Shifted Symplectic Lie Algebroids

Shifted symplectic Lie and $L_{\infty }$ algebroids model formal neighborhoods of manifolds in shifted symplectic stacks and serve as target spaces for twisted variants of the classical topological field theory defined by Alexandrov--Kontsevich--Schwarz--Zaboronsky. 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.

移位辛李代数胚和$L_{\infty}$代数胚为移位辛栈中流形的形式邻域建模，并作为由亚历山德罗夫 - 孔采维奇 - 施瓦茨 - 扎博隆斯基所定义的经典拓扑场论的扭曲变体的目标空间。在本文中，我们依据经典几何“高阶结构”，例如由$\Omega ^{2}$- gerbes扭曲的柯朗代数胚，对零移位、一位移和二位移辛代数胚及其高阶规范对称性进行分类。作为应用，我们从余维二循环中产生扭曲柯朗代数胚的新例子，并且我们对高阶结构的若干著名特征（例如扭曲、庞特里亚金类和张量积）给出辛解释。这些证明在$C^{\infty}$、全纯和代数设定下均有效，并且基于关于$L_{\infty}$代数胚及其微分形式的同伦理论的若干技术结果，这些结果可能具有独立的研究价值。