Taking the homological interpretation of (19) as starting point, a natural way to extend the Solomon functional is to make use of the bordism current $\mathcal{C}$ between an unobstructed Lagrangian $L$ and a fixed unobstructed reference Lagrangian $L_{0}$. We have $\partial\mathcal{C}=L-L_{0}$ as currents, and $\mathcal{C}$ comes from the universal family of holomorphic curves. Our proposed formula is

A few conceptual points are in order:

• •

  We emphasize that this depends not only on the underlying Lagrangian, but also on the potential $f_{L}$.

• •

  There is no need to pass to any universal cover in the space of Lagrangians, as in Solomon’s work (cf. section 2.8).

• •

  The topology of $L$ is no longer fixed, and in particular the Hamiltonian isotopy class may change.

• •

  We view (20) as a unification of the very different viewpoints of Solomon and Lotay-Pacini. In section 2.10 we suggested that this extended Solomon functional may be relevant for quantum tunneling amplitudes between the branes $L_{0}$ and $L$.

• •

  Suppose we vary the Lagrangian $L$ within a 1-parameter exact isotopy family of unobstructed Lagrangians $L_{t}$. The bordism currents $\mathcal{C}_{t}$ between $L_{t}$ and $L_{0}$ satisfy

  $$\mathcal{C}_{t_{2}}=\mathcal{C}_{t_{1}}+\cup_{t_{1}\leq t\leq t_{2}}L_{t}\text{ modulo exact $(n+1)$-dim currents},$$

  then the computation in Lem 3.3 proves the first variation formula for the Solomon functional

  $$\frac{d}{dt}\mathcal{S}(L_{t})=\int_{L_{t}}h_{t}\text{Im}(e^{-i\hat{\theta}}\Omega)$$

  (21)

  which is of course the defining feature of the Solomon functional. Consequently, the formula (20) extends Solomon’s definition in our exact setting, and fixes the multivaluedness problem (i.e. the need to pass to universal covers) in Solomon’s work.