Stability conditions, metrics, and compactifications

Question~No description for this link has significant implications to many longstanding and well-studied open questions. I will now describe two such implications. The first concerns \(D^b\operatorname{Coh}X)\), and spefically the case where \(X\) is a K3 surface. The second concerns a question in group theory that a priori has nothing to do with derived categories or even triangulated categories.

Both applications have to do with understanding symmetries of a triangulated category \(\mathcal{C}\). By a symmetry of \(\mathcal{C}\), I mean an exact auto-equivalence, namely an exact functor \[\phi \colon \mathcal{C} \to \mathcal{C}\] which is an equivalence of categories. The set of auto-equivalences of \(\mathcal{C}\) modulo natural isomorphisms forms a group under composition, say \(G\). The action of \(G\) on \(\mathcal{C}\) induces an action on \(\operatorname{Stab}(\mathcal{C})\). The first situation—a group acting on a category—is exotic; the second situation—a group acting on a manifold—is more familiar. So we may hope to say something about \(G\) and its action on \(\mathcal{C}\) by studying how it acts on \(\operatorname{Stab}(\mathcal{C})\). This is precisely how the applications arise.

I hope that I have convinced you that the question

is important if you care about derived categories, and even if you do not.

Let \(\mathcal{C}\) be a triangulated category. A stability condition on \(\mathcal{C}\) consists of two pieces of data:

1. A central charge, namely a homomorphism \[ Z \colon K(\mathcal{C}) \to \mathbf{C}\]
2. A slicing, namely a collection of abelian subcategories \(P(\phi)\) for every real number \(\alpha \in \mathcal{C}\).

The slicing satisfies the following properties:

1. \(P(\phi+1) = P(\phi)[1]\)
2. \(\Hom(P(\phi), P(\psi)) = 0\) if \(\phi > \psi\)
3. Harder Narasimhan property: For every \(x \in \mathcal{C}\), there exists a filtration \[ 0 = x_0 \to x_1 \to \cdots \to x_n = x \] whose sub-quotients \(a_{i}\) lie in \(P(\phi_i)\) with \[ \phi_0 > \phi_1 > \cdots > \phi_{n}.\]

The slicing and the central charge are compatible in the following sense. For every \(0 \neq x \in P(\phi)\), we have \[ Z(x) = m_{x} \cdot e^{i\pi\phi}\] for some \(m_{x} \in \mathbf{R}_{> 0}.\)

Plus, there is a local finiteness condition that I will not explain.

For \(\mathcal{C}\) arising in algebraic geometry, such as \(\mathcal{C} = D^b(\operatorname{Coh} X)\), we often require that the central charge is numerical. That is, the map \[ Z \colon K(\mathcal{C}) \to \mathbf{C}\] factors through the Chern character \[ K(\mathcal{C}) \to H^{*}(X, \mathcal{Z}).\] In effect, we replace \(K(\mathcal{C})\) by its image in cohomology called \(K^{\rm num}(\mathcal{C}).\)

The set of stability conditions on \(\mathcal{C}\) has a metric and hence a topology.

Theorem (Bridgeland 2007) — The map \[ \operatorname{Stab}(\mathcal{C}) \to \Hom(K(\mathcal{C}), \mathbf{C})\] defined by \[ (Z, P) \mapsto Z\] is a local homeomorphism.

Here, local means “local on the source”.

Let us begin by interpreting a stability condition as a metric. Now, for certain triangulated categories, like the Fukaya category of curves on a surface etc, several authors have proved an isomorphism between the stability manifold and the space of appropriate metrics on the surface (Dmitrov, Haiden, Katzarkov, Kontsevich, Bridgeland, Smith). This is excellent motivation to say that considering a stability condition as a metric is a good idea. The interpretation that I am about to give is for any triangulated category and is more direct.

We first recall the notion of a metric on a category. The following definition appears in [Lawvere]. I learned it from [Neeman].

You should think of a sequence of composable maps as a path in the category \[ x_0 \to x_1 \to \cdots \to x_{n}.\] The composition \(x_0 \to x_{n}\) is a “direct path”. The length of the direct path is no longer than the length of the broken path.

