Shifted symplectic Lie algebroids

TL;DR

This paper classifies shifted symplectic Lie algebroids and their higher gauge symmetries, connecting them to classical geometric structures and providing new examples and interpretations within shifted symplectic geometry.

Contribution

It offers a classification of shifted symplectic Lie algebroids and their symmetries, linking them to higher geometric structures like Courant algebroids and gerbes.

Findings

Classified zero-, one-, and two-shifted symplectic algebroids.

Produced new twisted Courant algebroids from codimension-two cycles.

Provided symplectic interpretations for features like twists and Pontryagin classes.

Abstract

Shifted symplectic Lie and $L_\infty$ algebroids model formal neighbourhoods of manifolds in shifted symplectic stacks, and serve as target spaces for twisted variants of classical AKSZ topological field theory. In this paper, we classify zero-, one- and two-shifted symplectic algebroids and their higher gauge symmetries, in terms of classical geometric "higher structures", such as Courant algebroids twisted by $\Omega^2$-gerbes. As applications, we produce new examples of twisted Courant algebroids from codimension-two cycles, and we give symplectic interpretations for several well known features of higher structures (such as twists, Pontryagin classes, and tensor products). The proofs are valid in the $C^\infty$, holomorphic and algebraic settings, and are based on a number of technical results on the homotopy theory of $L_\infty$ algebroids and their differential forms, which may be of independent interest.