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 $L_{0}$.