Now let \(\mathbf{C}\) be triangulated. In this case, it is useful to consider a special class of metrics.

In other words, in the diagram

the top and the bottom arrows have the same length. Thus, a translation invariant metric on \(\mathcal{C}\) is determined by the function \[ m(a) = \ell(0 \to a).\] Conversely, given \(m \colon \operatorname{Ob}(\mathcal{C}) \to \mathbf{R}_{\geq 0}\), we can define \(f \colon \operatorname{Mor}(\mathcal{C}) \to \mathbf{R}_{\geq 0}\) by setting \[ \ell(a \to b) = m(\operatorname{Cone}(a \to b)).\] Then \(\ell\) is, by construction, translation invariant. But when is \(\ell\) a metric? First of all, we must have \(m(0) = 0\), so that \(\ell\) vanishes on isomorphisms. Secondly, let us investigate when \(\ell\) satisfies the triangle inequality. Consider a sequence of morphisms \[ a \xrightarrow{f} b \xrightarrow{g} c.\] By the octahedral axiom, we have a distinguished triangle \[ \operatorname{Cone}(f) \to \operatorname{Cone}(g \circ f) \to \operatorname{Cone}(g) \xrightarrow{+1}. \] The condition \[ \ell(g \circ f) \leq \ell(f) + \ell(g)\] translates into \[ m(\operatorname{Cone}(g \circ f)) \leq m(\operatorname{Cone}(f)) + m(\operatorname{Cone}(g)).\] Thus, \(\ell\) will satisfy the triangle inequality for composable morphisms if \(m\) satisfies the triangle inequality for distinguished triangles.

As a result, we see that translation invariant metrics on \(\mathcal{C}\) are the same thing as functions \(m \colon \operatorname{Ob}(\mathcal{C}) \to \mathbf{R}_{\geq 0}\) such that in any distinguished triangle \[ a \to b \to c \xrightarrow{+1}, \] we have \[ m(b) \leq m(a) + m(c).\]

Let \(\mathbf{\tau}\) be a stability condition on \(\mathcal{C}\). Let me show you how \(\tau\) gives rise to a family of translation invariant metrics on \(\mathcal{C}\).

Define the function \(m_{\tau} \colon \operatorname{Ob}(\mathcal{C}) \to \mathbf{R}_{\geq 0}\), called the mass function, as follows. If \(x \in \mathcal{C}\) is \(\tau\) semi-stable, set \[ m_{\tau}(x) = |Z_{\tau}(x)|.\] In general, let \[ 0 = x_0 \to x_1 \to \dots \to x_n = x \] be the Harder-Narasimhan filtration of \(x\) with sub-quotients \(a_{1}, \dots, a_{n}\). Set \[ m_\tau(x) = \sum |Z_\tau(a_i)|.\]

Proposition – The mass function \(m_{\tau}\) satisfies the triangle inequality. That is, if \[ x \to y \to z \xrightarrow{+1}\] is a distinguished triangle, then \[ m_\tau(y) \leq m_\tau(x) + m_\tau(z).\]

As a result, \(m_{\tau}\) defines a translation invariant metric on \(\mathcal{C}\).

In fact, \(m_{\tau}\) is one of a family of metrics we can define using \(\tau\). Fix a positive real number \(q\). We define the \(q\)-mass function \(m_{\tau,q} \colon \operatorname{Ob}(\mathcal{C}) \to \mathbf{R}_{\geq 0}\) as follows. If \(x \in \mathcal{C}\) is semi-stable of phase \(\phi\), set \[ m_{\tau,q}(x) = |Z_\tau(x)| q^\phi.\] In general, let \[ 0 = x_0 \to x_1 \to \dots \to x_n = x \] be the Harder-Narasimhan filtration of \(x\) with sub-quotients \(a_{1}, \dots, a_{n}\) of phases \(\phi_{1}, \dots, \phi_{n}\). Set \[ m_{\tau,q}(x) = \sum |Z_{\tau}(a_{i})| q^{\phi_{i}}.\]

Theorem (Ikeda) – For any \(q > 0\), the mass function \(m_{\tau,q}\) satisfies the triangle inequality.

