## Triangulated categories of singularities and D-branes in Landau-Ginzburg models (2005)

Venue: | Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.):240–262 |

Citations: | 94 - 4 self |

### BibTeX

@ARTICLE{Orlov05triangulatedcategories,

author = {Dmitri Orlov},

title = {Triangulated categories of singularities and D-branes in Landau-Ginzburg models},

journal = {Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.):240–262},

year = {2005}

}

### Years of Citing Articles

### OpenURL

### Abstract

Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend

### Citations

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Algebraic geometry. Graduate Texts in
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- 1977
(Show Context)
Citation Context ...y free sheaves. It is clear for a closed subscheme while for an open subscheme U it follows from the fact that any coherent sheaf on U can be obtained as the restriction of a coherent sheaf on X (see =-=[12]-=-, ex.5.15). Denote by D b (coh(X)) (resp. D b (Qcoh(X)) ) the bounded derived categories of coherent (resp. quasi-coherent) sheaves on X . These categories have canonical triangulated structures. Sinc... |

342 | Homological algebra of mirror symmetry
- Kontsevich
- 1994
(Show Context)
Citation Context ...arities is proper then all Hom’s between objects are finite-dimensional vector spaces (Corollary 1.24). The investigation of such categories is inspired by the Homological Mirror Symmetry Conjecture (=-=[21]-=-). Works on topological string theory are mainly concerned with the case of N=2 superconformal sigma-models with a Calabi-Yau target space. In this case the field theory has two topologically twisted ... |

331 |
Algebraic K-Theory I
- Quillen
- 1973
(Show Context)
Citation Context ...t : P1 → Q0 such that f1 = q0t + sp1 and f0 = tp0 + q1s . It is clear that the category Pairw0 (W) is an exact category with respect to componentwise monomorphisms and epimorphisms (see definition in =-=[23]-=-). Remark 3.1. The remarkable fact is that such construction appeared many years ago in the paper [8] and is known for specialist in singular theory as a matrix factorization.The category DBw0 (W) ca... |

322 |
Residues and Duality
- Hartshorne
- 1966
(Show Context)
Citation Context ...A local noetherian ring A is called Gorenstein if A as a module over itself has a finite injective resolution.12 It can be shown that if A is Gorenstein than A is a dualizing complex for itself (see =-=[13]-=-). This means that A has finite injective dimension and the natural map M −→ RHom · (RHom · (M,A),A) is an isomorphism for any coherent A-module M and as consequence for any object from Db (coh(Spec(A... |

204 |
Methods of homological algebra
- GELFAND, MANIN
- 1996
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Citation Context ...nd triangulated categories 1.1. Triangulated categories and localization. In this section we remind definitions of a triangulated category and its localization which were introduced in [26] (see also =-=[10]-=-,[17],[18]). Let D be an additive category. The structure of a triangulated category on D is defined by giving of the following data: a) an additive autoequivalence [1] : D −→ D (it is called a transl... |

175 |
Higher algebraic K-theory of schemes and of derived categories, in ``The Grothendieck Festschrift
- Thomason, Trobaugh
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Citation Context ...f a perfect complex, which was introduced in [3]. A perfect complex is a complex of sheaves which locally is quasi-isomorphic to a complex of locally free sheaves of finite type ( a good reference is =-=[25]-=-). To any algebraic variety X one can attach the bounded derived category of coherent sheaves Db (coh(X)) . This category admits a triangulated structure. The derived category of coherent sheaves has ... |

105 | Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete - Gabriel, Zisman - 1967 |

104 |
Homological algebra on a complete intersection, with an application to group representations
- Eisenbud
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Citation Context ...ved for maximal Cohen-Macalay modules over local ring in [20], the construction of the section 3 is also known in the local theory of singularities as a matrix factorization, which is due to Eisenbud =-=[8]-=-, where using this construction MCM modules over local rings were described . 12 of singularities is proper then all Hom’s between objects are finite-dimensional vector spaces (Corollary 1.24). The i... |

