About Graded coverings of supermanifolds and their applications

Abstract: In geometry, the concept of a covering space is classical and well established. A familiar example is the universal covering \( p: \mathbb{R} \to S^1 \), given by \( t \mapsto \exp(it) \). Analogous constructions appear in algebra as well—for instance, in the theory of modules over rings, where one encounters flat or torsion-free coverings. Although they arise in different contexts, these notions share a common underlying idea: an object from a given category is covered by an object belonging to a smaller (or different) category in such a way that certain universal properties are satisfied.

In their paper "Super Atiyah classes and obstructions to splitting of supermoduli 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 their construction satisfies universal properties analogous to those found in other covering theories. In other words, this construction yields a covering of a supermanifold in the category of graded manifolds associated with the nontrivial homomorphism \( \mathbb{Z} \to \mathbb{Z}_2 \). Moreover, the space of infinite jets can also be viewed as a covering of a (super)manifold in the category of graded manifolds corresponding to the homomorphism \( \mathbb{Z} \times \mathbb{Z}_2 \to \mathbb{Z}_2 \), given by \( (m, \bar{n}) \mapsto \bar{n} \). (For ordinary manifolds, this homomorphism reduces to the trivial map \( \mathbb{Z} \to 0 \).)

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.

mathematical physicsalgebraic topologydifferential geometryrepresentation theorystatistics theory

Audience: researchers in the topic

( slides | video )

Comments: The seminar starts **30 minutes earlier than usual**.

We shall open ZOOM meeting at 12.45 for virtual coffee and close ZOOM at 14.30

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Prague-Hradec Kralove seminar Cohomology in algebra, geometry, physics and statistics

Series comments: Virtual coffee starts on Zoom already 15 minutes before the seminar.

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