research

What is a triangulated category?

Triangulated categories were invented independently by Puppe and Verdier in the 60s as an axiomatization of the properties of the stable homotopy category in algebraic topology and derived categories of abelian categories in homological algebra. A key idea: Every long exact sequence is the shadow of some exact triangle in a triangulated category.

An important development in the 80s was the discovery of perverse sheaves (Beilinson-Bernstein-Deligne) and t-structures, which relate triangulated categories and their abelian subcategories. The lesson is that a triangulated category usually contains many natural abelian subcategories (and even more unnatural ones), and this reflects interesting dualities – Riemann-Hilbert, Fourier-Mukai, quiver mutation, etc.

In the 90s, derived categories of coherent sheaves began to be studied as invariants of algebraic varieties (Bondal, Kapranov, Orlov, …) and connections with the representation theory of finite-dimensional algebras were found (Beilinson). In his 1994 ICM talk, Kontsevich revolutionized the subject by casting the Mirror Symmetry duality in String Theory as a conjectural equivalence of two triangulated categories attached to dual Calabi-Yau spaces: the derived category of coherent sheaves, and the (at the time obscure) Fukaya category of Lagrangian submanifolds. In physical terms, the triangulated category is the category of boundary conditions of a 2-d topological quantum field theory.

From the modern point of view, triangulated categories are not fundamental but the “1-categorical shadow” of a stable infinity-category (Lurie), which have a much more natural-looking definition. On a practical level, most constructions of infinity-categories (e.g. internal Hom, pullbacks) cannot be done on the level of triangulated categories. Pre-triangulated dg/A-infinity categories provide another language for stable infinity-categories in the linear case.

My own research on triangulated categories

Much of my own research is related to triangulated categories. To mention two examples not related to stability conditions:

• In joint work with Dimitrov, Katzarkov, and Kontsevich (Dynamical systems and categories) we introduce a notion of dynamical entropy of a functor from a triangulated category to itself and relate this to more classical notions of dynamical entropy measuring the complexity of a dynamical system. This also suggests a way of defining the dimension of a triangulated category with Serre functor.
• Together with Chang and Schroll (Braid group actions on branched coverings and exceptional collections) we found an example of a triangulated category where the action of the braid group on the set of full exceptional collections is not transitive, giving a first counter-example to a ’93 conjecture. Previous works had focused on proving transitivity in particular cases.

Why Bridgeland stability conditions?

(A biased view. See also Bayer-Macri, Smith)

Calibrated geometry. By the uniformization theorem, a Riemann surface has a “nicest” metric in its conformal equivalence class – the one of constant curvature. What about metrics on holomorphic bundles over Riemann surface? Here, “nicest” metric means: constant central curvature, a notion which is relative to a metric on the base surface. The Narasimhan-Seshadri theorem from the 60s says that we can find such a metric, and it is essentially unique, if (and only if) the bundle is stable: every strict subbundle has strictly smaller slope, which is defined as degree(E)/rank(E), or more generally polystable: a direct sum of stable bundles of the same slope.

The higher-dimensional generalization of this was proven by Donaldson and Uhlenbeck-Yau (DUY) in the 80s. “Nice metric” now means one satisfying the Hermitian Yang-Mills equation. The notion of degree, and thus of stability, now depends on a choice of Kähler class (or polarization in algebraic geometry). One needs to consider not just subbundles but also sub-coherent sheaves.

Around 2000, Thomas and Yau, motivated by mirror symmetry, proposed that the symplectic analog of the Hermitian Yang-Mills condition is the special Lagrangian condition on Lagrangian submanifolds of a Calabi-Yau manifold, and that some analog of the DUY theorem should be true where the category of coherent sheaves is replaced by the Fukaya category. Changing the metric on a holomorphic bundle then roughly corresponds to moving the Lagrangian submanifold by Hamiltonian isotopy. An immediate obstacle: While one can make sense of “subobjects”, and thus slope-stability, in any abelian category, the Fukaya category is not abelian but triangulated, where kernel and cokernel become intermixed into a single object: the cone.

A few years later, Bridgeland, motivated in part by work of the string theorist Douglas on stability of D-branes, introduced an axiomatic notion of stability in his seminal paper Stability conditions on triangulated categories. In view of this and his own work on special Lagrangian geometry, Joyce formulated an updated version of the Thomas-Yau conjectures, which roughly states that there exists a stability condition on the Fukaya category F(M) of a Calabi-Yau manifold M whose semistable objects are those objects of F(M) which can be represented by an object supported on a special Lagrangian submanifold. A possible strategy of proof involves the Lagrangian mean curvature flow (whose fixed points are special Lagrangians) with surgery, reminiscent of Perelman’s proof of the uniformization of 3-manifolds, but with added complications coming from holomorphic curves ending on the Lagrangian. An intricate story indeed!

Half-translation surfaces/quadratic differentials. This is essentially the special case of complex dimension one of the above story. One is dealing either with quadratic differentials on Riemann surface (complex analytic point of view) or half-translation surfaces (metric, dynamical point of view). Special Lagrangian submanifolds are just smooth geodesics with respect to the flat metric. Something very nice happens, beyond the general Joyce conjecture: moduli spaces of half-translation surfaces are identified with (components of) spaces of stability conditions on Fukaya-type categories of the surfaces. A first general result in this direction is due to Bridgeland-Smith, followed shortly by my own work with Katzarkov and Kontsevich base on an entirely different approach. More recently, much of this was generalized and unified in my joint work with Christ-Qiu.

Donaldson-Thomas invariants and wall-crossing. The story of Donaldson-Thomas invariants and their generalizations and that of Bridgeland stability conditions first intersect in celebrated work of Kontsevich-Soibelman who propose a way of “counting” semistable objects of a 3-Calabi-Yau triangulated A-infinity category. Needless to say, the “counting” here is rather sophisticated and involves things such as motivic Milnor fibers, ind-constructible stacks, and quantum tori. A very interesting property of the “categorical DT invariants” is that, while they depend on the choice of stability condition, one can calculate how they change along a path in the space of stability conditions by a wall-crossing formula. This is reminiscent of how the various power series expansions of a meromorphic functions at different points are related. As an example from my own work, one can construct a 3CY category with stability condition from any quadratic differential with simple zeros (generalizing Bridgeland-Smith) so that stable objects correspond to finite-length geodesics (closed loops and saddle connections). The wall-crossing formula then predicts how counts of such geodesics change as one varies the Riemann surface and/or quadratic differential.