1.1. Algebraic torus fibrations

As a first generalisation, consider the following set-up. Let $Y= \mathbb{C}^n$ , or a more general Liouville manifold, and $f \;:\; Y \to \mathbb{C}$ be a holomorphic map with $0$ as a regular value. Then we can construct a fibration $\pi \;:\; X \to Y$ by setting $X$ to be the smooth hypersurface:

\begin{equation*} X = \{ (x,y,\textbf{w}) ;\, xy = f(\textbf{w}) \} \; \subset \mathbb{C}^2\times Y \end{equation*}

The generic fibre of $\pi \;:\; X \to Y$ is isomorphic to $\mathbb{C}^*$ , and it degenerates to a node along the smooth hypersurface $D = \{ f(\textbf{w}) =0 \}$ . The space $X$ admits a Hamiltonian $S^1$ action given by rotating the fibres: $ (x,y, \textbf{w}) \mapsto (e^{i\theta }x,e^{-i\theta }y, \textbf{w})$ for $e^{i\theta } \in S^1$ .

More generally, suppose we have holomorphic line bundles $L_1,\ldots , L_r$ on $Y$ and sections $f_i\in \Gamma (L_i^{\otimes 2})$ . Assume that each hypersurface $D_i = \{ f_i=0 \}$ is smooth and that together they form a simple normal crossing divisor $D=\cup _{i=1}^r D_i$ . Then we can form the smooth space:

\begin{equation*} X = \{ (x_1,y_1,\ldots , x_r,y_r, \textbf{w}) \,|\, x_iy_i = f_i(\textbf{w}), \text{for all } i \}\;\;\subset \textrm {Tot}\left (\oplus _{i=1}^r L_i^{\oplus 2} \right ) \end{equation*}

The projection to $\textbf{w}$ defines a fibration $\pi \;:\; X \to Y$ whose generic fibre is isomorphic to $(\mathbb{C}^*)^r$ , with singular fibres appearing over the divisor $D$ . We also have a Hamiltonian action of an $r$ -dimensional torus $T$ on $X$ by rotating the fibres.

If $r\gt \dim Y$ , then the intersection of all the $D_i$ is empty, and it follows that the diagonal $U(1)\subset T$ acts freely. Then the quotient $X' = X/\mathbb{C}^*$ is again a manifold fibering over $Y$ , with generic fibre an algebraic torus, and singular fibres appearing over the same divisor $D$ . It carries a Hamiltonian action of the rank $r-1$ torus $T/U(1)$ . In fact, if $r=\dim + k$ , then there is a rank $k$ subtorus of $T$ , which acts freely, and we can quotient by any subtorus of it.

We’ll refer to these $X\to Y$ or $X'\to Y$ , with their Hamiltonian torus actions, as algebraic torus fibrations.

Conjecture A. Given an algebraic torus fibration $X\to Y$ , we have a quasi-equivalence:

\begin{equation*} \mathcal{W}_{T}(X)_{-\textbf{1}} \simeq \mathcal{W}(Y\setminus D) \end{equation*}

Here if we work with $\mathbb{Z}$ -graded categories, the grading on $Y \setminus D$ should be chosen so that it extends over $D$ . The meaning of the suffix $-\textbf{1}$ will be explained in the next section; for the moment, we ignore it.

If we delete the divisor $\pi ^{-1}D$ from $X$ , then what remains is just a principal $(\mathbb{C}^*)^r$ bundle over $Y\setminus D$ . So it is not too surprising that there should be a quasi-equivalence:

\begin{equation*}\mathcal{W}_T(X\setminus \pi ^{-1}D) \simeq \mathcal{W}(Y\setminus D)\end{equation*}

and indeed, this fits with a more general story about Hamiltonian reduction that we will discuss in the next section. However, $\mathcal{W}_T(X)$ should be a deformation of $\mathcal{W}_T(X\setminus \pi ^{-1}D)$ , since including the extra divisor will add terms to the $A_\infty$ structure. ConjectureA is the claim that this deformation is in fact trivial.

In Section 2, we will provide evidence for the conjecture using toric mirror symmetry; in some examples, this amounts to an (excessively indirect!) proof.

1.2. Hamiltonian reduction

Suppose we have a Hamiltonian $S^1$ action on a symplectic manifold $X$ . The information of the $S^1$ action appears in the Fukaya category as an invertible element:

\begin{equation*}s \in SH^*(X)\end{equation*}

This gives rise to a natural automorphism of the identity functor, so for each object $L\in \mathcal{W}(X)$ , it provides an automorphism $s_{L}\;:\; L\stackrel {\sim }{\longrightarrow } L$ . Precisely, $s_{L} = \mathcal{CO}^0(s) \in HF^0(L,L)$ where $\mathcal{C}\mathcal{O}^0$ is the zeroth order part of the unital algebra map $\mathcal{CO}\;:\; SH^*(X) \to HH^*(CF^*(L,L))$ called the closed-open string map.

This $s$ was originally constructed by Seidel [Reference Seidel25] for compact $X$ and as an invertible element in the quantum cohomology $QH^*(X)$ . When $X$ is exact, Seidel’s construction yields an element in symplectic cohomology $SH^*(X)$ instead [Reference Ritter24], but in either case, we can map it to Hochschild cohomology using the closed-open map.Footnote ²

In the previous section, we discussed the equivariant wrapped category $\mathcal{W}_{S^1}\!(X)$ as though it was a single category. In fact, for any given $\lambda \in \mathbb{C}^*$ , one can construct a version of the equivariant category

\begin{equation*}\mathcal{W}_{S^1}\!(X)_\lambda \end{equation*}

starting from those objects of $\mathcal{W}(X)$ such that $s_L= \lambda 1_{L}$ . Teleman [Reference Teleman30] refers to these categories as the ‘spectral components’ of the equivariant Fukaya category, and they will be the main focus of this paper.Footnote ³

For example, an $S^1$ -invariant Lagrangian $L$ (which is monotone, has minimal Maslov at least 2 and is equipped with a spin structure) provides an object of $\mathcal{W}_{S^1}\!(X)_{\pm 1}$ . But if we give $L$ a non-trivial local system, whose monodromy along $S^1$ orbits is $\lambda$ , then we have an object of $\mathcal{W}_{S^1}\!(X)_{\pm \lambda }$ .

