The inspiration behind most of this paper is of course Kontsevich’s mirror conjecture [K]. In particular, Kontsevich proposes that the graded vector spaces Ext^∗ and $HF^{*}$ should be isomorphic for mirror choices of bundles $E_{i}$ and graded Lagrangians $L_{i}$ (or more exotic objects in their derived categories)

$$HF^{*}(L_{2},L_{1})\cong\,\mathrm{Ext}^{*}(E_{2},E_{1});$$

this corresponds to the equality of (graded) morphisms on both sides. Here $HF^{*}$ is Floer cohomology [Fl] – a symplectic refinement of the intersection number of $L_{1}$ and $L_{2}$ – which can be $\mathbb{Z}$-graded for graded Lagrangians [S2], whenever it is defined [FO3], [Fu1]. (More precisely it is the cohomology of a chain complex built out of the free vector space generated by the intersection points, with the differential defined by counting holomorphic discs with boundary in the Lagrangians running from one intersection point to another.) In mirror symmetry, and so in this paper, one should only really consider those Lagrangians whose Floer cohomology is well defined [Fu1].

Thus the point of intersection of the $L_{1}$ and $L_{2}$ of the last section define the Floer cohomology $HF^{*}(L_{2},L_{1})\cong\mathbb{C}\,$, and the grading of [S2] is designed specifically so that $L_{1}\#L_{2}$ can be graded precisely when the relative gradings of the $L_{i}$ force the Floer cohomology to be concentrated in degree 1; $HF^{*}(L_{2},L_{1})=HF^{1}(L_{2},L_{1})$. We then think of the connect sum $L_{1}\#L_{2}$ as being mirror to the extension (3.13) defined by Ext${}^{1}(E_{2},E_{1})\cong\mathbb{C}\,$. Fukaya, Seidel, and perhaps others have also proposed that Lagrangian connect sum should be mirror to extensions [Fu2], [S3].

We also consider connect sums of Lagrangians intersecting at $n$ points $p_{i}$. Then the connect sum is not unique up to hamiltonian deformation: $H^{1}$ is added to the Lagrangian as loops between the intersection points, giving additional deformations of its hamiltonian isotopy class. The upshot is that there is a scaling of the neck of the connect sum at each intersection point; we denote any such resulting Lagrangian by $L_{1}\#L_{2}$. Since we insist on all intersection points having Floer (Maslov) index one (so that the connect sum can be graded), the Floer differential vanishes in this case, and these scalings define a class in $HF^{1}(L_{2},L_{1})$.

Deformations (up to those which are hamiltonian) as such a connect sum are given by the elements of

$$H^{1}(L_{1}\#L_{2})\cong H_{n-1}(L_{1}\#L_{2})$$

spanned by the $S^{n-1}$ vanishing cycles $S_{i}$ at the points of intersection $p_{i}\in L_{1}\cap L_{2}$. Given a particular connect sum, the deformation represented by $\sum_{i}a_{i}S_{i}$ simply scales the local gluing parameter in a Darboux chart around each $p_{i}$ by a factor $(1+a_{i})$ (here $a_{i}$ is considered to be infinitesimal). Since the sum of these spheres separates $L_{1}\#L_{2}$ into $L_{1}\backslash\cup\{p_{i}\}$ and $L_{2}\backslash\cup\{p_{i}\}$ and so is zero in homology

$$\sum_{i}[S_{i}]=\pm\partial[L_{1}\backslash\cup\{p_{i}\}]=\mp\partial[L_{2}\backslash\cup\{p_{i}\}]=0\in H_{n-1}(L_{1}\#L_{2}),$$

the infinitesimal deformation represented by $\sum_{i}S_{i}$ is zero (it is pure hamiltonian) and dividing out gives the projectivisation

$$\mathbb{P}(\oplus_{i}\mathbb{R}_{p_{i}}).$$

(4.1)

(Replace $\mathbb{R}$ by $\mathbb{C}\,$ when including flat bundles and their gluing parameters at the $p_{i}$s.) This explains the earlier claim that connect sums at one intersection point are uniquely defined up to hamiltonian deformations. More precisely, when holomorphic discs are taken into account and we consider only those Lagrangians whose Floer cohomology is defined [FO3], hamiltonian deformation classes of connect sums whose Floer cohomology can be defined should be parameterised by $\mathbb{P}(HF^{1}(L_{2},L_{1}))$. (On the mirror side isomorphism classes of extensions of $E_{2}$ by $E_{1}$ are parametrised by $\mathbb{P}\mathrm{\,Ext}^{1}(E_{2},E_{1})$.)