103 | Semiorthogonal decompositions for algebraic varieties, preprint MPI
- Bondal, Orlov
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Citation Context ...tively ample. ✷ Actually, this Lemma shows us that the category D b (coh(Z1)) has a semiorthogonal decomposition of the form 〈 p ∗ 1 D b (coh(X0)) ⊗ O Z1 (−1), p∗ 1 Db (coh(X0)) 〉 (for definition see =-=[4]-=-). It can be proved for any P1-bundle and moreover for the projectivization of any bundle. The proof for smooth base can be found in [22] and it works for any base. The second Lemma is also almost evi... |

93 |
Théorie des Intersections et Théorème de Riemann–Roch, volume 225
- Berthelot, Grothendieck, et al.
- 1971
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Citation Context ...braic and come from derived categories of coherent sheaves. Categories of this type will be central in this work. An important notion here is the concept of a perfect complex, which was introduced in =-=[3]-=-. A perfect complex is a complex of sheaves which locally is quasi-isomorphic to a complex of locally free sheaves of finite type ( a good reference is [25]). To any algebraic variety X one can attach... |

83 |
D-branes in Landau-Ginzburg models and algebraic geometry
- Kapustin, Li
- 2003
(Show Context)
Citation Context ...t is equal to multiplication by W . This ”twisting” also breaks Z-grading down to Z/2grading. The equivalence of this definition with the physics notion of B-branes in LG models was verified in paper =-=[16]-=- in the case of the usual quadratic superpotential W = x2 1 + · · · + x2n and physical arguments were given supporting Kontsevich’s proposal for a general superpotential. We establish a connection bet... |

62 |
On the derived category of a finite-dimensional algebra
- Happel
- 1987
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Citation Context ... a category which has the same objects as E and a morphism in E is the equivalence class f of a morphism f of E modulo the subgroup of morphisms factoring through an injective in E (see, for example, =-=[11, 18, 19]-=-). If E also has enough projectives (i.e. for each G ∈ E there is an admissible epimorphism P ։ G with projective P ), and the classes of projectives and injectives coincide, then E is called a Froben... |

58 |
Cohen-Macaulay modules on hypersurface singularities
- Knörrer
- 1987
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Citation Context ...elated to known results in local theory of singularities. For instance, the assertion of Theorem 2.1 is known as Knörrer periodicity and is proved for maximal Cohen-Macalay modules over local ring in =-=[20]-=-, the construction of the section 3 is also known in the local theory of singularities as a matrix factorization, which is due to Eisenbud [8], where using this construction MCM modules over local rin... |

53 | Chain Complexes and Stable Categories
- Keller
- 1990
(Show Context)
Citation Context ...lass of admissible epimorphisms is closed under composition and under base change by pullback along an arbitrary map G ′ → G . This definition is equivalent to the original definition of Quillen (see =-=[19]-=-). An object I ∈ E is injective (resp. P is projective) if the sequence Hom(E,I) → Hom(F,I) → 0 (resp. Hom(P,E) → Hom(P,G) → 0) is exact for each admissible monomorphism (resp. epimorphism). We say th... |

37 |
Projective bundles, monoidal transformations, and derived categories of coherent sheaves
- Orlov
- 1993
(Show Context)
Citation Context ... (coh(X0)) ⊗ O Z1 (−1), p∗ 1 Db (coh(X0)) 〉 (for definition see [4]). It can be proved for any P1-bundle and moreover for the projectivization of any bundle. The proof for smooth base can be found in =-=[22]-=- and it works for any base. The second Lemma is also almost evident. Lemma 2.7. Let i : Z ֒→ Y be a closed embedding of a Cartier divisor. Let E be a sheaf on Y such that its restriction to the comple... |