Now consider the Hamiltonian reduction $ X {/\!/}_{a} \,S^1$ at some regular value $a\in \mathbb{R}$ of the moment map $\mu$ . There is a Lagrangian correspondence

(1.5) \begin{align} \Gamma = \{ (x, [x] ), \; \mu (x)=a \} \; \subset X^{-}\times (X{/\!/}_{a} \,S^1) \end{align}

which induces a functor

(1.6) \begin{align} \mathcal{W}(X) \to \mathcal{W}(X{/\!/}_{a} \,S^1) \end{align}

via the theory of holomorphic quilts [Reference Fukaya6,Reference Lekili and Lipyanskiy14,Reference Ma’u, Wehrheim and Woodward19]. Moreover, since $\Gamma$ is $S^1$ -invariant, we can use it to define a functor in the equivariant setting. Teleman conjectures that this gives an equivalence

(1.7) \begin{equation} \mathcal{W}_{S^1}\!(X)_{\pm e^{a}} \cong \mathcal{W}(X{/\!/}_{a} \,S^1) \end{equation}

between the Fukaya category of the Hamiltonian reduction and the corresponding spectral component of the equivariant category. More generally, we can consider the spectral component at $\lambda = \pm e^{a+ ib}$ , and this will give the wrapped category of the quotient equipped with a B-field (or perhaps a more general bulk deformation). A theorem along these lines, for compact $X$ , has been announced by Fukaya [Reference Fukaya7].

Let us see what this point of view brings to our Example1.1. The moment map there is $\mu = |x|^2 - |y|^2$ . Any non-zero $a\in \mathbb{R}$ is a regular value of $\mu$ and produces the quotient $X{/\!/}_{a} \,S^1 \cong \mathbb{C}$ . Since $\mathcal{W}(\mathbb{C})\cong 0$ , the corresponding spectral component should be zero.

Given Remark1.4, our observation (1.2) should really be the claim that:

\begin{equation*}\mathcal{W}_{S^1}(X)_{-1} \cong \mathcal{W}(P) \end{equation*}

So the spectral component at $\lambda =-1$ is non-zero. This must correspond to the singular value $a=0$ of the moment map, where we cannot do symplectic reduction. Instead, we are simply deleting the singularities of the moment map fibre $\mu ^{-1}(0)$ and forming the quotient:

\begin{equation*}\mathcal{P} = \big (\mu ^{-1}(0) - (0,0,\pm 1)\big ) / S^1\end{equation*}

With this prescription, we are extending the equivalences

\begin{equation*}\mathcal{W}_{S^1}(X)_{-e^a} \cong \mathcal{W}(X{/\!/}_{a} \,S^1)\end{equation*}

from (1.7) to the singular value $a=0$ .

From this example, we draw the following general conclusion:

Conjecture B. Let $X$ be a Liouville manifold with a Hamiltonian $S^1$ -manifold such that the fixed locus $X^{S^1}$ has codimension four, and assume there are no finite non-trivial stabiliser groups. Let $\mu \;:\; X \to \mathbb{R}$ be the moment map, and let $a\in \mathbb{R}$ be a singular value in the interior of $\operatorname{Im}\mu$ . Then there is a quasi-equivalence

\begin{equation*}\mathcal{W}_{S^1}\!(X)_{- e^{a}} \cong \mathcal{W}(U/S^1) \end{equation*}

where $U$ is the smooth locus in $\mu ^{-1}(a)$ .

As stated, our conjecture may be too optimistic, as in general $\mathcal{W}(U/S^1)$ may have to be bulk deformed.Footnote ⁴ However, we will see many examples below where such a bulk deformation is absent.

Most of the discussion above generalises easily to Hamiltonian actions of a higher-rank torus $T=(S^1)^r$ . Such an action produces $r$ Seidel elements $s_1,\ldots , s_r$ , and hence a category $\mathcal{W}_T(X)_\lambda$ for each $\lambda \in (\mathbb{C}^*)^r$ .

For regular values of the moment map, the corresponding component $\mathcal{W}_T(X)_\lambda$ will be the Fukaya category of the corresponding symplectic quotient of $X$ , possibly with a bulk deformation. At singular values, our conjecture is that we should instead – at least in some situations – take the quotient of the smooth part of the moment map fibre.

The set-up considered in ConjectureA is a specialcase of this story. A moment map value $(a_1,\ldots ,a_r)$ is regular if every $a_i$ is non-zero, and then the corresponding Hamiltonian reduction is the base $Y$ . So taking the spectral component at the value $\lambda = (-\!e^{a_1},\ldots , -e^{a_r})$ will produce $\mathcal{W}_{S^1}\!(X)_\lambda \cong \mathcal{W}(Y)$ . At a more general $\lambda$ , provided each $\lambda _i$ lies off the unit circle, we will get $\mathcal{W}(Y)$ , possibly with a bulk deformation.

But suppose $\lambda _1=-1$ . Then the fibre of the moment map has singularities over the divisor $D_1$ , and our conjecture is that the spectral component is

\begin{equation*}\mathcal{W}_{S^1}\!(X)_\lambda \cong \mathcal{W}(Y\setminus D_1)\end{equation*}

bulk deformed by $(b_2,\ldots , b_r)$ . If several $\lambda _i$ are equal to $-1$ , then we must delete several divisors, and at the most special value

\begin{equation*} -\textbf{1} = (\!-\!1,\ldots , -1)\end{equation*}

we get the statement of Conjecture A.

Example 1.12. The simplest possible example of Conjectures A and B is to set $X=\mathbb{C}^2$ with the $S^1$ action $\theta \;:\; (x,y) \to (e^{i\theta }x, e^{-i\theta }y)$ . We can view it as an algebraic torus bundle $\pi \,:\,X \to Y=\mathbb{C}$ with $\pi (x,y)=xy$ , having a single singular fibre over $D=\{0\}$ . The Hamiltonian reduction at generic values of $\mu$ is $\mathbb{C}$ , and at the singular value $a=0$ , the quotient of the smooth locus is $\mathbb{C}^*$ . So we expect that

\begin{equation*}\mathcal{W}_{S^1}(X)_{-e^a}\cong \mathcal{W}(\mathbb{C})\cong 0 \end{equation*}

if $a\neq 0$ , but that $\mathcal{W}_{S^1}\!(X)_{-1}\cong \mathcal{W}(\mathbb{C}^*)$ , which is not zero.

