Libraries and resellers, please contact cust-serv ams. See our librarian page for additional eBook ordering options. Paul S. Research in string theory over the last several decades has yielded a rich interaction with algebraic geometry. In , the introduction of Calabi—Yau manifolds into physics as a way to compactify ten-dimensional space-time has led to exciting cross-fertilization between physics and mathematics, especially with the discovery of mirror symmetry in
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Libraries and resellers, please contact cust-serv ams. See our librarian page for additional eBook ordering options. Paul S. Research in string theory over the last several decades has yielded a rich interaction with algebraic geometry. In , the introduction of Calabi—Yau manifolds into physics as a way to compactify ten-dimensional space-time has led to exciting cross-fertilization between physics and mathematics, especially with the discovery of mirror symmetry in A new string revolution in the mids brought the notion of branes to the forefront.
As foreseen 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. 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.
These developments have led to a great deal of new mathematical work. 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.
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. 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.
We hope it will allow students and researchers who are familiar with the language of one of the two fields to gain acquaintance with the language of the other.
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. 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. The physical existence conditions for branes are then discussed and compared in the context of mirror symmetry, culminating in Bridgeland's definition of stability structures, and its applications to the McKay correspondence and quantum geometry.
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. 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.
Graduate students and research mathematicians interested in mathematical aspects of quantum field theory, in particular string theory and mirror symmetry. Know that ebook versions of most of our titles are still available and may be downloaded immediately after purchase.
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The interplay between mathematics and physics that has so often yielded rich fruit over the centuries has of late been much in evidence, what with the work of such figures as Witten and Kontsevich, to name but two obvious luminaries. Arguably this ecumenical view of things underlay, and underlies, the work of many others who are perhaps seen as pure mathematicians exclusively. It is entertaining to speculate about the reasons for the ensuing separation of the fields; perhaps the catastrophe of World War II and the various diasporas of European scholars are part of the reason, coupled with the increase in intrinsic difficulty in any number of areas of mathematics that precipitated a narrowing of specialties and a prevailing orientation to more compact problems and research programs. Adopting a more polemical point of view, perhaps, and borrowing the marvelous if somewhat controversial imagery of frogs and birds introduced by Freeman Dyson in the February issue of the Notices of the AMS , one might posit that, with birds not only being rarer than frogs but having been scattered by the events of the middle to late twentieth century, frogs had their way in mathematics — ergo, narrower specialties. But birds will be birds, and in due course borders between mathematical areas were vaulted by a few trailblazers, with Grothendieck possibly the most obvious example. By the way, both Douglas and Bridgeland are counted among the present authors.
Dirichlet Branes and Mirror Symmetry
Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. Moore and Graeme B. Wilson Published Mathematics. Research in string theory over the last several decades has yielded a rich interaction with algebraic geometry. In , the introduction of Calabi-Yau manifolds into physics as a way to compactify ten-dimensional space-time has led to exciting cross-fertilization between physics and mathematics, especially with the discovery of mirror symmetry in