Assuming automatic transversality, we can rewrite the Solomon functional (20) as a moduli space integral in terms of the notations introduced in section 3.5. Let $u:\Sigma\to X$ be a holomorphic polygon, with first order deformation vector fields $v_{1},\ldots v_{n-1}$, so we can define a holomorphic function $F$ via (33). In clockwise order on $\partial\Sigma$, we encounter the degree one intersections on $L$, an intersection $p\in CF^{0}(L,L_{0})$, the degree one self intersections on $L_{0}$, and an intersection $q\in CF^{0}(L_{0},L)$. As before, we fix the additive constant by $F(q)=0$.

In the following calculation, we will use the complex orientation on $\Sigma$, and the counterclockwise orientation on $\partial\Sigma$. The Solomon functional contains a term $-\int_{\mathcal{C}}\lambda\wedge\text{Im}(e^{-i\hat{\theta}}\Omega).$ Now $-\int_{\mathcal{C}}\lambda\wedge\Omega$ can be expressed as an integral of the following $(n-1)$-form over the $(n-1)$-dimensional moduli spaces $\mathcal{M}$ of holomorphic curves:

$$\int_{\Sigma}\lambda\wedge\Omega(\cdot,v_{1},\ldots v_{n-1})=\int_{\Sigma}\lambda\wedge dF.$$

Notice that since $u:\Sigma\to X$ is a holomorphic curve and $\Omega$ is an $(n,0)$-form, adjusting $v_{i}$ by a vector field tangent to $\Sigma$ does not change this integrand, and all $v_{i}$ must hit $\Omega$ instead of the 1-form $\lambda$. After integration by part,

$$\int_{\Sigma}\lambda\wedge dF=\int_{\Sigma}Fd\lambda-\int_{\partial\Sigma}F\lambda=\int_{\Sigma}F\omega-\int_{\partial\Sigma}F\lambda.$$

The Solomon functional contains another two terms $\int_{L}f_{L}\text{Im}(e^{-i\hat{\theta}}\Omega)$ and $-\int_{L_{0}}f_{L_{0}}\text{Im}(e^{-i\hat{\theta}}\Omega)$. Now $\int_{L}f_{L}\Omega$ can be expressed as an integral of the following $(n-1)$-form over the moduli spaces $\mathcal{M}$:

$$-\int_{\partial\Sigma\cap L}f_{L}\Omega(\cdot,v_{1},\ldots v_{n-1})=-\int_{\partial\Sigma\cap L}f_{L}dF.$$

The abused notation $\partial\Sigma\cap L$ means the part of $\partial\Sigma$ mapping to $L$ instead of $L_{0}$. Similarly $-\int_{L_{0}}f_{L_{0}}\Omega$ is the moduli space integral of the $(n-1)$-form

$$-\int_{\partial\Sigma\cap L_{0}}f_{L_{0}}\Omega(\cdot,v_{1},\ldots v_{n-1})=-\int_{\partial\Sigma\cap L_{0}}f_{L_{0}}dF.$$

The extra minus sign comes from the fact that $\partial\mathcal{C}$ sweeps out the cycle $-L_{0}$ instead of $L_{0}$.

Combining all the three contributions, the Solomon functional is the moduli space integral with integrand

$$\mathcal{I}=\text{Im}\int_{\Sigma}e^{-i\hat{\theta}}F\omega-\text{Im}\int_{\partial\Sigma\cap L}e^{-i\hat{\theta}}d(f_{L}F)-\text{Im}\int_{\partial\Sigma\cap L_{0}}e^{-i\hat{\theta}}d(f_{L_{0}}F).$$

The last two terms involve total derivatives, so can be integrated along boundary segments between the corner points, to yield

$$\begin{split}&-\text{Im}\int_{\partial\Sigma\cap L}e^{-i\hat{\theta}}d(f_{L}F)-\text{Im}\int_{\partial\Sigma\cap L_{0}}e^{-i\hat{\theta}}d(f_{L_{0}}F)\\ &=\text{Im}\sum_{\text{all corners}}e^{-i\hat{\theta}}Ff|^{+}_{-},\end{split}$$

(38)

where $f|^{+}_{-}$ stands for the difference of the potentials $f_{L_{+}}-f_{L-}$ at a Lagrangian intersection point, such that $\partial\Sigma$ moves from $L_{+}$ to $L_{-}$ in the clockwise direction. In the more general framework of Floer theory with Novikov coefficients, $f|^{+}_{-}$ have the interpretation as the Novikov exponents of these intersection points. The bounding cochain elements have $f|^{+}_{-}\geq 0$, while $p\in CF^{0}(L,L_{0}),q\in CF^{0}(L_{0},L)$ may have negative Novikov exponents.

