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Differential Geometry Through Supersymmetric Glasses
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Book Synopsis Differential Geometry Through Supersymmetric Glasses by : Andrei Smilga
Download or read book Differential Geometry Through Supersymmetric Glasses written by Andrei Smilga and published by World Scientific. This book was released on 2020-06-16 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: Back in 1982, Edward Witten noticed that classical problems of differential geometry and differential topology such as the de Rham complex and Morse theory can be described in a very simple and transparent way using the language of supersymmetric quantum mechanics. Since then, many research papers have been written on this subject. Unfortunately not all the results in this field known to mathematicians have obtained a transparent physical interpretation, even if this new physical technique has also allowed many mathematical results to be derived which are completely new, in particular, hyper-Kaehler and the so-called HKT geometry. But in almost 40 years, no comprehensive monograph has appeared on this subject. So this book written by an expert in supersymmetric quantum field theories, supersymmetric quantum mechanics and its geometrical applications, addresses this yearning gap.It comprises three parts: The first, GEOMETRY, gives basic information on the geometry of real, complex, hyper-Kaehler and HKT manifolds, and is principally addressed to the physicist. The second part 'PHYSICS' presents information on classical mechanics with ordinary and Grassmann dynamics variables. Besides, the author introduces supersymmetry and dwells in particular on the representation of supersymmetry algebra in superspace. And the last and most important part of the book 'SYNTHESIS', is where the ideas borrowed from physics are used to study purely mathematical phenomena.
Book Synopsis Differential Geometry Through Supersymmetric Glasses by : A. V. Smilga
Download or read book Differential Geometry Through Supersymmetric Glasses written by A. V. Smilga and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: "Back in 1982, Edward Witten noticed that classical problems of differential geometry and differential topology such as the de Rham complex and Morse theory can be described in a very simple and transparent way using the language of supersymmetric quantum mechanics. Since then, many research papers have been written on this subject. Unfortunately not all the results in this field known to mathematicians have obtained a transparent physical interpretation, even if this new physical technique has also allowed many mathematical results to be derived which are completely new, in particular, hyper-Kaehler and the so-called HKT geometry. But in almost 40 years, no comprehensive monograph has appeared on this subject. So this book written by an expert in supersymmetric quantum field theories, supersymmetric quantum mechanics and its geometrical applications, addresses this yearning gap. It comprises three parts: The first, GEOMETRY, gives basic information on the geometry of real, complex, hyper-Kaehler and HKT manifolds, and is principally addressed to the physicist. The second part "PHYSICS" presents information on classical mechanics with ordinary and Grassmann dynamics variables. Besides, the author introduces supersymmetry and dwells in particular on the representation of supersymmetry algebra in superspace. And the last and most important part of the book "SYNTHESIS", is where the ideas borrowed from physics are used to study purely mathematical phenomena"--
Book Synopsis Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics by : Yuri E. Gliklikh
Download or read book Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics written by Yuri E. Gliklikh and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 207 pages. Available in PDF, EPUB and Kindle. Book excerpt: The geometrical methods in modem mathematical physics and the developments in Geometry and Global Analysis motivated by physical problems are being intensively worked out in contemporary mathematics. In particular, during the last decades a new branch of Global Analysis, Stochastic Differential Geometry, was formed to meet the needs of Mathematical Physics. It deals with a lot of various second order differential equations on finite and infinite-dimensional manifolds arising in Physics, and its validity is based on the deep inter-relation between modem Differential Geometry and certain parts of the Theory of Stochastic Processes, discovered not so long ago. The foundation of our topic is presented in the contemporary mathematical literature by a lot of publications devoted to certain parts of the above-mentioned themes and connected with the scope of material of this book. There exist some monographs on Stochastic Differential Equations on Manifolds (e. g. [9,36,38,87]) based on the Stratonovich approach. In [7] there is a detailed description of It6 equations on manifolds in Belopolskaya-Dalecky form. Nelson's book [94] deals with Stochastic Mechanics and mean derivatives on Riemannian Manifolds. The books and survey papers on the Lagrange approach to Hydrodynamics [2,31,73,88], etc. , give good presentations of the use of infinite-dimensional ordinary differential geometry in ideal hydrodynamics. We should also refer here to [89,102], to the previous books by the author [53,64], and to many others.
