2.1 Preliminary wrapped Fukaya categories

Let $(W,\theta )$ denote a finite-type complete Liouville manifold (i.e. the completion of a Liouville domain $\mathring {W}$ by cylindrical ends). A Liouville domain $(\mathring {W},\theta )$ canonically determines a finite-type complete Liouville manifold $(W,\theta )$. A Lagrangian is cylindrical in $(W,\theta )$, respectively $(\mathring {W},\theta )$, if it is invariant under the Liouville vector field outside a compact set, respectively in a collar neighbourhood of the boundary. Note that any exact cylindrical Lagrangian for $(W, \theta )$ is exact Lagrangian isotopic, via the inverse Liouville flow, to a Lagrangian which is the completion of a cylindrical Lagrangian in $(\mathring {W}, \theta )$.

‘Preliminary’ is intended as in [Reference SeidelSei08b]. We will later restrict to Lagrangians which admit brane structures including a choice of $Spin$-structure and grading, yielding a category $\unicode{x1D4B2}(W, \theta )$ which is defined over $\mathbb {Z}$ and $\mathbb {Z}$-graded.

2.2 Actions of symplectomorphisms

An exact symplectomorphism of $(\mathring {W}, \theta )$ is a diffeomorphism $\phi : \mathring {W} \to \mathring {W}$ such that $\phi ^\ast \theta = \theta + df$, where $f$ need not have support in $\operatorname {Int}(\mathring {W})$ (and is formally defined on a small open neighbourhood of $\mathring {W}$ in $W$). Let $\operatorname {Symp} (\mathring {W})$ be the group of all symplectomorphisms of $\mathring {W}$, equipped with the $C^\infty$ topology. Let $\operatorname {Symp}_{\operatorname {ex}} (\mathring {W}, \theta )$ denote the subgroup of exact symplectomorphisms; the connected component of the identity in $\operatorname {Symp}_{\operatorname {ex}} (\mathring {W}, \theta )$ is $\operatorname {Ham} (\mathring {W}, \theta )$ the subgroup of Hamiltonian diffeomorphisms of $(\mathring {W}, \theta )$.

2.3 Gradings and orientations

We now assume that $2c_1(W, d \theta ) = 0 \in H^2 (W; \mathbb {Z})$, i.e. the bundle $(\Lambda ^{\dim {W}} T^\ast W)^{\otimes 2}$ is trivial (here $\dim$ denotes the complex dimension). Then $W$ admits a grading, i.e. a choice of fibrewise universal cover for the Lagrangian Grassmanian bundle $\operatorname {Lag} (W) \to W$, say $\widetilde {\operatorname {Lag}} (W) \to W$. The isomorphism classes of such covers are in one-to-one correspondence with trivialisations of $(\Lambda ^{\dim {W}} T^\ast W)^{\otimes 2} \simeq W \times \mathbb {C}$, and form an affine space over $H^1(W; \mathbb {Z})$ (see [Reference SeidelSei00, Lemma 2.2]).

Fix a grading on $W$. Given a Lagrangian $L \subset W$, the trivialisation $(\Lambda ^{\dim {W}} T^\ast W)^{\otimes 2} \simeq W \times \mathbb {C}$ determines a (homotopy class of) map $L \to \mathbb {C}^\ast$, and so a class in $H^1(L; \mathbb {Z})$, called the Maslov class of $L$. There exists a lift of $TL \subset \operatorname {Lag} (W)$ to $\widetilde {\operatorname {Lag}} (W)$ precisely when $L$ has Maslov class zero; a choice of such a lift is called a grading on $L$.

We can now upgrade to the ‘full’ version of the wrapped Fukaya category, which will be used for the rest of this article.

We will often suppress $\theta$ from the notation when it is clear from the context which primitive we are referring to.

Suppose that we are given $\phi \in \operatorname {Symp} (W)$. We say that $\phi$ is gradeable if under pullback by $\phi$, our choice of trivialisation of $(\Lambda ^{\dim {W}} T^\ast W)^{\otimes 2}$ is preserved; a grading for $\phi$ is then a choice of lift of the pullback map to a bundle automorphism of $\widetilde {\operatorname {Lag}}(W)$. We let $\operatorname {Symp}^{\operatorname {gr}} W$ denote the group of graded symplectomorphisms of $W$, and $\operatorname {Symp}^{\operatorname {gr}}_{\operatorname {ex}} (W, \theta )$ the subgroup of exact graded symplectomorphisms. From the discussion above there is an exact sequenceFootnote ¹ (see [Reference SeidelSei00, Lemma 2.4])

(2.1)\begin{equation} 1 \to \mathbb{Z} \to \operatorname{Symp}^{\operatorname{gr}} (W) \to \operatorname{Symp} (W) \to H^1(W; \mathbb{Z}). \end{equation}

If $\phi \in \operatorname {Symp}(W)$ is gradeable and has compact support, it comes with a preferred grading: the one which fixes $\widetilde {\operatorname {Lag}}(W)$ outside of the support of $\phi$. This implies that there is a splitting of the forgetful map from $\operatorname {Symp}_c^{\operatorname {gr}}(W)$ to the group of gradeable compactly supported symplectomorphisms.

The results of the previous subsection readily generalise as follows.

We then immediately get the following corollary.

The natural module action of symplectic cohomology on wrapped Floer cohomology shows that $\unicode{x1D4B2}(W)$ admits the structure of a category linear over $SH^0(W)$. From the discussion above, we immediately get that graded symplectomorphisms act on $\unicode{x1D4B2}(X)$ by $SH^0(W)$-linear equivalences.

3.1 Descriptions of the conifold smoothing $X$

Consider $\mathbb {C}^2\times \mathbb {C}^2\times \mathbb {C}^*$ with coordinates $(u_1,v_1,u_2,v_2,z)$, equipped with the standard product symplectic form $\omega = d\theta$. Here $\mathbb {C}^\ast$ has the symplectic form identifying it with $T^\ast S^1$. For concreteness, take the Kähler form with potential $|u_1|^2+ |u_2|^2+ |v_1|^2+|v_2|^2+(|z|^2 + 1/|z|^2)$. (The last term can be thought of as the restriction of the standard potential on $\mathbb {C}^2$ to $\{ zw =1\}$.)

