We now explain how to semi-heuristically understand the first variation formula (21) as a consequence of Prop. 3.4, from the perspective of the moduli space integral formula (39). We hope this viewpoint is better suited for generalization to compact Calabi-Yau settings.

Suppose we are given a 1-parameter exact isotopy of unobstructed exact immersed Lagrangians $L_{t}$, and we wish to calculate $\frac{d}{dt}\mathcal{S}_{L_{0}^{\prime}}(L_{t})$ at $t=0$. The change of reference Lagrangian formula (cf. Prop. 3.4) allows us to replace $\mathcal{S}_{L_{0}^{\prime}}(L_{t})$ by $\mathcal{S}_{L_{0}}(L_{t})$. The Lagrangian $L_{t}$ for $|t|\ll 1$ is approximately the graph of $tdh$ in $T^{*}L_{0}$ (understood in an immersed sense), for the Hamiltonian function $h=h_{t}|_{t=0}$ on $L_{0}$. The holomorphic discs between $L_{t}$ and $L_{0}$ for $|t|\ll 1$ have small energy of order $O(|t|)$, and are locally approximated by Morse trajectories of $h$. Write $X_{h}$ as the Hamiltonian vector field, namely $\omega(X_{h},\cdot)=dh$, then

$$\frac{d}{dt}|_{t=0}\int_{\Sigma}F\omega=\int_{\partial\Sigma\cap L_{0}}F\omega(\cdot,X_{h})=-\int_{\partial\Sigma\cap L_{0}}Fdh.$$

Now we examine $\sum_{\text{all corners}}Ff|^{+}_{-}$ for very small $t$. The Lagrangian intersections come in two types:

• •

  The corners $p\in CF^{0}(L_{t},L_{0})$ and $q\in CF^{0}(L_{0},L_{t})$ correspond to the local extrema of the Hamiltonian $h$.

• •

  Any self intersection $p_{i}$ between two local sheets $L_{+},L_{-}$ of $L_{0}$ can be paired with a very nearby self intersection of $p_{i}^{t}$ between two sheets $L_{+}^{t},L_{-}^{t}$ of $L_{t}$. The bounding cochain on $L_{t}$ is thus induced from the bounding cochain on $L_{0}$.

At the intersection points $p,q$,

$$\begin{cases}f|^{+}_{-}(q)=f_{L_{0}}(q)-f_{L_{t}}(q)=-th(q)+O(t^{2}),\\ f|^{+}_{-}(p)=f_{L_{t}}(p)-f_{L_{0}}(p)=th(p)+O(t^{2}).\end{cases}$$

The self intersections are usually not important here, because the smallness of energy prevents their appearance on $\partial\Sigma$, unless $f|^{+}_{-}(p_{i}^{t})=O(|t|)$, and $f|^{+}_{-}(p_{i})=0$, which is a rather nongeneric situation. When the self intersections do appear, the evolution of the potential under exact isotopy gives

$$(f_{L_{+}^{t}}-f_{L_{-}^{t}})(p_{i}^{t})=(f_{L_{+}}-f_{L_{-}})(p_{i})+\int_{0}^{t}(h_{+,\tau}-h_{-,\tau})(p_{i}^{\tau})d\tau=f|^{+}_{-}(p_{i})+th|^{+}_{-}(p_{i})+O(t^{2}),$$

where $h_{\pm}$ keeps track of the hamiltonian on the different sheets of $L_{0}$. Thus

$$f|^{+}_{-}(p_{i}^{t})=f_{L_{-}^{t}}(p_{i}^{t})-f_{L_{+}^{t}}(p_{i}^{t})=-f|^{+}_{-}(p_{i})-th|^{+}_{-}(p_{i})+O(t^{2}).$$

Here we have a tricky sign reversal, because if $L_{+}$ and $L_{-}$ are clockwise ordered on $\partial\Sigma$, then $L_{+}^{t}$ and $L_{-}^{t}$ are counterclockwise ordered. In summary,

$$\frac{d}{dt}|_{t=0}\sum_{\text{all corners}}Ff|^{+}_{-}(t)=\lim_{t\to 0}\{-Fh(q)+Fh(p)-\sum_{L_{0}-\text{self intersection corners}}Fh|^{+}_{-}(p_{i})\}.$$

Combining the above, and integrating by parts,

$$\begin{split}&\frac{d}{dt}|_{t=0}\left(\int_{\Sigma}F\omega+\sum_{\text{all corners}}Ff|^{+}_{-}(t)\right)\\ &=\lim_{t\to 0}\{-Fh(q)+Fh(p)-\sum_{L_{0}-\text{corners}}Fh|^{+}_{-}(p_{i})-\int_{\partial\Sigma\cap L_{0}}Fdh\}\\ &=\lim_{t\to 0}\{\int_{\partial\Sigma\cap L_{0}}hdF\}.\end{split}$$

Observe that for very small $t$, as the holomorphic curves vary in the $(n-1)$-dimensional moduli spaces $\mathcal{M}$, under the counterclockwise sign convention for $\partial\Sigma$, the boundary evaluation of $\partial\Sigma\cap L_{0}$ sweeps out the cycle $L_{0}$ (beware of the sign!), and any generic point on $L_{0}$ is swept out precisely once due to the Morse theory limiting description. Consequently, the moduli space integral

$$\lim_{t\to 0}\int_{\mathcal{M}}\{\int_{\partial\Sigma\cap L_{0}}hdF\}=\int_{L_{0}}h\Omega,$$

hence

$$\frac{d}{dt}|_{t=0}\int_{\mathcal{M}}\left(\int_{\Sigma}F\omega+\sum_{\text{all corners}}Ff|^{+}_{-}(t)\right)=\int_{L_{0}}h\Omega.$$

By the moduli integral formula (39) of the Solomon functional,

$$\frac{d}{dt}|_{t=0}\mathcal{S}(L_{t})=\frac{d}{dt}|_{t=0}\int_{\mathcal{M}}\mathcal{I}=\int_{L_{0}}h\text{Im}(e^{-i\hat{\theta}}\Omega).$$

This recovers the first variation formula (21).