This has led to the formulation of a notion of stability for objects in a derived category, contact with Kontsevich’s homological mirror symmetry conjecture, and . We present a justification on the conjecture on the mirror construction of D- branes in Aganagic-Vafa [2]. We apply the techniques employed in. PDF | This monograph builds on lectures at the Clay School on Geometry and String Theory that sought to bridge the gap between the languages of string .

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This book is suitable for graduate students and researchers with either a physics or mathematics background, who are interested in the interface between string theory and algebraic geometry. A natural sequel to the first Clay monograph on Mirror Symmetry, it presents the new ideas coming out of the interactions of string theory and algebraic geometry in a coherent logical context.

Readers will have to know category theory fairly well to appreciate why non-Abelian categories are more difficult to work with.

Dirichlet Branes and Mirror Symmetry

A Point’s point of view of stringy geometry – Aspinwall, Paul S. Product details Format Hardback pages Dimensions A new string revolution in the mids brought the notion of branes to the forefront. Abelian categories and derived categories: Hence one could conclude that the K-theory classification of D-branes is stronger than the classification by BPS states. There was a problem filtering reviews right now.

Dirichlet branes, homological symmetrry symmetry, and stability – Douglas, Michael R. Massless black holes and conifolds in string theory – Brames, Andrew Nucl. Goodreads is brane world’s largest site for readers with over 50 million reviews. Orbifold resolution by D-branes – Douglas, Michael R.

This implies the use of the Hodge star, which depends on the metric and is continuously valued. One difficulty in understanding all aspects of this work is that it requires being able to speak two different languages, the language of string theory and the language of algebraic geometry.

Two lectures on D-geometry and noncommutative geometry – Douglas, Michael R. The authors were not satisfied to tell their story twice, from separate mathematics and physics points of view.


The notion of a mapping cone comes from algebraic topology, where there is no notion of a kernel or cokernel in the homotopy category of topological spaces. Dirkchlet up a giveaway.

Dirichlet branes, homological mirror symmetry, and stability – INSPIRE-HEP

This has led to exciting new work, including the Strominger-Yau-Zaslow conjecture, which used the theory of branes to propose a geometric basis for mirror symmetry, the theory of stability conditions on triangulated categories, and a physical basis for the McKay correspondence.

If it is not Artinian, then some objects could decay into an infinite number of subjects on the line of marginal stability. This implies the need for D-branes at generic points in moduli space to have “sub” D-branes, which implies the need for a notion of “subobject” of an object in the category of D-branes. For a point u in the Kahler moduli space, E is ‘pi-semistable’ at u if and only if for every sub-brane E’ of E at u, one has phi E is greater than or equal to phi E’where phi E is related to the central charge Z E, u.

Dirichlet Branes and Mirror Symmetry

Top Reviews Most recent Top Reviews. Quantum Fields and Strings: However, kernels and cokernels between coherent sheaves will be coherent. The group of distinguished mathematicians and mathematical physicists who produced this monograph worked as a team to create a unique volume. The derived category goes beyond K-theory in that it keeps track of all massless fermionic open strings between a syjmetry of D-branes. Author s Product display: Representations of quivers can be nontrivial to find despite the simplicity of the quiver itself.

Discover Prime Book Box for Kids. The Clay School on Geometry and String Theory set out to bridge this gap, and this monograph builds on the expository lectures given there to provide an up-to-date discussion including subsequent developments. Read more Read less.

Graduate students and research mathematicians interested in mathematical aspects of quantum field theory, in particular string theory and mirror symmetry. The point is that not all triangles of maps are exact, but that any triangle isomorphic to a distinguished triangle is declared to be exact. Superschool on Derived Categories and D-branes: The K-theory classification classifies the allowed RR field strengths as well as the charges of stable D-branes.


The Best Books of An object E’ is a sub-object of an object E if there exists and injective homomorphism from E’ to E. After showing how notions of branes arose in string theory, it turns to an introduction to the algebraic geometry, sheaf theory, and homological algebra needed to define and work with derived categories.

They relate the ideas to active areas of research that include the McKay correspondence, topological quantum field theory, branea stability structures.

The Poincare bundle induces equivalences between the derived category of an abelian variety and its dual.

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Ships from and sold by Amazon. Speaking somewhat more formally, in the category of complexes an object and its resolution are not always isomorphic, but the complexes can be altered to repair this defect. A bounded t-structure is determined by its ‘heart’. This determines which elements in a derived category actually correspond to physical BPS D-branes. This is what derived categories do, and so every time the notion of a sub-object is needed, one can find a replacement that uses only the triangulated structure.

Ordering on the AMS Bookstore is limited to individuals for personal use only. In the homotopy category for example it is difficult to say when a sequence of morphisms is exact, and so there is no kernel or cokernel. K-theory and twisted K-theory: Clay Mathematics Monographs Volume: This review is based on a reading of chapters of the book.

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