Speaker
Description
In geometry, the notion of a covering space is classical and well estab-
lished. A familiar example is the universal covering: p : ℝ → S¹, given by
t ↦ exp(it). Analogous constructions also appear in algebra, for instance
in the theory of modules over rings, where one encounters flat or torsion-
free coverings. Despite arising in different contexts, these coverings share a
common underlying idea: an object from a given category is covered by ob-
jects belonging to a smaller (or different) category in such a way that certain
universal properties are satisfied.
In the paper “Super Atiyah classes and obstructions to splitting of su-
permoduli space”, Donagi and Witten introduced a construction of the first
obstruction class to the splitting of a supermanifold. Later we observed that
the infinite prolongation of the Donagi–Witten construction satisfies univer-
sal properties common for other coverings. In other words, this construction
yields a covering of a supermanifold in the category of graded manifolds asso-
ciated with the nontrivial homomorphism ℤ → ℤ₂. Furthermore, the space of
infinite jets can also be viewed as a covering of a (super)manifold in the cate-
gory of graded manifolds corresponding to the homomorphism ℤ × ℤ₂ → ℤ₂,
given by (m, n¯ ) ↦ n¯. (For ordinary manifolds, this homomorphism reduces
to the trivial map ℤ → 0.)
Our talk is devoted to the current state of the theory of graded coverings,
including the general framework, key examples, and a presentation of our
recent results.
