3.1 Bridgeland stability on D b ℱ ( M ) for M Calabi–Yau [03NM]

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3.1 Bridgeland stability on $D^{b}{\mathbin{\mathscr{F}}}(M)$ for $M$ Calabi–Yau

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold, with Kähler form $\omega$ , so that $(M,\omega)$ is a symplectic Calabi–Yau manifold. As in §2.5, we will consider the derived Fukaya category $D^{b}{\mathbin{\mathscr{F}}}(M)$ of $M$ , in the sense of Fukaya, Oh, Ohta and Ono [18, 20]. Objects of $D^{b}{\mathbin{\mathscr{F}}}(M)$ include triples $(L,E,b)$ , where $L$ is a compact, spin, graded Lagrangian in $M$ and $E\rightarrow L$ a rank one ${\mathbin{\mathbb{F}}}$ -local system such that $(L,E)$ has $HF^{*}$ unobstructed, and $b$ is a bounding cochain for $(L,E)$ .

Note in particular that not every compact, graded Lagrangian $L$ or brane $(L,E)$ yields an object of $D^{b}{\mathbin{\mathscr{F}}}(M)$ , but only those $(L,E)$ with $HF^{*}$ unobstructed. One of our themes will be that we expect Lagrangians $L$ with $HF^{*}$ unobstructed to be better-behaved from the point of view of Lagrangian MCF.

We hope to use special Lagrangians and Lagrangian MCF in $(M,J,g,\Omega)$ to define an additional structure on the triangulated category $D^{b}{\mathbin{\mathscr{F}}}(M)$ , a stability condition in the sense of Bridgeland [10] (see also Huybrechts [30]):

Definition 3.1.

Let ${\mathbin{\cal T}}$ be a triangulated category. A (Bridgeland) stability condition $(Z,{\mathbin{\cal P}})$ on ${\mathbin{\cal T}}$ consists of a group homomorphism $Z:K_{0}({\mathbin{\cal T}})\rightarrow{\mathbin{\mathbb{C}}}$ called the central charge, and full additive subcategories ${\mathbin{\cal P}}(\phi)\subset{\mathbin{\cal T}}$ for each $\phi\in{\mathbin{\mathbb{R}}}$ , satisfying the following properties:

(i)

If $A\in{\mathbin{\cal P}}(\phi)$ then $Z([A])=m(A)e^{i\pi\phi}$ for some $m(A)>0$ .
(ii)

For all $\phi\in{\mathbin{\mathbb{R}}}$ , ${\mathbin{\cal P}}(\phi+1)={\mathbin{\cal P}}(\phi)[1]$ .
(iii)

If $\phi_{1}>\phi_{2}$ and $A_{j}\in{\mathbin{\cal P}}(\phi_{j})$ then $\mathop{\rm Hom}\nolimits_{\mathbin{\cal T}}(A_{1},A_{2})=0$ .

(iv)

For each nonzero object $F\in{\mathbin{\cal T}}$ there is a finite sequence of real numbers $\phi_{1}>\phi_{2}>\cdots>\phi_{n}$ and a diagram in ${\mathbin{\cal T}}$

where the triangles are distinguished and $A_{j}\in{\mathbin{\cal P}}(\phi_{j})$ for $j=1,\ldots,n$ .

Objects in ${\mathbin{\cal P}}(\phi)$ for some $\phi\in{\mathbin{\mathbb{R}}}$ are called semistable.

The following conjecture extending Thomas [69] (perhaps excluding (c),(c $)^{\prime}$ ?) is folklore, known for years in some form to many in the Geometry and String Theory communities, and is mentioned briefly in Bridgeland [10, §1.4].

Conjecture 3.2.

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold, either compact or suitably convex at infinity, and $D^{b}{\mathbin{\mathscr{F}}}(M)$ the derived Fukaya category of $M$ in the sense of [18, 20]. Then there exists a natural Bridgeland stability condition $(Z,{\mathbin{\cal P}})$ on $D^{b}{\mathbin{\mathscr{F}}}(M)$ such that:

(a)

The central charge $Z$ is the composition of the natural maps

(3.1)

(b)

If $(L,E,b)\in D^{b}{\mathbin{\mathscr{F}}}(M)$ with $L$ special Lagrangian of phase $e^{i\pi\phi},$ so that $L$ has constant phase function $\theta_{L}=\pi\phi,$ then $(L,E,b)\in{\mathbin{\cal P}}(\phi)$ .
(c)

