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Journal Article

Shifted symplectic Lie algebroids


Safronov,  Pavel
Max Planck Institute for Mathematics, Max Planck Society;

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Pym, B., & Safronov, P. (2020). Shifted symplectic Lie algebroids. International Mathematics Research Notices, 2020(21), 7489-7557. doi:10.1093/imrn/rny215.

Cite as: http://hdl.handle.net/21.11116/0000-0007-0CCB-8
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.