Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities

By Dmitri Orlov. 2005

472 citations recorded by the source corpus.

In this paper we establish an equivalence between the category of graded D-branes of type B in Landau–Ginzburg models with homogeneous superpotential W and the triangulated category of singularities of the fiber of W over zero. The main result is a theorem that shows that the graded triangulated category of singularities of the cone over a projective variety is connected via a fully faithful functor to the bounded derived category of coherent sheaves on the base of the cone. This implies that the category of graded D-branes of type B in Landau–Ginzburg models with homogeneous superpotential W is connected via a fully faithful functor to the derived category of coherent sheaves on the projective variety defined by the equation W = 0.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/915234fe4d270f9cea8f937420cf356accea998e

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Citations in the corpus

The motive of a classifying space (2014, background): “For a triangulated subcategory E of a triangulated category T , the right orthogonal to E is the full subcategory E of all objects M such that Hom(N,M) = 0 for every N in E [44].”
Koszul equivalences in A ∞ -algebras (2008, methodology): “This is consistent with terminology for associative algebras: Orlov called the triangulated category Dfg(R) the derived category of the singularity of R [28].”
Phases Of N=2 Theories In 1+1 Dimensions With Boundary (2008, methodology): “Let us now describe the construction of [5].”
Triangle singularities, ADE-chains, and weighted projective lines (2012, methodology): “An L-graded version of Orlov’s theorem In the beginning of this section we allow arbitrary weight type.”
Nilpotent operators and weighted projective lines (2010, methodology): “Stable categories of (graded) Cohen-Macaulay modules play an important role in the analysis of singularities, see [3, 9, 10, 17].”
Homological mirror symmetry for Brieskorn–Pham singularities (2009, methodology): “The homotopy category HMF W of MF W is triangulated, and Calabi-Yau/Landau-Ginzburg correspondence is a theorem in this case: Theorem 4 (Orlov [15]).”