3.6 Towards a Bridgeland stability condition [04BP]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

3.6 Towards a Bridgeland stability condition

Joyce’s proposal and Bridgeland stability condition revisited

We now seek a better appreciation of the Bridgeland stability aspect of Joyce’s proposal (cf. section 2.1). To specify the Bridgeland stability condition on $D^{b}Fuk(X)$ , we need the central charge $Z(L)=\int_{L}\Omega$ , and all the subcategories $\mathcal{P}(\phi_{0}<\phi<\phi_{1})$ for any interval $(\phi_{0},\phi_{1})$ . Joyce’s proposal [41] strongly suggests two claims:

•

If an unobstructed Lagrangian brane $L$ has phase angle function $\theta\in(\pi\phi_{0},\pi\phi_{1})$ , then $L$ defines an object in the subcategory $\mathcal{P}(\phi_{0}<\phi<\phi_{1})$ generated by all stable objects with $\phi_{0}<\phi<\phi_{1}$ . This claim is because the infinite time limit of the LMCF should provide the stable objects which generate $L$ , and by the monotonicity of the Lagrangian angle (cf. section 4.1 below), we can predict a priori $\theta\in(\pi\phi_{0},\pi\phi_{1})$ for all these stable objects.
•

Any object in $\mathcal{P}(\phi_{0}<\phi<\phi_{1})$ can be generated by unobstructed Lagrangian branes with phase angle function $\theta\in(\pi\phi_{0},\pi\phi_{1})$ .

Thus one can simply define $\mathcal{P}(\phi_{0}<\phi<\phi_{1})$ to be the subcategory of the derived Fukaya category (suitably enlarged to allow for immersed and singular objects) generated by all unobstructed Lagrangian branes $L$ with $\theta\in(\pi\phi_{0},\pi\phi_{1})$ , and then reconstruct $\mathcal{P}(\phi^{\prime})$ as the intersection of all $\mathcal{P}(\phi_{0}<\phi<\phi_{1})$ for all $\phi_{0}<\phi^{\prime}<\phi_{1}$ . Such a definition would make the Thomas-Yau proposal nearly tautological, and the difficult part of Joyce’s proposal is to verify this indeed defines a Bridgeland stability. In fact, by the discussions in section 2.1, 2.2, the only formidable part is the Harder-Narasimhan decomposition, for which Joyce’s LMCF provides the conjectural mechanism.

There are two primary applications of the Thomas-Yau-Joyce proposal to keep in mind:

•

The existence of special Lagrangians is important for geometric measure theory. A definition of stability conditions along the above lines is too tautological to be useful.
•

Defining special Lagrangian DT invariants is important for mirror symmetry (cf. section 2.4). Knowing the existence of a Bridgeland stability condition on $D^{b}Fuk(X)$ is of great theoretic significance in view of Kontsevich and Soibelman’s framework [46][47], but without a more Floer theoretic characterization it would lack computability.

Thus even if Joyce’s conjectures can be proved along the lines in [41], it is still desirable to have a Floer theoretic characterization of the Bridgeland stability condition. We first revisit Theorem 3.21 in the light of the Thomas-Yau-Joyce conjectural picture, but without assuming the automatic transversality and positivity condition.

Conjecture 3.31.

Suppose we have almost calibrated exact Lagrangian objects $L_{1},L_{2},L$ , fitting into a distinguished triangle $L_{1}\to L\to L_{2}\to L_{1}[1]$ , and satisfies the destabilizing condition

\hat{\theta}_{1}=\arg\int_{L_{1}}\Omega>\hat{\theta}_{2}=\arg\int_{L_{2}}\Omega.

Then the phase angle inequality (31) follows. In particular, the derived category class of $L$ admits no special Lagrangian representative.

Proof.

