Formal exponentials and linearisations of QP-manifolds

TL;DR

This paper introduces formal exponential maps for graded manifolds, linking them to Grothendieck connections, and applies this framework to linearize QP-manifolds, revealing their local L-infinity algebra structures.

Contribution

It defines formal exponential maps for graded manifolds, relates them to Grothendieck connections, and applies these concepts to linearize QP-manifolds at points.

Findings

Formal exponential maps are uniquely determined by flat Grothendieck connections.

A large class of formal exponentials can be recovered via tangent bundle connections.

Linearizing QP-manifolds yields local L-infinity algebra structures with invariant inner products.

Abstract

We define formal exponential maps for any graded manifold as maps from the formal tangent bundle (that we also define) into the graded manifold. We show that each such map uniquely determines and is determined by its associated Grothendieck connection, which is shown to be flat, and to furnish a resolution of the ring of functions. We then show how a recent construction involving the data of a connection on the tangent bundle recovers a large class of formal exponentials in our definition. As an application, we use a formal exponential map to linearise a QP-manifold at a point. This gives the formal tangent space at each point the structure of an $L_\infty$-algebra with invariant inner product.