If one wishes to build Floer theory for Lagrangian currents by mimicking the smooth case constructions, then one immediately runs into a large number of severe difficulties.

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  For exact embedded Lagrangians, the self Floer cohomology of a Lagrangian is isomorphic to the singular cohomology: $HF^{*}(L,L)\simeq H^{*}(L)$. Now in the light of Almgren’s big regularity theorem, our best hope is that in the variational argument we only encounter codimension two singularities in the Lagrangian. We have no right to assume the topology of the Lagrangian is fixed in the variational framework. The homology groups $H_{n-m}(L)$ for $m\geq 1$ are highly unstable under varifold/current convergence if codimension two singularities can form, so for $m\geq 1$ we do not expect a direct geometric definition of $HF^{m}(L,L)$ for Lagrangian currents, that possesses any reasonable continuity property under convergence.

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  The standard way to set up Floer theory between two Lagrangians is to consider the transverse intersection points as the generators of the Floer complex, and counts of holomorphic strips as differentials between generators. This viewpoint depends heavily on the differential topology of the Lagrangians, which runs into troubles for Lagrangian currents, where tangent spaces only need to exist almost everywhere in a measure theoretic sense.

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  Once Lagrangian intersections are not well behaved, we cannot define the bounding cochains supported at intersection points in the usual way.

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  Parallel transport along local systems may break down.

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  It is unclear how to define (relative) spin structures on Lagrangian currents.

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  Standard Floer theory depends heavily on transversality arguments based on differential topology, which is lost on Lagrangian currents.

In short, a direct geometric construction of the $A_{\infty}$ structure is unlikely for Lagrangian currents.

One natural idea is that we only develop Floer theory for sufficiently smooth Lagrangians (eg. immersed Lagrangians, isolated $T^{2}$-cones, etc), and formally treat Lagrangian currents using approximation by smooth objects. Suppose $L_{i}$ are sufficiently smooth Lagrangian branes in the same $D^{b}Fuk(X)$ class, and $L_{i}\to L$ in the varifold/current topology, and assume the brane structures provide a Cauchy sequence in some appropriate sense, then one formally declare the Lagrangian current $L$ as carrying an object in the same $D^{b}Fuk(X)$ class. A weak Lagrangian brane would then tautologically be an equivalence class of Cauchy sequences. The same Lagrangian current may in principle support many different formal brane structures, not necessarily all in the same derived category class.

In this perspective, weak Lagrangian branes are indirect constructions, whose properties amount to quantitative properties of sufficiently smooth Lagrangians that can be bounded in terms of a priori quantities such as the distance on the branes, the flat norm on the currents, the Hausdorff distance between the Lagrangians, etc.

One concrete notion of distance is as follows (cf. [35, Definition 2.2], see also [10, section 5]). We can look for the $\alpha,\beta$ representing generators in $HF^{0}(L,L^{\prime})$ and $HF^{0}(L^{\prime},L)$ with cohomological compositions equal to the identity; in the almost calibrated case $CF^{-1}(L,L^{\prime})=0$, so $\alpha,\beta$ are unique up to scaling. Since all bounding cochains and $A_{\infty}$ products have non-negative Novikov exponents, and the sum of Novikov exponents add up to zero, we must have some negative Novikov exponent for $\alpha$ or $\beta$. In our context, the Novikov exponent amounts to $(f_{L}-f_{L^{\prime}})(p)$ at $p\in CF^{0}(L,L^{\prime})$ and $(f_{L^{\prime}}-f_{L})(q)$ at $q\in CF^{0}(L^{\prime},L)$. The quantity

$$-\min\{\text{Novikov exponents among all intersection points of $\alpha,\beta$}\}$$

provides a candidate notion of distance $d(L,L^{\prime})$ between Lagrangian branes. Notice this distance bounds the energy of the holomorphic discs with boundary on $L,L^{\prime}$. Given three objects $L,L^{\prime},L^{\prime\prime}$, by considering the composition of the generators, it is easy to deduce $d(L,L^{\prime\prime})\leq d(L,L^{\prime})+d(L^{\prime},L^{\prime\prime})$.

Does this notion of distance have any precompactness property? Namely, given a sequence of sufficiently smooth Lagrangian objects $L_{i}\in\mathcal{L}$, (eg. a minimizing sequence for the Solomon functional), and assuming the Lagrangian potentials are uniformly bounded, then up to making gauge equivalent choices of local systems and bounding cochains, when can we extract a Cauchy subsequence?

One may also ask if the weak Lagrangian branes agree with ordinary Lagrangian branes in the case of smooth immersed Lagrangians:

It is interesting to ask if any Floer theoretic geometric construction may be performed on Lagrangian currents at all. While the $A_{\infty}$-category structure on the Fukaya category may not necessarily be robust under varifold/current convergence of Lagrangians, only a subset of the structures are essential to the Thomas-Yau conjecture:

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  The notion of derived Fukaya category classes.

