Normal forms of $\mathbb Z$-graded $Q$-manifolds

TL;DR

This paper establishes local and global normal form results for $Q$-structures on $ ext{Z}$-graded $Q$-manifolds, highlighting how their structures are concentrated along the zero-locus of their curvatures, especially in the Koszul--Tate case.

Contribution

It provides new normal form theorems for $Q$-structures on $ ext{Z}$-graded manifolds, including a local splitting theorem and analysis of curvature concentration.

Findings

Normal forms for $Q$-structures are characterized both locally and globally.

Structures are concentrated along the zero-locus of their curvatures.

A local splitting theorem is established.

Abstract

Following recent results of A.K. and V.S. on $\mathbb Z$-graded manifolds, we give several local and global normal forms results for $Q$-structures on those, i.e. for differential graded manifolds. In particular, we explain in which sense their relevant structures are concentrated along the zero-locus of their curvatures, especially when the negative part is of Koszul--Tate type. We also give a local splitting theorem.