1.1

Lemma 2.3 states incorrectly that if $\mathring{W}$ is a Liouville domain, the inclusion of exact symplectomorphisms into all symplectomorphisms is a homotopy equivalence. Mamaev explains the following counterexample.

The error in our proof of [Reference Keating and SmithKS24, Lemma 2.3] occurs in the application of results of [Reference Bourgeois, Ekholm and EliashbergBEE12, § 1], arising from different normalisation conventions for the symplectic form and its primitive in the cylindrical end in their and our work.

1.2

Let $(\mathring{W}, \theta)$ be a Liouville domain with completion W. [Reference Keating and SmithKS24, Corollary 2.4] asserts that the group $\pi_0\operatorname{Symp}(\mathring{W})$ acts on the ‘preliminary’ (i.e. ungraded) wrapped Fukaya category of the completion $\mathcal{W}_{\mathrm{pr}}(W)$ . There are several ways to see this is true. We give an argument below.

1.3

We explain a proof of [Reference Keating and SmithKS24, Corollary 2.4] by reducing to [Reference Keating and SmithKS24, Lemma 2.2].

The argument of [Reference Bourgeois, Ekholm and EliashbergBEE12, Lemma 1.1], which uses the model $( \partial \mathring{W} \times [0,\infty), d(e^s\cdot \alpha))$ , constructs the symplectic isotopy as the time-1 flow of a vector field of the form $e^{-s} \widetilde{X}$ , where $\widetilde{X}$ is independent of s (but not ‘vertical’ with respect to the cylindrical end; it is denoted by $\widetilde{V}$ in loc. cit.). Now consider the diffeomorphism $f: \partial \mathring{W} \times [0,\infty) \to \partial \mathring{W} \times [0,\infty)$ given by $(x, s) \mapsto (x,t = e^{s})$ . The vector field we want is the image of $e^{-s} \widetilde{X}$ under Df; now observe that this is of the claimed form. (Note that the scalar function $\chi$ can, and in some cases such as that of Remark 1.2 must, change sign.)

Take a symplectomorphism $\psi \in \mathrm{Symp}(\mathring{W},\theta)$ of a Liouville subdomain, and let $\hat\psi$ be the symplectomorphism of W obtained by completing this conically (conjugating with the Liouville flow). We now have two different Liouville structures on W: as the completion of $(\mathring{W}, \theta)$ , with Liouville vector field Z; and as the completion of $(\mathring{W}, ({\psi}^{-1})^\ast\theta)$ , with Liouville vector field $Z_{\psi} :=\hat{\psi}_\ast Z$ . Let $\sigma_t$ be the isotopy of W such that $\sigma_1 \circ \hat\psi$ is exact, which is provided by Lemma 1.3 using the second of the Liouville structures on W.

We infer that $\sigma_1(\partial \mathring{W})$ simultaneously has contact type with respect to the pair of Liouville forms

(1.1) \begin{equation} ((\hat\psi)^{-1})^*\theta \quad \mathrm{and} \quad ((\sigma_1\circ\hat\psi )^{-1})^*\theta \end{equation}

by Lemma 1.4, respectively, tautologically. By pulling back data, the map $\hat\psi$ defines an equivalence

\[\mathcal{W}(W,\theta) \to \mathcal{W}(W,((\sigma_1\circ\hat\psi)^{-1})^*\theta). \]

The argument of the proof of [Reference Keating and SmithKS24, Lemma 2.2], using an isotopy associated to an interpolation of the contact forms of (1.1), constructs an equivalence

\[\mathcal{W}_{\mathrm{pr}}(W,((\sigma_1\circ\hat\psi)^{-1})^*\theta) \to \mathcal{W}_{\mathrm{pr}}(W,(\hat\psi^{-1})^*\theta). \]

We now repeat the argument for the hypersurface $\partial\mathring{W}$ (rather than its $\sigma_1$ -image) and run [Reference Keating and SmithKS24, Lemma 2.2] again to pass from $\mathcal{W}_{\mathrm{pr}}(W,(\hat\psi^{-1})^*\theta) $ back to $\mathcal{W}_{\mathrm{pr}}(W,\theta)$ . Overall this constructs an action of $\psi$ on $\mathcal{W}_{\mathrm{pr}}(W,\theta)$ . The choices made throughout the construction belong to (highly) connected spaces, so this descends to an action of the component group $\pi_0\mathrm{Symp}(\mathring{W})$ as desired.