Consider the invariant Lagrangian torus $L=\{|x|=|y|=1\}$ , and equip it with the canonical spin structure and some choice of $\mathbb{C}^*$ local system. Then we have an object of $\mathcal{W}_{S^1}\!(X)_\lambda$ where $\lambda$ is the monodromy along $S^1$ orbits in $L$ . Following the discussion after Conjecture A, we could delete $\pi ^{-1}D$ and consider just the $\mathbb{C}^*$ bundle $\pi \;:\; (\mathbb{C}^*)^2\to \mathbb{C}^*$ . In this space, $L$ is exact, and since the $S^1$ action is free, we can compute

\begin{equation*}HF_{S^1}(L,L) = HF(L/S^1, L/S^1) = H^*(S^1)\end{equation*}

regardless of the local system. This is consistent with the equivalence (1.7) that says

\begin{equation*}\mathcal{W}_{S^1}\!(X\setminus \pi ^{-1}D)_{-e^a} \cong \mathcal{W}(\mathbb{C}^*)\end{equation*}

for all values of $a$ . But if we do the computation in $X$ instead, this answer gets deformed by holomorphic discs. Indeed, the moduli space of discs bounded by $L$ is diffeomorphic to $S^1 \sqcup S^1$ , and the $S^1$ action is free on each component [Reference Seidel26, Lem. 2.16]. Moreover, the boundary classes of the two $S^1$ -families of discs differ by an orbit of a point in $L$ ,Footnote ⁵ and their contributions cancel each other exactly when $\lambda = -1$ .

So if $\lambda \neq -1$ , then this object is isomorphic to zero in $\mathcal{W}_{S^1}\!(X)_\lambda$ , but if $\lambda =-1$ , we have a whole family of non-zero objects parameterised by the remaining monodromy $\nu \in \mathbb{C}^*$ of the local system.Footnote ⁶ This supports the claim that $\mathcal{W}_{S^1}\!(X)_{-1}\cong \mathcal{W}(\mathbb{C}^*)$ but that $\mathcal{W}_{S^1}\!(X)_\lambda \cong 0$ for all other values of $\lambda$ . It seems that the deformation

\begin{equation*}\mathcal{W}_{S^1}\!(X\setminus \pi ^{-1}D)_\lambda \leadsto \mathcal{W}_{S^1}\!(X)_\lambda \end{equation*}

is trivial exactly for the special value $\lambda =-1$ .

Following Remark 1.11, we can look in particular at the components $\mathcal{W}_{S^1}\!(X)_{-e^{ib}}$ for $b\neq 0$ . These all appear to be zero and hence non-trivial bulk deformations of $\mathcal{W}_{S^1}\!(X)_{-1}$ .

In Example 2.3, we will show that all these claims are consistent with mirror symmetry.

In fact, we haven’t found any examples for Conjecture B, as stated, that go beyond the set-up of Conjecture A. But we do have a couple of rather speculative examples where we drop the condition that $X$ is Liouville – see Section 2.5.

1.3. Mirror symmetry

Given a Hamiltonian action $S^1\curvearrowright X$ , the Seidel element makes the wrapped Fukaya category $\mathcal{W}(X)$ – at least at the level of homology – linear over the ring $\mathbb{C}[s, s^{-1}]$ . This can be understood as part of a more general story of ‘topological group actions’ on categories envisaged by Teleman [Reference Teleman30] (see Section 1.6).

Now suppose that $X$ is a mirror to an algebraic variety $\breve {X}$ . Then, since $\mathcal{W}(X)=D^b(\breve {X})$ , the mirror to $s$ must be an invertible element $\sigma$ in:

(1.13) \begin{equation} HH^0(D^b(\breve {X})) = \Gamma (\mathcal{O}_{\breve {X}}) \end{equation}

If we have an $S^1$ action on the symplectic side, then on the mirror, we have a function $\sigma \,:\,\breve {X}\to \mathbb{C}^*$ .

More generally, $X$ might be mirror to a Landau-Ginzburg model $(\breve {X}, \breve {W})$ . Then $\mathcal{W}(X)$ is equivalent to the category of matrix factorizations $\textrm {MF}(\breve {X}, \breve {W})$ , but still a function $\sigma \,:\,\breve {X}\to \mathbb{C}^*$ does provide a natural automorphism of this category, so a possible mirror to the $S^1$ action on $X$ .

‘Conjecture’ C. Suppose we have a Hamiltonian $S^1$ action on a Liouville manifold $X$ . Suppose $X$ has a mirror Landau-Ginzburg model $(\breve {X}, \breve {W})$ , and that the $S^1$ action is mirror to a function $\sigma \;:\; \breve {X}\to \mathbb{C}^*$ . Then for every $\lambda \in \mathbb{C}^*$ , we have an equivalence

\begin{equation*}\mathcal{W}_{S^1}\!(X)_\lambda \;\cong \; \textrm {MF}\!\big (\breve {Z}_\lambda , \breve {W}|_{\breve {Z}_\lambda } \big ) \end{equation*}

where $\breve {Z}_\lambda \subset \breve {X}$ denotes the hypersurface $\sigma ^{-1}(\lambda )$ .

The generalisation to higher-rank torus actions is obvious.

This claim will be central to all the mirror symmetry evidence that we provide for our other conjectures. However, it is not really a precise conjecture because we have not specified what we mean by ‘mirror’. In particular, it’s not enough to just assume that $\mathcal{W}(X)\cong \textrm {MF}(\breve {X}, \breve {W})$ , as we shall see in Section 1.6.