The conifold smoothing $X$ is the symplectic submanifold defined by the equations

(3.1)\begin{equation} X = \{u_1v_1 = z-1, \ u_2v_2 = z+1\}. \end{equation}

It is the fibre $X=X_{1,-1}$ over $(+1,-1) \in \ (\mathbb {C}^* \times \mathbb {C}^*) \backslash \Delta$ of a family of symplectic submanifolds

(3.2)\begin{equation} X_{a,b} = \{u_1v_1 = z-a, \ u_2v_2 = z-b\}. \end{equation}

Let $\pi : X_{a,b} \to \mathbb {C}^*$ denote the projection to the $z$ coordinate. This is a Morse–Bott–Lefschetz fibration, meaning that $d\pi$ has transversely non-degenerate critical submanifolds. The smooth fibres are isomorphic to $\mathbb {C}^*\times \mathbb {C}^* = T^*T^2$, and there are singular fibres $(\mathbb {C}\vee \mathbb {C}) \times \mathbb {C}^*$, respectively $\mathbb {C}^* \times (\mathbb {C}\vee \mathbb {C})$, over $a$, respectively $b$.

Proof. Paths emanating from $+ 1\in \mathbb {C}^*$, respectively $-1 \in \mathbb {C}^\ast$, have associated Morse–Bott thimbles $S^1\times D^2$, respectively $D^2\times S^1$. These restrict to Lagrangian $T^2 {\rm s}$ in nearby fibres. Let $\mu$ be the moment map generating the $T^2$ action. Then each thimble is the intersection of $\mu ^{-1}(0)$ with the preimage of an open subset of $\gamma$, and we immediately see that the thimbles match without any corrections. We get

\[ S_\gamma = \mu^{-1}(0) \cap \pi^{-1} (\gamma). \]

To see that the $S_\gamma$ are all spheres, first note that if $\gamma$ is the upper or lower unit circle, then $S_\gamma$ is presented via the standard genus one Heegaard splitting of the sphere: the vanishing cycles give the standard generators of $\mathbb {Z}^2 = H_1(T^2;\mathbb {Z})$. Now for a general matching path, the Morse–Bott monodromies act trivially on the homology of the fibre $T^*T^2$, hence preserving the fact that the two vanishing cycles at the ends of the path span $\mathbb {Z}^2 = H_1(T^2;\mathbb {Z})$.

Recall that the compact core (or ‘skeleton’) of a Liouville manifold $W$ with Liouville vector field $Z$ is the locus $\bigcup _{K\subset W} \bigcap _{t>0} \phi _Z^{-t}(K)$, where we take the union over relatively compact codimension-zero subdomains with smooth boundary transverse to $Z$. When $W$ is finite type, so the completion of a fixed subdomain $K$, this is just $\bigcap _{t>0} \phi _Z^{-t}(K)$, and can be understood as the locus of points which do not flow to infinity under the Liouville vector field.

Proof. It suffices to consider $X=X_{1,-1}$. Recall that for any Kähler potential $\rho$, the Liouville vector field dual to the Liouville form $-d^c \rho$ is the gradient vector field of $\rho$ with respect to the Kähler metric. We will use the Kähler potential $\rho = |u_1|^2+ |u_2|^2+ |v_1|^2+|v_2|^2+2(|z|^2 + 1/|z|^2)$, where the factor of 2 is benign but simplifies some calculations. We have Liouville form $\theta =-d^c \rho$; let $Z$ be the Liouville vector field dual to $\theta$.

At any point away from the singular locus of $\pi$, the tangent space splits as a direct sum of the tangent space to the fibre $\pi ^{-1}(\operatorname {pt})$ and its symplectic orthogonal. This induces a decomposition of the Liouville vector field, say $Z = Z_F + Z_B$.

Let $i: F \hookrightarrow X$ be the inclusion of a fibre. At any smooth point of $i(F)$, the vector field $Z_F$ is the dual (using $i^\ast \omega$) to $i^\ast \theta$. In addition, we have that $i^\ast \theta = i^\ast (-d^c(|u_1|^2+ |u_2|^2+ |v_1|^2+|v_2|^2))$. In particular, a standard calculation shows that $Z_F$ vanishes whenever $|u_i| = |v_i|$ for both $i$, and that its positive flow (on a fixed fibre) scales $\log (|u_i| / |v_i|)$ for both $i$.

Symplectic parallel transport intertwines the fibrewise $T^2$ action, preserving the ratios $|u_1| / |v_1|$ and $|u_2|/ |v_2|$. Thus, the flow of $Z$ itself, while not fibre-preserving, acts by scaling $\log (|u_i| / |v_i|)$ for both $i$. In particular, any point in $X$ which does not satisfy $|u_i| =|v_i|$ for both $i$ flows off to infinity, and so the compact core is contained in the subset $K = \{ |u_i| = |v_i|, i=1, 2 \} \subset X$. (Note this also applies to points in the annuli of critical points with $|u_i| \neq |v_i|$ for one $i$.)

It remains to analyse the Liouville flow on $K$. Using $T^2$-equivariance, one sees that the degenerate 2-form $\omega |_K$ is pulled back from a symplectic form on the base, say $\omega _B$. Restricted to $K$, the Kähler potential is

\[ \rho (z) = 2|z-1| + 2|z+1| + 2(|z|^2 + 1/|z|^2). \]

The Liouville flow $Z=Z_B$ is the gradient vector field of $\rho (z)$ with respect to the restriction of the Kähler metric to $K$. This is the same as the pullback of the gradient flow of $\rho (z)$ on $\mathbb {C}^\ast$, using the metric associated to $\omega _B$.

Irrespective of the metric used for geodesic flow, standard calculus techniques show that the real function $\rho (z)$ has two minima, for $z = \pm 1$, and two saddles at opposite points on the imaginary axis, with modulus the real root of $x^3+x^2-1$ (approximately 0.75).

