nLab
definite globalization of WZW term

Contents

Idea

Given a WZW term L\mathbf{L} (a differential cocycle) on some VV, and given a VV-manifold XX, a definite globalization of L\mathbf{L} over VV is a WZW term L X\mathbf{L}^X on XX which is suitably locally equivalent to L\mathbf{L}. In particular the curvature form of L X\mathbf{L}^X is a definite form on XX, definite on the curvature form of the local model L\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 L\mathbf{L} induce definite parameterizations, namely parameterization of the restriction L inf\mathbf{L}^{inf} of L\mathbf{L} to the infinitesimal disk in VV, over the infinitesimal disk bundle of XX. These in turn correspond to G-structures for GG the homotopy stabilizer group of L inf\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 ( 2n,dp idq i)(\mathbb{R}^{2n}, \mathbf{d}p_i \wedge \mathbf{d}q^i).

Super pp-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.

References

Revised on July 31, 2017 04:36:26 by David Corfield (91.88.197.172)