bundles

# Contents

## Definition

###### Definition

For $(X,\omega)$ a (pre-)symplectic manifold such that $\omega$ is an integral form, a prequantum line bundle is any line bundle $P \to X$ with connection $\nabla$ on $X$ such that

$\omega = F_\nabla$

is the curvature 2-form of $\nabla$.

###### Remark

Choosing a prequantum line bundle is the first step in the geometric quantization of $(X, \omega)$.

###### Remark

In cohomology, a choice of prequantum line bundle corresponds to a lift from curvature 2-forms to ordinary differential cohomology $H^2(X)_{diff}$ through the curvature projection

$H^2(X)_{diff} \stackrel{F}{\to} \Omega^2_{int}(X) \,.$

The above definition has an immediate generalization to n-plectic geometry.

###### Definition

For $(X,\omega)$ an n-plectic manifold such that $\omega$ is an integral form, a prequantum circle n- bundle is any circle n-bundle with connection $(P \to X, \nabla)$ such that

$\omega = F_\nabla$

is the curvature $(n+1)$-form of $\nabla$.

###### Remark

In cohomology, a choice of prequantum circle $n$-bundle corresponds to a lift from curvature $(n+1)$-forms to ordinary differential cohomology $H^{n+1}(X)_{diff}$ through the curvature projection

$H^{n+1}(X)_{diff} \stackrel{F}{\to} \Omega^{n+1}_{int}(X) \,.$

extended prequantum field theory

$0 \leq k \leq n$(off-shell) prequantum (n-k)-bundletraditional terminology
$0$differential universal characteristic maplevel
$1$prequantum (n-1)-bundleWZW bundle (n-2)-gerbe
$k$prequantum (n-k)-bundle
$n-1$prequantum 1-bundle(off-shell) prequantum bundle
$n$prequantum 0-bundleaction functional

## References

Lecture notes with more details are in the section Lagrangians and Action functionals of

Revised on September 15, 2013 18:15:45 by Urs Schreiber (89.204.137.43)