Book Synopsis Differential Geometry For Physicists And Mathematicians: Moving Frames And Differential Forms: From Euclid Past Riemann by : Jose G Vargas
Download or read book Differential Geometry For Physicists And Mathematicians: Moving Frames And Differential Forms: From Euclid Past Riemann written by Jose G Vargas and published by World Scientific. This book was released on 2014-03-06 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a book that the author wishes had been available to him when he was student. It reflects his interest in knowing (like expert mathematicians) the most relevant mathematics for theoretical physics, but in the style of physicists. This means that one is not facing the study of a collection of definitions, remarks, theorems, corollaries, lemmas, etc. but a narrative — almost like a story being told — that does not impede sophistication and deep results.It covers differential geometry far beyond what general relativists perceive they need to know. And it introduces readers to other areas of mathematics that are of interest to physicists and mathematicians, but are largely overlooked. Among these is Clifford Algebra and its uses in conjunction with differential forms and moving frames. It opens new research vistas that expand the subject matter.In an appendix on the classical theory of curves and surfaces, the author slashes not only the main proofs of the traditional approach, which uses vector calculus, but even existing treatments that also use differential forms for the same purpose.
Book Synopsis Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional by : Enno Keßler
Download or read book Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional written by Enno Keßler and published by Springer. This book was released on 2019-08-29 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book treats the two-dimensional non-linear supersymmetric sigma model or spinning string from the perspective of supergeometry. The objective is to understand its symmetries as geometric properties of super Riemann surfaces, which are particular complex super manifolds of dimension 1|1. The first part gives an introduction to the super differential geometry of families of super manifolds. Appropriate generalizations of principal bundles, smooth families of complex manifolds and integration theory are developed. The second part studies uniformization, U(1)-structures and connections on Super Riemann surfaces and shows how the latter can be viewed as extensions of Riemann surfaces by a gravitino field. A natural geometric action functional on super Riemann surfaces is shown to reproduce the action functional of the non-linear supersymmetric sigma model using a component field formalism. The conserved currents of this action can be identified as infinitesimal deformations of the super Riemann surface. This is in surprising analogy to the theory of Riemann surfaces and the harmonic action functional on them. This volume is aimed at both theoretical physicists interested in a careful treatment of the subject and mathematicians who want to become acquainted with the potential applications of this beautiful theory.
Download or read book Witten Index written by Andrei Smilga and published by . This book was released on 2024-09-27 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is devoted to vacuum structure of supersymmetric quantum mechanical and field theories. The Witten Index (the title of book) is a powerful theoretical tool, which allows one to find out whether supersymmetry breaks down spontaneously in a given theory or not. This is the main physical interest of this notion, but the latter has also many beautiful purely mathematical connotations. It represents a variant of the so-called equivariant index introduced by Cartan back in 1950 and is closely related to the Atiyah-Singer index. In his previous book "Differential Geometry through Supersymmetric Glasses", World Scientific, 2020, the author showed how the supersymmetric language allows one to describe, in a rather transparent way, some known facts of differential geometry and also derive new results in this field. This book is mostly addressed to experts in quantum field theory, but the first three chapters has an introductory textbook nature and can be read by a non-expert. In Chapters 4 and 5, the general aspects of the Witten index are explained and the relationship with pure mathematical problems is elucidated. Chapters 6, 7, 8 are devoted to four-dimensional supersymmetric gauge theories: pure supersymmetric Yang-Mills theories in Chapter 6, the theories including a nonchiral (Chapter 7) and chiral (Chapter 8) matter. Chapter 9 is devoted to the so-called maximal supersymmetric quantum mechanics obtained by a dimensional reduction of the 10-dimensional supersymmetric Yang-Mills theory. In Chapter 10, the numbers of supersymmetric vacua in 3-dimensional supersymmetric Yang-Mills-Chern-Simons theories is calculated. Finally, in Chapter 11, the author discusses some relatives of the Witten index, such as the indices for the 4-dimensional theories compactified on S3 x R, rather than 4-torus or the so-called Cecolli-Fendley-Intriligator-Vafa index.
Book Synopsis Supersymmetry and Noncommutative Geometry by : Wim Beenakker
Download or read book Supersymmetry and Noncommutative Geometry written by Wim Beenakker and published by Springer. This book was released on 2015-11-04 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work the question whether noncommutative geometry allows for supersymmetric theories is addressed. Noncommutative geometry has seen remarkable applications in high energy physics, viz. the geometrical interpretation of the Standard Model, however such a question has not been answered in a conclusive way so far.The book starts with a systematic analysis of the possibilities for so-called almost-commutative geometries on a 4-dimensional, flat background to exhibit not only a particle content that is eligible for supersymmetry, but also have a supersymmetric action. An approach is proposed in which the basic `building blocks' of potentially supersymmetric theories and the demands for their action to be supersymmetric are identified. It is then described how a novel kind of soft supersymmetry breaking Lagrangian arises naturally from the spectral action. Finally, the above formalism is applied to explore the existence of a noncommutative version of the minimal supersymmetric Standard Model.This book is intended for mathematical/theoretical physicists with an interest in the applications of noncommutative geometry to supersymmetric field theories.
