the moduli integral notations. Then (40) reduces to the two claims:

We first explain Claim 3.36. First, we calculate the derivatives of $F$. Let $y_{1},\ldots y_{n}$ be local coordinates on $\tilde{\mathcal{M}}$, so $\frac{\partial}{\partial y_{i}}$ can be identified as first order deformations of holomorphic curves. The local coordiates on $\Sigma$ are denoted as $s,t$. We can write $F=\sum F_{i}(-1)^{i-1}dy_{1}\wedge\ldots\stackrel{{\scriptstyle\mbox{\large$\frown$}}}{{dy_{i}}}\ldots dy_{n}$, such that along $\Sigma$,

$$\partial_{s}F_{i}=(-1)^{i-1}\Omega(\frac{\partial}{\partial s},\frac{\partial}{\partial y_{1}},\ldots\stackrel{{\scriptstyle\mbox{\large$\frown$}}}{{\frac{\partial}{\partial y_{i}}}},\ldots\frac{\partial}{\partial y_{n}}),\quad\partial_{t}F_{i}=(-1)^{i-1}\Omega(\frac{\partial}{\partial t},\frac{\partial}{\partial y_{1}},\ldots\stackrel{{\scriptstyle\mbox{\large$\frown$}}}{{\frac{\partial}{\partial y_{i}}}},\ldots).$$

The holomorphic volume form satisfies $d\Omega=0$, whence

$$\sum_{i=1}^{n}\partial_{s}\partial_{i}F_{i}=\partial_{s}(\Omega(\frac{\partial}{\partial y_{1}},\ldots,\frac{\partial}{\partial y_{n}})),\quad\sum_{i=1}^{n}\partial_{t}\partial_{i}F_{i}=\partial_{t}(\Omega(\frac{\partial}{\partial y_{1}},\ldots,\frac{\partial}{\partial y_{n}})).$$

Notice $\Omega(\frac{\partial}{\partial y_{1}},\ldots,\frac{\partial}{\partial y_{n}})$ vanishes at the corners due to the decay of the first order deformation vector fields, and comparing with the additive normalization convention on $F$, we find

$$\sum_{i=1}^{n}\partial_{i}F_{i}=\Omega(\frac{\partial}{\partial y_{1}},\ldots,\frac{\partial}{\partial y_{n}}).$$

(41)

In particular, at all the corner points $\sum_{i}\partial_{i}F_{i}=0$. The term $f|^{+}_{-}$ at the corners are independent of the moduli space parameters, so the only contribution to $d\mathcal{I}$ comes from the $\text{Im}\int_{\Sigma}e^{-i\hat{\theta}}F\omega$ term in formula (39).

We calculate

$$\sum_{i=1}^{n}\partial_{i}\int_{\Sigma}F_{i}\omega=\int_{\Sigma}\sum_{i}\partial_{i}F_{i}\omega+\int_{\Sigma}\sum_{i}F_{i}\partial_{i}\omega.$$

Here $\partial_{i}\omega$ is the Lie derivative of the symplectic form $\omega$ with respect to the vector field $\frac{\partial}{\partial y_{i}}$, which by Cartan’s formula is

$$\partial_{i}\omega=d(\omega(\frac{\partial}{\partial y_{i}},\cdot))+\iota_{\frac{\partial}{\partial y_{i}}}d\omega=d(\omega(\frac{\partial}{\partial y_{i}},\cdot)).$$

Thus

$$\int_{\Sigma}\sum_{i}F_{i}\partial_{i}\omega=\int_{\partial\Sigma}\sum_{i}F_{i}\omega(\frac{\partial}{\partial y_{i}},\cdot)-\int_{\Sigma}\sum_{i}dF_{i}\wedge\omega(\frac{\partial}{\partial y_{i}},\cdot).$$

Notice that $\partial\Sigma$ and $\frac{\partial}{\partial y_{i}}$ are both tangent to the Lagrangian boundary, so the $\partial\Sigma$ integrand vanishes. We are left with

$$\begin{split}&\sum_{i=1}^{n}\partial_{i}\int_{\Sigma}F_{i}\omega=\int_{\Sigma}\sum_{i}\partial_{i}F_{i}\omega-\int_{\Sigma}\sum_{i}dF_{i}\wedge\omega(\frac{\partial}{\partial y_{i}},\cdot)\\ =&\int_{\Sigma}\Omega(\frac{\partial}{\partial y_{1}},\ldots,\frac{\partial}{\partial y_{n}})\omega+\sum_{i}(-1)^{i-1}\omega(\frac{\partial}{\partial y_{i}},\cdot)\wedge\Omega(\cdot,\frac{\partial}{\partial y_{1}},\ldots\stackrel{{\scriptstyle\mbox{\large$\frown$}}}{{\frac{\partial}{\partial y_{i}}}},\ldots\frac{\partial}{\partial y_{n}}).\end{split}$$

Here we used the definition of $F_{i}$ via $\partial_{s}F_{i},\partial_{t}F_{i}$, and the formula (41) for $\sum_{i}\partial_{i}F_{i}$. We contract the identity $\omega\wedge\Omega=0$ with $\frac{\partial}{\partial y_{1}},\ldots,\frac{\partial}{\partial y_{n}}$. When two $\frac{\partial}{\partial y_{i}}$ hit $\omega$, the $\Omega$ term will be contracted only $(n-2)$ times, which produces a $(2,0)$-form vanishing identically on the holomorphic curve $\Sigma$. When at most one $\frac{\partial}{\partial y_{i}}$ hits $\omega$, we obtain the above integrand. In effect, the integrand vanishes identically:

$$\sum_{i=1}^{n}\partial_{i}\int_{\Sigma}F_{i}\omega=0,$$

which then implies $d\mathcal{I}=0$.

We next explain Claim 3.37. In general, the compactified moduli space has many boundary strata corresponding to disc bubbling and disc splitting.

To see this, we need to understand how $\mathcal{I}$ (and notably $F$) behaves near the boundary of the moduli space. Recall that when the holomorphic disc is degenerating to several disc components, then under transversality conditions, the cokernel of the extended linearized Cauchy-Riemann operator vanishes, and for small fixed gluing parameters, the kernel elements (i.e. first order deformations) are up to small perturbation obtained by gluing the kernel elements from the degenerate disc components. The perturbation effect tends to zero as we approach the moduli space boundary. Now the kernel elements from different disc components have essentially disjoint supports, so unless we have at least $(n-1)$ kernel elements supported on one disc component such that $\Omega(\cdot,v_{1},\ldots v_{n-1})$ does not vanish identically, we will have $dF=0$ for the moduli boundary strata, so that $F=\text{const}$ along $\Sigma$, whence $\mathcal{I}=0$ by Remark 3.16. This shows Claim 3.38. We comment that this phenomenon is closely related to the fact that many moduli boundary strata do not contribute to the boundary of the bordism current $\mathcal{C}$ due to support reasons (cf. section 3.1).

On the boundary strata, the only contributions to the integral $\mathcal{I}$ come from the $(n-1)$-dimensional moduli spaces. The role of the holomorphic curves of virtual dimension zero, is to provide the counting factors, in a manner entirely analogous to section 3.1.2. Most contributions cancel out due to the Mauer-Cartan equation on the bounding cochains, and the closedness of the $HF^{0}$ generators. The remaining contributions produce the RHS in Claim 3.37.