Suppose we have two Lagrangians $L_{1},\,L_{2}$ hamiltonian isotoped to intersect transversally in a finite number of points. We will work at one of these points $p$. There we can pick a local Darboux chart with coordinates $(x_{i},y_{i})$ and symplectic form $\sum_{i}dx_{i}\wedge dy_{i}$ such that $L_{2}=\{y_{i}=0\}$ is the $x$-axes, and

$$L_{1}=\{y_{i}=\tan(\alpha)x_{i}\}$$

(3.1)

for some $\alpha\in(0,\pi)$. (It would be more usual to use $\alpha=\pi/2$, of course, but that situation can be moved to this one by an obvious symplectic (shear) transformation).

Using $z_{i}=x_{i}+iy_{i}$ coordinates to set up the obvious isomorphism to $\mathbb{C}\,^{n}$, $L_{1}$ and $L_{2}$ are at

$$L=\{z_{j}=re^{i\alpha}a_{j}\,:\,r\in[0,\infty),\,\mathbf{a}=(a_{j})_{j=1}^{n}\in S^{n-1}\subset\mathbb{R}^{n}\subset\mathbb{C}\,^{n}\},$$

where $\alpha$ is set to zero to give $L_{2}$.

So given a curve $\gamma$ in $\mathbb{C}\,$, we define a Lagrangian

$$L_{\gamma}=\gamma.S^{n-1}=\{z_{j}=\gamma a_{j}\,:\,\mathbf{a}=(a_{j})_{j=1}^{n}\in S^{n-1}\subset\mathbb{R}^{n}\subset\mathbb{C}\,^{n}\}.$$

Then $L_{2}$ is represented by $\gamma_{2}=[0,\infty)\subset\mathbb{C}\,$, $L_{1}$ by $\gamma_{1}=e^{i\alpha}[0,\infty)\subset\mathbb{C}\,$, and $L_{1}\cup L_{2}$ by the V-shaped union of these curves.

In this notation the Lagrangian connect sum $L_{1}\#L_{2}$ is represented by any smoothing $\gamma=:\gamma_{1}\#\gamma_{2}$ of $\gamma_{1}\cup\gamma_{2}$ staying inside the cone $\{re^{i\beta}\,:\,r>0,\beta\in[0,\alpha]\}$ which is $\gamma_{1}\cup\gamma_{2}$ outside a compact set, and a smooth curve cutting off the cone at the origin. (So here $\gamma$ is not a connect sum of the curves $\gamma_{i}$ in the topological sense; we only use the notation because the resulting Lagrangians are topological connect sums.)

Now suppose we are in the special case that we can choose $\alpha$ such that in these coordinates, $\Omega\arrowvert_{p}$ takes the standard form $dz_{1}\ldots dz_{n}$. Then the phase function of the Lagrangian $L_{\gamma}$ associated to a curve $\gamma$ is easily calculated to be $\theta(\gamma^{\prime})+(n-1)\theta(\gamma)+N\pi$ for any $N\in\mathbb{Z}$ (where $\theta(z)\in(-\pi,\pi]$ is the phase of a complex number $z=re^{i\theta(z)}$). Orienting $\gamma_{2}$ such that $\gamma_{2}^{\prime}$ is a positive real number, and choosing $L_{2}$ to have phase 0, corresponds to choosing $N=0$ and so grading $L_{1}$ by

$$(-\pi+\alpha)+(n-1)\alpha=n\alpha-\pi.$$

(3.2)

In particular, choosing $\alpha=\pi/n$, and setting, for any $c>0$,

$$\gamma^{\ }_{c}=\{re^{i\theta}\,:\,r^{n}=c\sin(n\theta),\ \theta\in(0,\pi/n)\}$$

gives a SLag $L_{\gamma}$ which has a grading of phase zero, asymptotic to $L_{1}$ and $L_{2}$ at infinity, and this is precisely the local model of the example of Joyce, Harvey and Lawlor used so extensively in [Th].

While this is not strictly of the form $\gamma_{1}\#\gamma_{2}$ as defined above (it is only asymptotic to the $\gamma_{i}$, not equal outside a compact set), by taking $c$ small we can make it as close as we like to such a connect sum, all in the same hamiltonian deformation class, and the construction of Joyce is indeed a hamiltonian deformation of a connect sum as claimed in [Th].

We plot these SLag curves $\gamma_{c}\subset\mathbb{C}\,$ in Figure 1 as the light lines, converging as $c\to 0$ to the V-shaped $\gamma_{1}\cup\gamma_{2}$ (with $\alpha=\pi/n$). Then the dark lines depict connect sums $L_{1}\#L_{2}$ for $\phi(L_{2})=0$ and $\phi(L_{1})=\pm\epsilon$. If $\phi(L_{1})<0$, the stable case as described in [Th], then we can choose the connect sum such that the phase of $L_{1}\#L_{2}$ varies monotonically between its values on $L_{1}$ and $L_{2}$, i.e. between $-\epsilon$ and $\epsilon$.

