nLab definite globalization of WZW term

Idea

Given a WZW term $\mathbf{L}$ (a differential cocycle) on some $V$, and given a $V$-manifold $X$, a definite globalization of $\mathbf{L}$ over $V$ is a WZW term $\mathbf{L}^X$ on $X$ which is suitably locally equivalent to $\mathbf{L}$. In particular the curvature form of $\mathbf{L}^X$ is a definite form on $X$, definite on the curvature form of the local model $\mathbf{L}$.

Hence a definite globalization of a WZW term may be thought of as a higher prequantization of a definite form.

Definite gobalizations of WZW terms $\mathbf{L}$ induce definite parameterizations, namely parameterization of the restriction $\mathbf{L}^{inf}$ of $\mathbf{L}$ to the infinitesimal disk in $V$, over the infinitesimal disk bundle of $X$. These in turn correspond to G-structures for $G$ the homotopy stabilizer group of $\mathbf{L}^{inf}$.

Examples

Geometric pre-quantization

By the Darboux theorem for line bundles, every prequantization of a symplectic manifold is automatically a definite globalization of some fixed pre-quantization of $(\mathbb{R}^{2n}, \mathbf{d}p_i \wedge \mathbf{d}q^i)$.

Super $p$-branes on curved super-spacetimes

The equations of motion of supergravity theories typically imply that the WZW curvatures of the relevant super p-brane sigma models on super Minkowski spacetime extend as definite forms over the super-spacetime. Hence the full WZW term defining the super p-brane sigma model needs to be a definite globalization over super-spacetime of the local model over super-Minkowski spacetime.

Related concepts

• higher Cartan geometry

References

• section “definite forms” in differential cohomology in a cohesive topos