(Heuristic) Consider the Harder-Narasimhan decomposition (4) of $L_{1}$ :

0=\mathcal{E}_{0}\to\mathcal{E}_{1}\to\ldots\to\mathcal{E}_{N}=L_{1},

fitting into the distinguished triangles

\mathcal{E}_{i-1}\to\mathcal{E}_{i}\to L_{i}^{\prime}\to\mathcal{E}_{i-1}[1],

where $L_{i}^{\prime}$ represents an object in $\mathcal{P}(\phi_{i})$ , with $\phi_{1}>\phi_{2}\ldots>\phi_{N}$ . Since by assumption $L_{1}$ is almost calibrated, we have $L_{1}\in\mathcal{P}(-\frac{1}{2}<\phi<\frac{1}{2})$ , hence $-\frac{1}{2}<\phi_{i}<\frac{1}{2}$ . Since the central charges satisfy

Z(L_{1})=\sum_{1}^{N}Z(L_{i}^{\prime}),\quad\arg Z(L_{i})=\pi\phi_{i},

we must have $\hat{\theta}_{1}\leq\pi\phi_{1}$ . The conjectural description of the Bridgeland stability condition requires that $L_{i}^{\prime}$ has a special Lagrangian representative with constant Lagrangian phase $\pi\phi_{i}$ . A weaker requirement which suffices for us is that there exists a representative $L_{i}^{\prime}$ with Lagrangian angle function $\theta_{L_{i}^{\prime}}$ satisfying the oscillation bound $|\theta_{L_{i}^{\prime}}-\pi\phi_{i}|<\epsilon$ for any given $\epsilon>0$ . It is expected that this flexibility allows one to assume sufficient smoothness on the Lagrangian.

By combining the distinguished triangles, we obtain a new distinguished triangle

L_{1}^{\prime}\to L\to L^{\prime\prime}\to L_{1}^{\prime}[1].

Here $L_{1}^{\prime},L,L^{\prime\prime}$ are all almost calibrated. Suppose for contradiction that $\sup_{L}\theta_{L}<\hat{\theta}_{1}$ . Then $\sup_{L}\theta_{L}<\pi\phi_{1}$ , and we can arrange $\sup_{L}\theta_{L}<\inf_{L_{1}^{\prime}}\theta_{L_{1}^{\prime}}$ . The Floer degree formula (63) implies $CF^{0}(L_{1}^{\prime},L)=0$ , and in particular $HF^{0}(L_{1}^{\prime},L)=0$ . The distinguished triangle splits: $L^{\prime\prime}\simeq L\oplus L_{1}^{\prime}[1]$ . Since $L^{\prime\prime}$ is almost calibrated, it lies in $\mathcal{P}(-\frac{1}{2}<\phi<\frac{1}{2})$ , and so must $L_{1}^{\prime}[1]$ . But $L_{1}^{\prime}\in\mathcal{P}(\phi_{1})$ implies $L_{1}^{\prime}[1]\in\mathcal{P}(\phi_{1}+1)$ . Since $\phi_{1}+1>\frac{1}{2}$ , we know $\mathcal{P}(\phi_{1}+1)\cap\mathcal{P}(-\frac{1}{2}<\phi<\frac{1}{2})=\emptyset$ , contradiction. This proves $\sup_{L}\theta_{L}\geq\hat{\theta}_{1}$ , subject to the conjectural existence of the Bridgeland stability condition.

A very similar argument, beginning with the Harder-Narasimhan decomposition of $L_{2}$ , would show $\inf_{L}\theta_{L}\leq\hat{\theta}_{2}$ . ∎

Remark 3.14.

In the above argument, once we achieved $CF^{0}(L_{1}^{\prime},L)=0$ , there is a different way to proceed. We reinterpret the distinguished triangle $L_{1}^{\prime}\to L\to L^{\prime\prime}\to L_{1}^{\prime}[1]$ as an isomorphism in $D^{b}Fuk(X)$ between $L$ and a twisted complex built from $L_{1}^{\prime}\cup L^{\prime\prime}$ . This would give rise to a bordism current constructed from the universal family of holomorphic curves, with $\partial\mathcal{C}=L-L_{1}^{\prime}-L^{\prime\prime}$ . However, $CF^{0}(L_{1}^{\prime},L)=0$ implies that no holomorphic curve contributing to $\mathcal{C}$ passes from $L_{1}^{\prime}$ to $L$ in the clockwise direction of $\partial\Sigma$ . The twisted complex structure on $L_{1}^{\prime}\cup L^{\prime\prime}$ forbids the passage from $L_{1}^{\prime}$ to $L^{\prime\prime}$ in the clockwise direction of $\partial\Sigma$ . Thus if the holomorphic curve has any boundary portion on $L_{1}^{\prime}$ , its entire boundary would lie on $L_{1}^{\prime}$ , which cannot happen in the almost calibrated setting.