If, however, $\phi(L_{1})>0$, the unstable case in [Th], then the phase of $L_{1}\#L_{2}$ must initially decrease to move away from $L_{1}$ before decreasing to reach $L_{2}$ (i.e. $\gamma$ must cross the light lines one way then the other), giving a phase function which necessarily goes outside the range $(0,\epsilon)$ (see Figure 1). This will be important to us later – under mean curvature flow we expect the phase function $\theta$ to evolve to a constant in the stable case (under the heat equation (2.5)) and to a Heaviside step function (with values $0$ and $\epsilon$) in the unstable case. This does not then contradict the maximum principle as the unstable case has the described non-monotonic phase.

While this defines the symplectic connect sum in general by means of our Darboux chart, the analysis of phases depended on a choice of complex structure. In the general case we can still fix $\theta_{p}(L_{2})=0$, without loss of generality, by rotating $\Omega$. We can then pick local complex coordinates $z_{i}=x_{i}+iy_{i}$ such that $\Omega_{p}=dz_{1}\ldots dz_{n}$ and, at the level of tangent spaces at $p$, (the tangent space to) $L_{2}$ is at $y_{i}=0\ \forall i$. $L_{1}$ will be

$$L_{1}=\{z_{i}=re^{i\alpha_{i}}\},$$

(3.3)

for some $\alpha_{i}$s that are no longer necessarily all the same. We are now connect summing Lagrangians of pointwise phase $0$ and $\sum_{i=1}^{n}(\alpha_{i})-\pi$ (compare (3.2)). To do so we of course have to pick different coordinates as in the original definition above (3.1) and proceed as before; therefore the resulting phase function will not be as simple as before – it is not pulled back from $\gamma$ but will vary over the $S^{n-1}$ fibres. Its average phase over the $S^{n-1}$s will have a similar form to that in Figure 1, however, and in the case of all the $\alpha_{i}$s being the same we get the earlier simpler picture.

The dependence of the hamiltonian deformation class of $L_{1}\#L_{2}$ on the choice of scale of the neck at each intersection point was described in ([Th] Section 4) (in particular if there is only one intersection point the class is uniquely defined). We should also point out that the graded connect sum (when it exists) is also independent of hamiltonian deformations of $L_{1}$ and $L_{2}$. While the $L_{i}$s intersect transversely this is clear; we need only understand what happens in crossing the codimension one wall of Lagrangians intersecting in a double point (i.e. creating or cancelling two intersection points). But it will be clear from the definition of grading below that two such points must have grading differing by one, and so the connect sum along both of them cannot be graded (3.5).

The Floer cohomology group $HF^{*}(L_{2},L_{1};\mathbb{C}\,)$ [FO3] is the cohomology of a cochain complex made from a copy of $\mathbb{C}\,$ for each intersection point of two graded Lagrangians in general position. The differential is defined by counting holomorphic strips, with boundary in the Lagrangians, running from one intersection point to another. It is a symplectic refinement of the topological intersection theory of $L_{1},\,L_{2}$, and as such is invariant only under hamiltonian deformations of the $L_{i}$. What is important to us is the grading of a particular transverse intersection point, as defined in [S2], [FO3].

While this can be defined completely topologically, it is most easily (and equivalently) defined via a complex structure. Again we work at the level of tangent spaces, pick local coordinates and, without loss of generality, take $L_{2}$ to have phase 0 and to be the $x$-axes: $L_{2}=\{y_{i}=0\}$. Write $L_{1}$ as

$$L_{1}=\{z_{i}=re^{i\alpha_{i}}\},$$

where the $\alpha_{i}$s are all in $(0,\pi)$. Then $\sum\alpha_{i}=\theta_{p}(L_{1})$ mod $\pi$, and the following integer is the definition of the Floer index of the point $p$:

$$\ind_{p}(L_{2},L_{1}):={1\over\pi}\left(\sum\alpha_{i}-\theta_{p}(L_{1})\right).$$

(3.4)

Notice therefore that $\ind_{p}(L_{2},L_{1})+\ind_{p}(L_{1},L_{2})=n$. Applying the definition (3.4) to the connect sums defined in the last section (for which $\theta_{p}(L_{1})=\sum\alpha_{i}-\pi$), we recover a result of Seidel [S2]:

$$L_{1}\#L_{2}\text{\emph{ exists as a graded connect sum if and only if }}\ind_{p}(L_{2},L_{1})=1.$$

(3.5)

(The only if part follows from the independence of gradings and the Floer index from the complex structure; we may therefore pick the complex structure locally to have the form of the local model above.) Given $L_{1}$ there is at most one choice of the grading on $L_{2}$ such that $\ind_{p}(L_{2},L_{1})=1$ at all intersection points $p$, so that $L_{1}\#L_{2}$ can be graded.

In fact connect sums $L_{1}\#L_{2}$ whose own Floer cohomology is well defined [FO3] should correspond to elements of $HF^{1}(L_{2},L_{1})$, mirror to the fact that extensions of sheaves $0\to E_{1}\to E\to E_{2}\to 0$ correspond to elements of Ext${}^{1}(E_{2},E_{1})$, as discussed in [Th].

We can also deal with the connect sums mentioned in [Th] which are relative versions of the above construction; $(n-r)$-dimensional connect sums carried out in a smooth family over an $r$-dimensional base. Then the same Floer index can be defined; there are now $r$ angles between the Lagrangians that are zero, and $(n-r)$ whose signs can be computed to get the Floer index. (Signs cannot change over the family since the intersection of the Lagrangians fibres over the base of the family with fibres of constant dimension; an angle going to zero would cause a fibre dimension to increase.)