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

Discussion of prequantized (and polarized) symplectic manifolds in the context of cobordism rings and Thom spectra is in

• Jack Morava, Cobordism of symplectic manifolds and asymptotic expansions, talk at the conference in honor of S.P. Novikov’s 60th birthday (arXiv:9908070)

Revised on July 16, 2015 01:24:33 by Urs Schreiber (82.69.72.163)