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.

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).

Floer theoretic foundations are much more complicated beyond the exact setting, and the foundations concerning the open-closed string map in the immersed Fukaya category setting are not fully written out in the literature. Nonetheless, due to the interest of the topic, we shall offer some speculations about how the Solomon functional formula (39) generalizes to the compact almost Calabi-Yau setting. Our local systems will have coefficients in $\mathbb{R},\mathbb{Q}$, i.e. the parallel transport in the local system have only Novikov exponent zero components. This convention is somewhat more restrictive than [7][81][82].

First, we recall the role of Novikov coefficients in Floer theory. All Floer cochain spaces $CF^{*}$ are modules over the Novikov field

$$\Lambda=\{\sum_{i}a_{i}T^{\lambda_{i}}:\quad\lambda_{i}\in\mathbb{R},\lambda_{1}<\lambda_{2}<\ldots\to+\infty\},$$

and $a_{i}\in\mathbb{Q}$ or $\mathbb{R}$ depending on the coefficient field choice.⁴³⁴³ 43 We do not know if the Fukaya category can be defined over integers in general. One should not confuse the coefficients $a_{i}$ with the Novikov exponents $\lambda_{i}$. Typically $a_{i}$ are rational numbers related to counting, while $\lambda_{i}$ are real numbers related to the energy. There are a few conventions to define $A_{\infty}$-operations. Let $L$ be a compact immersed Lagrangian with transverse self intersections. In the Morse model [81], the self Floer cochain space $CF^{*}(L,L)$ is generated by the Morse critical points on $L$, and the ordered self intersections (twisted by local system hom and orientation factors as usual). The Fukaya $A_{\infty}$-algebra is a collection of Novikov-multilinear operations

$$m_{k}:CF^{*}(L,L)\otimes\ldots CF^{*}(L,L)\to CF^{*}(L,L)[2-k]$$

defined by counting holomorphic treed discs $u:\Sigma\to X$ (cf. [81, Definition 3.1]), weighted by the holonomy and orientation factors, and an energy factor $T^{E(u)}$. Very roughly, the domain $\Sigma$ have surface parts (which consist of discs, and spheres attached to them), and tree parts connecting the disc boundaries. Then $u$ is a holomorphic map with Lagrangian boundary on the surface parts, and Morse gradient flowlines on the tree parts. The role of $CF^{*}$ elements is to specify the limiting behaviour of the Morse flowlines, and the Lagrangian self intersections on $\partial\Sigma$. The energy $E(u)=\int_{\Sigma}\omega$ is the sum of $\int u^{*}\omega$ on all the surface parts of $\Sigma$.

A nontrivial fact is that (after complicated perturbation schemes, or virtual techniques) this gives rise to a curved $A_{\infty}$-algebra structure [81]. The most important new feature, absent in the exact case, is that the disc bubbling can occur at points of $L$, which are not necessarily self intersection points. The domain disc splits into two discs, attached at a boundary node. This phenomenon is compensated by considering two discs joined by a gradient flowline segment, whose length shrinks to zero, producing the same nodal discs in the degeneration limit. With the appropriate weights and orientations taken into account, these two effects would cancel algebraically. On the other hand, the length parameter of the tree parts can tend to infinity, causing the Morse gradient flow line to break, a phenomenon which contributes to the boundary of the one dimensional moduli spaces, reflected algebraically in the $A_{\infty}$-relations.

Similar to the exact immersed case (cf. Appendix 6.2), the bounding cochains are $b\in CF^{1}(L,L)$ elements satisfying the nonnegative Novikov exponent requirement, and the Mauer-Cartan equation

$$m_{0}+m_{1}(b)+m_{2}(b,b)+\ldots=0.$$

Generally speaking, the sum is infinite, but after truncating the Novikov series at any given high energy, only finitely many terms appear due to Gromov compactness, so the sum makes formal sense. As usual, the Lagrangian with bounding cochain structures are called unobstructed.