Possibly after a small perturbation of $\omega _B$ (and so of the metric on the base), there is a unique trajectory (arc) from each saddle to each minimum, which can be parametrised to have strictly monotone angular coordinate. Put together, the two arcs emanating from one saddle will give the matching path for one core, and the other two arcs will give the matching path for the other core. (The constructed vector field is not necessarily Morse; that can either be achieved by a further small perturbation of the Kähler potential, alternatively one can note that the definition of the skeleton or compact core does not rely on the Morse condition; see, e.g., [Reference StarkstonStar18] for a general discussion of skeleta in a Morse–Bott setting.)

It follows that $X$ is the Weinstein completion of a Weinstein domain obtained by plumbing two copies of the disc cotangent bundle $D^*S^3$ along a Hopf link.

An affine variety determines a (complete) Weinstein symplectic manifold canonically up to exact symplectomorphism [Reference SeidelSei08a, § 4b]. We will give two further descriptions of $X$ by giving two further descriptions of the affine variety (3.1).

Recall that if $Z \subset \mathbb {C}^2$ is a Zariski-open subset and $f: Z \to \mathbb {C}$ is a regular function, the spinning of $f$ is the threefold

(3.3)\begin{equation} \{(x,y,u,v) \in Z \times \mathbb{C}^2 \mid f(x,y) = uv\}. \end{equation}

This is a fibration over $Z$ by affine conics $\mathbb {C}^*$, with singular fibres $\mathbb {C}\vee \mathbb {C}$ along $f^{-1}(0) \subset Z$.

The subset $Z$ contains an immersed Lagrangian $2$-sphere, obtained from parallel transport around the unit circle in the $z$-plane, cf. [Reference SeidelSei, § 11]. Viewing this as a union of two Lefschetz thimbles fibred over the upper and lower half-circles with common boundary in the fibre $z=1$, the immersed $2$-sphere naturally ‘spins’ to the compact core $S_0 \cup _{\mathrm {Hopf}} S_1$ of $X$ described above, i.e. the matching $3$-spheres are obtained as $S^1$-fibrations over the thimbles with fibres degenerating over the boundary of the $2$-disc.

Our final description of $X$ is as a log Calabi–Yau threefold. We will not use this directly, but it may be of independent interest. Consider $\mathbb {P}^1 \times \mathbb {P}^1 \times \mathbb {P}^1$, which we view as the trivial $\mathbb {P}^1\times \mathbb {P}^1$-bundle over the base $\mathbb {P}^1$. We blow up this variety along the (disjoint) curves $\mathbb {P}^1 \times \{pt\} \times \{1\}$ and $\{pt\} \times \mathbb {P}^1 \times \{-1\}$ to obtain a variety $\overline {X}$ with an induced map $\overline {X} \to X \to \mathbb {P}^1$. Let $D$ denote the divisor given by the union of two smooth fibres of $\overline {X} \to \mathbb {P}^1$ together with the subset of the toric boundary comprising its four components isomorphic to the first Hirzebruch surface $\mathbb {F}_1$, see Figure 1.

Lemma 3.5 The space $X$ is symplectically equivalent to the symplectic completion of $\overline {X}\backslash D$.

Figure 1.

The toric compactification of $X$; blow up the thickened black edges on the cube for $\mathbb {P}^1\times \mathbb {P}^1 \times \mathbb {P}^1$. This slices off wedges of the corresponding moment polytope. Two of the four $\mathbb {F}_1$-boundary components of the result have been shaded.

Proof. In an affine chart $\mathbb {C}_p \times \mathbb {C}_q \times \mathbb {C}^*_z$ we blow up $\{p=1=z\}$ and $\{q=1=-z\}$. Taking $[\lambda,\mu ] \in \mathbb {P}^1$ and $[\tilde {\lambda }, \tilde {\mu }] \in \mathbb {P}^1$ to be homogeneous coordinates on two different copies of $\mathbb {P}^1$, the blow-up is given by

\[ (p-1)/(z-1) = \lambda / \mu; \ (q-1)/(z+1) = \tilde{\lambda} / \tilde{\mu}. \]

Letting $u_1 = q-1$, $u_2 = p-1$, $v_1 = \tilde {\mu }/\tilde {\lambda }$ and $v_2 = \mu /\lambda$ gives

\[ u_1v_1 = z+1, \ u_2v_2 = z-1; \]

since $z\in \mathbb {C}^*$ none of $u_i, v_j$ can be infinite. Now remove the irreducible divisor from the projective blow-up where any of the $u_i, v_j$ coordinates do become infinite. This meets the general fibre in its toric boundary; each special fibre is again toric, isomorphic to $(\mathbb {P}^1\times \mathbb {P}^1) \cup _{\mathbb {P}^1 \times \{pt\}} (\mathbb {P}^1\times \mathbb {P}^1)$, and the divisor meets this in a hexagon of lines (one does not remove the entire line of intersection of the two components, but only its intersection with the rest of the boundary).

3.2 Exact Lagrangian submanifolds

As remarked previously in Lemma 3.2, any matching path $\gamma$ between the two critical values in the base of the Morse–Bott–Lefschetz fibration $X \to \mathbb {C}^*$ defines a Lagrangian sphere $S_\gamma$ in the total space $X$. We will be particularly interested later in the collection of matching spheres $\{S_i\}_{i\in \mathbb {Z}}$ corresponding to the matching paths given in Figure 2.

Figure 2.

Lagrangian matching spheres $S_i$ (with matching paths $\gamma _i$) and Lagrangian discs $L_i$ (see Theorem 5.1).

Suppose we are given an embedded $S^1 \subset \mathbb {C}^\ast \backslash \{ -1, 1 \}$ in the base of the Morse–Bott–Lefschetz fibration, avoiding the critical values. By similar considerations to those in Lemmas 3.1 and 3.2, parallel transporting the $T^2$ vanishing cycle about the $S^1$ gives a Lagrangian $T^3$. This $T^3$ will be exact precisely when the $S^1$ in the base is Hamiltonian isotopic to the unit circle in $\mathbb {C}^\ast$. (Recall here that we are using the standard Kähler form for $\mathbb {C}^\ast$ as an affine manifold.) For any choice of such circle containing all of $\{ -1, 0 , 1 \}$, call the associated exact torus $T$.