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  The notion of distinguished triangles, within the class of Lagrangians $\mathcal{L}$. This is the categorical shadow of the phenomenon that Lagrangians can be broken into several components under weak limits.

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  The central charge function.

The central charge is of numerical nature, and is continuous under convergence in the current topology. A key feature of $L,L^{\prime}$ lying in the same derived category class is that there is a bordism current $\mathcal{C}$ constructed from holomorphic curves, such that $\partial\mathcal{C}=L-L^{\prime}$. Likewise for distinguished triangles $L_{1}\to L\to L_{2}\to L_{1}[1]$ in the weak regularity setting, a key expected property is that there should be a bordism current between $L$ and $L_{1}+L_{2}$, constructed from families of holomorphic curves.

The basic idea is to take the bordism current $\mathcal{C}_{i}$ with $\partial\mathcal{C}_{i}=L_{i}-L_{i}^{\prime}$, constructed from the universal family of holomorphic curves, and attempt to extract the limit as currents. This could be morally viewed as a version of Gromov compactness for families. As rather strong evidence, in the quantitatively almost calibrated setting we derived uniform energy bound for holomorphic curves contributing to $\mathcal{C}_{i}$, by proving the potential clustering property (cf. section 5.2.1, and Prop. 3.41). If we work with Fukaya category over the integers, the bordism currents $\mathcal{C}_{i}$ would be integral currents, and we can hope to extract limit by some compactness argument. The problem is that we do not know $\mathcal{C}_{i}$ have uniform mass upper bounds. Morever, it is an interesting question how to formulate the parametrized family structure of the bordism current in the geometric measure theory language.

While Lagrangian intersections, bounding cochains, spin structures, local systems etc. do not make sense directly on Lagrangian currents, the bordism current has a chance to make sense, and encodes substantial information. For instance, the orientations of the moduli spaces reflect the spin structures, and the weighting factors for the moduli spaces encode the combined effect of bounding cochain elements and the parallel transport along the local system.

We mentioned in Remark 3.5 that when more than two Lagrangians are present, Floer theory would also produce $(n+2)$-dimensional currents whose boundary exhibit homological relations between the $(n+1)$-dimensional bordism currents. Such ‘bordisms between bordisms’ may encode further information about the triangulated category.

We consider quantitative almost calibrated Lagrangians. We mentioned above that $HF^{m}(L,L)$ for $m\geq 1$ is problematic, by analogy with singular cohomology. On the other hand, $H^{0}(L)\simeq H_{n}(L)$ is much more robust compared to higher cohomologies, in the sense that the fundamental cycle of $L$ can deform in a continuous way, under topological changes such as the shrinking of a codimension two cycle. Continuing with the analogy, we expect the geometric information in $HF^{0}$ behaves more continuously under current/varifold limits than the higher degree Floer groups. This is compatible with the fact that the bordism current $\mathcal{C}$ between $L,L^{\prime}$ encodes the compositions $\alpha\circ\beta=1_{L^{\prime}}$ and $\beta\circ\alpha=1_{L}$, with $\alpha\in HF^{0}(L,L^{\prime})$ and $\beta\in HF^{0}(L^{\prime},L)$, and we expect bordism currents have some continuity properties under varifold/current convergence.

The same underlying geometric Lagrangian can conceivably support many different objects in the Fukaya category. A possible source of this problem is a sequence of immersed Lagrangians $L_{i}$ converging to a multiple of a Lagrangian current $L$. The underlying Lagrangian current contains only the support information and the multiplicity, which can be imagined as the number of sheets in $L_{i}$. Much geometric information, however, is not captured this way:

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  Take two Lagrangians $L_{1},L_{2}$ which are both $C^{\infty}$ close to a given immersed Lagrangian $L^{\prime}$, but whose Lagrangian potentials differ by approximately a constant. In the limit $L_{1}\cup L_{2}\to 2L^{\prime}$ as currents, but the potential information is lost. On the other hand, the potential clustering property can restore this information.

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  Immersed Lagrangians may be nontrivial (branched) covers over other immersed Lagrangians. When this happens, the monodromy information is not remembered by the underlying current. On the other hand, it is conceivable that some (generalized) local system data can restore this information.

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  Let $Q$ be a closed smooth manifold. Abouzaid [3] showed that the wrapped Fukaya category of the cotangent bundle $T^{*}Q$ is generated by any cotangent fibre $T_{q}^{*}Q$, and the wrapped Floer cochain complex of $T_{q}^{*}Q$ is $A_{\infty}$-equivalent to $C_{-*}(\Omega_{q}Q)$ for the based loop space $\Omega_{q}Q$. In particular, for any (compact, embedded, exact) Lagrangian $L\subset T^{*}Q$, the Floer cohomologies $HW^{*}(T_{q}^{*}Q,L)$ and $HF^{*}(L,L)$ are representations of $H_{-*}(\Omega_{q}Q)$. This cotangent bundle case can be viewed as the local model of Lagrangians contained in a small neighbourhood of a given embedded Lagrangian.

It is interesting to ask how much of such information can still make sense for Lagrangian currents.