Introduction to graded geometry

Abstract

This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and exhibit their graded local basis. The paper also reviews some correspondences between differential Z-graded manifolds and algebraic structures.

15 pages, to appear in European Journal of Mathematics

References in corpus (5)

• Supergroupoids, double structures, and equivariant cohomology
• The Atiyah class of a dg-vector bundle
• Atiyah classes and dg-Lie algebroids for matched pairs
• ${\mathbb{Z}}_2^n$-Supergeometry I: Manifolds and Morphisms
• The category of ${\mathbb{Z}}_2^n$-supermanifolds

Cited by in corpus (10)

• $L_{\infty }$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism
• $L_{\infty }$-Algebras, the BV Formalism, and Classical Fields
• Global Theory of Graded Manifolds
• The Schwarz-Voronov Embedding of ${\mathbb{Z}}_2^n$-Manifolds
• Monoidally graded manifolds
• Graded Generalized Geometry
• Connections Adapted to Non-Negatively Graded Structures
• Threefold Nature of Graded Vector Bundles
• Extended Field Theories as higher Kaluza-Klein theories
• $H$-covering of a supermanifold