nLab
prequantum line bundle

Context

Geometric quantization

Bundles

Contents

Definition

Definition

For (X,ω)(X,\omega) a (pre-)symplectic manifold such that ω\omega is an integral form, a prequantum line bundle is any line bundle PXP \to X with connection \nabla on XX such that

ω=F \omega = F_\nabla

is the curvature 2-form of \nabla.

Remark

Choosing a prequantum line bundle is the first step in the geometric quantization of (X,ω)(X, \omega).

Remark

In cohomology, a choice of prequantum line bundle corresponds to a lift from curvature 2-forms to ordinary differential cohomology H 2(X) diffH^2(X)_{diff} through the curvature projection

H 2(X) diffFΩ int 2(X). H^2(X)_{diff} \stackrel{F}{\to} \Omega^2_{int}(X) \,.

The above definition has an immediate generalization to n-plectic geometry.

Definition

For (X,ω)(X,\omega) an n-plectic manifold such that ω\omega is an integral form, a prequantum circle n- bundle is any circle n-bundle with connection (PX,)(P \to X, \nabla) such that

ω=F \omega = F_\nabla

is the curvature (n+1)(n+1)-form of \nabla.

Remark

In cohomology, a choice of prequantum circle nn-bundle corresponds to a lift from curvature (n+1)(n+1)-forms to ordinary differential cohomology H n+1(X) diffH^{n+1}(X)_{diff} through the curvature projection

H n+1(X) diffFΩ int n+1(X). H^{n+1}(X)_{diff} \stackrel{F}{\to} \Omega^{n+1}_{int}(X) \,.

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

References

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

Revised on September 15, 2013 18:15:45 by Urs Schreiber (89.204.137.43)