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.