3.3 Gradings

As $X$ is an affine complete intersection in Euclidean space, we have that $2c_1(X) = 0$, and the isomorphism classes of gradings on $X$ form an affine space over $H^1(X; \mathbb {Z}) \cong \mathbb {Z}$. Our convention is that throughout this paper we work with a fixed, distinguished grading for $X$: the unique one for which the exact Lagrangian torus $T$ defined above has Maslov class zero (this will also be true of any other tori fibred over $S^1$s containing $0$ in the base of the Morse–Bott–Lefschetz fibration).

We now want to fix gradings for the $S_i$. Fix an arbitrary grading for $S_0$. For $i \neq 0$, we want to grade $S_i$ as follows: first, note that we can make fibrewise Morse perturbations such that $S_0$ and $S_i$ intersect transversally in $4|i|$ points. One can check that there is a choice of grading for $S_i$ such that: for $i>0$, as generators for the Floer complex $CF^\ast (S_0, S_{-i})$, there are $i+1$ intersection points with grading zero; $2i$ points with grading one; and $i-1$ points with grading two; and for $i<0$, the same is true for $CF^\ast (S_i, S_0)$.

We fix an arbitrary grading on the Lagrangian torus $T$.

3.4 Symplectomorphisms of $X$: first considerations

Let $(\mathring {X}, \theta )$ be a Liouville domain such that $(X, \theta )$ is its associated completion. Following the notation in § 2.2, let $\operatorname {Symp}(\mathring {X})$ denote the group of all symplectomorphisms of $\mathring {X}$ equipped with the $C^\infty$ topology, and let $\operatorname {Symp}_{\operatorname {ex}}(\mathring {X}, \theta )$ be the subgroup of exact symplectomorphisms. We define graded versions $\operatorname {Symp}^{\operatorname {gr}}(\mathring {X})$ and $\operatorname {Symp}^{\operatorname {gr}}_{\operatorname {ex}} (\mathring {X})$ similarly.

By Lemma 2.8, the inclusion $\operatorname {Symp}_{\operatorname {ex}}^{\operatorname {gr}}(\mathring {X}, \theta ) \hookrightarrow \operatorname {Symp}^{\operatorname {gr}}(\mathring {X})$ is a weak homotopy equivalence. We define $\operatorname {Symp}^{\operatorname {gr}}(X)$ to mean $\operatorname {Symp}^{\operatorname {gr}} (\mathring {X})$, and $\operatorname {Symp}_{\operatorname {ex}}^{\operatorname {gr}} (X)$ to mean $\operatorname {Symp}_{\operatorname {ex}}^{\operatorname {gr}} (\mathring {X}, \theta )$. (Similarly for ungraded versions.) For any other Liouville domain $(\mathring {X}', \theta )$ with completion $(X, \theta )$, there is a natural weak homotopy equivalence $\operatorname {Symp}^{\operatorname {gr}}( \mathring {X}) \to \operatorname {Symp}^{\operatorname {gr}}(\mathring {X}')$, and similarly for exact maps. In particular, the group $\pi _0 \operatorname {Symp}^{\operatorname {gr}}(X)$ is independent of choices. By Corollary 2.9, $\pi _0 \operatorname {Symp}^{\operatorname {gr}}(X)$ acts on the Fukaya category $\unicode{x1D4B2}(X)$; this readily restricts to an action on $\unicode{x2131}(X)$.

Let $\operatorname {Symp}_c (X)$ denote the group of compactly supported symplectomorphisms of $X$.

It is then immediate that $\pi _0 \operatorname {Symp}_c(X)$ acts on $\unicode{x1D4B2}(X)$ and $\unicode{x2131}(X)$.

There is a well-known source of compactly supported symplectomorphisms: for any matching path $\gamma$ in the base of the Morse–Bott–Lefschetz fibration, the Dehn twist $\tau _{S_\gamma } \in \pi _0 \operatorname {Symp}_c X$. By [Reference Smith and WemyssSW23, Lemma 4.13], $\tau _{S_\gamma }$ is the monodromy associated to a full right-handed twist in the path $\gamma \subset \mathbb {C}^\ast$. More precisely, suppose we parametrise the full right-handed twist as a one-parameter family of compactly supported diffeomorphisms of $\mathbb {C}^\ast$, giving a map $\rho : [0,1] \times \mathbb {C}^\ast \to \mathbb {C}^\ast$. See Figure 3 for an illustration.

Figure 3.

For a choice of $\gamma$, visualisation of the associated full right-handed twist $\rho$, as an isotopy of $\mathbb {C}^\ast$.

Then $\tau _{S_\gamma }$ can be interpreted as the monodromy of the family $X_{\rho (t,-1), \rho (t,1)}$, $t \in [0,1]$, with the notation from (3.2). We will revisit this in Lemma 3.10.

3.5 The $\operatorname {PBr}_3$ action

We start with some generalities on the pure braid group $\operatorname {PBr}_3$. We will usually think of it as a subgroup of the fundamental group of the space of configurations of two points in $\mathbb {C}^*$, with basepoint $\{ -1, 1 \} \subset \mathbb {C}^\ast$; equivalently, it is naturally isomorphic to $\pi _1 ((\mathbb {C}^\ast )^2 \backslash \Delta )$.

By [Reference Margalit and McCammondMM09, Theorem 2.3], we have a presentation for $\operatorname {PBr}_3$ with three generators $R_1, R_2$ and $R_3$, given by the full twists described in Figure 4, and the sole relation

\[ R_1 R_2 R_3 = R_2 R_3 R_1 = R_3 R_1 R_2. \]

The element $R_1 R_2 R_3$ generates the centre $Z(\operatorname {PBr}_3) \cong \mathbb {Z}$. Geometrically, it corresponds to an inverse Dehn twist in a boundary $S^1$.

Figure 4.