This motivates the following definition, whose precise meaning depends on the conjectural enlargement of the derived Fukaya category by incorporating singular Lagrangian objects:

Definition 3.32.

Let $L$ be an almost calibrated exact Lagrangian brane representing a class in the (suitably enlarged) derived Fukaya category. Suppose for any almost calibrated exact Lagrangian objects $L_{1},L_{2}$ fitting into a distinguished triangle $L_{1}\to L\to L_{2}\to L_{1}[1]$ , we always have

\hat{\theta}_{1}=\arg\int_{L_{1}}\Omega\leq\hat{\theta}_{2}=\int_{L_{2}}\Omega,\quad\text{resp. }\hat{\theta}_{1}=\arg\int_{L_{1}}\Omega<\hat{\theta}_{2}=\int_{L_{2}}\Omega,

then we say $L$ is Thomas-Yau semistable (resp. strictly stable). If $L$ fails to be Thomas-Yau semistable, we say it is Thomas-Yau unstable.

We have attributed this definition to Thomas-Yau [65][66], since it is in their spirit that stability conditions should be Floer theoretic conditions to be tested on the distinguished triangles, and that one should restrict attention only to almost calibrated Lagrangians. We now argue that if Joyce’s conjectural Bridgeland stability exists with its expected properties, then its semistable objects should agree with Thomas-Yau semistability.

Conjecture 3.33.

An almost calibrated exact Lagrangian brane $L$ defines a semistable object in $D^{b}Fuk(X)$ under Joyce’s Bridgeland stability, if and only if it is Thomas-Yau semistable.

Proof.

(Heuristic) If the derived category class of $L$ is semistable in Joyce’s sense, then we can choose an optimal representative which is a special Lagrangian, or at least has phase oscillation arbitrarily small. By conjecture 3.31, we cannot have any destabilizing distinguished triangle, i.e. $L$ is Thomas-Yau semistable.

Conversely, if $L$ is not a semistable object in Joyce’s sense, then from its Harder-Narasimhan decomposition we can produce a destabilizing distinguished triangle $L_{1}\to L\to L_{2}\to L_{1}[1]$ , with almost calibrated $L_{1},L_{2}$ , which violates Thomas-Yau semistability. ∎

Having discussed the semistable objects, it is interesting to see what Joyce’s LMCF picture suggests about the Harder-Narasimhan decomposition.

Conjecture 3.34.

Assume further that the Kähler metric on $X$ is Calabi-Yau. Suppose $L$ is an almost calibrated exact Lagrangian brane in $D^{b}Fuk(X)$ , with Harder-Narasimhan decomposition (4)

0=\mathcal{E}_{0}\to\mathcal{E}_{1}\to\ldots\to\mathcal{E}_{N}=L,

fitting into the distinguished triangles

\mathcal{E}_{i-1}\to\mathcal{E}_{i}\to L_{i}\to\mathcal{E}_{i-1}[1],

such that $L_{i}\in\mathcal{P}(\phi_{i})$ with $\phi_{1}>\ldots>\phi_{N}$ . We have $\hat{\theta}_{i}=\pi\phi_{i}=\arg\int_{L_{i}}\Omega$ . Then the phase angle inequality (35) and the volume lower bound (36) hold.

Proof.

(Heuristic) In Joyce’s conjectural program, the Harder-Narasimhan decomposition is constructed by running the LMCF $(L_{t})$ starting from the unobstructed Lagrangian $L$ , and take the infinite time limit (6) to obtain the limiting special Lagrangians $L_{1},\ldots,L_{N}$ with angles $\hat{\theta}_{1}>\hat{\theta}_{2}>\ldots>\hat{\theta}_{N}$ , assuming $L_{1},\ldots L_{N}$ have enough regularity to be admitted as objects of $D^{b}Fuk(X)$ . It is expected that $L_{1},\ldots L_{N}$ generate $L$ in $D^{b}Fuk(X)$ via (4).

