Triangulated categories of singularities and D-branes in Landau-Ginzburg models

By Dmitri Orlov. 2003

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In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a connection of these categories with D-branes in Landau-Ginzburg models It seems that two different types of categories 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. 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 bounded 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 D(coh(X)) . This category admits a triangulated structure. The derived category of coherent sheaves has a triangulated subcategory Perf(X) formed by perfect complexes. If the variety X is smooth then any coherent sheaf has a finite resolution of locally free sheaves of finite type and the subcategory of perfect complexes coincides with the whole of D(coh(X)). But for singular varieties this property is not fulfilled. We introduce a notion of triangulated category of singularities DSg(X) as the quotient of the triangulated category D(coh(X)) by the full triangulated subcategory of perfect complexes Perf(X) . The category DSg(X) reflects the properties of the singularities of X and ”does not depend on all of X ”. For example we prove that it is invariant with respect to a localization in Zariski topology (Proposition 1.14). The category DSg(X) has good properties when X is Gorenstein. In this case, if the locus of singularities is complete 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 sigmamodels with a Calabi-Yau target space. In this case the field theory has two topologically twisted versions: Amodels and B-models. The corresponding D-branes are called A-branes and B-branes. The 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

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

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Citations in the corpus

Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “The generic fibre of the deformation is called the associated noncommutative LandauGinzburg model (borrowing language from [30]).”
Extended canonical algebras and Fuchsian singularities (2006, background): “For a variety X Orlov investigated in [29] the triangulated category DSg(X) of the singularities of X defined as the quotient of the bounded derived category D(coh(X)) of coherent sheaves modulo the full subcategory of perfect complexes.”
Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, neutral): “Recall that the B-model of (Cn+(2) , W) is described by Orlov's triangulated category of singularities Dsing(W-1 (0)) (see [12]).”
Mirror symmetry for log Calabi-Yau surfaces I (2011, background): “g potential. Let B + iωbe a complexiﬁed Ka¨hler form on Y and DπF(Y) the Fukaya category of (Y,B+iω). Let s= exp(2πi(B+iω)) and X= Xs. Let DSing(X,W) be the derived category of singularities of W: X→C[Or04]. Conjecture 0.16. Assume −KY is ample. Then DπF(Y) ≃DSing(X,W). Remark 0.17. There is ongoing”
Triangulated categories of singularities and equivalences between Landau-Ginzburg models (2005, positive): “For example it is invariant with respect to a localization in Zariski topology ([18], and Prop.”
Topological Correlators in Landau-Ginzburg Models with Boundaries (2003, background): “It turns out one can make a very precise mathematical statement about how this localization occurs, i.e. one can describe the category of B-branes entirely in terms of the geometry of the critical level set [14].”
Derived Categories (2016, background): “Orlov [117], and this was later shortened to the singularity category of A.”
A Beginner’s Introduction to Fukaya Categories (2013, background): “While this conjecture was initially stated for compact Calabi-Yau manifolds (and recently proved for the quintic 3-fold by Sheridan [46]), it also holds (and is often easier to prove) for noncompact manifolds (in which case one should consider the wrapped Fukaya category), and outside of the Calabi-”
Coherent analogues of matrix factorizations and relative singularity categories (2011, methodology): “One generalizes the above definition, replacing free modules with projective modules [18, 27], with locally free sheaves [29], and finally with coherent sheaves [20].”
Gorenstein categories, singular equivalences and finite generation of cohomology rings in recollements (2014, background): “For instance, if the variety X is smooth, then the singularity category Dsg(X) is trivial but this is not true in general [50].”
D-branes and K-theory in 2D topological field theory (2006, background): “In this setting, we find without difficulty that the endomorphisms of the identity morphism are given, just as in the topological example above, by the Hochschild complex Ĉ = {A→ A⊗ A→ A⊗ A⊗ A→ . . . }, whose cohomology is H∗(X ; ∧∗ TX).”
Phases Of N=2 Theories In 1+1 Dimensions With Boundary (2008, background): “The category D(B)/P(B) is also discussed more recently in [101] where it is called the category of singulariries.”
On the Fukaya category of a Fano hypersurface in projective space (2013, background): “(equivalence of the homotopy categories is proven in [28], and quasi-equivalence as DG categories then follows from the results of [23]).”
Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations (2010, background): “[25]) with the shift functor induced from the functor T : MF(R,w) → MF(R,w) given by T (E, δE) = (E[1],−δE), T (f) = f [1], for f ∈ Homw(Ē, F̄ ).”