1.4

The proof of the graded version of Corollary 2.4, namely Corollary 2.9, is exactly analogous but working throughout with graded maps, noting that one can canonically transport gradings along the isotopy $\sigma_t$ considered above.

1.5

There are two related viewpoints on circumventing the error in [Reference Keating and SmithKS24, Lemmas 2.3 and 2.8] which some readers might also find helpful.

1. (1) It is likely that one could prove Corollary 2.4, and its graded cousin, by using ‘dissipative’ almost complex structures and geometric boundedness arguments in the vein of [Reference GromanGro23, Reference Ganatra, Pardon and ShendeGPS24]. For instance, the reverse isoperimetric inequality immediately implies compactness for families of holomorphic polygons with Lagrangian boundary conditions that are disjoint at infinity, even for non-cylindrical Lagrangians obtained from pulling back by a symplectomorphism which is not strongly exact.

2. (2) Whilst the discussion in [Reference Keating and SmithKS24, § 2] concerns a general Liouville domain, it is worth pointing out that recourse to these results can be entirely circumvented for the specific examples considered in the main body of [Reference Keating and SmithKS24], that is, for the conifold smoothing W (and manifolds obtained from that by subcritical handle addition). Indeed, as pointed out in [Reference Keating and SmithKS24, Remark 3.11], in our particular case the non-compactly supported symplectomorphisms that we encounter can be deformed to be strongly exact, that is, satisfying that on the completion W one has $\phi^*\alpha = \alpha + dH$ for a compactly supported function H. The ability to achieve strong exactness uses the special feature that W is an affine variety, and the non-compactly supported maps arise from parallel transport in a family of affine varieties which admit a compatible family of compactifications at infinity. It is classical that strongly exact symplectomorphisms act on the wrapped category.

2.1

The proof of Lemma 6.11 of op. cit. has a gap; we do not know if the claimed result is true. The reference [Reference Seidel and SmithSS10] proves the weaker fact that there is a pair of Hamiltonian symplectomorphisms $\Phi_0$ and $\Phi_1$ , obtained by integrating a pair of different exact Lagrangian isotopies, both satisfying (i), one of which satisfies (ii) and one of which satisfies (iii). That is, we have that the normal bundles of the isotoped stabilised Lagrangians satisfy $\nu_{\Phi_0(\hat{L}_0)} = \mathbb{R}^{n_{\mathrm{anti}}+r}$ whilst $\nu_{\Phi_1(\hat{L}_1)} = i\mathbb{R}^{n_{\mathrm{anti}}+r}$ . The key point is that in [Reference Seidel and SmithSS10, § 3.5] the Lagrangians called there $\tilde{L}_0$ and $\tilde{L}_1$ are obtained as the images of the naive stabilisations of $L_0$ and $L_1$ under two different exact Lagrangian isotopies. That translates to the weaker version of Lemma 6.11 given above.

2.2

Definition 6.12 is defining $\Upsilon$ -constrained perturbation data for the pair $(\Phi_0(\hat{L}_0),\Phi_1(\hat{L}_1))$ (rather than the pair $(\Phi(\hat{L}_0), \Phi(\hat{L}_1))$ ; the definition is otherwise unchanged. Lemma 6.13 and its proof are correct as written.

Corollary 6.14 is correct when interpreted as saying that one can achieve simultaneous regularity for solutions to equation (6.2) in $\hat{M}$ and its fixed locus for the pair $(\Phi_0(\hat{L}_0),\Phi_1(\hat{L}_1))$ .

2.3

We note that these two Lagrangians are exact isotopic to the ‘naive’ stabilisations $L_0 \times \mathbb{R}^r$ and $L_1 \times i\mathbb{R}^r$ , and they meet in the fixed locus $M^{\iota}$ exactly as $L_0$ and $L_1$ do. However, the stabilised Lagrangians $(\Phi_0(\hat{L}_0),\Phi_1(\hat{L}_1))$ could have additional non-fixed intersection points. In [Reference Seidel and SmithSS10, 3.5], the construction of an equivariant regular J allows such a change to the set of non-fixed intersections. Thus, whilst Lemma 6.8 of the paper is correct, we do not know how to satisfy its hypotheses.

2.4

The paragraph after Corollary 6.14 should read as follows. The upshot of the discussion is that by replacing

\[(M,L_0,L_1,\iota)\]

by $(\hat{M}, \Phi_0(\hat{L}_0), \Phi_1(\hat{L}_1), \hat{\iota})$ , the Floer complexes $CF(\Phi_0(\hat{L}_0),\Phi_1(\hat{L}_1))$ and $CF(L_0,L_1)$ are equivariantly chain homotopic (so in particular have the same cohomology), and the subcomplexes corresponding to invariant intersection points are naturally identified. Moreover, the former complex admits an equivariant regular J.

