To motivate a guess at the correct definition of stability for Lagrangians, we expand on an example of Lawlor and Joyce ([J] Sections 6 and 7, building on work of [Ha], [L]; see also a similar example in [SV] that is studied in [TY]), explaining its relevance to mirror symmetry, and giving a simple example in algebraic geometry that mirrors it.

First define the pointwise phase $\theta$ of a submanifold $L$: we may write

$$\Omega\arrowvert_{L}=e^{i\theta}\mathrm{\,vol\,}$$

where vol is the Riemannian volume form on $L$ induced by Yau’s Ricci-flat metric [Y] on $W$. Thus vol provides a (local) orientation for $L$, and reversing its sign alters the phase $\theta$ by $\pi$. A SLag is a Lagrangian with constant phase $\theta$.

At first sight $\theta$ is multiply-valued; we always choose it to be a fixed single-valued function to $\mathbb{R}$, lifting $e^{i\theta}:\,L\to S^{1}$ and thus providing the Lagrangian with a grading as introduced by Kontsevich [K], [S2]. Thus we only consider Lagrangians of vanishing Maslov class – for a Calabi-Yau this is the winding class $\pi_{1}(L)\to\pi_{1}(S^{1})$ of the phase map

$$L\stackrel{{\scriptstyle e^{i\theta}\,}}{{\longrightarrow}}S^{1},$$

which of course vanishes for a SLag. (The definition of grading in [K], [S2] is topological and uses the universal $\mathbb{Z}$-cover of the bundle of Lagrangian Grassmannians; here we first pass to the $\mathbb{Z}/2$ orientation cover of the Grassmannian, choosing an orientation of our Lagrangians, and then use a complex structure to pass to the universal $\mathbb{Z}$-cover of this. The two definitions are of course equivalent.)

Similarly we can define a kind of average phase $\phi=\phi(L)$ of a submanifold (or homology class) $L\subset W$ by

$$\int_{L}\Omega=A\,e^{i\phi(L)},$$

for some real number $A$; we then use Re $(e^{-i\phi(L)}\Omega\arrowvert_{L})$ to orient $L$. Reversing the sign of $A$ alters the phase by $\pi$ and reverses the orientation. Again for a graded Lagrangian $L=(L,\theta)$, and we will always implicitly assume a grading, $\phi(L)$ is canonically a real number (rather than $S^{1}$-valued). Shifting the grading $[\,2n\,]:\,\theta\mapsto\theta+2n\pi$ gives a similar shift to the phase $\phi(L)$.

The terminology comes from the fact that if there is a submanifold in the same homology class as $L$ that is SLag with respect to some rotation of $\Omega$, then it is with respect to $e^{-i\phi(L)}\Omega$. Slope, which we define as

$$\mu(L):=\tan(\phi(L))=\frac{1}{\int_{L}\mathrm{Re\,}\Omega}\int_{L}\,\mathrm{Im}\,\Omega,$$

is defined independently of grading, is monotonic in $\phi$ in the range $(-\pi/2,\pi/2)$, and is invariant under change of orientation $\phi\mapsto\phi\pm\pi$. This agrees with the slope of a straight line SLag in the case of $T^{2}$, as featured in [PZ], and we think of it as mirror to the slope of a mirror sheaf, as is shown for tori in [PZ] (see [DFR] for corrections in higher dimensions away from the large complex structure limit).

Joyce describes examples of SLags which we interpret as follows. We have a family of Calabi-Yau 3-folds $W^{t}$ as $t$ ranges through (a small open subset of) the moduli space of complex structures on $W$ with fixed symplectic structure. That is, the holomorphic 3-form $\Omega^{t}$ varies with $t$, but the Kähler form $\omega$ is fixed. We also have a family of SLag homology 3-spheres $L_{1}^{t},\ L_{2}^{t}\subset W^{t}$ such that $L^{t}_{1}$ and $L^{2}_{t}$ intersect at a point. If we choose a rotation of $\Omega^{t}$ such that $L_{2}^{t}$ always has phase $\phi_{2}^{t}\equiv 0$ (this is possible locally at least; in the family described later it will have to be modified slightly), then we are interested as $t$ varies only in the complex number

$$\int_{L_{1}^{t}}\Omega^{t}=R^{t}e^{\phi^{t}_{1}}$$

and its polar phase $\phi=\phi^{t}_{1}$; we plot this (i.e. the projection from the complex structure moduli space to $\mathbb{C}\,$ via this map) in Figure 1.

Then in Joyce’s example, for $\phi<0$ (and $R^{t}>0$) there is a SLag $L^{t}$ (of some phase $\phi^{t}$) in the homology class $[L^{t}]=[L^{t}_{1}]+[L^{t}_{2}]$, such that as $\phi\uparrow 0$, this degenerates to a singular union of SLags of the same phase $L^{t}=L^{t}_{1}\cup L^{t}_{2}$ and then disappears for $\phi>0$.