Book Synopsis Advances in Discrete Differential Geometry by : Alexander I. Bobenko
Download or read book Advances in Discrete Differential Geometry written by Alexander I. Bobenko and published by Springer. This book was released on 2016-08-12 with total page 441 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. The authors take a closer look at discrete models in differential geometry and dynamical systems. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Nevertheless, the difference between the corresponding smooth curves, surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics. This book is written by specialists working together on a common research project. It is about differential geometry and dynamical systems, smooth and discrete theories, and on pure mathematics and its practical applications. The interaction of these facets is demonstrated by concrete examples, including discrete conformal mappings, discrete complex analysis, discrete curvatures and special surfaces, discrete integrable systems, conformal texture mappings in computer graphics, and free-form architecture. This richly illustrated book will convince readers that this new branch of mathematics is both beautiful and useful. It will appeal to graduate students and researchers in differential geometry, complex analysis, mathematical physics, numerical methods, discrete geometry, as well as computer graphics and geometry processing.
Author :Iskander Asanovich Taĭmanov Publisher :European Mathematical Society ISBN 13 :9783037190500 Total Pages :224 pages Book Rating :4.1/5 (95 download)
Book Synopsis Lectures on Differential Geometry by : Iskander Asanovich Taĭmanov
Download or read book Lectures on Differential Geometry written by Iskander Asanovich Taĭmanov and published by European Mathematical Society. This book was released on 2008 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential geometry studies geometrical objects using analytical methods. Like modern analysis itself, differential geometry originates in classical mechanics. For instance, geodesics and minimal surfaces are defined via variational principles and the curvature of a curve is easily interpreted as the acceleration with respect to the path length parameter. Modern differential geometry in its turn strongly contributed to modern physics. This book gives an introduction to the basics of differential geometry, keeping in mind the natural origin of many geometrical quantities, as well as the applications of differential geometry and its methods to other sciences. The text is divided into three parts. The first part covers the basics of curves and surfaces, while the second part is designed as an introduction to smooth manifolds and Riemannian geometry. In particular, Chapter 5 contains short introductions to hyperbolic geometry and geometrical principles of special relativity theory. Here, only a basic knowledge of algebra, calculus and ordinary differential equations is required. The third part is more advanced and introduces into matrix Lie groups and Lie algebras the representation theory of groups, symplectic and Poisson geometry, and applications of complex analysis in surface theory. The book is based on lectures the author held regularly at Novosibirsk State University. It is addressed to students as well as anyone who wants to learn the basics of differential geometry.
Book Synopsis Differential Geometry and Relativity by : M. Cahen
Download or read book Differential Geometry and Relativity written by M. Cahen and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 303 pages. Available in PDF, EPUB and Kindle. Book excerpt: On the occasion of the sixtieth birthday of Andre Lichnerowicz a number of his friends, many of whom have been his students or coworkers, decided to celebrate this event by preparing a jubilee volume of contributed articles in the two main fields of research marked by Lichnerowicz's work, namely differential geometry and mathematical physics. Limitations of space and time did not enable us to include papers from all Lichnerowicz's friends nor from all his former students. It was equally impossible to reflect in a single book the great variety of subjects tackled by Lichnerowicz. In spite of these limitations, we hope that this book reflects some of the present trends of fields in which he worked, and some of the subjects to which he contributed in his long - and not yet finished - career. This career was very much marked by the influence of his masters, Elie Cartan who introduced him to research in mathematics, mainly in geometry and its relations with mathematical physics, and Georges Darmois who developed his interest for mechanics and physics, especially the theory of relativity and electromagnetism. This par ticular combination, and his personal talent, made of him a natural scientific heir and continuator of the French mathematical physics school in the tradition of Henri Poincare. Some of his works would even be best qualified by a new field name, that of physical ma thematics: branches of pure mathematics entirely motivated by physics.
Book Synopsis Differential Geometry in the Large by : Owen Dearricott
Download or read book Differential Geometry in the Large written by Owen Dearricott and published by Cambridge University Press. This book was released on 2020-10-22 with total page 401 pages. Available in PDF, EPUB and Kindle. Book excerpt: From Ricci flow to GIT, physics to curvature bounds, Sasaki geometry to almost formality. This is differential geometry at large.