The generators for $\operatorname {PBr}_3$.

Quotienting by the centre, we get isomorphisms

\[ \operatorname{PBr}_3 \simeq \operatorname{PBr}_3 / Z(\operatorname{PBr}_3) \times Z(\operatorname{PBr}_3) \simeq (\mathbb{Z} \ast \mathbb{Z}) \times \mathbb{Z}. \]

This decomposition as a direct product of groups is of course not unique: for instance, we could have $(\mathbb {Z}\langle R_i \rangle \ast \mathbb {Z} \langle R_j \rangle ) \times \mathbb {Z} \langle R_1 R_2 R_3 \rangle$ for any $i \neq j$.

Given any matching path $\gamma$ between $-1$ and $+1$ in $\mathbb {C}^\ast$, consider the full twist $t_\gamma \in \operatorname {PBr}_3$. We want to characterise the subgroup generated by these twists.

Proof. Given any $\gamma$, it is immediate that $t_\gamma \in \ker \{ \operatorname {PBr}_3 \xrightarrow {(\textrm{forget}-1, \textrm{forget}\,1)} \operatorname {PBr}_2 \times \operatorname {PBr}_2 \}$, and so the inclusion one way is clear.

Now suppose that we are given a pure braid

\[ \psi \in K = \ker \biggl\{ \operatorname{PBr}_3 \xrightarrow{(\textrm{forget }b_{-1},\textrm{ forget }b_1)} \operatorname{PBr}_2 \times \operatorname{PBr}_2 \biggr\}. \]

To analyse this, first consider simply forgetting $b_1$. For any element in the kernel, $b_{-1}$ and $b_0$ can be simultaneously pulled taut, so that all the braiding is done by $b_1$. This gives a short exact sequence of groups:

\[ 0 \to \mathbb{Z} \ast \mathbb{Z} \to \operatorname{PBr}_3 \xrightarrow{\textrm{forget }b_1} \mathbb{Z} \to 0, \]

where the kernel is identified with $\pi _1 (\mathbb {C} \backslash \{ -1, 0 \} ) \simeq \mathbb {Z} \ast \mathbb {Z}$. In terms of braids, in the notation of Figure 4, this is naturally generated by $R_2$ and $R_3$.

Putting things together, we have

\[ K = \ker \biggl\{ \mathbb{Z}\langle R_2 \rangle \ast \mathbb{Z} \langle R_3 \rangle \xrightarrow{\textrm{forget }b_{-1}} \operatorname{PBr}_2 \biggr\}. \]

A pure braid $\psi \in \mathbb {Z}\langle R_2 \rangle \ast \mathbb {Z} \langle R_3 \rangle$ is in $K$ if and only if the power of $R_2$ in the abelianisation of $\mathbb {Z}\langle R_2 \rangle \ast \mathbb {Z} \langle R_3 \rangle$ vanishes. To conclude, consider the infinite cyclic cover of the wedge of two circles with group of deck transformations $\mathbb {Z} \langle R_2 \rangle$. The image of $\pi _1$ of this cover in $\mathbb {Z}\langle R_2 \rangle \ast \mathbb {Z} \langle R_3 \rangle$ is $K$; we now see that it is a $\mathbb {Z}^{\ast \infty }$ with generators $R_2^{-i} R_3 R_2^{i} = t_i$, $i \in \mathbb {Z}$.

Mapping $R^{-i}_2 R_3 R^{i}_2$ to the Dehn twist $\tau _{S_i}$ gives a map $\operatorname {PBr}_3^c \to \pi _0 \operatorname {Symp}_c X$; this is compatible with the description of $\tau _{S_i}$ as the monodromy of the full twist in $\gamma _i$ from [Reference Smith and WemyssSW23, Lemma 4.13]. We will now upgrade this to a representation of the whole of $\operatorname {PBr}_3$ as symplectomorphisms; in order to do this, we have to enlarge the group we are working in to include non-compactly supported exact symplectomorphisms.

Lemma 3.10 There is a map

(3.4)\begin{equation} \operatorname{PBr}_3 \to \pi_0 \operatorname{Symp}_{\operatorname{ex}} X \end{equation}

extending the composition $\operatorname {PBr}_3^c \to \pi _0 \operatorname {Symp}_c X \to \pi _0 \operatorname {Symp}_{\operatorname {ex}} X$. This is compatible with the action of $\operatorname {PBr}_3$ on the collection of matching paths: if $\gamma$ is a matching path, $R \in \operatorname {PBr}_3$, and $\gamma ' = R \cdot \gamma$, then the image of $R$ in $\pi _0 \operatorname {Symp}_{\operatorname {ex}} X$ takes $S_\gamma$ to $S_{\gamma '}$ (up to Hamiltonian isotopy).

Proof. Consider $\mathbb {C}^4 \times \mathbb {C}^\ast \times (\mathbb {C}^\ast )^2$ with coordinates $(u_1, v_1, u_2, v_2, z, a, b)$ and its standard Kähler form. Consider the smooth affine subvariety

(3.5)\begin{equation} \unicode{x1D4B3} = \{ u_1 v_1 = z-a, u_2 v_2 = z-b \}. \end{equation}

Projection to the final two coordinates $(a,b)$ defines a symplectic fibre bundle $\unicode{x1D4B3} \to (\mathbb {C}^\ast )^2$. The fibre over $(a, b) \in (\mathbb {C}^\ast )^2 \backslash \Delta$ is the space $X_{a,b}$ from (3.2), and the family has threefold ordinary double points along $\Delta = \{ a=b\}$.