Most importantly, where $L^{t}$ exists as a smooth SLag ($\phi<0$) we have the slope (and phase) inequality

$$\mu^{t}_{1}<\mu^{t}_{2},\qquad\mathrm{i.e.\ \,}\phi=\phi^{t}_{1}<\phi^{t}_{2}\equiv 0;$$

(3.3)

at $t=0,\ L^{t}$ becomes the singular union of $L^{t}_{1}$ and $L^{t}_{2}$, with

$$\mu^{t}_{1}=\mu^{t}_{2}\ (\phi^{t}_{1}=\phi^{t}_{2});$$

then there is no SLag in $L$’s homology class for

$$\mu^{t}_{1}>\mu^{t}_{2}\ (\phi^{t}_{1}>\phi^{t}_{2}),$$

though there is a Lagrangian, of course – the symplectic structure has not changed. Though we have been using slope $\mu$ in order to strengthen the analogy with the mirror (bundle) situation, from now on we shall use only the phase (lifted to $\mathbb{R}$ using the grading). While each is monotonic in the other for small phase (as $\tan\phi=\mu$), slope does not see orientation as phase does; reversing orientation adds $\pm\pi$ to the phase but leaves $\mu$ unchanged. This is related to the fact that we should really be working with complexes and so forth on the mirror side (the bundle analogy is too narrow) and changing orientation has no mirror analogue in terms of only stable bundles; it corresponds to shifting (complexes of) bundles by one place in the derived category. While slopes of bundles cannot go past infinity (without moving degree in the derived category at least), for Lagrangians they certainly can, and phase $\phi$ continues monotonically upwards as its slope $\tan\phi$ becomes singular and then negative.

Importantly, we can think of the various SLags as independent of time when thought of as Lagrangians in the fixed symplectic manifold $W^{t}$:

Thus for $\phi>0$ we consider the $L^{t}$s as the same as Lagrangian submanifolds (up to hamiltonian deformation) in the fixed symplectic manifold $W^{t}$; it is only the SLag representative that changes as $\Omega^{t}$ varies. We think of this as mirror to a fixed holomorphic bundle in a fixed complex structure, with varying HYM connection as the mirror Kähler form changes.

There is a notion of connect summing Lagrangian submanifolds intersecting in a single point (probably due to Polterovich) – see for instance Appendix A of [S1] – which we claim gives the smoothings $L^{t}$ of the singular $L^{0}=L_{1}\cup L_{2}$. This follows by comparing the local models [J], [S1] for the Lagrangians; see [TY] where it is studied in more detail for a related purpose, and our conventions (slightly different from those of [S2]) are described. While topologically we are just connect summing $L_{1}$ and $L_{2}$ by removing a small 3-ball containing the intersection point from each and gluing the resulting boundary $S^{3}$s (there are two ways, depending on orientation), symplectically the construction does not explicitly use orientations of the submanifolds. (Effectively we are using their relative orientation – the canonical orientation of the sum of the tangent spaces of $L_{1},\ L_{2}$ at the intersection point given by the symplectic structure.)

Giving $L_{1}$ and $L_{2}$ in that order produces a Lagrangian, well defined up to hamiltonian isotopy (this will be shown in Section 4 in more generality; see (4.1)),

$$L_{1}\#L_{2},$$

with the singular union $L_{1}\cup L_{2}$ a limit point in the hamiltonian isotopy class, which is not itself hamiltonian isotopic to $L_{1}\#L_{2}$ (we have seen a family of hamiltonian deformations which has limit $L_{1}\cup L_{2}$, but the deformations are singular at this limit).

There is also an obvious notion of graded connect sum, which is in fact what we shall always mean by $\#$. There is a unique grading on $L_{1}$ compatible with a fixed grading on $L_{2}$ such that we can give a (continuous) grading to the smoothing $L_{1}\#L_{2}$. In the case of connect summing at multiple intersection points (Section 4) there is at most one such grading; in general the graded connect sum may not exist.

In $n$ dimensions, if $L_{1}$ and $L_{2}$ are graded such that $L_{1}\#L_{2}$ exists, then on reversing the order of the $L_{i}$, the graded sum that exists is

$$L_{2}\#(L_{1}[2-n])\quad\text{in the homology class}\quad[L_{2}]+(-1)^{n}[L_{1}].$$

(3.7)

Here $L[m]$ means the graded Lagrangian $L$ with its grading changed by adding $m\pi$ to $\theta$, and the homology class of $L_{1}\#L_{2}$ is $[L_{1}]+[L_{2}]$ using the orientations on the $L_{i}$s induced by the gradings.

