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
(see also Chern-Weil theory, parameterized homotopy theory)
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
Discussion of prequantized (and polarized) symplectic manifolds in the context of cobordism rings and Thom spectra is in