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.

• •

  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.

• •

  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.

• •

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

• •

  Parallel transport along local systems may break down.

• •

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

• •

  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.