Now on the one hand, $\pi _1 ((\mathbb {C}^\ast )^2 \backslash \Delta )$ is naturally the pure braid group $\operatorname {PBr}_3$. On the other hand, given any path $\gamma : [0,1] \to (\mathbb {C}^\ast )^2 \backslash \Delta$ in the smooth locus, we can construct parallel transport maps over $\gamma$ which are globally defined and strongly exact, by the argument of [Reference KeatingKea15, Lemma 2.2]. Symplectic monodromy then defines the representation (3.4). The fact that the fibres over points of $\Delta$ have a single ordinary double point implies that the monodromy around a path which is a meridian to $\Delta$ (oriented anticlockwise as the boundary of a disc positively transverse to $\Delta$ at one point) is a Dehn twist (see also [Reference Smith and WemyssSW23, Lemma 4.13]). This implies that (3.4) extends the previously defined $\operatorname {PBr}_3^c \to \pi _0 \operatorname {Symp}_c X$. It is then standard, cf. [Reference Khovanov and SeidelKS02], that the action is compatible with the natural action on matching paths.

Let $W^{2n}$ be a symplectic manifold with $c_1(W)=0$, and fix a compatible almost complex structure $J$. A choice of holomorphic volume form $\Theta _J \in \Gamma (\Lambda _{\mathbb {C}}^n T^*W)$ defines a distinguished fibrewise universal cover $\widetilde {\operatorname {Lag}}(W) \to \operatorname {Lag}(W)$ of the fibre bundle $\operatorname {Lag}(W) \to W$ whose fibre is the Lagrangian Grassmannian $Gr_{Lag}(T_wW) \cong U(n)/O(n)$. Explicitly, we have a map

\[ \Theta_J : \operatorname{Lag}(W) \to S^1, \quad (v_1,\ldots,v_n) \mapsto \Theta_J(v_1\wedge \cdots \wedge v_n) / | \Theta_J(v_1\wedge \cdots \wedge v_n)| \]

and we consider the bundle $\widetilde {\operatorname {Lag}}(W) \to \operatorname {Lag}(W)$ with fibre the pairs $(\Lambda,t) \in \operatorname {Lag}(W) \times \mathbb {R}$ with $\Theta _J(\Lambda ) = e^{i\pi t}$.

Proof. Recall the space $X_{a,b}$ from (3.2) is given by $\{u_1v_1=z-a, u_2v_2=z-b\} \subset \mathbb {C}^4\times \mathbb {C}^*$. Write $f_1=u_1v_1-z+a$ and $f_2 = u_2v_2-z+b$. Following Griffiths, there is a holomorphic volume form on $X_{a,b}$, sometimes symbolically written as

\[ \frac{d \log z \wedge d u_1 \wedge d u_2 \wedge d v_1 \wedge d v_2} {d f_1 \wedge d f_2}. \]

We spell this out on charts. On the chart $\{ u_1 \neq 0 \neq u_2 \}$, the holomorphic volume form is given by

\[ d \log z \wedge d u_1 \wedge d u_2 \cdot \frac{1}{ \partial f_1 / \partial v_1} \cdot \frac{1}{\partial f_2 / \partial v_2} = d\log(z) \wedge d\log(u_1) \wedge d\log(u_2). \]

On the charts $\{ v_1 \neq 0 \neq u_2 \}$, $\{ u_1 \neq 0 \neq v_2 \}$ and $\{ v_1 \neq 0 \neq v_2 \}$, we get similar expressions (up to signs) by replacing $u_1$ with $v_1$, and / or $u_2$ with $v_2$. Together these charts cover all of $X$ except the locus where $u_1 = v_1 = 0$ or $u_2 = v_2 = 0$. If $u_1 = v_1 = 0$, we must have $z=a \in \mathbb {C}^\ast$ and $u_2v_2\neq 0$. Working in the chart $\{ z \neq 0 \neq u_2 \}$, the holomorphic volume form is given by

\[ - \frac{1}{z} \cdot\ d u_1 \wedge d u_2 \wedge dv_1 \cdot \frac{1}{ \partial f_1 / \partial z} \cdot \frac{1}{\partial f_2 / \partial v_2} = \frac{1}{z} \cdot \frac{1}{u_2} d u_1 \wedge d u_2 \wedge dv_1. \]

Up to signs, it has a similar expression on the chart $\{ z \neq 0 \neq u_1 \}$. Altogether, our charts now cover $X$.

This existence of our holomorphic volume form implies that the family $\unicode{x1D4B3} \to \operatorname {Conf}_2(\mathbb {C}^*)$ over configuration space from (3.5) admits a relative holomorphic volume form. Via the preceding construction, this defines a fibration $\unicode{x1D4B3}^{\infty } \to \unicode{x1D4B3}$ with fibre the universal cover $\widetilde {U(n)/O(n)}$. A choice of Ehresmann connection for this fibration enables one to lift (not necessarily uniquely) the parallel transport maps for $\unicode{x1D4B3} \to \operatorname {Conf}_2(\mathbb {C}^*)$ to parallel transport maps for $\unicode{x1D4B3}^{\infty } \to \operatorname {Conf}_2(\mathbb {C}^*)$, which shows, in particular, that these symplectomorphisms admit coherent gradings.

Choice of gradings for the action of $PBr_3$. We now fix a preferred lift $\operatorname {PBr}_3 \to \pi _0 \operatorname {Symp}_{\operatorname {ex}}^{\operatorname {gr}}(X)$, by specifying the lifts of each of the three generators $R_1, R_2$ and $R_3$. First, the image of $R_3$ is the Dehn twist $\tau _{0} = \tau _{S_0}$; we give it the preferred grading for a compactly supported symplectomorphism. Second, we grade the image of $R_2$, say $\lambda$, by asking for $\lambda ^i S_0 = S_{-i}$ as graded objects. Finally, to choose the grading for the image of $R_1$, say $\rho$, recall that $R_1 R_2 R_3$ generates the centre of $\operatorname {PBr}_3$, and corresponds to the inverse Dehn twist in a boundary parallel circle; in particular, the product $\rho \circ \lambda \circ \tau _{S_0}$ takes $S_0$ to itself as an ungraded object. Fixing a grading on $\rho$ is the same as fixing the shift of $S_0$ under $\rho \circ \lambda \circ \tau _{S_0}$; we choose this to be $[1]$.

