These phenomena are similar to wall-crossing in bundle theory on the complex side – in a real one-parameter family of Kähler forms, for fixed complex structure, stable holomorphic bundles for $t>0$ can become semistable at $t=0$ and unstable for $t<0$.

An example that mirrors Joyce’s is the following. Suppose we have two stable bundles (or coherent sheaves) $E_{1}$ and $E_{2}$ with

$$\mathrm{Ext}^{1}(E_{2},E_{1})\cong\mathbb{C}\,.$$

This is $H^{1}(E_{1}\otimes E_{2}^{*})$ in the case of bundles and is the mirror [K] of the one dimensional Floer cohomology $HF^{*}(L_{2},L_{1})\cong\mathbb{C}\,$ that is defined by the single intersection point of $L_{1}$ and $L_{2}$ (see Section 4 for more details of this, and an explanation of why we are dealing with Ext¹ and $HF^{1}$ here). We then form $E$ from this extension class

$$0\to E_{1}\to E\to E_{2}\to 0.$$

(3.13)

Take a family of Kähler forms $\omega^{t}$ such that $\mu^{t}(E_{2})-\mu^{t}(E_{1})$ is the same sign as $t$ (here $\mu^{t}(F)=c_{1}(F)\,.\,(\omega^{t})^{n-1}/$ rk $(F)$ is the slope of $F$ with respect to $\omega^{t}$). Supposing that the $E_{i}$ are stable for all $t\in(-\epsilon,\epsilon)$, we claim that $E$ is stable for sufficiently small $t>0$, while it is destabilised by $E_{1}$ for $t\leq 0$. Without loss of generality take $\mu^{t}(E_{2})=\mu$ fixed, and $\mu^{t}(E_{1})=\mu-t$. As $E_{2}$ is stable, for $t$ sufficiently small there are no subsheaves of $E_{2}$ of slope greater than $\mu-t$, so for any stable destabilising subsheaf $F$ of $E$, the composition

$$F\hookrightarrow E\to E_{2}$$

cannot be an injection (unless it is an isomorphism, but (3.13) does not split. So $F\cap E_{1}\neq 0$, and the quotient $Q=F/(F\cap E_{1})$ has slope $\mu(Q)>\mu(F)>\mu-t$ by the stability of $F$ and instability of $E$. But $Q$ injects into $E_{2}$, which we know is impossible.

In the 2-dimensional case, by Serre duality Ext${}^{1}(E_{1},E_{2})\cong\,$Ext${}^{1}(E_{2},E_{1})^{*}\cong\mathbb{C}\,$on $K3$ or $T^{4}$, so for $t<0$ we can instead form an extension

$$0\to E_{2}\to E^{\prime}\to E_{1}\to 0,$$

(3.14)

to give a new bundle $E^{\prime}$ which is also stable, and has the same Mukai vector

$$v(E^{\prime})=v(E_{1})+v(E_{2});$$

compare (3.7). At $t=0$ we take the (polystable) bundle

$$E_{1}\oplus E_{2}.$$

This is because the semistable extension (3.13) no longer admits a Hermitian-Yang-Mills metric, but $E_{1}\oplus E_{2}$ does. Also, the algebraic geometry of the moduli problem shows that while a semistable bundle gets identified in the moduli space with the other (“S-equivalent”) sheaves in the closure of its gauge group orbit, there is a distinguished representative of its equivalence class – the polystable direct sum (of the Jordan-Hölder filtration, which here is $E_{1}\oplus E_{2}$).

Thus, while the HYM connections vary, the bundle has only 3 different holomorphic structures – for $t>0,\ t=0,$ and $t<0$. Put another way (to spell out the analogy with the Lagrangians $L^{t},\ L_{1},\ L_{2}$) as $\omega_{t}$ varies with $t>0$ we take different points in a fixed complexified gauge group orbit, and at $t=0$ we take as limit point something in a different orbit that is nonetheless in the closure of the $t>0$ (and $t<0$) orbit. The stable deformations of the polystable $E_{1}\oplus E_{2}$ (which we are thinking of as the mirror of the singular union $L_{1}\cup L_{2}$, of course) are precisely (3.13) for $t>0$ and (3.14) for $t<0$.

In the 3-fold case, however, Serre duality gives Ext${}^{2}(E_{1},E_{2})\cong\,$Ext${}^{1}(E_{2},E_{1})^{*}\cong\mathbb{C}\,$ instead, and so no stable extension (3.14). In fact one would expect there to be no stable bundle with the right Chern classes; instead the one dimensional Ext² gives us a complex $E^{\prime}$ in the derived category $D^{b}(M)$ fitting into an exact sequence of complexes

$$0\to E_{2}\to E^{\prime}\to E_{1}[-1]\to 0,$$

where $E_{1}[-1]$ is $E_{1}$ shifted in degree by one place to the right as a complex. This has Mukai vector

$$v(E^{\prime})=v(E_{2})-v(E_{1}),$$

compare (3.7). Thus, just as in the case of SLags, as we pass through $t=0$ there is no natural stable object on the other side in the same homology class in 3 dimensions (though there is in 2 dimensions) and so an element of the appropriate moduli space disappears.

In fact, as in the Lagrangian example, the natural stable object on the other side of the wall is $E_{2}$ if we consider monodromy. The mirror of the symplectic Dehn twists of above are described in [ST] (in the case that the bundles $E_{i}$ are spherical in the sense of [ST]: Ext${}^{k}(E_{i},E_{i})\cong H^{k}(S^{n};\mathbb{C}\,)$; this is the natural mirror analogue of the $L_{i}$s being spheres). These are the twists $T_{E_{1}}$ of [ST] on the derived category of the Calabi-Yau that act on the extension bundle $E$ of (3.13) to give precisely the extension (3.14),

$$T_{E_{1}}^{2}E=E^{\prime}$$

(compare (3.10)), as a short calculation using [ST] shows. Similarly

$$T_{E_{1}}E=E_{2},$$

the analogue of (3.12). (In both of these calculations it is important to calculate this monodromy in the derived category; in the $K3$ case the action of $T_{E_{1}}^{2}$ is trivial on K-theory and cohomology, and we cannot distinguish between (3.13) and (3.14), but they are very different as holomorphic bundles and as elements of the derived category.)

The mirror wall crossing, with a SLag splitting into two and then disappearing, is interpreted in [DFR] (and in [SV] in a different case) as the state it represents decaying as we reach a point of ‘marginal stability’. Despite this dealing with only SLags (and so with only a priori stable Lagrangians in our mathematical sense of stability), this suggestive language does in fact have something to say about the stability, in our sense of group actions, of (non-special) Lagrangians, by considering the nodal limit $L_{1}\cup L_{2}$ to be a semistable Lagrangian.

Thus the Lagrangian $L_{1}\#L_{2}$ (which always exists as a Lagrangian as the complex structure varies with fixed Kähler form) becomes semistable at $t=0$ and is represented by something in a different orbit of the hamiltonian deformation symmetry group (but in the closure of the original orbit), and is unstable for $t<0$ so exists there only as a Lagrangian and not as a SLag. This, and the bundle analogue described above, leads us to think of the Lagrangian $L_{1}$ as destabilising $L=L_{1}\#L_{2}$ when $\phi(L_{1})\geq\phi(L_{2})$. This motivates the now obvious definition of stability in Section 5; first we explain more about the connections to mirror symmetry, and generalisations to connect sums at more intersection points.