nLab prequantum line bundle

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

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

Related concepts

• prequantized Lagrangian correspondence

• prequantum circle n-bundle

• local prequantum field theory

• fiber bundles in physics

• circle bundle, circle n-bundle with connection

extended prequantum field theory

$0 \leq k \leq n$	(off-shell) prequantum (n-k)-bundle	traditional terminology
$0$	differential universal characteristic map	level
$1$	prequantum (n-1)-bundle	WZW bundle (n-2)-gerbe
$k$	prequantum (n-k)-bundle
$n-1$	prequantum 1-bundle	(off-shell) prequantum bundle
$n$	prequantum 0-bundle	action functional

• geometric quantization, higher geometric quantization

References

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

• geometry of physics

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)