This is closely related, as we shall see, to Joyce’s obstruction, and the lack of it in dimension 2 (Lemma 3.6). In 2 dimensions, $L_{1}\#L_{2}$ and $L_{2}\#L_{1}$ are in the same homology class, though by a result of Seidel [S1] not in general in the same hamiltonian isotopy class,

$$L_{1}\#L_{2}\not\approx L_{2}\#L_{1},$$

importantly (we use $\approx$ to denote equivalence up to hamiltonian deformations). For $t>0$ in the above family $L^{t}$ is in the constant hamiltonian deformation class of $L_{1}\#L_{2}$, for $t<0$ it is in the different class of $L_{2}\#L_{1}$, and at $t=0$ it is $L_{1}\cup L_{2}$ – in neither class but in the closure of both. (For complex $t$ the symplectic structure is no longer constant like it is for $t\in\mathbb{R}$, as one can see by following through the hyperkähler rotation; thus we do not get a contradiction to the above statement by going round $t=0$ in $\mathbb{C}\,$.) In 3 dimensions, however, the corresponding obvious choice for a SLag on the other side of the $\pi^{t}_{1}=0$ wall, $L_{2}\#L_{1}[-1]$, is in the wrong homology class.

In the case that the $L_{i}$ are Lagrangian spheres we can see this by going round the wall

$$\phi(L_{1}^{t})=0\simeq\phi(L_{2}^{t}),$$

(3.8)

and using monodromy. In the 2-dimensional $K3$ or $T^{4}$ case this works as follows.

Consider a disc in complex structure moduli space over which the family of Kähler $K3$ surfaces (with constant Kähler form) degenerates at the origin to a $K3$ with an ordinary double point (ODP) with the Lagrangian $L_{1}\cong S^{2}$ as vanishing cycle. A local model is the standard Kähler structure on $x^{2}+y^{2}+z^{2}=u$, over the parameter $u$ in the unit disc in $\mathbb{C}\,$. Now base-changing by pulling back to the double cover in $u$, $u\mapsto u^{2}$, we get the 3-fold

$$x^{2}+y^{2}+z^{2}=u^{2},$$

with a 3-fold ODP which has a small resolution at the origin putting in a holomorphic sphere resolving the central $K3$ fibre $u=0$. Choosing a nowhere-zero holomorphic section $\Omega_{u}$ of the fibrewise $(2,0)$-forms (using the fact that the relative canonical bundle of either family is trivial), this restricts to zero on the exceptional $\mathbb{P}^{1}$ (which is homologous to the vanishing cycle $L_{1}$). Therefore the function

$$\int_{L_{1}}\Omega_{u}$$

(3.9)

has a simple zero at $u=0$, i.e. it vanishes to order 1 in $u$. (The same expression vanished only as $\sqrt{u}$ in the original family with the singular fibre, and as such its sign was not well defined; this is because the class $[L_{1}]$ was defined globally only up to the monodromy $T_{L_{1}}[L_{1}]=-[L_{1}]$, i.e. up to a sign. In our new family the monodromy action $T^{2}_{L_{1}}$ is trivial on homology so it makes sense to talk about the homology class $[L_{1}]$ in any fibre, and (3.9) is single valued.)

Then our loop of complex structures is given by taking the loop $u=e^{it}$ and setting $\Omega^{t}=\Omega_{e^{it}}$. Pulling back the Kähler form from the original family, we get a locally trivial fibre bundle of symplectic manifolds over the circle whose monodromy is the Dehn twist $T_{L_{1}}^{2}$ (since the monodromy round the un-base-changed loop is $T_{L_{1}}$ [S1]). As the family no longer has a singular fibre this monodromy is trivial as a diffeomorphism, but it is a result of [S1], [S2] that as a symplectic automorphism it is non-trivial. This is possible because although the family is a locally trivial bundle of symplectic manifolds over the punctured disc, over $u=0$ the symplectic form becomes degenerate since it was pulled back via the resolution map.

Measuring $[L_{1}]$ against $\Omega_{u}$ as in (3.9) we see a principle familiar in physics (in issues of ‘marginal stability’, and taught to me by Eric Zaslow) – we detect a monodromy, like the degree 1 map $S^{1}\to\mathbb{C}\,^{\!\times}$ given by $t\mapsto\int_{L_{1}}\Omega^{t}$ in this example, by counting wall crossing where a certain real part (here $\int_{L_{1}}\,\mathrm{Im}\,\Omega^{t}$, or the phase $\phi_{1}^{t}$) hits $0\simeq\phi^{t}_{2}$.