(Dubious, probably false as stated.) Suppose we enlarge the definition of $D^{b}{\mathbin{\mathscr{F}}}(M)$ so that it contains ‘as many Lagrangians $L$ as possible for which $HF^{*}$ can be defined’, including immersed Lagrangians as in §2.6, and some classes of singular Lagrangians. Then every isomorphism class of objects in ${\mathbin{\cal P}}(\phi)$ for any $\phi\in{\mathbin{\mathbb{R}}}$ contains a unique representative $(L,E,b)$ with $L$ a (possibly immersed or singular) special Lagrangian of phase $e^{i\pi\phi}$ .

Part (c) requires the inclusion of badly singular Lagrangians in $D^{b}{\mathbin{\mathscr{F}}}(M),$ which may not be feasible. Here is an alternative which may work with $D^{b}{\mathbin{\mathscr{F}}}(M)$ containing only more mildly singular Lagrangians:

(c $\boldsymbol{)}{}^{\prime}$

(Still dubious.) Suppose we enlarge $D^{b}{\mathbin{\mathscr{F}}}(M)$ so that it contains ‘sufficiently many Lagrangians $L$ for which $HF^{*}$ can be defined’, including immersed and some singular Lagrangians. Then for any $\epsilon>0$ and $\phi\in{\mathbin{\mathbb{R}}},$ every isomorphism class of objects in ${\mathbin{\cal P}}(\phi)$ contains a representative $(L,E,b)$ whose phase function $\theta_{L}$ maps $\theta_{L}:L\rightarrow(\pi\phi-\epsilon,\pi\phi+\epsilon)$ .

Remark 3.3.

(i) The enlargement of $D^{b}{\mathbin{\mathscr{F}}}(M)$ envisaged in (c),(c $)^{\prime}$ adds more objects to $D^{b}{\mathbin{\mathscr{F}}}(M)$ , but it need not change $D^{b}{\mathbin{\mathscr{F}}}(M)$ up to equivalence.

An example of the kind of enlargement the author has in mind is including immersed Lagrangians in $D^{b}{\mathbin{\mathscr{F}}}(M)$ , as in §2.6. We have embedded and immersed derived Fukaya categories $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm em}\subset D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm im}$ , but if every immersed Lagrangian $(L,E,b)$ in $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm im}$ is equivalent to a twisted complex of embedded Lagrangians, then $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm em}\simeq D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm im}$ .

For many applications in symplectic topology, one only really cares about $D^{b}{\mathbin{\mathscr{F}}}(M)$ up to equivalence, so adding extra geometric objects to $D^{b}{\mathbin{\mathscr{F}}}(M)$ in this way is unnecessary. But for Conjecture 3.2(c),(c $)^{\prime}$ , it is vital — if an isomorphism class in ${\mathbin{\cal P}}(\phi)$ contains a unique special Lagrangian representative $(L,E,b)$ , and $L$ happens to be immersed, then restricting to embedded Lagrangians would make Conjecture 3.2(c) false. Similarly, we will see that the programme of long-time existence for Lagrangian MCF we outline below must take place in an enlarged category of Lagrangians to have any chance of working.

(ii) The uniqueness of $(L,E,b)$ in its isomorphism class in Conjecture 3.2(c), provided it exists, should be proved as in Thomas and Yau [70, Th. 4.3].

Note however that Thomas and Yau’s method does not exclude the possibility that $L^{\prime}\rightarrow L$ and $L^{\prime\prime}\rightarrow L$ are non-isomorphic $k$ -fold multiple covers of a non-simply-connected special Lagrangian $L$ in $M$ for $k>1$ , with $(L^{\prime},E^{\prime},b^{\prime})\cong(L^{\prime\prime},E^{\prime\prime},b^{\prime\prime})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ . A good uniqueness statement in Conjecture 3.2(c) may be that the special Lagrangian integral current in Geometric Measure Theory induced by $L$ is unique, so that in the case above the special Lagrangian integral currents of both $L^{\prime},L^{\prime\prime}$ would be $kL$ .

(iii) There may be a way to construct the expected Bridgeland stability conditions on $D^{b}{\mathbin{\mathscr{F}}}(M)$ in examples (though initially without proving that semistable objects are represented by special Lagrangians) using Mirror Symmetry.