The framework for setting up the Fukaya algebra of a single immersed Lagrangian, also assigns meanings to Floer cohomologies between two immersed Lagrangians with bounding cochain structures. Suppose $\alpha\in CF^{0}(L,L^{\prime})$ and $\beta\in CF^{0}(L^{\prime},L)$ represent elements in $HF^{0}(L,L^{\prime})$ and $HF^{0}(L^{\prime},L)$ whose cohomological compositions are the identities. The $\alpha,\beta$ are in generally represented by infinite series in the Novikov variable $T$, where some Novikov exponents may be negative, and may bot be bounded above, but at least they are bounded from below depending on $\alpha,\beta$. We now speculate that there is a bordism current $\mathcal{C}$ with $\partial\mathcal{C}=L-L^{\prime}$, constructed from the universal families of treed holomorphic discs over the $(n-1)$-dimensional moduli spaces $\mathcal{M}$. The monomial summands of $\alpha,\beta$ and the bounding cochain elements prescribe the corners of the treed holomorphic discs, and monomials with different Novikov exponents are viewed as independent contributions to $\mathcal{M}$ and $\mathcal{C}$. Beyond the almost calibrated case, one would also need to incorporate degree $-1$ self intersections as usual. We think the moduli spaces that contribute to $\mathcal{C}$ would satisfy the Novikov exponent condition

$$\int_{\Sigma}\omega+\sum_{\text{all corners}}\text{Novikov exponent}=0.$$

(42)

Here the corners include the monomial summands of $\alpha,\beta$ (or the degree $-1$ self intersections as appropriate), and the bounding cochains at the degree one self intersections/Morse critical points of $L,L^{\prime}$. Since all Novikov exponents at the bounding cochains are non-negative (not so at $\alpha,\beta$, and the degree $-1$ self intersections!), this condition would impose an energy upper bound on the holomorphic treed discs depending on $\alpha,\beta$, whence only finitely many moduli spaces contribute to the bordism current.

Now the moduli space integral formula (39) formally makes sense almost verbatim, ignoring all virtual perturbation nuances. For first order deformations $v_{1},\ldots v_{n-1}$ of the holomorphic treed discs, we can define $F$ on the domain $\Sigma$ via the 1-form $dF=\Omega(\cdot,v_{1},\ldots v_{n-1})$. On the surface parts of $\Sigma$, we would obtain a holomorphic function $F$ by complex integrability as usual (which must be constant on the holomorphic sphere components by the Liouville theorem), while on the tree parts, there is no obstruction for the 1-form to be exact. Next, we replace the appearance of $f|^{+}_{-}$ in (39) by the Novikov exponents of the monomial summands at the corners, to define the moduli integrand $\mathcal{I}$. The term $\int_{\Sigma}F\omega$ is understood to only involve integration on the surface parts of $\Sigma$. The Solomon functional $\mathcal{S}(L)$ still has the form $\int_{\mathcal{M}}\mathcal{I}$. Notice that adding a constant to $F$ would not change the moduli integrand $\mathcal{I}$, thanks to (42).

In the absence of the Lagrangian potential, the Novikov exponents of $\alpha,\beta$ are no longer canonically fixed. Suppose we replace $\alpha$ by $T^{\mu}\alpha$, and $\beta$ by $T^{-\mu}\beta$, for some $\mu\in\mathbb{R}$. This would affect the moduli integrand $\mathcal{I}$, by the amount

$$\mu\text{Im}\{e^{-i\hat{\theta}}(F(p)-F(q))\},$$

where $p,q$ stand for the components of $\alpha,\beta$. By analogy with the exact case, we expect

$$\text{Im}\left(e^{-i\hat{\theta}}\int_{\mathcal{M}}F(p)-F(q)\right)=\text{Im}\left(e^{-i\hat{\theta}}\int_{L}\Omega\right)=0,$$

whence $\mathcal{S}(L)$ is independent of $\mu$.

Once the foundations are in place, we expect

The slogan is that Floer theory should fix the multivaluedness problem of the Solomon functional (cf. section 2.8). As a more technical observation, once the change of reference Lagrangian formula (22) is established, one can remove the assumption for $L$ to be transverse to $L_{0}$, using a perturbation $L_{0}^{\prime}$ of $L_{0}$.