Remarks

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There is an obvious notion of a flux homomorphism for isotopies of smooth Lagrangians, taking a deformation to an element of $H^{1}(L;\mathbb{R})$ (and linearising to give the usual deformation theory of Lagrangians). Namely, take a deformation $\Phi_{t}(L)$ through a vector field $X_{t},\ t\in[0,1]$ to the one form

$$\int_{0}^{1}(X_{t}{\mbox{\,}\_\hskip-4.2679pt\shortmid\hskip 1.5pt}\omega)dt\in H^{1}(L;\mathbb{R}).$$

Alternatively, the homomorphism takes a loop $\gamma\subset L$, tracing out the 2-cycle $f(\gamma\times[0,1])$ in $W$ under the isotopy, to the real number $\int_{\gamma\times[0,1]}\omega$. (See Chapter 10 of [MS] for the analogous map for symplectomorphisms.) If the isotopy $\Phi_{t}$ is hamiltonian, the flux is zero; the converse is also easily proved using the methods of ([MS] Theorem 10.12): we may assume without loss of generality that the 1-form $\int_{0}^{1}X_{t}{\mbox{\,}\_\hskip-4.2679pt\shortmid\hskip 1.5pt}\omega$ is identically zero in $\Omega^{1}(L)$. [To see this, write the 1-form as $d\phi$, and compose the deformation with the time one map of the hamiltonian flow with vector field $d\phi{\mbox{\,}\_\hskip-4.2679pt\shortmid\hskip 1.5pt}\omega^{-1}$; this does not alter the flux in $H^{1}(L;\mathbb{R})$ or the property of being hamiltonian.] Then let $\Sigma^{s}$ be the closed 1-form on $L$ defined by

$$\Sigma^{s}=\int_{0}^{s}X_{t}{\mbox{\,}\_\hskip-4.2679pt\shortmid\hskip 1.5pt}\omega dt,$$

and let $\Psi^{s}_{t}$ be the corresponding flow through time $t$. Then the flow $\phi_{t}=\Psi^{t}_{t}\circ\Phi_{t}$ is the corresponding hamiltonian flow from $\Phi_{0}(L)$ to $\Phi_{1}(L)$; see [MS].

Thus it is not too hard to check if two Lagrangians are hamiltonian deformations of each other, at least through smooth Lagrangians, if we know they are deformations of each other as Lagrangians. This second condition, however, is harder to test, as the results of [S1] demonstrate.

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As mentioned in the last section, holomorphic discs are crucial in both mirror symmetry and Floer cohomology; thus one should perhaps restrict attention in the above definition to those Lagrangians whose Floer cohomology is defined [FO3],

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As pointed out to me by Conan Leung, this definition and the resulting conjecture below may only be reasonable close to the large complex structure limit point where the mirror symmetric arguments used to motivate the conjecture are most valid.