Example 1.15. Consider

\begin{equation*}X = \mathbb{C}^2 \setminus \{ xy=1 \}\end{equation*}

equipped with the restriction of the standard symplectic form on $\mathbb{C}^2$ . This is a log-Calabi Yau surface, which is known to be self-mirror ([Reference Pascaleff22]). It is an important example because it is the simplest example of a symplectic manifold admitting a Lagrangian torus fibration that has a singular fibre. We write

\begin{equation*} \breve {X} = \mathbb{C}^2 \setminus \{ \breve {x}\breve {y}=1 \} \end{equation*}

for the mirror.

Now add a Hamiltonian $S^1$ action on $X$ by $e^{i\theta }(x,y) = (e^{i\theta }x, e^{-i\theta }y)$ . Using some toric mirror symmetry (see Example 2.5), one can argue that on $\breve {X}$ , this becomes the non-vanishing function:

\begin{equation*} \sigma = -1-\breve {x}\breve {y} \end{equation*}

If $\lambda \in \mathbb{C}^*$ with $\lambda \neq -1$ , then $\sigma ^{-1}(\lambda )\cong \mathbb{C}^*$ . So the claim is that $\mathcal{W}_{S^1}\!(X)_\lambda \cong D^b(\mathbb{C}^*)$ . But for $\lambda =-1$ , we are claiming that $\mathcal{W}_{S^1}\!(X)_{-1}$ should be equivalent to the derived category of the node $\breve {Z}_{-1}= \{\breve {x}\breve {y}=0\}$ .

We can also relate these spectral components to symplectic reduction as we discussed in the previous section. For a regular moment map value $a\in \mathbb{R} \setminus 0$ , Fukaya-Teleman tell us (1.7) that

\begin{equation*}\mathcal{W}_{S^1}\!(X)_{-e^a}\cong \mathcal{W}(X{/\!/}_{a}\, S^1) = \mathcal{W}(\mathbb{C} \setminus 1)\end{equation*}

which is consistent with our mirror symmetry claims since $\mathbb{C}\setminus 1$ is indeed mirror to $\mathbb{C}^*$ . At the singular value $a=0$ , we can apply our Conjectures A or B, which tell us to delete the singularity from $\mu ^{-1}(0)$ before we take the quotient. The result is $\mathbb{C} \setminus \{0, 1\}$ , which is the pair-of-pants,Footnote ⁷ and this is indeed a mirror to the node $\breve {Z}_{-1}$ .

We can also move off the real line and set $\lambda = -e^{a+ib}$ . If $a\neq 0$ , this has no effect; all bulk deformations of $\mathbb{C}\setminus \{1\}$ are trivial (there are no $SH^2$ classes to deform with) and $\mathcal{W}(\mathbb{C}\setminus \{ 1 \}) \cong D^b(\mathbb{C}^*)$ , consistent with the mirror. But this is not true for $a=0$ , and we can see from the mirror that $\mathcal{W}_{S^1}\!(X)_{-e^{ib}}$ should be a non-trivial deformation of $\mathcal{W}_{S^1}\!(X)_{-1}$ (c.f. Remark 1.11 again).

Note that the two functions $\breve {W}$ and $\sigma$ on the mirror $\breve {X}$ are playing very different roles, and this becomes clearer when we discuss grading. To give $\mathcal{W}(X)$ a $\mathbb{Z}$ -grading, we must choose a grading structure, which is a homotopy class of sections of $LGr(TX)$ . On the mirror, this usuallyFootnote ⁸ corresponds to a choice of ‘R-charge’, that is, a $\mathbb{C}^*$ action on $\breve {X}$ for which $\breve {W}$ has weight 2. The function $\sigma$ on the other hand must have weight zero.

The presence of R-charge alters the grading on Hochschild cohomology, thereby destroying the equality (1.13), even in the case $\breve {W}=0$ . So in principle, the mirror to $S^1\curvearrowright X$ might be a $\sigma$ with some higher-order terms given by polyvector fields. But we will not encounter any such examples in this paper.

1.4. Recovering the non-equivariant category

Given $S^1\curvearrowright X$ , we have for each $\lambda \in \mathbb{C}^*$ a spectral component $\mathcal{W}_{S^1}\!(X)_\lambda$ of the equivariant Fukaya category. These categories have some important extra structure; they are linear over the ground ring of $S^1$ -equivariant cohomology:

\begin{equation*}H^\bullet _{S^1}(pt) = \mathbb{C}[t], \quad \deg t =2 \end{equation*}

This structure is built into the construction, and all $A_\infty$ structure maps respect it. It is therefore possible to take the fibre of $\mathcal{W}_{S^1}\!(X)_\lambda$ at $t=0$ . The result is a subcategory

\begin{equation*}\mathcal{W}_{S^1}\!(X)_\lambda |_{t=0} \subset \mathcal{W}(X) \end{equation*}

of the ordinary wrapped Fukaya category of $X$ ; it is the full subcategory of objects $L$ with $s_{L}=\lambda 1_{L}$ .

It seems reasonable to expect that $\mathcal{W}_{S^1}\!(X)_\lambda$ is generated by invariant Lagrangians, provided that we include the non-compact ones. Moreover, in some examples $\mathcal{W}(X)$ itself can be generated by invariant Lagrangians; this happens, for instance, in Example 1.1 since the cotangent fibre at one of the two fixed points is invariant and generates $\mathcal{W}(T^*S^2)$ . In this situation, it may be possible to recover the non-equivariant category $\mathcal{W}(X)$ from a single spectral component $\mathcal{W}_{S^1}\!(X)_\lambda$ of the equivariant category.

Now suppose we have the set-up of Conjecture A, with (for simplicity) $r=1$ . So we have a rank 1 algebraic torus fibration $\pi \;:\; X \to Y$ degenerating over a divisor $D\subset Y$ . The conjecture is that:

\begin{equation*} \mathcal{W}_{S^1}\!(X)_{-1} \cong \mathcal{W}(Y\setminus D)\end{equation*}

Since the category on the left is linear over $\mathbb{C}[t]$ , the category on the right should be too.