3.6 $K$-theoretic actions of symplectomorphism groups

We have that $K(\unicode{x1D4B2} (X)) = \mathbb {Z}^2$ generated by two cotangent fibres $F_0$ and $F_1$, by for instance [Reference Ganatra, Pardon and ShendeGPS24, Theorem 1.13] or [Reference Chantraine, Dimitroglou Rizell, Ghiggini and GolovkoCDGG24]. This is naturally identified with $H_3(X, \partial _\infty X; \mathbb {Z})$, where $\partial _\infty X$ denotes the conical end of $X$. Moreover, there is a pairing

(3.6)\begin{equation} K(\unicode{x2131}(X)) \times K(\unicode{x1D4B2}(X)) \to \mathbb{Z}, \quad (K, L) \mapsto \chi(HF^\ast (L,K)). \end{equation}

Evaluating this on fibres and zero-sections shows that $\mathbb {Z}^2 = \langle S_0, S_1 \rangle \to K(\unicode{x2131}(X))$ is injective. (Note we do not have an explicit description for $K(\unicode{x2131}(X))$.)

The pairing (3.6) descends to a non-degenerate pairing $K_{\textrm {num}} (\unicode{x2131}(X)) \times K(\unicode{x1D4B2}(X)) \to \mathbb {Z}$. This implies that $K_{\textrm {num}}(\unicode{x2131}(X)) = \mathbb {Z}^2 = \langle S_0,S_1 \rangle$. This is naturally identified with $H_3 (X; \mathbb {Z})$, and the non-degenerate pairing

\[ K_{\textrm{num}} (\unicode{x2131} (X)) \times K ( \unicode{x1D4B2}(X)) \to \mathbb{Z} \]

is identified with the intersection pairing

\[ H_3 (X; \mathbb{Z}) \times H_3 (X, \partial_\infty X; \mathbb{Z}) \to \mathbb{Z}. \]

Both $\operatorname {Symp}^{\operatorname {gr}}(X)$ and $\operatorname {Symp}_c(X)$ (with preferred gradings) act on $K(\unicode{x2131}(X))$ and $K(\unicode{x1D4B2}(X))$ compatibly with this pairing. This implies that there is an induced action on the numerical Grothendieck group. On the other hand, as the intersection pairing of $S_0$ with both $S_0$ and $S_1$ vanishes, and similarly for $S_1$, we see that the standard map $H_3 (X; \mathbb {Z}) \to H_3(X, \partial _\infty X; \mathbb {Z})$ has zero image. The following is now immediate.

For arbitrary compactly supported maps, we have the following.

Proof. Let $\phi \in \operatorname {Homeo}_c (X)$ be a compactly supported homeomorphism. Considering the induced action on the long exact sequence for the pair $(X, \partial _\infty X)$, we get a commutative diagram:

(3.7)\begin{equation} \begin{split} \matrix{ \ldots {\longrightarrow} & H_3( \partial_\infty X) \longrightarrow & H_3( X) \longrightarrow & H_3( X, \partial_\infty X) \longrightarrow & \ldots \\ &{\phi_\ast} \unicode{x2193} &{\phi_\ast} \unicode{x2193} &{\phi_\ast} \unicode{x2193} \\ \ldots \longrightarrow & H_3( \partial_\infty X) \longrightarrow & H_3( X) \longrightarrow & H_3( X, \partial_\infty X) \longrightarrow & \ldots } \end{split} \end{equation}

where all coefficient rings are $\mathbb {Z}$. By assumption, $\phi _\ast : H_3(\partial _\infty X) \to H_3(\partial _\infty X)$ is the identity. On the other hand, recall that in this case the map $H_3(X) \to H_3(X, \partial _\infty X)$ has zero image. It is then immediate that $\phi _\ast : H_3(X) \to H_3(X)$ is the identity.

Lemma 3.17 will also imply that there are infinity many orbits of Lagrangians spheres in $X$; see Corollary 7.2 for a more general statement. On the other hand, by the proof of Proposition 3.9, we see that all of the $S_\gamma$ are contained in the orbits of the $S_i$, $i \in \mathbb {Z}$, under the action of the group generated by Dehn twists in the $S_i$.

6.1 Twists act trivially on numerical $K$-theory

Fix a spherical object $S \in D\unicode{x1D4B2}(X)$ and consider the associated spherical twist $T_S$. By [Reference PomerleanoPom21, Lemma 5.46] we can assume that the category $D\unicode{x1D4B2}(X)$ of perfect modules over $\unicode{x1D4B2}(X)$ has a strictly $SH^0(X)$-linear chain model. The total evaluation morphism $\operatorname {Hom}(S,\bullet )\otimes S \to \bullet$ is then $SH^0$-linear, and the cone inherits an $SH^0(X)$-linear structure. This means that $T_S$ is linear as an autoequivalence. By Theorem 5.8, we know that $T_S$ is an element of $\operatorname {Auteq}_{SH^0} D\unicode{x1D4B2}(X) \cong \operatorname {Auteq} \unicode{x1D49F} \cong \operatorname {PBr}_3$.

Proof. Under the equivalence $\Upsilon : D\unicode{x1D4B2}(X) \to D(Y)$ of Theorem 5.1, $S$ corresponds to some complex $\mathcal {E}_S$ of sheaves on $Y$. As in the proof of Lemma 5.3, the argument in the proof of [Reference Bayer, Craw and ZhangBCZ17, Theorem 7.2.1] shows that $\mathcal {E}_S$ must have proper support in $Y$ (in the sense of [Reference HuybrechtsHuy06, Definition 3.8], meaning the support is the union of the supports of the cohomology sheaves): otherwise it would have infinite-rank morphisms in some degree with the tilting object in $D(Y)$, which violates the definition of being spherical. As $\mathcal {E}_S$ is also indecomposable, it must have connected support, which by the classification of proper subvarieties of $Y$ implies that the support is either $C$ or a single point. By [Reference HuybrechtsHuy06, Lemma 4.5], the support must be $C$ and so, in fact, $\mathcal {E}_S$ belongs to the category $\unicode{x1D49F} \subset D(Y)$. It follows that $S$ belongs to the subcategory of $\unicode{x1D4B2}(X)$ generated by $S_0$ and $S_1$ and, in particular, belongs to the compact Fukaya category $\unicode{x2131}(X) \subset \unicode{x1D4B2}(X)$.

