nLab prequantum 0-bundle



Geometric quantization



Quantum field theory


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In higher differential geometry the notion of connection on a bundle and in particular that of circle bundle with connection is refined to a tower of notions of circle n-bundles with connection for all nn \in \mathbb{N}. In particular also the degenerate case of n=0n = 0 is defined and fits into this tower: a 0-bundle is simply a function with values in the given structure group (e.g. the circle group U(1)U(1)).

Moreover, for a prequantum field theory defined by an extended Lagrangian there is such a circle (n-k)-bundle with connection for each closed manifold of dimension kk, called the prequantum (n-k)-bundle. For n=kn = k this is the action functional of the theory. Hence the action functional may be thought of as the prequantum 0-bundle of an extended prequantum field theory.

extended prequantum field theory

0kn0 \leq k \leq n(off-shell) prequantum (n-k)-bundletraditional terminology
00differential universal characteristic maplevel
11prequantum (n-1)-bundleWZW bundle (n-2)-gerbe
kkprequantum (n-k)-bundle
n1n-1prequantum 1-bundle(off-shell) prequantum bundle
nnprequantum 0-bundleaction functional


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

Created on January 5, 2013 at 19:56:26. See the history of this page for a list of all contributions to it.