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Braid group actions on derived categories of coherent sheaves
Paul Seidel,Richard P. Thomas +1 more
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In this article, the authors give a construction of braid group actions on coherent sheaves on a variety of manifolds and show that these actions are always faithful when the manifold is an elliptic curve.Abstract:
This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when $\dim X \geq 2$, our braid group actions are always faithful.
We describe conjectural mirror symmetries between smoothings and resolutions of singularities that lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.read more
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Mutation in triangulated categories and rigid Cohen–Macaulay modules
Osamu Iyama,Yuji Yoshino +1 more
TL;DR: In this paper, the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality was introduced.
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Stability conditions on $K3$ surfaces
TL;DR: In this article, the authors describe a connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K3 surface. But their analysis is restricted to the case where the stable sheaves are coherent.
Journal ArticleDOI
D-branes, categories and N=1 supersymmetry
TL;DR: In this article, it was shown that boundary conditions in topological open string theory on Calabi-Yau (CY) manifolds are objects in the derived category of coherent sheaves, as foreseen in the homological mirror symmetry proposal of Kontsevich.
Book
A theory of generalized Donaldson-Thomas invariants
Dominic Joyce,Yinan Song +1 more
TL;DR: In this article, generalized Donaldson-Thomas invariants are defined for all classes of coherent sheaves, and they are equal to $DT^\alpha(\tau)$ when it is defined.
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Introduction to $A$-infinity algebras and modules
TL;DR: These are expanded notes of four introductory talks on $A_\infty$-algebras, their modules, and their derived categories as mentioned in this paper, which can be found in Table 1.
References
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Journal ArticleDOI
Mirror symmetry is T duality
TL;DR: In this paper, it was argued that every Calabi-Yau manifold X with a mirror Y admits a family of supersymmetric toroidal 3-cycles and that the moduli space of such cycles together with their flat connections is precisely the space Y.
Book ChapterDOI
Homological Algebra of Mirror Symmetry
TL;DR: Mirror symmetry was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeros).
Book
Methods of Homological Algebra
Sergei Gelfand,Yuri I. Manin +1 more
TL;DR: In this article, the authors introduce homotopic algebra and define the notion of simplicial sets, derived categories and derived functors, and triangulated categories for homotopy algebra.