(Here we can no longer choose the phase of $\Omega$ such that $\phi^{t}_{2}=\phi(L^{t}_{2})\equiv 0$ in the whole family, as the homology class of $L_{2}$ is not preserved in the family:

$$[T_{L_{1}}^{2}L_{2}]=[L_{2}]+2[L_{1}].$$

However, for a sufficiently small loop about the ODP, i.e. for $\big|\int_{L_{1}}\Omega\big|$ sufficiently small, this will not affect us much and we can write $\phi^{t}_{2}\simeq 0$: we are only interested in topological information like winding numbers and $\phi^{t}_{1}$ crossing the wall at $\phi^{t}_{2}\simeq 0$, which are unaffected by small perturbations.)

So instead of going through the $\phi(L_{1}^{t})=\phi(L_{2}^{t})\simeq 0$ wall we can go round it. If the loop is sufficiently small we do not encounter any more walls where the homology class $[L_{1}]+[L_{2}]$ can be split into classes of the same phase to possibly make the SLag a singular union of distinct SLags of equal phase. For instance the wall at phase 0 does not extend past $u=0$ to phase $\phi^{t}_{1}=\pi$ (even though there $\mu^{t}_{1}=0$) – the phase of $L_{1}$ is not zero but $\pi$, and is only zero for $L_{1}$ with the opposite orientation, so it does not exist as a SLag (e.g. in the hyperkähler rotated situation, we are saying there is no complex curve in $L_{1}$’s homology class to possibly make $L$ the nodal union of $L_{1}$ and something else, there is only an anti-complex curve). So we really can go round the wall; it ends at $u=0$.

So this monodromy description shows that on the other $t\uparrow 2\pi$ side of the wall the SLag deforming $L_{2}\cup L_{1}$ is in the hamiltonian deformation class

$$T_{L_{1}}^{2}L\,=\,T_{L_{1}}^{2}(L_{1}\#L_{2})\,=\,T_{L_{1}}^{2}(T_{L_{1}}^{-1}L_{2})\,\approx\,T_{L_{1}}L_{2}\,\approx\,L_{2}\#L_{1},$$

(3.10)

as claimed (for the above equalities see [S1], [S2]).

Notice that the alternative connect sum description of the above Lagrangian

$$L_{2}\#L_{1}\,=\,T_{L_{1}}^{2}(L_{1}\#L_{2})\,\approx\,T_{L_{1}}^{2}(L_{1})\#T_{L_{1}}^{2}(L_{2})\,\approx\,L_{1}[-2]\#T_{L_{1}}^{2}(L_{2}),$$

(3.11)

does not violate the phase inequality to (3.3), as

$$-2\pi+\epsilon\approx\phi(L_{1}[-2])<\phi(T_{L_{1}}^{2}(L_{2}))\simeq 0.$$

This is why it is important here to keep track of gradings – assigning the phase $\epsilon$ to $\phi(T_{L_{1}}^{2}(L_{1}))$ would give the opposite inequality, but one would not be able to form the above graded connect sum without also shifting the phase of $T_{L_{1}}^{2}(L_{2})$ by $-2\pi$.

The 3-fold case (which Dominic Joyce has also, independently, considered) is slightly different; we need only take a single Dehn twist $T_{L_{1}}$ corresponding to the local family

$$x^{2}+y^{2}+z^{2}=u,$$

over $u\in\mathbb{C}\,$ to get a winding number one loop in the phase of $L_{1}$. This is because

$$T_{L_{1}}L_{1}\approx L_{1}[1-n]$$

in dimension $n$, so in 3 dimensions the homology class $[L_{1}]$ is preserved instead of being reversed. The corresponding picture is displayed in Figure 2.

Again there is a SLag on the other side of the $\phi=0$ wall, but it is in the wrong homology class $[L_{2}]$:

$$T_{L_{1}}L\approx L_{2}.$$

(3.12)

Analogously to (3.11) this has a number of decompositions as connect sums induced by monodromy,

$$T_{L_{1}}(L_{1}\#L_{2})\,\approx\,L_{1}[-2]\#(L_{2}\#(L_{1}[-1]))\,\approx\,L_{2}\,\approx\,(L_{1}\#L_{2})\#(L_{1}[\,1\,]),$$

none of which violate the phase inequality (3.3). The only other obvious choice for a (S)Lag on the other side of the $\phi=0$ wall (given the $K3$ result) is $T_{L_{1}}^{2}(L_{1}\#L_{2})\approx L_{2}\#(L_{1}[-1])$; this however is also in the wrong homology class, and in any case does violate (3.3) and so, by Joyce’s analysis, should not be represented by a SLag. Thinking of $T_{L_{1}}^{2}$ as rotating through $-4\pi$ in Figure 2, it is at roughly $-3\pi$ that the phase inequality (3.3) gets violated, and the $-\pi$ rotation of $L_{2}$ splits as a SLag into the union of the $-\pi$ rotations of $(L_{1}\#L_{2})$ and $L_{1}[\,1\,]$: these both have phase approximately zero.