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

Discussion of prequantized (and polarized) symplectic manifolds in the context of cobordism rings and Thom spectra is in

  • Jack Morava, Cobordism of symplectic manifolds and asymptotic expansions, talk at the conference in honor of S.P. Novikov’s 60th birthday (arXiv:9908070)

Revised on July 16, 2015 01:24:33 by Urs Schreiber (82.69.72.163)