As a result, the function \(m_{\tau,q}\) define a family of translation invariant metrics on \(\mathcal{C}\). We will mostly be interested in the case \(q = 1\), but it is important to remember the added flexibility provided by varying the \(q\).

Consider a sequence of maps \[ 0 = x_{0} \to x_{1} \to \dots \to x_{n} = x.\] Recall that we view this is a path from \(0\) to \(x\). If this sequence of maps is the Harder-Narasimhan filtration of \(x\), then, by construction, this path is a geodesic path for the metric given by \(m_{\tau,q}\). That is, the length of the arrow \(0 \to x\) is the sum of lengths of the arrows \(x_{i} \to x_{i+1}\). This property, however, does not characterise HN filtrations—there can be many other geodesic paths to \(x\). Nevertheless, we have the following.

Proposition (BDL) — Let \(x\) be \(\tau\)-stable. For the metric \(m_{\tau, q}\), the path \(0 \to x\) is an indivisible geodesic path (cannot be broken non-trivially). Also, this indivisibility property characterises the stable objects.

As a result, if \(m_{\tau, q} = m_{\tau',q}\), then \(\tau\) and \(\tau'\) have the same stable objects. But we cannot necessarily say that \(\tau = \tau'\). There are examples where \(m_{\tau,q} = m_{\tau',q}\) for distinct \(\tau\) and \(\tau'\). But here, the flexibility of being able to vary \(q\) comes handy.

Theorem (BDL) — If \(m_{\tau,q} = m_{\tau',q}\) for two (or more) distinct values of \(q\), then \(\tau = \tau'\).

So, although one metric may not recover the stability condition, the family definitely does.

Aside — Which (translation invariant) metrics come from stability conditions? This is a great question!

In particular situations, stronger results hold. For example, let \(\mathcal{C} = \mathcal{C}_{\Gamma}\) be the 2-CY category associated to a graph \(\Gamma\). Then \(\mathcal{C}\) has a so-called “standard \(t\)-structure”, and \(\operatorname{Stab}(\mathcal{C})\) has a distinguished connected component containing stability conditions that induce the standard \(t\)-structure.

Theorem (BDL) — Let \(\tau, \tau'\) be stability conditions in the distinguished component of \(\operatorname{Stab}(\mathcal{C}_{\Gamma})\). Then \(m_{\tau} = m_{\tau'}\) implies that \(\tau'\) is a shift of \(\tau\).

Treating a stability condition as a metric allows us to construct a compactification of the space of stability conditions by emulating a construction in geometric group theory due to Thurston. This construction compactifies the Teichmuller space of a surface. It has been hugely benefitial in studying the mapping class group, and we can hope to reap similar benefits for auto-equivalence groups of categories.

Let me quickly recall Thurston’s construction. Fix a closed orientable surface \(M\) of genus \(g \geq 2\). The Teichmuller space \(T(M)\) is the space of hyperbolic metrics of constant curvature \(-1\) on \(M\). This is a finite dimensional non-compact manifold (turns out to be homeomorphic to \(\mathbf{R}^{6g-6}\)). oThurston compactifies \(T(M)\) by first mapping it to an infinite projective space. Let \(S\) be the collection of isotopy classes of simple closed curves on \(M\). Consider the infinite projective space \[ \P^S = (\mathbf{R}^{S} - 0) / \text{Scaling}.\] We have a map \[ i \colon T(M) \to \P^S\] defined as follows. Let \(s \in S\) be a curve, up to isotopy. Let \(\ell_\mu(s)\) be the length of \(s\) with respect to \(\mu\) (that is, the minimum length of a representative of the isotopy class). Set \[ i(\mu) = [\ell_\mu].\]

Theorem (Thurston) —

1. The map \(i\) is injective. In fact, it is a homeomorphism onto its image.
2. The closure of the image of \(i\) is compact.
3. The closure is isomorphic to a closed ball. Its boundary sphere has a modular interpretation as the space of projective measured foliations.

Using the metrics \(m_{\tau,q}\), we can emulate the same construction as before.

