geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
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
is the curvature 2-form of $\nabla$.
Choosing a prequantum line bundle is the first step in the geometric quantization of $(X, \omega)$.
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
The above definition has an immediate generalization to n-plectic geometry.
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
is the curvature $(n+1)$-form of $\nabla$.
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
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 |
Lecture notes with more details are in the section Lagrangians and Action functionals of