Crucially, the Floer complexes $CF(\Phi_0(\hat{L}_0),\Phi_1(\hat{L}_1))$ and $CF(L_0,L_1)$ are not globally identified, they are only globally chain homotopic. Nonetheless, it remains true that

(2.1) \begin{equation} \operatorname{HF}(L_0,L_1) = \operatorname{HF}(\Phi_0(\hat{L}_0), \Phi_1(\hat{L}_1)) \end{equation}

by using an easy Künneth theorem to see that $ \operatorname{HF}(L_0,L_1) = \operatorname{HF}(L_0\times \mathbb{R}^r, L_1\times i\mathbb{R}^r)$ and then using Hamiltonian invariance of Floer cohomology to identify the latter with $\operatorname{HF}(\Phi_0\hat{L}_0, \Phi_1\hat{L}_1)$ . The isomorphism of (2.1) holds over $\mathbb{Z}$ .

2.5

At this point, the localisation theorem implies that

(2.2) \begin{equation} \mathrm{rank} \operatorname{HF}(\Phi_0(\hat{L}_0), \Phi_1(\hat{L}_1);\mathbb{Z}/2) \geq \mathrm{rank} \operatorname{HF}((\Phi_0(\hat{L}_0))^{fix}, (\Phi_1(\hat{L}_1))^{fix};\mathbb{Z}/2)\end{equation}

since we are in an equivariantly transverse situation with stable normal polarisations. Finally,

\[\operatorname{HF}((\Phi_0(\hat{L}_0))^{fix}, (\Phi_1(\hat{L}_1))^{fix};\mathbb{Z}/2)= \operatorname{HF}(\hat{L}_0^{fix}, \hat{L}_1^{fix};\mathbb{Z}/2) = \operatorname{HF}(L_0^{fix},L_1^{fix};\mathbb{Z}/2) \]

since the Hamiltonian isotopies $\Phi_j$ were both the identity on the fixed locus, condition (i) from Lemma 6.11.

2.6

In Lemma 6.15, the conclusion of the Lemma should be that there is a rank inequality

(2.3) \begin{equation} \mathrm{rank} \operatorname{HF}(L_0, L_1;\mathbb{Z}/2) \geq \mathrm{rank} \operatorname{HF}(L_0^G,L_1^G;\mathbb{Z}/2)\end{equation}

and not that there is a cochain homotopy equivalence as claimed. Note that we now never use hypothesis (4) in the statement. To prove (2.3), first use the stable normal trivialisation for $\iota_1$ to show (as an instance of (2.2)) that

(2.4) \begin{equation}\mathrm{rank}\operatorname{HF}(L_0, L_1;\mathbb{Z}/2) \geq \mathrm{rank} \operatorname{HF}(L_0^{\iota_1}, L_1^{\iota_1};\mathbb{Z}/2).\end{equation}

Then the existence of the second stable normal trivialisation gives a rank inequality

(2.5) \begin{equation}\mathrm{rank} \operatorname{HF}(L_0^{\iota_1}, L_1^{\iota_1};\mathbb{Z}/2) \geq \mathrm{rank}\operatorname{HF}(L_0^G,L_1^G;\mathbb{Z}/2).\end{equation}

Proposition 6.5 of op.cit asserts that

(2.6) \begin{equation}\mathrm{dim} \operatorname{HF}(S_a,S_b;\mathbb{Z}/2) = 4+4 I (\gamma_a,\gamma_b).\end{equation}

However, the argument described above, in particular the two inequalities (2.4) and (2.5), together only establish the weaker fact that

(2.7) \begin{equation}\mathrm{dim} \operatorname{HF}(S_a,S_b;\mathbb{Z}/2) \geq 4+4 I (\gamma_a,\gamma_b).\end{equation}

so the proposition should be corrected to give that inequality.

2.7

The whole purpose of the localisation argument is to prove Proposition 6.5, which is the only thing used elsewhere in the paper, namely in the proof of Proposition 6.18. The proof of Proposition 6.18 only uses the rank inequality which is the conclusion of the corrected argument: it uses that there is a topological lower bound on the Floer rank coming from a geometric intersection number. Both the statement and proof of Proposition 6.18 are unaffected, as are all subsequent arguments.