Recall that we have an action of \(\mathbf{C}\) on \(\operatorname{Stab}(\mathcal{C})\). The number \(z = a+ib\) acts by scaling the central charge by \(e^{a+ib}\) and shifting the slicing by \(b\). Set \[ \operatorname{PStab} = \operatorname{Stab} / \mathbf{C}.\] The action is properly discontinuous, so the quotient is also a manifold.

Let \(S \subset \mathcal{C}\) be the set of isomorphism classes of objects of \(\mathcal{C}\). We have a map \[ i \colon \operatorname{PStab}(\mathcal{C}) \to \P^{S}\] defined by \[ i \colon \tau \mapsto [m_{\tau}].\] More generally, we can fix \(q_{1}, \dots, q_{n}\) and consider \[ i \colon \operatorname{PStab}(\mathcal{C}) \to \P^{S} \times \dots \times \P^{S},\] defined by \[ i \colon \tau \mapsto ([m_{\tau,q_1}], \dots, [m_{\tau,q_n}]).\]

Theorem (BDL) —

1. Suppose \(S\) contains a (classical) generator of \(\mathcal{C}\). Then the image of \(i\) is compact.
2. For \(n \geq 2\), the map \(i\) is injective. For the quiver categories \(\mathcal{C}_{\Gamma}\), the map \(i\) is injective for \(n = 1\).

The remaining task is to understand the geometry of the closure, for example, its boundary. For this, it is nice to be able to take a smaller \(S\), one whose masses we understand. Fortunately, there is considerable flexibilility in choosing \(S\). The theorem remains true if we restrict \(S\) to contain:

• objects with no negative endomorphisms,
• objects in the heart of some bounded t-structure,
• objects that are stable with respect to some stability condition,
• for \(\mathcal{C}_{\Gamma}\), just the spherical objects.

• Barbara Bolognese: (partial) completion of Stab using the metric inherited from the central charge.
• Broomhead, Pauksztello, Ploog, Woolf: Partial compactification of stability manifolds by massless semistable objects

• Haiden, Katzarkov, Kontsevich “Flat surfaces and stability structures”
• Bridgeland, Smith “Quadratic differentials”

We now explore which mass functions appear on the boundary of PStab. We first recall the picture from geometry. For simplicity, we consider the map \(m_{q}\) for \(q = 1\) and assume that the set \(S\) consists of indecomposable objects that have no endomorphisms of negative degree.

In Teichmuller theory, the following points appear on the boundary. Let \(a\) be a simple closed curve (up to isotopy). Define a functional on the set of simple closed curves by \[ b \mapsto |a \cap b|.\] This is an unsigned count of intersections. (We have to make sure that we choose isotopy representatives for \(a\) and \(b\) are such that the intersection number is the minimum.)

There is a compelling categorical analogue of the intersection function above. Henceforth, we will assume that all objects of our set \(S\) have the property that \(x\) is indecomposable and \(\operatorname{hom}^{i}(x,x) = 0\) for \(i < 0\). Given a spherical object \(a\), define the functional \(\overline \operatorname{hom}_a\) as follows.

Theorem (BDL) — Let \(\mathcal{C}\) be a (hom-finite, k-linear) triangulated category and \(a \in \mathcal{C}\) a spherical object that is semi-stable in a stability condition. Then \(\mathrm{homBar}(a)\) lies on the boundary of \(\operatorname{PStab}(\mathcal{C})\). More precisely, if \(a\) is semi-stable for \(\tau\), then \[ \lim_{n \to \infty} \sigma_{a}^{n} \tau = [\mathrm{homBar}(a)].\]

• The A_n category.
• The standard heart.
• The braid group action.
• All spherical objects are in one braid orbit.
• All stability conditions are in the braid orbit of a standard stability condition.
• Draw the final picture for the A2 compactification.
• Raise the question – what are the irrational points on the boundary?

• Important to understand the global geometry of Stab.
• Constructing a compactification goes a long way towards understading the global geometry.
• Good idea to view stability conditions as metrics, and understand their limits as limiting metrics.

• Nielsen-Thurston classification