Landau-Ginzburg Models, D-branes, and Mirror Symmetry

1. Triangulated categories 1.1. Definition of triangulated categories and their localizations. Triangulated categories appeared in algebra and geometry as generalization and formalization of notions of derived and homotopy categories. If we are given an abelian category A, we can consider its homotopy category H * (A) and then pass to its derived category D * (A), where * is reserved for {b, +, -, ∅} depending which complexes we consider: bounded, bounded above, bounded below, or unbounded. The objects of H * (A) are appropriate complexes of objects of A and morphisms are morphisms of complexes up to homotopy. The objects of D * (A) are the same as those of H * (A), but we have to invert all quasi-isomorphisms, i.e. all morphisms of complexes that induce isomorphisms on cohomology. In other words we can obtain the derived category as a localization of the homotopy category with respect to the class of all quasi-isomorphisms:

It is evident that there is a canonical embedding A ֒→ D * (A), which sends an object A ∈ A to the complex • • • → 0 → A → 0 → • • • with one nontrivial term in degree 0. Both categories H * (A) and D * (A) have natural triangulated structures. Definition 1.1 ( [44]). Let T be an additive category. The structure of a triangulated category on T is defined by giving of the following data: a) an additive autoequivalence [1] : T -→ T (it is called a shift functor or a translation functor), b) a class of exact (or distinguished) triangles:

which must satisfy the set of axioms Verdier T1-T4.

T1. a) For each object X the triangle X id -→ X -→ 0 -→ X [1] is exact. b) Each triangle isomorphic to an exact triangle is exact.

-u [1] -→ Y [1] is exact. T3. For any two exact triangles and two morphisms f, g the diagram below

can be completed to a morphism of triangles by a morphism h : Z → Z ′ . T4. For each pair of morphisms

x [1] —-→ Z ′ [1] where the first two rows and the two central columns are exact triangles.

This definition is useful and has many applications to algebra, geometry, topology, and even physics. In particular, any homotopy category H * (A) and any derived category D * (A) have natural triangulated structures. Now we recall the definition of a localization of categories. Let C be a category and let Σ be a class of morphisms in C. It is well-known that there is a large category C[Σ -1 ] and a functor Q : C → C[Σ -1 ] which is universal among the functors making the elements of Σ invertible. The category C[Σ -1 ] has a good description if Σ is a multiplicative system.

A family of morphisms Σ in a category C is called a multiplicative system if it satisfies the following conditions: M1. all identical morphisms id X belongs to Σ ; M2. the composition of two elements of Σ belong to Σ ; M3. any diagram X ′ s ←-X u -→ Y, with s ∈ Σ can be completed to a commutative square

with t ∈ Σ (the same when all arrows reversed);

. It can be checked that C[Σ -1 ] is a category and there is a quotient functor

which inverts all elements of Σ and it is universal in this sense.

Let D be a triangulated category and N ⊂ D be a full triangulated subcategory. Denote by Σ(N ) a class of morphisms s in D embedding into an exact triangle X s -→ Y -→ N -→ X [1] with N ∈ N . It can be checked that Σ(N ) is a multiplicative system. We define the quotient category D/N := D[Σ(N ) -1 ].

We endow the category D/N with a translation functor induced by the translation functor in the category D.

Lemma 1.2. The category D/N becomes a triangulated category by taking for exact triangles such that are isomorphic to the images of exact triangles in D. The quotient functor Q : D -→ D/N annihilates N . Moreover, any exact functor F : D -→ D ′ of triangulated categories for which F (X) ≃ 0 when X ∈ N factors uniquely through Q.

Example 1.3 (Rings and Modules). Let A be a ring. We can consider the abelian category Mod -A of all (right) A -modules and take the unbounded derived category D(Mod -A). This is a triangulated category with arbitrary direct sums. We can also consider a full triangulated subcategory of D(Mod -A) that is called a triangulated category of perfect complexes and consists of all bounded complexes of projective A -modules of finite type. We denote it as Perf(A). It is not a derived category of any abelian category, but it is a derived category of the exact category of projective modules of finite type (see [21] for definition).

If the ring A is noetherian, then we can also consider the bounded derived category D b (mod -A) of (right) A -modules of finite type. It contains the triangulated category of perfect complexes Perf(A) and they are equivalent when A has a finite global dimension.

Example 1.4 (Schemes). The most important example of a derived or triangulated category comes from a given scheme (X, O X ). In this case it is natural to consider an abelian category of sheaves of O X -modules and an abelian category of quasi-coherent sheaves Qcoh(X). If X is noetherian then we can also con

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