Affiliation:
1. Mathematical Institute and Jesus College, University of Oxford, UK
2. Max-Planck-Institut Für Mathematik, Bonn, Germany
Abstract
Abstract
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.
Funder
Engineering and Physical Sciences Research Council
Publisher
Oxford University Press (OUP)
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