Kontsevich’s Homological Mirror Symmetry Conjecture [44] roughly says that Calabi–Yau $m$ -folds should exist in ‘mirror pairs’ $M,\check{M}$ for which there should be an equivalence of triangulated categories

D^{\pi}{\mathbin{\mathscr{F}}}(M)\simeq D^{b}\mathop{\rm coh}(\check{M}),

(3.2)

where $D^{b}\mathop{\rm coh}(\check{M})$ is the derived category of coherent sheaves on $\check{M}$ . (Really $\check{M}$ should be defined over the Novikov ring $\Lambda_{\rm nov}$ .)

Kontsevich [44] proved (3.2) when $M$ is an elliptic curve (a Calabi–Yau 1-fold). Seidel [63] proved it for $M$ a quartic surface in ${\mathbin{\mathbb{CP}}}^{3}$ (a Calabi–Yau 2-fold), and Sheridan [65] proved it for $M$ a smooth Calabi–Yau $m$ -fold hypersurface in ${\mathbin{\mathbb{CP}}}^{m+1}$ for $m\geqslant 3$ . If (3.2) holds then stability conditions on $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ are equivalent to stability conditions on $D^{b}\mathop{\rm coh}(\check{M})$ . But derived categories of coherent sheaves are generally better understood than derived Fukaya categories.

Bridgeland stability conditions on $D^{b}\mathop{\rm coh}(M)$ are defined by Bridgeland [10, Ex. 5.4] for $M$ a Calabi–Yau 1-fold and [11] for $M$ an algebraic $K3$ surface (a Calabi–Yau 2-fold). Assuming a conjecture on ‘Bogomolov–Gieseker type inequalities’, Bayer, Macrì and Toda [7] construct Bridgeland stability conditions on $D^{b}\mathop{\rm coh}(M)$ for $M$ a Calabi–Yau 3-fold; the conjecture is proved by Macioca and Piyaratne [48, 49] when $M$ is an abelian 3-fold.

Combining the two, one may be able to construct examples of Bridgeland stability conditions on $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ for $M$ a Calabi–Yau 1-fold, 2-fold or 3-fold.

The next definition and conjecture give an alternative formulation of stability which is much closer to Thomas’ definition [69, Def. 5.1]:

Definition 3.4.

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold, either compact or suitably convex at infinity, and $D^{b}{\mathbin{\mathscr{F}}}(M)$ the derived Fukaya category of $M$ , enlarged as in Conjecture 3.2 to include immersed Lagrangians, and maybe also some classes of singular Lagrangians. As in Remark 2.24, we may take all objects in $D^{b}{\mathbin{\mathscr{F}}}(M)$ to be of the form $(L,E,b)$ , we do not need twisted complexes.

Suppose $\alpha\in{\mathbin{\mathbb{R}}}$ is such that $[\Omega]\cdot[L]\notin e^{i\pi\alpha}\cdot(0,\infty)$ for all $(L,E,b)$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ , where $[\Omega]\in H^{m}(M;{\mathbin{\mathbb{C}}})$ and $[L]\in H_{m}(M;{\mathbin{\mathbb{Z}}})$ . As there are only countably many such homology classes $[L]$ , this holds for generic $\alpha\in{\mathbin{\mathbb{R}}}$ . Write ${\mathbin{\cal A}}_{\alpha}$ for the full subcategory of $D^{b}{\mathbin{\mathscr{F}}}(M)$ with objects $(L,E,b)$ such that the phase function $\theta_{L}$ of $L$ maps $L\rightarrow(\pi\alpha,\pi(\alpha+1))$ . Write ${\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ for the full subcategory of objects in $D^{b}{\mathbin{\mathscr{F}}}(M)$ isomorphic to an object of ${\mathbin{\cal A}}_{\alpha}$ , so that ${\mathbin{\cal A}}_{\alpha},{\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ are equivalent categories with ${\mathbin{\cal A}}_{\alpha}\subset{\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}\subset D^{b}{\mathbin{\mathscr{F}}}(M)$ .

We have ${\mathbin{\cal A}}_{\alpha}[1]\!=\!{\mathbin{\cal A}}_{\alpha+1}$ and ${\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}[1]\!=\!{\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha+1}$ . The condition on $\alpha$ is to avoid taking phases in a half-open interval $(\pi\alpha,\pi(\alpha+1)]$ , which could cause problems. If $(L,E,b)\in{\mathbin{\cal A}}_{\alpha}$ , then $L$ is almost calibrated (has phase variation less than $\pi$ ).

