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.