nLab geometric quantization of non-integral forms

Idea

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…).

References

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

• Alan Weinstein, Cohomology of symplectomorphism groups and critical

  values of hamiltonians_, Math. Z. 201 (1989), 75—82

and reviewed in section II 2.5 of

• Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization

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

• Christoph Wockel, Non-integral geometric prequantisation, talk at HSLR II (2012)

Mentioning of prequantization of non-integral forms is also in section 3.2.1 of

• Yoshiaki Maeda, Peter Michor, Takushiro Ochiai, Akira Yoshioka (eds.), From Geometry to Quantum Mechanics: In Honor of Hideki Omori