Using the almost calibrated condition, we see that every $(L,E,b)\in{\mathbin{\cal A}}_{\alpha}$ has a unique global phase $\phi(L)\in(\pi\alpha,\pi(\alpha+1))$ with $\int_{L}\Omega=Re^{i\phi(L)}$ for $R>0$ , as in Thomas [69, §3]. If $(L^{\prime},E^{\prime},b^{\prime})\in{\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ then $(L^{\prime},E^{\prime},b^{\prime})\cong(L,E,b)$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ for some $(L,E,b)\in{\mathbin{\cal A}}_{\alpha}$ , and $\int_{L^{\prime}}\Omega=\int_{L}\Omega=Re^{i\phi(L)}$ , where $\phi(L)$ is independent of the choice of $(L,E,b)$ . Thus we may define $\phi(L^{\prime})=\phi(L)$ for $(L^{\prime},E^{\prime},b^{\prime})\in{\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ .

In a similar way to Thomas [69, Def. 5.1], we say that a nonzero object $(L,E,b)$ in ${\mathbin{\cal A}}_{\alpha}$ or ${\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ is stable (or semistable) if there is no distinguished triangle

(3.3)

in $D^{b}{\mathbin{\mathscr{F}}}(M)$ with $(L_{1},E_{1},b_{1}),(L_{2},E_{2},b_{2})$ nonzero objects in ${\mathbin{\cal A}}_{\alpha}$ or ${\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ such that $\phi(L_{1})\geqslant\phi(L_{2})$ (or $\phi(L_{1})>\phi(L_{2})$ ).

Conjecture 3.5.

In Definition 3.4, ${\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ is the heart of a bounded t-structure on $D^{b}{\mathbin{\mathscr{F}}}(M),$ and so ${\mathbin{\cal A}}_{\alpha},{\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ are abelian categories, and (3.3) becomes a short exact sequence in ${\mathbin{\cal A}}_{\alpha}$ or ${\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ . Furthermore, the Bridgeland stability condition $(Z,{\mathbin{\cal P}})$ on $D^{b}{\mathbin{\mathscr{F}}}(M)$ in Conjecture 3.2 may be described as follows: $Z$ is defined by (3.1), and ${\mathbin{\cal P}}(\alpha)=\emptyset,$ and for each $\beta\in(\alpha,\alpha+1),$ ${\mathbin{\cal P}}(\beta)$ is the full subcategory of semistable objects $(L,E,b)$ in ${\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ with $\phi(L)=\pi\beta$ .

Note that (semi)stability in Definition 3.4 is equivalent to slope (semi)stability on the (conjecturally abelian) categories ${\mathbin{\cal A}}_{\alpha},{\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ , with slope function

\mu(L,E,b)=\frac{-\cos\pi\alpha\int_{L_{\vphantom{l}}}\mathop{\rm Re}\Omega-\sin\pi\alpha\int_{L}\mathop{\rm Im}\Omega}{-\sin\pi\alpha\int_{L}\mathop{\rm Re}\Omega+\cos\pi\alpha\int_{L}\mathop{\rm Im}\Omega}\,,

since $\phi(L)=\tan^{-1}(\mu(L,E,b))+\pi\alpha+\frac{\pi}{2}$ . Thomas’ analogue of (3.3) is to require $L_{1},L_{2}$ to intersect transversely at one point $p$ , and $L$ to be Hamiltonian isotopic to the Lagrangian connect sum $L_{1}\#L_{2}$ at $p$ . Equation (3.3) is more general, e.g. it does not imply that $L$ is diffeomorphic to $L_{1}\#L_{2}$ . It would be nice to state the relationship between $L$ and $L_{1},L_{2}$ geometrically rather than categorically.

As in §2.5, there are two versions $D^{b}{\mathbin{\mathscr{F}}}(M)\subseteq D^{\pi}{\mathbin{\mathscr{F}}}(M)$ of the derived Fukaya category, where $D^{b}{\mathbin{\mathscr{F}}}(M)$ has objects twisted complexes in ${\mathbin{\mathscr{F}}}(M)$ , and $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ has objects direct summands of objects in $D^{b}{\mathbin{\mathscr{F}}}(M)$ . By Remark 2.22, for immersed Lagrangians we do not need to add twisted complexes, so we can take all objects in $D^{b}{\mathbin{\mathscr{F}}}(M)$ to be of the form $(L,E,b)$ .

