nLab
prequantum 0-bundle

Context

Geometric quantization

: Lagrangians and Action functionals + Geometric Quantization

Prerequisites

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Prequantum field theory

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    • prequantum 1-bundle = , regular, = lift of to

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Geometric quantization

Applications

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Bundles

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Context

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Classes of bundles

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Universal bundles

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Presentations

Examples

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Constructions

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Quantum field theory

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Physics

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Surveys, textbooks and lecture notes

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Contents

Idea

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.

0kn0 \leq k \leq n(off-shell) traditional terminology
00
11
kk
n1n-1(off-shell)
nn

References

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.