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)
In higher geometric prequantization (see there for more details for the moment) the notion of prequantum circle bundle is refined to that of a prequantum circle n-bundle with connection for all $n \in \mathbb{N}$.
Given an E-infinity ring $E$ and an infinity-representation
we have the associated infinity-bundle
This is the higher analog of the prequantum line bundle, the higher prequantum line bundle.
See at motivic quantization for more on this.
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 |