We wrote Conjecture 3.2 using $D^{b}{\mathbin{\mathscr{F}}}(M)$ , since the extra objects in $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ are not geometric, and our programme does not make sense for them. For example, the map $K_{0}(D^{b}{\mathbin{\mathscr{F}}}(M))\rightarrow H_{m}(M;{\mathbin{\mathbb{Z}}})$ in (3.1) is not defined for $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ , as we cannot associate a homology class to a direct summand of $(L,E,b)$ .

However, if $D^{b}{\mathbin{\mathscr{F}}}(M)$ has a Bridgeland stability condition, then it has a bounded t-structure, and so by Huybrechts [30, Rem. 1.15] it is idempotent complete. Thus Conjecture 3.2 or Conjecture 3.5 imply:

Conjecture 3.6.

In the situation of Conjecture 3.2, the enlarged version of $D^{b}{\mathbin{\mathscr{F}}}(M)$ with objects $(L,E,b)$ for $L$ a possibly singular, compact, immersed, graded Lagrangian is idempotent complete. Hence $D^{\pi}{\mathbin{\mathscr{F}}}(M)\simeq D^{b}{\mathbin{\mathscr{F}}}(M),$ and we can take all objects of $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ to be geometric, of the form $(L,E,b)$ .

Remark 3.7.

A partial verification of Conjecture 3.6 in the case $M=T^{2}$ is provided by Haug [28]. He defines a version of the derived Fukaya category $D^{b}{\mathbin{\mathscr{F}}}(T^{2})$ in which the objects are twisted complexes built out of pairs $(L,E)$ for $L$ a compact, spin, graded, embedded Lagrangian in $T^{2}$ , and $E\rightarrow L$ a local system, and proves that $D^{b}{\mathbin{\mathscr{F}}}(T^{2})$ is idempotent complete.

Haug remarks [28, §1] that for $T^{2}$ , including local systems $E\rightarrow L$ has the effect of making $D^{b}{\mathbin{\mathscr{F}}}(T^{2})$ idempotent complete, and that $D^{b}{\mathbin{\mathscr{F}}}(T^{2})$ would not be idempotent complete if we took objects to be twisted complexes of Lagrangians $L$ rather than pairs $(L,E)$ . This shows that including local systems $E\rightarrow L$ in objects $(L,E,b)$ is necessary for our programme, since otherwise Conjecture 3.6 and hence Conjecture 3.2 would be false even for $M=T^{2}$ . We will see in §3.4 how nontrivial local systems are needed for some kinds of surgeries.

Haug’s definition of $D^{b}{\mathbin{\mathscr{F}}}(T^{2})$ is not quite the same as ours. He does not include bounding cochains $b$ in his objects $(L,E)$ (the simplicity of dimension 1 permits this). He fixes ${\mathbin{\mathbb{F}}}={\mathbin{\mathbb{C}}}$ . His local systems $E\rightarrow L$ [28, §3.1.1] are not ${\mathbin{\mathbb{F}}}$ -local systems, as in §2.5, but $\Lambda_{\rm nov}$ -local systems of arbitrary finite rank, such that (roughly) the eigenvalues of $\mathop{\rm Hol}\nolimits(\nabla_{E})$ lie in ${\mathbin{\mathbb{F}}}^{*}\subset\Lambda_{\rm nov}^{*}$ to leading order.

I expect this should be related to our definition of $D^{b}{\mathbin{\mathscr{F}}}(T^{2})$ as follows. In dimension 1, the combination of a rank one ${\mathbin{\mathbb{F}}}$ -local system $E\rightarrow L$ and a bounding cochain $b$ is essentially equivalent to a rank one $\Lambda_{\rm nov}$ -local system $E_{\rm nov}\rightarrow L$ satisfying Haug’s condition, where the holonomies satisfy $\mathop{\rm Hol}\nolimits(\nabla_{E_{\rm nov}})[\gamma]=\mathop{\rm Hol}\nolimits(\nabla_{E})[\gamma]\cdot e^{\int_{\gamma}b}$ for $[\gamma]\in\pi_{1}(L)$ . Also, I expect that for $T^{2}$ , considering rank one local systems $E\rightarrow L$ on immersed Lagrangians has a similar effect to considering higher rank local systems $E\rightarrow L$ on embedded Lagrangians.