We would then expect that the resulting connect sum has a canonical homomorphism from $L_{1}$; that is there should be a canonical element

$$\mathrm{id}_{L_{1}}\in HF^{0}(L_{1},L_{1}\#L_{2})$$

for any graded Lagrangians $L_{i}$ for which the graded connect sum exists. While a local model suggests this is true (see for instance [TY]), a complete proof is still not available. This homomorphism we think of as expressing $L_{1}$ as a subobject of $L_{1}\#L_{2}$; i.e. as giving an injection. It should be emphasised that subobject does not make sense in a triangulated category such as the derived Fukaya category of Lagrangians; in the context of the derived category of sheaves, subobject only makes sense for an abelian category such as that of the sheaves themselves (i.e. complexes with cohomology in degree zero only). What we are proposing is that it also makes sense in the category of (complexes of sheaves mirror to) graded Lagrangians, and is vital to make definitions of stability (which involve such subobjects). While there are now more Homs to consider, in particular those of higher order (i.e. Homs to Lagrangians shifted in phase by some $2\pi n$), the targets of these Homs have higher phase and so do not disturb the definition of stability below – this is seemingly a huge piece of luck that means we can extend the stability condition for bundles to all Lagrangians. For similar reasons, the many connect sum decompositions of the $L_{i}$s given in the last section also do not destabilise them.

There are other operations, however, which can also be thought of as Ext¹-type extensions. For instance, taking the product of a single Lagrangian curve $L_{1}$ in $T^{2}$ with a (graded) connect sum $L_{2}\#L_{3}$ in another $T^{2}$, we get a Lagrangian $L_{1}\times(L_{2}\#L_{3})$ in $T^{4}$ which is some kind of extension of the Lagrangians $L_{1}\times L_{2}$ and $L_{1}\times L_{3}$ in $T^{4}$. Supposing that the $L_{i}$s are mirror to some (complexes of) sheaves $E_{i}$, and that the connect sum $L_{2}\#L_{3}$ is mirror to an extension represented by an element $e\in$ Ext${}^{1}(E_{3},E_{2})$. Then by the Künneth formula for sheaf cohomology, we see that $L_{1}\times(L_{2}\#L_{3})$ is indeed mirror to an extension

$$\mathrm{id}\,\otimes e\in\mathrm{Hom\,}(E_{1},E_{1})\otimes\mathrm{Ext}^{1}(E_{3},E_{2})=\mathrm{Ext}^{1}(E_{1}\boxtimes E_{3},E_{1}\boxtimes E_{2}),$$

and so this sort of relative connect sum (which is not $\#$ on $T^{4}$: $L_{1}\times L_{2}$ and $L_{1}\times L_{3}$ do not intersect transversely) should also be considered.

So we consider Lagrangians $L_{1},\,L_{2}$ intersecting cleanly (see e.g. [S1] Definition 2.1), that is $N=L_{1}\cap L_{2}$ is a smooth submanifold, and $TN=TL_{1}\arrowvert_{N}\cap TL_{2}\arrowvert_{N}$. Basic results of Weinstein allow us to identify a neighbourhood of $N$ with a neighbourhood of the zero section $N$ in $T^{*}N\oplus E$, where the total space of $T^{*}N$ has its canonical symplectic structure, and

$$E\equiv(TL_{1}\arrowvert_{N})/TN\,\oplus\,(TL_{2}\arrowvert_{N})/TN$$

is the annihilator, under the symplectic form, of $TN\subset TX\arrowvert_{N}$ (to which the symplectic form therefore descends, making $E$ a symplectic bundle).

Choosing a metric on $E$, compatible with its symplectic structure, such that its transverse subbundles $(TL_{1}\arrowvert_{N})/TN,\ (TL_{2}\arrowvert_{N})/TN$ are orthogonal, we can now perform the family connect sum of these, over the base $N$, since the local model in [S1] is $O(n)$ invariant. As before we insist that this can be compatibly graded again denote it by $\#$; given a grading on $L_{1}$ there will be at most one grading on $L_{2}$ such that this graded relative connect sum exists.

It should be noted that although such a clean intersection could be hamiltonian isotoped to be transverse, the resulting intersection points would not necessarily all be of Floer/Maslov index one, and so the pointwise graded connect sum could not be formed at every point; we would end up with an immersed Lagrangian. Studying which immersed Lagrangians should be included in the Fukaya category, and which embedded Lagrangians they should be considered equivalent to, is an important part of mirror symmetry and will need to be better understood to refine our conjecture. For instance forming extensions of bundles which also have nonzero homorphisms between them would appear to be mirror to forming connect sums between graded Lagrangians at index one intersection points, leaving the index zero intersection points immersed. In general one would like to consider two objects of the Fukaya category to be equivalent if their Floer cohomologies with any other objects are the same. This would include hamiltonian deformation equivalence, but also more exotic equivalences for immersed Lagrangians (thanks to Paul Seidel for pointing this out to me). A start in understanding the Floer cohomology of immersed Lagrangians is [Ak]; in the present paper we are largely ignoring singularities.