The spherical twist $T_S$ acting on $\unicode{x2131}(X)$ fits into an exact triangle

(6.1)

and, hence, acts on (numerical) $K$-theory by the Picard–Lefschetz-type transformation

\[ A \mapsto A - \chi(S,A)[S], \]

where $\chi (S,A) \in \mathbb {Z}$ is the Euler characteristic of $\operatorname {Hom}_{\unicode{x2131}}(S,A)$. Since the intersection pairing on $K_{\textrm {num}}(\unicode{x2131}(X)) = H_3(X)$ vanishes, $\chi$ vanishes identically.

Lemma 6.3 combined with Corollary 4.11 then implies that $T_S$ lies in $\mathbb {Z}^{\ast \infty } \times 2\mathbb {Z} \subset (\mathbb {Z} \ast \mathbb {Z}) \times \mathbb {Z} \cong \operatorname {PBr}_3$.

6.2 A localisation argument

The key input for Theorem 6.2 is the following analogue, in our setting, of a well-known theorem [Reference Khovanov and SeidelKS02] due to Khovanov and Seidel, which localises the computation of Floer cohomology to the fixed locus of a finite group action.

For matching paths $\gamma _a, \gamma _b$ in the base $\mathbb {C}^*$ of the Morse–Bott–Lefschetz fibration $\pi : X\to \mathbb {C}^*$, we let $I(\gamma _a, \gamma _b)$ denote the minimal (unsigned, i.e. geometric) interior intersection number between $\gamma _a$ and $\gamma _b$, relative to the their end points. Here isotopies of paths are not allowed to cross the puncture; and ‘interior’ means that we are not including the end points in our count.

Proposition 6.5 Suppose $S_a, S_b \in \mathcal {S}$ correspond to matching paths $\gamma _a, \gamma _b \subset \mathbb {C}^*$. Then the Floer homology of $S_a$ and $S_b$ with coefficients in $\mathbb {Z}/2$ satisfies

\[ \dim \operatorname{HF}(S_a, S_b;\mathbb{Z}/2) = 4 + 4 I(\gamma_a, \gamma_b). \]

The proof of Proposition 6.5 has two ingredients: the existence of a $\mathbb {Z}/2 \times \mathbb {Z}/2$ action on $X$, preserving the Lagrangians, and with the property that there are no holomorphic discs in the fixed point locus; and an equivariant transversality argument which allows us to compare Floer theory in $X$ with Floer theory in that fixed point locus. We describe the involutions and the reduction to existence of an equivariantly transversal $J$ in this section; the existence of such a $J$ will be broken down into §§ 6.3–6.5.

Recall $X$ is given explicitly by

\[ X = \{ u_1 v_1 = z-1, \, u_2 v_2 = z+1 \} \subset \mathbb{C}^4 \times \mathbb{C}^\ast. \]

Here $X$ carries two commuting involutions: $\iota _1$, trading $u_1$ and $v_1$; and $\iota _2$, trading $u_2$ and $v_2$. Let $G = \mathbb {Z}/2\times \mathbb {Z}/2$ be the group generated by the $\iota _j$.

The common fixed point locus of both $\iota _1$ and $\iota _2$, say $X^{\text {inv}}$, has equation $\{ p^2 = z-1, \, q^2 = z+1 \} \subset \mathbb {C}^2 \times \mathbb {C}^{\ast }$. This is smooth, and Weinstein with the restriction of the form from $X$. (Topologically, $X^{\text {inv}}$ is a sphere with six punctures.)

Projection $X^{\text {inv}} \to \mathbb {C}^*$ to the $z$ coordinate, i.e. to the base of the Morse–Bott–Lefschetz fibration, defines a fourfold cover of $\mathbb {C}^\ast$, branched over $\pm 1$. Note that the branching pattern is as follows: say the sheets are labelled by $(\pm p, \pm q)$. Then above $+1$, the sheets with the same value of $q$ meet; and above $-1$, the sheets with the same value of $p$ meet.

For any matching path $\gamma$, $S_\gamma$ meets $X^{\text {inv}}$ cleanly in an embedded $S^1$, say $s_\gamma$ (which is itself a matching sphere in $X^{\text {inv}}$). The circle $s_{\gamma }$ is a fourfold cover of $\gamma$ under projection to $z$.

Following [Reference Fathi, Laudenbach and PoénaruFLP79, Reference Khovanov and SeidelKS02], we say that matching paths $\gamma _a$ and $\gamma _b$ intersect minimally if they intersect transversally, and the following holds: assume we are given any two points $z_{-} \neq z_+$ in $\gamma _a \cap \gamma _b$ which are not both end points of the matching paths, and two arcs $\alpha \subset \gamma _a$, $\beta \subset \gamma _b$ ending on $z_{-}, z_+$ such that those arcs have no other intersection points. Then if $K$ is the connected component of $\mathbb {C} \backslash (\gamma _a \cup \gamma _b)$ bounded by $\alpha \cup \beta$, and $K$ is topologically an open disc, then it must contain the puncture $\{ 0 \}$.

Using [Reference Fathi, Laudenbach and PoénaruFLP79, Exposé III], we can assume that after isotopy rel end points, the matching paths $\gamma _a$ and $\gamma _b$ intersect minimally, in $I(\gamma _a, \gamma _b)$ points. Any isotopy of a matching path $\gamma$ in the base (rel end points) is covered by a Hamiltonian isotopy of $S_\gamma$ in $X$ and a Hamiltonian isotopy of $s_\gamma$ in $X^{\text {inv}}$. Thus, after Hamiltonian isotopy in $X$, we may assume $S_a$ and $S_b$ intersect transversally in $4+4I(\gamma _a, \gamma _b)$ points, all fixed by both $\iota _j$.

In the terminology of Khovanov and Seidel, this is saying that $S_a$ and $S_b$ satisfy condition (S2) of [Reference Khovanov and SeidelKS02, Lemma 5.14]: there are no bigons between $S_a$ and $S_b$.