The Schwarz-Voronov Embedding of ${\mathbb Z}_{2}^{n}$-Manifolds

Symmetry Integrability and Geometry-Methods and Applications

article

Andrew James Bruce¹  Eduardo Ibarguengoytia¹  Norbert Poncin¹ 

[1] Mathematics Research Unit, University of Luxembourg

关键词: Hecke groups;    Chazy equations;    Painlev´e analysis;   

DOI  :  10.3842/SIGMA.2020.002

来源: National Academy of Science of Ukraine

【 摘 要 】

Informally, ${\mathbb Z}_2^n$-manifolds are 'manifolds' with ${\mathbb Z}_2^n$-graded coordinates and a sign rule determined by the standard scalar product of their ${\mathbb Z}_2^n$-degrees. Such manifolds can be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant differences, in particular in integration theory. In this paper, we reformulate the notion of a ${\mathbb Z}_2^n$-manifold within a categorical framework via the functor of points. We show that it is sufficient to consider ${\mathbb Z}_2^n$-points, i.e., trivial ${\mathbb Z}_2^n$-manifolds for which the reduced manifold is just a single point, as 'probes' when employing the functor of points. This allows us to construct a fully faithful restricted Yoneda embedding of the category of ${\mathbb Z}_2^n$-manifolds into a subcategory of contravariant functors from the category of ${\mathbb Z}_2^n$-points to a category of Fréchet manifolds over algebras. We refer to this embedding as the Schwarz-Voronov embedding . We further prove that the category of ${\mathbb Z}_2^n$-manifolds is equivalent to the full subcategory of locally trivial functors in the preceding subcategory.

【 授权许可】

Unknown