vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A density on a manifold of dimension $n$ is a function that to each point assigns an infinitesimal volume (in general signed, and possibly degenerate), hence a volume of $n$-hypercubes in the tangent space at that point. A positive definite density is equivalently a volume element (or a volume form on an oriented manifold).
For $X$ a manifold its 1-density bundle is the real line bundle associated to the principal bundle underlying the tangent bundle by the 1-dimensional representation of the general linear group
A section of the $1$-density bundle on $X$ is called a $1$-density on $X$.
This is the general object against which one has integration of functions on $X$.
More generally, for $s \in \mathbb{R} - \{0\}$ an $s$-density is a section of the line bundle which is associated to the principal bundle by the representation
The parameter $s$ is called the weight of the density. In particular for $s = 1/2$ one speaks of half-densities.
We earlier spoke of a density (of weight $1$) $\rho$ as a measure of volume, but in application to physics a density on spacetime (or space) might as easily be a measure of some other extensive quantity $Q$ (say, mass). We then call $\rho$ the $Q$-density (say, mass density); the integral of $\rho$ over a region $R$ is the amount of $Q$ in $R$.
Relative to a nondegenerate notion of volume given by another density $vol$, the ratio $\rho/vol$ is a scalar field, an intensive quantity which is often also referred to as the density. But $\rho$ itself is more fundamental in the geometry of physics.
In the context of geometric quantization one considers spaces of sections of line bundles (“prequantum line bundles”) and tries to equip these with an inner product given by pointwise pairing followed by integration over the base such as to then complete to a Hilbert space.
One can define the integration against a fixed chosen measure, but more canonical is to instead form the tensor product of the prequantum line bundle with the bundle of half-densities. The compactly supported sections of that tensor bundle can then naturally be integrated. This is sometimes called the “canonical Hilbert space” construction (e.g. (Bates-Weinstein)).
The following table lists classes of examples of square roots of line bundles
A textbook account is for instance on p. 29 of
Discussion of half-densities in the context of geometric quantization is in
Last revised on May 10, 2024 at 10:01:39. See the history of this page for a list of all contributions to it.