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| Author | : Paul Seidel |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 142 |
| Release | : 2015-06-26 |
| Genre | : Mathematics |
| ISBN | : 1470410974 |
The author proves Kontsevich's form of the mirror symmetry conjecture for (on the symplectic geometry side) a quartic surface in C .
| Author | : Anton Kapustin |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 281 |
| Release | : 2009 |
| Genre | : Mathematics |
| ISBN | : 3540680292 |
An ideal reference on the mathematical aspects of quantum field theory, this volume provides a set of lectures and reviews that both introduce and representatively review the state-of-the art in the field from different perspectives.
| Author | : Raf Bocklandt |
| Publisher | : Cambridge University Press |
| Total Pages | : 403 |
| Release | : 2021-08-19 |
| Genre | : Mathematics |
| ISBN | : 110848350X |
Introduction to homological mirror symmetry from the point of view of representation theory, suitable for graduate students.
| Author | : Ricardo Castano-Bernard |
| Publisher | : Springer |
| Total Pages | : 445 |
| Release | : 2014-10-07 |
| Genre | : Mathematics |
| ISBN | : 3319065149 |
The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the “tropical” approach to Gromov-Witten theory and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry. The central ideas first appeared in the work of Maxim Kontsevich (1993). Roughly speaking, the subject can be approached in two ways: either one uses Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow picture, further developed by Kontsevich and Soibelman) or one uses Lefschetz fibrations of symplectic manifolds (suggested by Kontsevich and further developed by Seidel). Tropical Geometry studies piecewise-linear objects which appear as “degenerations” of the corresponding algebro-geometric objects.
| Author | : Kentaro Hori |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 954 |
| Release | : 2003 |
| Genre | : Mathematics |
| ISBN | : 0821829556 |
This thorough and detailed exposition is the result of an intensive month-long course on mirror symmetry sponsored by the Clay Mathematics Institute. It develops mirror symmetry from both mathematical and physical perspectives with the aim of furthering interaction between the two fields. The material will be particularly useful for mathematicians and physicists who wish to advance their understanding across both disciplines. Mirror symmetry is a phenomenon arising in string theory in which two very different manifolds give rise to equivalent physics. Such a correspondence has significant mathematical consequences, the most familiar of which involves the enumeration of holomorphic curves inside complex manifolds by solving differential equations obtained from a ``mirror'' geometry. The inclusion of D-brane states in the equivalence has led to further conjectures involving calibrated submanifolds of the mirror pairs and new (conjectural) invariants of complex manifolds: the Gopakumar-Vafa invariants. This book gives a single, cohesive treatment of mirror symmetry. Parts 1 and 2 develop the necessary mathematical and physical background from ``scratch''. The treatment is focused, developing only the material most necessary for the task. In Parts 3 and 4 the physical and mathematical proofs of mirror symmetry are given. From the physics side, this means demonstrating that two different physical theories give isomorphic physics. Each physical theory can be described geometrically, and thus mirror symmetry gives rise to a ``pairing'' of geometries. The proof involves applying $R\leftrightarrow 1/R$ circle duality to the phases of the fields in the gauged linear sigma model. The mathematics proof develops Gromov-Witten theory in the algebraic setting, beginning with the moduli spaces of curves and maps, and uses localization techniques to show that certain hypergeometric functions encode the Gromov-Witten invariants in genus zero, as is predicted by mirror symmetry. Part 5 is devoted to advanced topi This one-of-a-kind book is suitable for graduate students and research mathematicians interested in mathematics and mathematical and theoretical physics.
| Author | : Noriko Yui, Helena Verrill, and Charles F. Doran |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 324 |
| Release | : |
| Genre | : Duality (Mathematics) |
| ISBN | : 9780821871577 |
"This book is a testimony to the BIRS Workshop, and it covers a wide range of topics at the interface of number theory and string theory, with special emphasis on modular forms and string duality. They include the recent advances as well as introductory expositions on various aspects of modular forms, motives, differential equations, conformal field theory, topological strings and Gromov-Witten invariants, mirror symmetry, and homological mirror symmetry. The contributions are roughly divided into three categories: arithmetic and modular forms, geometric and differential equations, and physics and string theory. The book is suitable for researchers working at the interface of number theory and string theory."--BOOK JACKET.
