An essential aspect of $Coh(X^{\vee})$ is that new bundles can be constructed from extensions of known bundles $E_{1},E_{2}$, namely

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

Such extension sequences are classified by the complex vector space $\text{Ext}^{1}(E_{2},E_{1})$. Due to the $\mathbb{C}^{*}$-scaling, the choice of $E$ is parametrised by the projective space $\mathbb{P}(\text{Ext}^{1}(E_{2},E_{1}))$. In general, extensions are not symmetric in $E_{1}$ and $E_{2}$. The extensions

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

are classified by $\text{Ext}^{1}(E_{1},E_{2})$, which is a quite different space. Extensions can also be viewed as distinguished triangles in $D^{b}Coh(X^{\vee})$.

In the mirror picture, exact sequences do not make a priori sense, but one can talk about distinguished triangles, whose geometric sources are the graded Lagrangian connected sums $L_{1}\#L_{2}$, fitting into a distinguished triangle

$$L_{1}\to L_{1}\#L_{2}\to L_{2}\to L_{1}[1].$$

Such distinguished triangles are classified by $HF^{1}(L_{2},L_{1})$.³²³² 32 The caveat is that unlike bundles, the neck length of the Lagrangian connected sum cannot be arbitrarily large. Again there is a scaling symmetry related to the neck size of the Lagrangian connected sum, and there is an asymmetry between $L_{1}\#L_{2}$ and $L_{2}\#L_{1}$.

This analogy is a prime example of homological mirror symmetry. On either side, only pure complex geometry/pure symplectic geometry appears.

Wall crossing in the categorical context refers to the following phenomenon when the stability condition varies in a 1-parameter family, with central charges $Z_{t}$. For $t>0$, the objects $E_{1},E_{2},E$ in the extension sequence are all stable, so necessarily

$$\arg Z_{t}(E_{1})<\arg Z_{t}(E)<\arg Z_{t}(E_{2}),\quad Z_{t}(E)=Z_{t}(E_{1})+Z_{t}(E_{2}).$$

At $t=0$, the phase angles become equal, and for $t<0$, the phase angle inequality is reversed, and $E$ becomes unstable. This phase alignment occurs on a codimension one locus in the space of stability condition, and thus they are called walls. Every extension sequence potentially gives rise to a wall, and the walls can be dense in general.

A notable special case is $\mu$-stability of bundles for a 1-parameter family of Kähler classes $[\omega_{t}]$, and the slopes $\mu(E_{1}),\mu(E_{2}),\mu(E)$ become equal precisely for $t=0$. On one side of the wall, the HYM connections exist on $E_{1},E_{2},E$, and on the other side $E$ becomes unstable and no longer admits any HYM connection.

Thomas [65] interpreted a gluing construction of Joyce as the mirror analogue of the wall crossing phenomenon for bundles.³³³³ 33 It is quite remarkable that Thomas and Yau knew before Bridgeland, that stability conditions make sense categorically beyond $\mu$-stability, and the mirror of the hypothetical special Lagrangian stability condition does not need to be $\mu$-stability. In the simplest case, let $n\geq 3$, fix a symplectic structure $\omega$, and vary the holomorphic volume form $\Omega_{t}$ in a 1-parameter family while keeping the almost Calabi-Yau condition. Let $L^{1}_{t},L^{2}_{t}$ be smooth special Lagrangians with respect to $\Omega_{t}$ with phase $\theta_{t}^{1}$ and $\theta_{t}^{2}$, intersecting transversely at precisely one point $p_{t}$, with Floer degree $\mu_{L_{t}^{2},L_{t}^{1}}(p_{t})=1$, such that $\theta_{t}^{2}-\theta_{t}^{1}$ increases past zero at $t=0$, and in particular $\theta_{0}^{1}=\theta_{0}^{2}$. Thus at $t=0$, the tangent planes can be put into the standard form inside $\mathbb{C}^{n}$:

$$T_{p}L_{0}^{1}=(e^{i\phi_{1}},\ldots e^{i\phi_{n}})\mathbb{R}^{n},\quad T_{p}L_{0}^{2}=\mathbb{R}^{n},\quad 0<\phi_{k}<\pi,\quad\sum\phi_{k}=\pi,$$

$$\omega=\frac{\sqrt{-1}}{2}\sum dz_{k}\wedge d\bar{z}_{k},\quad e^{-i\theta_{0}^{1}}\Omega=a^{n}dz_{1}\wedge\ldots dz_{n},\quad a>0.$$

³⁴³⁴ 34 The constant $a$ comes from the almost Calabi-Yau structure, and $a=1$ in the Calabi-Yau case.

This provides the appropriate framing data to topologically glue in a Lawlor neck $L_{\phi,A}$ (cf. section 2.3) to desingularize $L_{t}^{1}\cup L_{t}^{2}$ for small $t$, so that the glued Lagrangian has the topology $L_{t}^{1}\#L_{t}^{2}$. Joyce [44, Thm 9.10] shows that for each small $t>0$, the glued Lagrangian can be perturbed into a special Lagrangian of phase $\theta_{t}\approx\theta_{0}^{1}$. As $t\to 0$, these special Lagrangians converge as currents to $L_{t}^{1}\cup L_{t}^{2}$, while for $t<0$ this gluing strategy does not produce any new special Lagrangian.

The central charges are

$$Z(L_{t}^{i})=\int_{L_{t}^{i}}\Omega_{t}^{i}=R_{t}^{i}e^{\sqrt{-1}\theta_{t}^{i}},\quad Z(L_{t}^{1}\#L_{t}^{2})=Z(L_{t}^{1})+Z(L_{t}^{2})=R_{t}e^{\sqrt{-1}\theta_{t}}.$$

In particular $R_{t}^{1}\sin(\theta_{t}^{1}-\theta_{t})=-R_{t}^{2}\sin(\theta_{t}^{2}-\theta_{t})$. The asymmetry between $t>0$ and $t<0$ in Joyce’s gluing construction, comes from an approximate formula for the Lawlor neck parameter valid for small $t$ ³⁵³⁵ 35 For a heuristic short derivation see [42, section 6]. Beware that Joyce’s Lagrangian connected sum has the opposite convention.

$$R_{t}^{1}\sin(-\theta_{t}^{1}+\theta_{t})=a^{m}A>0.$$

Thus $\theta_{t}^{1}<\theta_{t}$ is needed in the gluing construction. This is strongly reminiscent of a Bridgeland stability condition. What happens when $t$ crosses zero can be interpreted as wall crossing.

To summarize, the mirror analogy of wall crossing phenomenon is of categorical nature. However, it goes beyond homological mirror symmetry as soon as it involves the stability conditions, and the new problems involve analytical aspects, beyond purely topological issues. The similarity between the Joyce gluing and the bundle case is a strong motivation for Thomas and Yau. However, the Joyce analysis is only valid in a perturbative regime, and what is missing here is a non-perturbative understanding of when the $D^{b}Fuk(X)$ class of the Lagrangian connected sum can admit special Lagrangians.