Geometry Meets Supersymmetry

This book is devoted to description of different geometric structures, including the classical structures of differential geometry like de Rham complex, Dolbeaux complex and Dirac complex and also some structures that have not attracted yet an attention of mathematicians in the language of supersymmetric quantum mechanics. The book is addressed both to physicists and mathematicians. The first part is addressed mainly to physicists and describes basic properties of smooth manifolds of different type: real, complex, Kaehler, hyperkaehler and HKT. The second part is addressed to mathematicians and describes basic properties of classical and quantum mechanical systems, including supersymmetric systems with Grassmann dynamic variables. The third part is called Synthesis: we show how the physical methods allow one to describe in a simple way and understand many nontrivial geometric facts. For example, the famous Atiyah-Singer theorem admits a rather natural and simple supersymmetric interpretation. This book is an updated and expanded version of the book Differential geometry through supersymmetric glasses published in 2020 by World Scientific. New material on hyperkaehler geometry and its supersymmetric description and on the gauge fields in CPn manifolds is added.