26 |
Rational singularities and almost split sequences
- Auslander
- 1986
(Show Context)
Citation Context ...e introduced in [26] (see also [10],[17],[18]). Let D be an additive category. The structure of a triangulated category on D is defined by giving of the following data: a) an additive autoequivalence =-=[1]-=- : D −→ D (it is called a translation functor), b) a class of exact (or distinguished) triangles: X u −→ Y which must satisfy the set of axioms T1–T4. v −→ Z w −→ X[1], T1. a) For each object X the tr... |

22 |
Catégories dérivées
- Verdier
- 1977
(Show Context)
Citation Context ...Singularities and triangulated categories 1.1. Triangulated categories and localization. In this section we remind definitions of a triangulated category and its localization which were introduced in =-=[26]-=- (see also [10],[17],[18]). Let D be an additive category. The structure of a triangulated category on D is defined by giving of the following data: a) an additive autoequivalence [1] : D −→ D (it is ... |

8 |
Almost split sequences for rational double points, Trans
- Auslander, Reiten
- 1987
(Show Context)
Citation Context ...P to the object P[1] = ( P0 �� −p0 �� P1 i.e. it changes the order of the modules and signs of the morphisms, and takes a morphism f = (f0,f1) to the morphism f[1] = (f1,f0) . We see that the functor =-=[2]-=- is the identity functor. −p1 For any morphism f : P → Q from the category Pairw0 (W) we define a mapping cone C(f) as an object such that c0 = C(f) = ⎛ ⎝ q0 ( Q1 ⊕ P0 f1 0 −p1 ⎞ �� c1 c0 ⎠ , c1 = ) ,... |

6 |
The Auslander-Reiten quiver of a simple curve singularity
- Dieterich, Wiedemann
- 1986
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Citation Context ...zn 0 +z2 1 + · · ·+z2 2k which is given on C2k+1 . (The descriptions of these categories and another categories which is connected with another Dynkin diagram are known and can be found in the papers =-=[1, 2, 6]-=-, where the technique of Auslander-Reiten sequences is used.) The superpotential W has only one singular point over 0 . By Theorem 3.9 the category of B-branes DB0(W ′) is equivalent to the triangulat... |

6 |
Derived categories and their uses, Chapter of the Handbook
- Keller
- 1996
(Show Context)
Citation Context ...lated categories 1.1. Triangulated categories and localization. In this section we remind definitions of a triangulated category and its localization which were introduced in [26] (see also [10],[17],=-=[18]-=-). Let D be an additive category. The structure of a triangulated category on D is defined by giving of the following data: a) an additive autoequivalence [1] : D −→ D (it is called a translation func... |

5 |
Vanishing cycles and mutation, arXiv:math.SG/0007115; More about vanishing cycles and mutation
- Seidel
(Show Context)
Citation Context ...gories can be associated with singularities (or singularities of maps). Categories of the first type are connected with vanishing cycles and closely related to the categories which were introduced in =-=[24]-=- for symplectic Picard-Lefschetz pencils. Categories of the second type are purely algebraic and come from derived categories of coherent sheaves. Categories of this type will be central in this work.... |

2 |
D-branes, Categories and N=1 Supersymmetry, J.Math.Phys
- Douglas
(Show Context)
Citation Context ... mirror symmetry should interchange these two classes of D-branes. From the mathematical point of view the category of B-branes on a Calabi-Yau is the derived category of coherent sheaves on it ([21],=-=[7]-=-). As a candidate for a category of A-branes on Calabi-Yau manifolds so-called Fukaya category has been proposed. Its objects are, roughly speaking, Lagrangian submanifolds equipped with flat vector b... |

1 |
The comparison theorem, appendix to Buchweitz
- Buchweitz
- 1987
(Show Context)
Citation Context ...seful discussions. I also thank Nikolai Tyurin for reading a preliminary draft of the paper and making a number of valuable comments. I am grateful to Duco van Straten who drew my attention to papers =-=[8, 20, 5]-=-. I would like to thank Max-Planck-Institut für Mathematik for the hospitality during the writing of this paper. This work is done under partial financial support of the Russian Foundation for Basic R... |