A basic feature of LMCF in Calabi-Yau manifolds is that the Lagrangian angle satisfies a heat equation (cf. section 4.1), so $\sup_{L_{t}}\theta$ is nonincreasing in time (resp. $\inf_{L_{t}}\theta$ is nondecreasing). Comparing the initial time with the infinite time limit, this suggests $\sup\theta_{L}\geq\hat{\theta}_{1}$ and $\inf\theta_{L}\leq\hat{\theta}_{N}$ .

Morever, if the ambient metric is Calabi-Yau, then LMCF is a special case of mean curvature flow, so the volume functional decreases in time. This monotonicity is not affected by the surgeries in Joyce’s LMCF. Thus

\text{Vol}(L)\geq\sum_{1}^{N}\text{Vol}(L_{i})=\sum_{1}^{N}|Z(L_{i})|=\sum_{1}^{N}|\int_{L_{i}}\Omega|.

Under the Calabi-Yau metric, the volume of the Lagrangian $L$ agrees with the $J$ -volume:

\text{Vol}(L)=\int_{L}e^{-i\theta}\Omega=\int_{L}|\Omega|.

so (36) follows. ∎

Remark 3.15.

Analogously in the context of HYM connections, if a holomorphic bundle $E$ is unstable, then its Harder-Narasimhan decomposition provides a lower bound on the Yang-Mills energy of any Chern connection on $E$ compatible with $\bar{\partial}_{E}$ , which improves the topological energy bound. This type of phenomenon is common in Kähler geometry, for instance it also happens in the context of K-stability. These topics are covered in the introduction of [30].

In conclusion, Joyce’s conjectural picture suggests that if a Lagrangian object is unstable, then it satisfies certain angle and volume inequalities which quantitatively forbids it to be a special Lagrangian, and these obstructions detect the features of the Harder-Narasimhan decomposition. This should be compared with Theorem 3.26, which contains the main features of the obstructions, but makes no a priori reference to the LMCF or special Lagrangian representatives. The cost is that Theorem 3.21 and 3.26 require the positivity condition as an extra hypothesis.

Almost calibrated case: categorical predictions of the Joyce picture

A complete characterization of Bridgeland stability conditions on a triangulated category, known since the inception of the subject [13, Prop 5.3], is that $\mathcal{P}(\phi_{0}<\phi\leq\phi_{0}+1)$ defines an abelian subcategory (‘the heart of a bounded $t$ -structure’), and the central charge function on this abelian subcategory satisfies the Harder-Narasimhan condition.

In the Thomas-Yau-Joyce picture, $\mathcal{P}(-\frac{1}{2}<\phi\leq\frac{1}{2})$ essentially is the same as the subcategory of almost calibrated Lagrangians, if we assume there is no special Lagrangian of phase $\frac{\pi}{2}$ , which holds as long as the discrete set of values of $\Omega$ -periods on $H_{n}(X,\mathbb{Z})$ miss the phase angle $\frac{\pi}{2}$ . Then this picture would predict almost calibrated Lagrangians to form an abelian category, which morever generate the entire derived Fukaya category using the shift operator. Morally, this is asserting that there are sufficiently many almost calibrated Lagrangians, which is evidently very deep since constructing geometric Lagrangian objects is known to be a difficult problem in symplectic topology. Another deep prediction of the existence of Bridgeland stability condition [41, conjecture 3.6], is that the derived Fukaya category $D^{b}Fuk(X)$ (after incorporating immersed and singular Lagrangians with local systems) is automatically idempotent complete, so agrees with $D^{\pi}Fuk(X)$ . These predictions, if correct, are very interesting structural results on the Fukaya category, but at the moment they are controversial.

In Chapter 5, we will set up a variational framework to find special Lagrangian representatives of $D^{b}Fuk(X)$ classes under the assumption of Thomas-Yau semistability. By restricting only to the subcategory of almost calibrated Lagrangians, our program evades these difficult structural claims on the entire $D^{b}Fuk(X)$ . It would thus not have the same strength as the Joyce program, nor is it subject to the same falsification criteria.