Everything works rather simply on $T^{2}$; Grayson [G], building on work of Gage, Hamilton and others (e.g. [GH]), has shown that mean curvature flow for curves (of Maslov class zero) converges to straight lines and so we get the mirror symmetric analogue of Atiyah’s classification [At] of sheaves on an elliptic curve – they are basically all sums of stable sheaves. The only exceptions are the non-trivial extensions of certain sheaves by themselves; these correspond to thickenings of the corresponding special Lagrangian (giving fat SLags, as they are known in Britain, or multiply-wrapped cycles in physics speak).

We give an example to demonstrate why one cannot form smooth unstable Lagrangians on $T^{2}$ in Figure 3. First, giving $L_{1}$ and $L_{2}$ the gradings such that their phases are 0 and $\pi/4$, we expect $L_{1}\#L_{2}$ to be stable, and indeed we see it is hamiltonian deformation equivalent to the slope $1/2$ SLag mirror to the stable extension $E$ of $\mathscr{O}$ by $\mathscr{O}(p)$ (where $p$ is a basepoint of $T^{2}$ with corresponding line bundle mirror to the diagonal SLag drawn).

If one then tries to form an unstable SLag $L_{2}\#L_{1}$, the graded connect sum does not exist – the phase would become discontinuous. To form $L_{2}\#L_{1}$ we see from the diagram that we have to take the phase of $L_{1}$ to be $\pi$, thus reversing its orientation, and in fact forming $L_{2}\#(L_{1}[1])$. Then the stability inequality (3.3) is not violated, and in fact this Lagrangian is stable and hamiltonian deformation equivalent to the SLag in $T^{2}$ represented by the vertical edge of the square (and so drawn with a little artistic license in Figure 3). Under the mirror map this corresponds to replacing the extension Ext¹ class by a Hom (as we have shifted complexes of sheaves by one place) and taking the cone of this in the derived category; this is the cokernel $\mathscr{O}_{p}$ of Figure 3.

As pointed out to me by Markarian and Polishchuk, one can play with lots of pictures of connect sums on tori to recover descriptions of certain moduli of sheaves, their special cycles (for instance where one connect-sum neck parameter goes to zero), and so forth, giving results similar to some of those in [FO].

This example can be extended to show that we cannot form the graded connect sum $L_{1}\#L_{2}$ of any two Lagrangians (via a class in $HF^{*}(L_{1},L_{2})$) if $\phi(L_{1})>\phi(L_{2})$. Namely, replace $L_{1}$ and $L_{2}$ by their hamiltonian deformation equivalent SLag representatives, which are straight lines of constant phase $\theta_{i}=\phi(L_{i})$. As Figure 3 shows, $L_{1}\#L_{2}$ can be compatibly graded about an intersection point if and only if we have the local inequalities

$$\theta_{2}>\theta_{1}>\theta_{2}-\pi.$$

Thus we require $\phi(L_{2})>\phi(L_{1})$. (We will explain this kind of phenomenon more generally in [TY] in terms of the grading on Floer cohomology.) Each intersection point is Floer coclosed since the Floer grading is the same as the relative orientation of the Lagrangians, mod 2, and this is the same at each intersection point of the straight lines. So each possible connect sum of the SLags defines a class in $HF^{*}$, and any other connect sum, defined on hamiltonian deformations of $L_{1}$ and $L_{2}$ by a class in $HF^{*}$, will be hamiltonian deformation equivalent to the appropriate connect sum of the SLags, and so satisfy the same phase inequality.

If two smooth Lagrangians have the same phase then their representative SLags will either be the same or disjoint parallel SLags. Either way there are no connect sums (though as mentioned above to account for the mirror symmetry of bundles one should also include non-trivial thickenings of SLags in the Fukaya category).

So unstable Lagrangians do not exist, and by the result of [G] mentioned earlier, the conjecture is true on $T^{2}$.

Thus complex dimension 1 is too simple – in trying to make the phase of one Lagrangian become larger than the phase of another, the two must cross, thus reversing their relative orientations and changing the order of the connect sum. Far more complicated phenomena arise in 2 and 3 dimensions, however.