There is an obvious guess for what this extra structure on $\mathcal{W}(Y\setminus D)$ is. Indeed, there is a class

\begin{equation*}\tau \in SH^2(Y \setminus D)\end{equation*}

corresponding to a simple Reeb orbit going around the divisor $D$ once. It is sometimes called the Borman-Sheridan class [Reference Tonkonog32]. With this choice of $\tau$ , the category $\mathcal{W}(Y\setminus D)$ becomes linear over $\mathbb{C}[t]$ .Footnote ⁹

If correct, this implies that the fibre $\mathcal{W}(Y\setminus D)|_{t=0}$ agrees with the subcategory of $\mathcal{W}(X)$ generated by objects $L$ such that $s_L = -\!\operatorname{1}_L$ . If this is the whole of $\mathcal{W}(X)$ – which looks roughly equivalent to asking that $\mathcal{W}(X)$ is generated by invariant Lagrangians – then it follows that:

(1.19) \begin{equation} \mathcal{W}(Y\setminus D)|_{t=0}\;\cong \; \mathcal{W}(X) \end{equation}

We expect this to hold whenever the wrapped category of $Y$ is zero, which will be true if $Y$ is a subcritical Weinstein manifold such as $\mathbb{C}^n$ . In particular, it should work for Example 1.1. In the next section, we’ll discuss this in more detail, including the mirror picture.

When it holds, the equivalence (1.19) opens a route to proving some new instances of homological mirror symmetry, by bootstrapping up from a theorem about $Y$ to a theorem about $X$ .

Remark 1.20. In our discussion after Conjecture A, we noted that the deformation

\begin{equation*} \mathcal{W}_{S^1}\!(X\setminus \pi ^{-1} D)_{-1} \;\leadsto \; \mathcal{W}_{S^1}\!(X)_{-1} \end{equation*}

must be trivial. But now we can refine this: the $A_\infty$ -structure is not deforming, but the $\mathbb{C}[t]$ structure is. On the quotient, we are starting with the trivial $\mathbb{C}[t]$ structure on $\mathcal{W}(Y \setminus D)$ and deforming it to the non-trivial one where $t$ acts as $\tau$ . The class $\tau$ itself relates to the formal deformation of $\mathcal{W}(Y \setminus D)$ to the relative category $\mathcal{W}(Y,D)$ , which is the subject of our next section.

Remark 1.21. A more precise version of “Conjecture” C would also include $\mathbb{C}[t]$ structures. If the mirror to $S^1\curvearrowright X$ is $\sigma \;:\; \breve {X} \to \mathbb{C}^*$ , then each category $D^b(\breve {Z}_\lambda )$ is automatically linear over $\mathbb{C}[t]$ , because the presentation of $\breve {Z}_\lambda$ as a hypersurface provides a choice of $t\in HH^2(\breve {Z}_\lambda )$ (it’s the class of the deformation of $\breve {Z}_{\lambda }$ that we get by varying $\lambda$ to first order). This should agree with the $\mathbb{C}[t]$ structure on $\mathcal{W}_{S^1}\!(X)_\lambda$ .

If the mirror is an LG model, then we again have an automatic choice of

\begin{equation*}t\in HH^2\big (\textrm {MF}(\breve {Z}_\lambda , \breve {W}|_{\breve {Z}_\lambda })\big )\end{equation*}

but now $t$ might be recording that the superpotential $\breve {W}|_{\breve {Z}_\lambda }$ is varying with $\lambda$ , rather than the hypersurface itself. Or they might both be deforming.

1.5. Relative Fukaya categories

Suppose again we have the set-up of Conjecture A with $r=1$ . On the base space $Y$ , we can consider the relative wrapped Fukaya category:

\begin{equation*}\mathcal{W}(Y, D)\end{equation*}

This category is, by construction, linear over a power series ring $\mathbb{C}[[h]]$ where $\deg h = 0$ . The relative Fukaya category was first considered by Seidel in [Reference Seidel27]. The objects are exact Lagrangians in $Y \setminus D$ , which may be non-compact but must stay away from $D$ ; that is, they should not asymptotically approach to $D$ .

Setting $h=0$ in the relative category recovers the full subcategory of $\mathcal{W}(Y\setminus D)$ spanned by these same Lagrangians. If instead we invert $h$ , we get the ordinary wrapped Fukaya category $\mathcal{W}(Y)$ , with $h$ essentially becoming the Novikov parameter (c.f. [Reference Lekili and Perutz16, Lemma 1.2]). So the relative category captures the ‘total space’ of the formal deformation with central fibre $\mathcal{W}(Y\setminus D)$ and generic fibres $\mathcal{W}(Y)$ .

Now consider the space $X$ . The $S^1$ action makes $\mathcal{W}(X)$ linear over the ring $\mathbb{C}[s^{\pm 1}]$ . At a generic value $\lambda$ of $s$ , one expects that $\mathcal{W}_{S^1}\!(X)_\lambda$ is equivalent to $\mathcal{W}(Y)$ , possibly with a bulk deformation. At $\lambda =-1$ , our Conjecture A is that $\mathcal{W}_{S^1}\!(X)_{-1} \cong \mathcal{W}(Y\setminus D)$ .

Comparing the previous two paragraphs, we reach the following conclusion:

If we express this in terms of mirror symmetry as in Section 1.3, then we are claiming the following. Suppose $X$ is mirror to $(\breve {X}, \breve {W})$ and the $S^1$ action is mirror to a function $\sigma :\breve {X} \to \mathbb{C}^*$ . Then $\mathcal{W}(Y, D)$ will be equivalent to the category of matrix factorizations on the formal scheme obtained by completing $\breve {X}$ along the divisor $\breve {Z}_{-1}= \sigma ^{-1}(\!-\!1)$ .

Perhaps the most interesting situation for this conjecture is the case when $\mathcal{W}(Y)\cong 0$ , as we discussed in the previous section. In this case, we expect from (1.7) that $\lambda =-1$ is the only non-zero spectral component of $\mathcal{W}_{S^1}\!(X)$ .

On the mirror side, this is the statement that

\begin{equation*}\textrm {MF}\!\big (\breve {Z}_{\lambda }, \breve {W}|_{\breve {Z}_{\lambda }}\big )\cong 0 \quad \quad \mbox{for all } \lambda \neq -1\end{equation*}

which says that the restriction of $\breve {W}$ to $\breve {Z}_\lambda$ has no critical points if $\lambda \neq -1$ . It follows that $\textrm {Crit}(\breve {W})$ is contained (set-theoretically) in $\breve {Z}_{-1}$ , and hence completing $\breve {X}$ at $\breve {Z}_{-1}$ has no effect on the category of matrix factorizations.Footnote ¹⁰ Equivalently, the subcategory $\textrm {MF}_{\breve {Z}_{-1}}(\breve {X}, \breve {W})$ of matrix factorizations supported on $\breve {Z}_{-1}$ is in fact the whole of $\textrm {MF}(\breve {X}, \breve {W})$ .

