Differential Geometry through Supersymmetric Glasses

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.Contents: Geometry:Real ManifoldsComplex ManifoldsHyper-Kähler and HKT ManifoldsPhysics:Dynamical Systems with and without Grassmann VariablesSupersymmetryPath Integrals and the Witten IndexSuperspace and SuperfieldsSynthesis:Supersymmetric Description of the de Rham ComplexSupersymmetric Description of the Dolbeault ComplexSigma Models with Extended SupersymmetriesTaming the Zoo of ModelsHK and HKT through Harmonic GlassesGauge Fields on the ManifoldsAtiyah-Singer Theorem
Readership: Graduate students and researchers interested in theoretical and mathematical physics.Supersymmetry;Differential Geometry;Sigma Models;Complex Manifolds;Hyper-Kaehler Manifolds00