geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
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Given a presymplectic form $(X, \omega)$ (hence any closed differential 2-form), a prequantization of it in the traditional sense is a choice of circle bundle with connection on $X$ whose curvature 2-form is $\omega$. Since the circle group $U(1)$ is equivalent, as a smooth ∞-group to the 2-group coming from the crossed module $(\mathbb{Z} \hookrightarrow \mathbb{R})$, such a lift exists whenever the periods of $\omega$ are integral, hence are in the inclusion of the integers into the real numbers $\mathbb{Z} \hookrightarrow \mathbb{R}$.
But conversely this means that for any closed differential 2-form $\omega$ there is a connection on a 2-bundle on a $(\Gamma \hookrightarrow \mathbb{R})$-principal 2-bundle, where $\Gamma \hookrightarrow \mathbb{R}$ is the inclusion of the discrete group of periods of $\omega$.
If here $\Gamma \hookrightarrow \mathbb{R}$ is a global multiple of the canonical inclusion $\mathbb{Z} \hookrightarrow \mathbb{R}$ then there is of course an isomorphism $(\mathbb{R}/\Gamma) \simeq (\mathbb{R}/\mathbb{Z}) = U(1)$. This identification coresponds to a choice of Planck's constant (see there).
If however $\Gamma$ is not finitely generated, then the smooth 2-group $(\Gamma \to \mathbb{R})$ is not equivalent to a smooth 1-group, and hence this is a genuine case of higher geometric prequantization. The upshot being that while not every closed 2-form has an ordinary prequantization, it always does have one in higher geometric prequantization, at least if we admit to choose the structure 2-group accordingly.
These considerations are currently mostly motivated purely mathematically. But the claim is that there useful physical applications (… eventually to be added here…).
The observation that non-integral closed 2-forms can be prequantized by diffeological principal bundles for the diffeological quotient of $\mathbb{R}$ by the subgroup of periods is due to
values of hamiltonians_, Math. Z. 201 (1989), 75—82
and reviewed in section II 2.5 of
The remark that the non-manifold quotient is usefully thought of as regarded instead in higher geometric prequantization by prequantum principal 2-bundles was made in
Mentioning of prequantization of non-integral forms is also in section 3.2.1 of
Last revised on September 1, 2013 at 20:20:51. See the history of this page for a list of all contributions to it.