# 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 $J^kM×\mathbb{R}$?

-
I understand prequantization to be a thing that you do to a symplectic manifold. What's the symplectic structure on a jet bundle? (It's the cotangent bundle, not the tangent bundle, that has a symplectic structure.) – Qiaochu Yuan Oct 31 '13 at 18:11
The term "contact structure" is usually used in mathematics to refer to the structure on the first jet bundle, with coordinates $(x,y,p)$, given by $0=dy-p \, dx$, where $x$ is a coordinate on $M$ and $(x,y)$ coordinates on the 0-jet bundle. However, many mathematicians working with higher order jet bundles refer to the tautological Pfaffian system on the $k$-jet bundle as a "contact structure". In the first sense, a contact structure on $X$ gives a symplectic structure on $X \times \mathbb{R}$. In the second sense, it doesn't. If $n=2$ and $k=2$ then the jet bundle is even dimensional. – Ben McKay Nov 1 '13 at 8:48
Why do you believe there is a way to do prequantization of a jet bundle, and what do you hope to do with it? In any case, it's unlikely that anybody else has worked this out, so you have an opportunity here to introduce a new idea and show how it can be used. – Deane Yang Nov 3 '13 at 19:02
My idea was decomposing $J^kM×\mathbb{R}$ as $$J^kM×\mathbb{R} = \mathbb{R^2} \oplus T^*M \oplus Sym^2T^*M \oplus\cdots \oplus Sym^kT^*M.$$ and find a way for it. – Hassan Jolany Nov 3 '13 at 19:13
Deane Yang, SO, thats why , I posted a series of questions concerning this question. But I am at my withs end for finding a method for prequantizing $J^kM×\mathbb{R}$ – Hassan Jolany Nov 3 '13 at 19:16

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

• 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

• 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.

-
I think Hossan was wondering about a prequantization the "contact" structure on the bare jet bundle, without any Lagrangian on it (or a vanishing Lagrangian). That would be a somewhat stranger construction, as discussed a bit in the comments to the question. – Igor Khavkine Nov 4 '13 at 10:33
Does this use higher order jets? – Deane Yang Nov 4 '13 at 12:11
Yes Igor Khavkine@, I completely impressed :) – Hassan Jolany Nov 4 '13 at 16:02
So quite generally for any closed (n+1)-form on anything, there is a canonical notion of pre-quantization and it is given by lifting to a cocycle in Deligne cohomology of degree (n+1). Covariant phase space methodology provides such forms on (higher order, yes) jet bundles. The multi-Legendre transform to this has mostly or exclusively been studied for first order jets only, but I suppose that's not a matter of principle. Only in this transformed case have I worked out that the induced L-infinity Poisson bracket encodes the equations of motions and the currents correctly. – Urs Schreiber Nov 4 '13 at 16:54