Returning to the A-side, we conclude that if $\mathcal{W}(Y)\cong 0$ , then completing $\mathcal{W}(X)$ at $s=-1$ has no effect. Equivalently, $\mathcal{W}(X)$ is generated by objects $L$ such that $s_L= -\operatorname{1}_L$ . So (1.19) holds, and Conjecture E becomes the claim that $\mathcal{W}(Y,D)$ and $\mathcal{W}(X)$ are equivalent.

To understand this claim in more detail, we draw the following diagram:

(1.25)

Here the categories along the top row are all supposed to be equivalent, as are the three categories along the bottom. The vertical arrows are an adjunction.

On the B-side, the adjunction is obvious; it’s given by pushing forward or pulling back along the inclusion $i\;:\; \breve {Z}_{-1}\hookrightarrow \breve {X}$ . On the symplectic manifold $X$ , only the upward functor is obvious – it’s the functor from Section 1.4, which takes an invariant Lagrangian in $\mathcal{W}_{S^1}(X)_{-1}$ and regards it as an object of $\mathcal{W}(X)$ instead. The adjoint to this functor is less clear.

Curiously, on the base $Y$ , it is the downward functor, which is obvious – just set the deformation parameter $h$ to zero – and the upward functor, which is a little more surprising.

Our conjectured equivalence $G\;:\; \mathcal{W}(Y\setminus D) \stackrel {\sim }{\longrightarrow } \mathcal{W}_{S^1}\!(X)_{-1}$ should roughly be given by the Lagrangian correspondence $\Gamma$ associated to $\mu ^{-1}(0)$ ; see (1.5). So, at least approximately, it sends a Lagrangian $\Lambda$ to the invariant Lagrangian $\pi ^{-1}(\Lambda )\cap \mu ^{-1}(0)$ .

On the top line, the functor $F\;:\; \mathcal{W}(X) \stackrel {\sim }{\longrightarrow } \mathcal{W}(Y, D)$ needs to be adjoint to $G$ , so it’s given by the same correspondence $\Gamma$ but viewed as a functor in the opposite direction. To apply this functor to a Lagrangian $L\subset X$ , we must perturb until $L$ intersects $\mu ^{-1}(0)$ transversely, and then we project $L\cap \mu ^{-1}(0)$ down to $Y$ .

Example 1.27. Revisiting Example 1.1, Conjecture E claims that we have an equivalence:

(1.28) \begin{equation} \mathcal{W}(T^*S^2) \cong \mathcal{W}(\mathbb{C}, \{\pm 1\}) \end{equation}

In this example, we can compute the effects of all functors in (1.25) quite explicitly.

Take a straight arc in $\mathbb{C}$ from either puncture to infinity, that is, either $\Lambda _2$ or $\Lambda _3$ in Figure 2. Under $G$ , they lift to the cotangent fibres at the two fixed points. These objects are not isomorphic in $\mathcal{W}_{S^1}\!(X)_{-1}$ , but they become isomorphic when we pass to $\mathcal{W}(X)$ since all cotangent fibres are isomorphic in that category. If we want to now apply the functor $F$ , we must take a cotangent fibre at a generic point so that it intersects $\mu ^{-1}(0)$ transversely, and the result is the arc $\Lambda _4$ . So the ‘surprising’ functor $\mathcal{W}(Y\setminus D)\to \mathcal{W}(Y, D)$ appearing in the third column of (1.25) sends both $\Lambda _2$ and $\Lambda _3$ to $\Lambda _4$ .

As a test of our claimed equivalence (1.28), we can compute $\textrm {Hom}_{\mathcal{W}(Y,D)}(\Lambda _4,\Lambda _4)$ directly using the wrapped Floer complex. The result is

\begin{equation*} \frac {\mathbb{C}\langle u,v\rangle [h]}{(u^2,v^2)}, \ \ \ du = h\cdot 1, dv = h \cdot 1, \ \ |u|=|v|=-1 \end{equation*}

and it is easy to see that this is quasi-isomorphic to $\mathbb{C}[x]$ with $|x|=-1$ , which is indeed the endomorphism algebra of a cotangent fibre in $\mathcal{W}(T^*S^2)$ .

If we apply $G$ to the arc $\Lambda$ from Example 1.1, then we get $S^2$ , as an object of $\mathcal{W}_{S^1}\!(X)_{-1}$ . But applying $F$ to $S^2 \in \mathcal{W}(X)$ does not produce $\Lambda$ ; no arc ending at $1$ or $-1$ can be in the image of $F$ . In fact, $F(S^2)$ is a figure-0eight encircling both punctures.

Figure 2.

Lagrangians in $\mathbb{C} \setminus \{\pm 1\}$ .

Remark 1.29. The ‘surprising’ functor $\mathcal{W}(Y\setminus D) \to \mathcal{W}(Y, D)$ formally is given by setting the Borman-Sheridan class $\tau$ to zero. We can get a more explicit understanding by passing through the mirror.

On the B-side, the composition $i_*i_*$ acts on objects as

\begin{equation*}i^*i_*\!:\; \mathcal{E} \;\mapsto \; \big [ \mathcal{E} \stackrel {t_{\mathcal{E}}}{\longrightarrow } \mathcal{E}[2] \big ]\end{equation*}

where $t_{\mathcal{E}}$ is the $HH^2$ class from Remark 1.21 evaluated on the object $\mathcal{E}$ . On the A-side, this corresponds to taking a Lagrangian $L \subset Y\setminus D$ and forming the twisted complex

\begin{equation*} \big [ L \stackrel {\tau _L}{\longrightarrow } L[2] \big ] \end{equation*}

where $\tau _L$ is the image of $\tau$ under the closed-open map $\mathcal{CO}^0 \;:\; SH^*(\mathcal{W}(Y \setminus D)) \to HF(L,L)$ . But we claim that this twisted complex can be represented by an object $\tilde {L}$ , which does not asymptote to $D$ , so it can be viewed as object of the relative category $\mathcal{W}(Y, D)$ . On the level of objects, we believe this is our functor. It’s easy to see that this consistent with our discussion of the objects $\Lambda , \Lambda _2$ and $\Lambda _3$ in the example above.