| Author | : Juan M Maldacena |
| Publisher | : World Scientific |
| Total Pages | : 572 |
| Release | : 2005-07-12 |
| Genre | : Science |
| ISBN | : 9814479845 |
Intended mainly for advanced graduate students in theoretical physics, this comprehensive volume covers recent advances in string theory and field theory dualities. It is based on the annual lectures given at the School of the Theoretical Advanced Study Institute (2003) a traditional event that brings together graduate students in high energy physics for an intensive course given by leaders in their fields. The first lecture by Paul Aspinwall is a description of branes in Calabi–Yau manifolds, which includes an introduction to the modern ideas of derived categories and their relation to D-branes. Juan Maldacena's second lecture is a short introduction to the AdS/CFT correspondence with a short discussion on its plane wave limit. Tachyon condensation for open strings is discussed in the third lecture by Ashoke Sen while Eva Silverstein provides a useful summary of the various attempts to produce four-dimensional physics out of string theory and M-theory in the fourth lecture. Matthew Strassler's fifth lecture is a careful discussion of a theory that has played a very important role in recent developments in string theory — a quantum field theory that produces a duality cascade which also has a large N gravity description. The sixth lecture by Washington Taylor explains how to perform perturbative computations using string field theory. The written presentation of these lectures is detailed yet straightforward, and they will be of great use to both students and experienced researchers in high energy theoretical physics. Contents:D-Branes on Calabi-Yau Manifolds (P S Aspinwall)Lectures on AdS/CFT (J M Maldacena)Tachyon Dynamics in Open String Theory (A Sen)TASI/PITP/ISS Lectures on Moduli and Microphysics (E Silverstein)The Duality Cascade (M J Strassler)Perturbative Computations in String Field Theory (W Taylor) Readership: Graduates, academics and researchers in high energy, particle, theoretical and mathematical physics. Keywords:String Theory;M-Theory;Supersymmetry;Field Theory;AdS/CFTKey Features:An ongoing series of lecture notes featuring an intensive course of advanced learning in high energy physics
| Author | : Jonathan Block |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 506 |
| Release | : 2012 |
| Genre | : Mathematics |
| ISBN | : 0821872958 |
The nature of interactions between mathematicians and physicists has been thoroughly transformed in recent years. String theory and quantum field theory have contributed a series of profound ideas that gave rise to entirely new mathematical fields and revitalized older ones. The influence flows in both directions, with mathematical techniques and ideas contributing crucially to major advances in string theory. A large and rapidly growing number of both mathematicians and physicists are working at the string-theoretic interface between the two academic fields. The String-Math conference series aims to bring together leading mathematicians and mathematically minded physicists working in this interface. This volume contains the proceedings of the inaugural conference in this series, String-Math 2011, which was held June 6-11, 2011, at the University of Pennsylvania.
| Author | : Fabrizio Catanese |
| Publisher | : Springer |
| Total Pages | : 363 |
| Release | : 2008-04-17 |
| Genre | : Mathematics |
| ISBN | : 3540782796 |
Modern approaches to the study of symplectic 4-manifolds and algebraic surfaces combine a wide range of techniques and sources of inspiration. Gauge theory, symplectic geometry, pseudoholomorphic curves, singularity theory, moduli spaces, braid groups, monodromy, in addition to classical topology and algebraic geometry, combine to make this one of the most vibrant and active areas of research in mathematics. It is our hope that the five lectures of the present volume given at the C.I.M.E. Summer School held in Cetraro, Italy, September 2-10, 2003 will be useful to people working in related areas of mathematics and will become standard references on these topics. The volume is a coherent exposition of an active field of current research focusing on the introduction of new methods for the study of moduli spaces of complex structures on algebraic surfaces, and for the investigation of symplectic topology in dimension 4 and higher.
| Author | : M. Dickmann |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 129 |
| Release | : 2015-10-27 |
| Genre | : Commutative rings |
| ISBN | : 1470414686 |
In this monograph the authors extend the classical algebraic theory of quadratic forms over fields to diagonal quadratic forms with invertible entries over broad classes of commutative, unitary rings where is not a sum of squares and is invertible. They accomplish this by: (1) Extending the classical notion of matrix isometry of forms to a suitable notion of -isometry, where is a preorder of the given ring, , or . (2) Introducing in this context three axioms expressing simple properties of (value) representation of elements of the ring by quadratic forms, well-known to hold in the field case.
