pre-quantization of Jet bundle We know that the notion of Jet bundle $J^kM×\mathbb{R}$, is generalization of cotangent bundle. What is the prequantization of $J^kM×\mathbb{R}$?
 A: In local covariant n-dimensional field theory, the jet bundle canonically supports a "pre n-plectic structure", namely that represented by the "pre-symplectic current density" which arises from the boundary term in the variation of the Lagrangian, in the sense of the variational bicomplex (for instance as introduced in Zuckerman, see the references on the nLab at covariant phase space).
Alternatively, the affine dual (first) jet bundle supports a Hamiltonian version of this Lagrangian picture in the form of what traditionally is called a multisymplectoc structure, the the reference on the nLab at multisymplectic geometry.
In either case, one can ask for a higher pre-quantizatioon of the given closed $(n+1)$-form on (dual) jet space in the sense of a lift of the $(n+1)$-form to a Deligne cocycle in degree (n+2), hence to a circle n-bundle with connection whose curvature $(n+1)$-form is the given $(n+1)$-from. This higher geometric pre-quantization as such is discussed in 


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*Fiorenza, Rogers, Schreiber, Higher geometric prequantum theory (arXiv:1304.0236)


The application of this to the higher prequantization of jet bundles in covariant field theory is in section 1.2.11 of


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*Schreiber, Differential cohomology in a cohesive infinity-topos (arXiv:1310.7930)


There is discussed in particular how the higher automorphisms of the higher prequantization, covering diffeos of the base (the jet bundle), have as higher Lie algebra the higher Poisson bracket Lie algebra and how this encodes the covariant Hamilton-de Donder-Weyl form of the Euler-Lagrange equations of motion.