Note that if $L$ does not have an end converging to $D$ , then we should have $\tau _L=0$ , and so the result of applying the functor should be just $L \oplus L[1]$ . On the mirror, this is what happens when $\mathcal{E}=i^*\mathcal F$ for some $\mathcal F\in \textrm {MF}(\breve {X}, \breve {W})$ .

Remark 1.30. We could also consider the relative category:

\begin{equation*}\mathcal{W}(X, \pi ^{-1} D)\end{equation*}

This will be linear over $\mathbb{C}[[h]]$ , and it should also have a $\mathbb{C}[s, s^{-1}]$ structure coming from the $S^1$ action. We believe that these two structures are essentially the same, and this encapsulates many of our claims. See Example 2.3 for an example (on the mirror side) of what we mean by this.

1.6. Topological $S^1$ actions on categories and the KRS 2-category

The structures that we have encountered can be understood more clearly using the framework of ‘topological group actions’ on categories sketched by Teleman in [Reference Teleman30]. In this section, we’ll give a brief explanation of these ideas, restricting only to the case of topological $S^1$ actions. The generalisation to actions of higher-rank tori is easy; the non-abelian case is considerably deeper, but we won’t need it.

To first approximation, a topological $S^1$ action on a category $\mathcal{C}$ is a choice of natural automorphism $s\;:\; \operatorname{1}_{\mathcal{C}} \to \operatorname{1}_{\mathcal{C}}$ of the identity functor, or equivalently a $\mathbb{C}[s, s^{-1}]$ structure on $\mathcal{C}$ . We saw in Section 1.3 that the Seidel element provides this structure on $\mathcal{C}=\mathcal{W}(X)$ when $X$ is a Hamiltonian $S^1$ -manifold. On the mirror side, a non-vanishing function on $\breve {X}$ provides this structure on $\mathcal{C}=D^b(\breve {X})$ .

A heuristic general explanation for this definition goes as follows. Imagine that $\mathcal{C}$ is a topological category (or $\infty$ -category) with a space of objects $Ob_{\mathcal{C}}$ . Then the space of automorphisms of any object $x\in Ob_{\mathcal{C}}$ is, by definition, the based loop space $\Omega _xOb_{\mathcal{C}}$ . A ‘topological $S^1$ action’ means an action $S^1 \curvearrowright Ob_{\mathcal{C}}$ , and then for any object $x$ , we have an automorphism $s_x:x\stackrel {\sim }{\longrightarrow } x$ given by the orbit of $x$ under the $S^1$ action.

If we have a topological $S^1$ action on $\mathcal{C}$ , we can then ask about the equivariant category. A natural definition of an ‘equivariant structure’ on an object $x$ is a choice of contracting homotopy $h$ of the loop $s_x$ (this is analogous to the definition of an equivariant sheaf). Such an $h$ induces – up to homotopy – an $S^1$ action on $\Omega _xOb_{\mathcal{C}}$ , and then the automorphism space of $(x,h)$ in the equivariant category will be something like the homotopy quotient $\Omega _xOb_{\mathcal{C}}\times _{S^1} ES^1$ . Hence, if we linearise by replacing spaces with their cohomology, we expect the equivariant category to be linear over $H^\bullet (BS^1)=\mathbb{C}[t]$ .

More algebraically, this discussion suggests that the equivariant category should be the fibre of $\mathcal{C}$ at the point $s=1$ . And indeed, the fibre category is always linear over $\mathbb{C}[t]$ .Footnote ¹¹ More generally, we could take the fibre at some value $s=\lambda$ and interpret this as a ‘twisted equivariant category’. This looks like it could be the correct framework for the equivariant Fukaya categories $\mathcal{W}_{S^1}\!(X)_\lambda$ and the mirror symmetry picture from Section 1.3.

Unfortunately (as we warned in Remark 1.16), this story is too simplistic.

That example cannot be $\mathbb{Z}$ -graded, but the next one can.

On the mirror side, these two examples are quite similar: the mirror superpotential $\breve {W}$ has no critical points, but the restriction of $\breve {W}$ to a hypersurface $\breve {Z}_\lambda$ does have critical points.Footnote ¹² So $\textrm {MF}(\breve {W})\cong 0$ but $\textrm {MF}(\breve {W}|_{\breve {Z}_\lambda })\ncong 0$ . This is obviously a general phenomenon and illustrates sharply why $\textrm {MF}(\breve {W}|_{\breve {Z}_\lambda })$ cannot be a fibre of $\textrm {MF}(\breve {W})$ .

Teleman’s solution to this puzzle uses the Kapustin-Rozansky-Saulina 2-category. This is a conjectural construction from 3D topological field theory (Rozansky-Witten theory) that associates a 2-category to a holomorphic symplectic manifold $M$ .

An object of this 2-category ${K\!RS}(M)$ is a choice of holomorphic Lagrangian $L\subset M$ , together with a category $\mathcal{C}_L$ , which is linear over $L$ , that is, which carries an action of the tensor category $Coh(L)$ . In particular, each $L$ defines an object by setting $\mathcal{C}_L = Coh(L)$ . These special objects should generate ${K\!RS}(M)$ , so using the Yoneda embedding, we can view a general object $\mathfrak{X}$ as an operation

\begin{equation*}L \mapsto \textrm {Hom}_{{K\!RS}(M)}(L, \mathfrak{X})\end{equation*}

which associates a (1-)category to each Lagrangian. We need this perspective for our version of equivariant mirror symmetry (see below). It also has the advantage that we can immediately handle objects $\mathfrak{X}$ supported on singular Lagrangians, by viewing them as an operation on smooth Lagrangians.

