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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The Geometrization Conjecture John Morgan. Nonlinear instantons from supersymmetric p-branes – Marino, Marcos et al. The book continues with detailed treatments of the Strominger—Yau—Zaslow conjecture, Calabi—Yau metrics and homological mirror symmetry, and discusses more recent physical developments. As mirdor by Kontsevich, these turned out to have mathematical counterparts in the derived category of coherent sheaves on an algebraic variety and the Fukaya category of a symplectic manifold.

Kahler cone substructure – Sharpe, Eric R. Write a customer review. In this context, a binding process brnes be viewed as a formation, wherein two D-branes can bind together via a potentially tachyonic open string. The book first introduces the notion of Dirichlet brane in the context of topological quantum field theories, and then reviews the basics of string theory.

Learn symmetrry about Amazon Prime. K theory and Ramond-Ramond charge – Minasian, Ruben et al. Libraries and resellers, please contact cust-serv ams.

Book ratings by Goodreads. 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. They also explore the ramifications and current state of the Strominger-Yau-Zaslow conjecture. This site is also available in the following languages: In a derived category the morphisms do not have symnetry or cokernels, and so they are ‘additive’ but not Abelian.

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Fukaya categories may not have a triangulated structure, so there is a need to add “potentially stable” A-branes to the Fukaya category so that it becomes triangulated.

The authors explain how Kontsevich’s conjecture is equivalent to the identification of two dirichlwt categories of Dirichlet branes. Share your thoughts with other customers. In a derived category then, quasi-isomorphisms are viewed as isomorphisms.

## Mathematics > Algebraic Geometry

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. Giving a fairly detailed overview of mirror symmetry that emphasizes both its mathematical and physical aspects, this book should be accessible to readers who are familiar with topological quantum field theory, superstring theory, and the highly esoteric mathematical constructions used in these fields.

This is supposed to classify the subset of RR field strengths that can exist in the absence of D-branes up to equivalence by large gauge dirichleh. Instead, theirs is a unified presentation offered in a way that both mathematicians and physicists can follow, without having all of the foundations of both subjects at their immediate disposal. There are several important vranes and concepts needed to understand the content of this book, and some of the main ones include: Related to quivers are the famous Kac-Moody algebras in that if Q is a quiver then the set of dimension vectors of indecomposable representations of Symmefry does not depend on the orientation of arrows in Q, and the dimension vectors of indecomposable representations correspond to positive roots of the corresponding root system.

Symmetrt Actionable Analytics for the Web. However, kernels and cokernels between coherent sheaves will be coherent. AmazonGlobal Ship Orders Internationally. B UTTG This implies that there are no short exact sequences, and to compensate for this, ‘distinguished triangles’ of maps are brought in symmefry a way of shifting complexes up and down.

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. 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. Showing of 1 reviews. 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.

This is different from the situation in K-theory, where a brane-antibrane pair cancels if all open strings to them cancel out of the Q-homology, i.

### [] SYZ Mirror Symmetry for Dirichlet Branes

We can notify you when this item is back in stock. Hence it will be generically irrational, conflicting with the fact that Chern characters in K-theory are always rational. 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.

D-brane stability and monodromy – Aspinwall, Paul S. This motivates the construction of the bounded derived category of coherent sheaves, with the category of coherent sheaves being a subcategory.

These developments have led to a great deal of new mathematical work.

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Research in string theory over the last several decades has yielded a rich interaction with algebraic geometry.

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