bundles

# Contents

## Definition

###### Definition

For $\left(X,\omega \right)$ a 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

$\omega ={F}_{\nabla }$\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 $\left(X,\omega \right)$.

###### Remark

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

${H}^{2}\left(X{\right)}_{\mathrm{diff}}\stackrel{F}{\to }{\Omega }_{\mathrm{int}}^{2}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$H^2(X)_{diff} \stackrel{F}{\to} \Omega^2_{int}(X) \,.

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

###### Definition

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

$\omega ={F}_{\nabla }$\omega = F_\nabla

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

###### Remark

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

${H}^{n+1}\left(X{\right)}_{\mathrm{diff}}\stackrel{F}{\to }{\Omega }_{\mathrm{int}}^{n+1}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$H^{n+1}(X)_{diff} \stackrel{F}{\to} \Omega^{n+1}_{int}(X) \,.

extended prequantum field theory

$0\le k\le n$(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 March 21, 2013 23:02:53 by Urs Schreiber (89.204.138.71)