For topological $S^1$ actions, the relevant holomorphic symplectic manifold is the shifted cotangent bundle:

\begin{equation*}M = T^\vee [\!-\!2]\mathbb{G}_m = \textrm {Spec} \mathbb{C}[s^{\pm 1}, t] \end{equation*}

In the $\mathbb{Z}_2$ -graded world, this is symplectic. In the $\mathbb{Z}$ -graded world, it is shifted-symplectic; the variable $t$ has degree 2.

The latter is much easier because there are very few (smooth) holomorphic Lagrangians. Indeed, we can interpret the grading as a $\mathbb{C}^*$ action on $T^\vee \mathbb{G}_m$ , which scales the fibres, and then the only $L$ ’s that are invariant under this action are the zero section and the cotangent fibres themselves. Moreover, these Lagrangians intersect each other in a very simple way. So in the $\mathbb{Z}$ -graded world, an object of ${K\!RS}(T^\vee [\!-\!2]\mathbb{G}_m)$ should be the following data:

1. (i) A $\mathbb{C}[s^{\pm 1}]$ -linear category $\mathcal{C}$ (this is the category associated to the zero section).

2. (ii) A $\mathbb{C}[t]$ -linear category $\mathcal{D}_\lambda$ for each $\lambda \in \mathbb{C}^*$ (this is the category associated to the cotangent fibre at $\lambda$ ).

3. (iii) For each $\lambda$ , an embedding

   \begin{equation*}\mathcal{C}|_\lambda \;\hookrightarrow \; \mathcal{D}_\lambda \end{equation*}
   where the source is the fibre of $\mathcal{C}$ at $s=\lambda$ , and the image is the subcategory of $\mathcal{D}_\lambda$ consisting of objects supported at $t=0$ .

4. (iv) For each $\lambda$ , an embedding

   \begin{equation*}(\mathcal{D}_\lambda )|_0 \;\hookrightarrow \; \mathcal{C}\end{equation*}
   where the source is the fibre of $\mathcal{D}_\lambda$ at $t=0$ and the image is the subcategory of $\mathcal{C}$ consisting of objects supported at $s=\lambda$ .

One of Teleman’s insights is that a ‘category with a topological $S^1$ -action’ is really an object of ${K\!RS}(T^\vee [\!-\!2]\mathbb{G}_m)$ . In the $\mathbb{Z}$ -graded world, we can view it as a system of categories as described above.

On the B-side, we get an example by taking a smooth variety $X$ with a function $\sigma \;:\; X \to \mathbb{C}^*$ and setting

\begin{equation*}\mathcal{C}=D^b(X) \quad \quad \mbox{and}\quad \quad \mathcal{D}_\lambda = D^b(Z_\lambda )\end{equation*}

where $Z_\lambda = \sigma ^{-1}(\lambda )$ . Here (iii) becomes the fact that the fibre of $D^b(X)$ at $s=\lambda$ is $\textrm {Perf}(Z_\lambda )$ , which is also the subcategory of $D^b(Z_{\lambda })$ on which $t$ acts nilpotently. And (iv) is the fact that setting $t=0$ in $D^b(Z_\lambda )$ recovers the subcategory $D^b_{Z_\lambda }\!(X)\subset D^b(X)$ of objects supported on $Z_\lambda$ .

More generally, we can take an LG model $(X, W)$ , with R-charge, equipped with a function $\sigma \;:\; X \to \mathbb{C}^*$ . Then we set $\mathcal{C}=\textrm {MF}(X, W)$ and $\mathcal{D}_\lambda = \textrm {MF}(Z_\lambda , W|_{Z_\lambda })$ .

On the A-side, we should get an example by taking a graded Hamiltonian $S^1$ -manifold $X$ and setting:

\begin{equation*}\mathcal{C} = \mathcal{W}(X)\quad \quad \mbox{and}\quad \quad \mathcal{D}_\lambda =\mathcal{W}_{S^1}\!(X)_\lambda \end{equation*}

If all fibres of the moment map are regular, we should be able to replace each $\mathcal{W}_{S^1}\!(X)_\lambda$ by

\begin{equation*}\mathcal{W}(X{/\!/}_{\log |\lambda |} S^1, \,\arg (\pm \lambda ))\end{equation*}

as discussed in Section 1.2. In the situation of Conjecture A, we should get an example if we set $\mathcal{C}= \mathcal{W}(X)$ , set $\mathcal{D}_\lambda = \mathcal{W}(Y)$ for $\lambda \neq -1$ , and set $\mathcal{D}_{-1}= \mathcal{W}(Y\setminus D)$ .

Remark 1.34. If we are $\mathbb{Z}$ -graded, and each category $\mathcal{D}_\lambda$ is smooth and proper, then $t$ must be torsion (since $\deg (t)=2$ ). It follows that (iii) must be an equivalence $\mathcal{C}|_\lambda \stackrel {\sim }{\longrightarrow } \mathcal{D}_\lambda$ . In this situation, the category $\mathcal{C}$ determines everything, and the simpler story described earlier in this section is enough (c.f. [Reference Teleman30, Remark 3.2]). In the KRS 2-category, we just have the category $\mathcal{C}$ , supported on the Lagrangian $\{t=0\}$ .

But Example 1.32 demonstrates that the simpler story fails under just the assumption of $\mathbb{Z}$ -grading alone. In that example, we have $\mathcal{D}_{-1} = \mathcal{W}(\mathbb{C}^*)$ , and we claim that the $\mathbb{C}[t]$ structure is just the co-ordinate function, that is,

\begin{equation*}\mathcal{D}_{-1} \cong \textrm {Spec} \mathbb{C}[t^{\pm 1}]\end{equation*}

(see Example 2.3). So $\mathcal{D}_{-1}\ncong 0$ , but it contains no objects supported at $t=0$ . This object of ${K\!RS}(T^\vee [\!-\!2]\mathbb{G}_m)$ is supported on the incomplete Lagrangian $\{s=-1, t\neq 0\}$ .

If we leave the $\mathbb{Z}$ -graded world, we see that the “the equivariant Fukaya category” should really be a construction that associates a category to any Lagrangian $L\subset T^\vee \mathbb{G}_m$ (with the spectral components just being the special case when $L$ is the cotangent fibre). It would be very interesting to understand this construction.