Book Synopsis Geometry of Supersymmetric Gauge Theories, Including an Introduction to BRS Differential Algebras and Anomalies by : François Gieres
Download or read book Geometry of Supersymmetric Gauge Theories, Including an Introduction to BRS Differential Algebras and Anomalies written by François Gieres and published by . This book was released on 1988 with total page 189 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Visual Differential Geometry and Forms by : Tristan Needham
Download or read book Visual Differential Geometry and Forms written by Tristan Needham and published by Princeton University Press. This book was released on 2021-07-13 with total page 530 pages. Available in PDF, EPUB and Kindle. Book excerpt: An inviting, intuitive, and visual exploration of differential geometry and forms Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton’s geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner. Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss’s famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein’s field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell’s equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan’s method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book. Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.
Book Synopsis Applied Differential Geometry: A Modern Introduction by : Vladimir G Ivancevic
Download or read book Applied Differential Geometry: A Modern Introduction written by Vladimir G Ivancevic and published by World Scientific. This book was released on 2007-05-21 with total page 1346 pages. Available in PDF, EPUB and Kindle. Book excerpt: This graduate-level monographic textbook treats applied differential geometry from a modern scientific perspective. Co-authored by the originator of the world's leading human motion simulator — “Human Biodynamics Engine”, a complex, 264-DOF bio-mechanical system, modeled by differential-geometric tools — this is the first book that combines modern differential geometry with a wide spectrum of applications, from modern mechanics and physics, via nonlinear control, to biology and human sciences. The book is designed for a two-semester course, which gives mathematicians a variety of applications for their theory and physicists, as well as other scientists and engineers, a strong theory underlying their models.
Book Synopsis Differential Geometry and Relativity Theory by : RichardL. Faber
Download or read book Differential Geometry and Relativity Theory written by RichardL. Faber and published by Routledge. This book was released on 2017-10-19 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differentilil Geometry and Relativity Theory: An Introduction approaches relativity asa geometric theory of space and time in which gravity is a manifestation of space-timecurvature, rathe1 than a force. Uniting differential geometry and both special and generalrelativity in a single source, this easy-to-understand text opens the general theory of relativityto mathematics majors having a backgr.ound only in multivariable calculus and linearalgebra.The book offers a broad overview of the physical foundations and mathematical details ofrelativity, and presents concrete physical interpretations of numerous abstract concepts inRiemannian geometry. The work is profusely illustrated with diagrams aiding in the understandingof proofs and explanations. Appendices feature important material on vectoranalysis and hyperbolic functions.Differential Geometry and Relativity Theory: An Introduction serves as the ideal textfor high-level undergraduate couues in mathematics and physics, and includes a solutionsmanual augmenting classroom study. It is an invaluable reference for mathematicians interestedin differential and IUemannian geometry, or the special and general theories ofrelativity
Book Synopsis Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics by : Yuri E. Gliklikh
Download or read book Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics written by Yuri E. Gliklikh and published by Springer. This book was released on 1996-08-31 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: The geometrical methods in modem mathematical physics and the developments in Geometry and Global Analysis motivated by physical problems are being intensively worked out in contemporary mathematics. In particular, during the last decades a new branch of Global Analysis, Stochastic Differential Geometry, was formed to meet the needs of Mathematical Physics. It deals with a lot of various second order differential equations on finite and infinite-dimensional manifolds arising in Physics, and its validity is based on the deep inter-relation between modem Differential Geometry and certain parts of the Theory of Stochastic Processes, discovered not so long ago. The foundation of our topic is presented in the contemporary mathematical literature by a lot of publications devoted to certain parts of the above-mentioned themes and connected with the scope of material of this book. There exist some monographs on Stochastic Differential Equations on Manifolds (e. g. [9,36,38,87]) based on the Stratonovich approach. In [7] there is a detailed description of It6 equations on manifolds in Belopolskaya-Dalecky form. Nelson's book [94] deals with Stochastic Mechanics and mean derivatives on Riemannian Manifolds. The books and survey papers on the Lagrange approach to Hydrodynamics [2,31,73,88], etc. , give good presentations of the use of infinite-dimensional ordinary differential geometry in ideal hydrodynamics. We should also refer here to [89,102], to the previous books by the author [53,64], and to many others.
Book Synopsis Advances in Differential Geometry and General Relativity by : John K. Beem
Download or read book Advances in Differential Geometry and General Relativity written by John K. Beem and published by American Mathematical Soc.. This book was released on 2004 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume consists of expanded versions of invited lectures given at The Beemfest: Advances in Differential Geometry and General Relativity (University of Missouri-Columbia) on the occasion of Professor John K. Beem's retirement. The articles address problems in differential geometry in general and in particular, global Lorentzian geometry, Finsler geometry, causal boundaries, Penrose's cosmic censorship hypothesis, the geometry of differential operators with variable coefficients on manifolds, and asymptotically de Sitter spacetimes satisfying Einstein's equations with positive cosmological constant. The book is suitable for graduate students and research mathematicians interested in differential geometry.
