Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



A density on a manifold of dimension nn is a function that to each point assigns an infinitesimal volume (in general signed, and possibly degenerate), hence a volume of nn-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 XX a manifold its 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 given by the determinant homomorphism

det:GL(n)GL(1)Aut Vect( 1). det \;\colon\; GL(n) \to GL(1) \simeq Aut_{Vect}(\mathbb{R}^1) \,.

A section of the density bundle on XX is called a density on XX.

This is the general object against which one has integration of functions on XX.

More generally, for s{0}s \in \mathbb{R} - \{0\} an ss-density is a section of the line bundle which is associated via the determinant to the power of ss:

det s:GL(n)GL(1)Aut Vect( 1). det^{s} \;\colon \; GL(n) \to GL(1) \simeq Aut_{Vect}(\mathbb{R}^1) \,.

The parameter ss is called the weight of the density. In particular for s=1/2s = 1/2 one speaks of half-densities.

Properties and applications

Physical interpretation

We earlier spoke of a density (of weight 11) ρ\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 QQ (say, mass). We then call ρ\rho the QQ-density (say, mass density); the integral of ρ\rho over a region RR is the amount of QQ in RR.

Relative to a nondegenerate notion of volume given by another density volvol, the ratio ρ/vol\rho/vol is a scalar field, an intensive quantity which is often also referred to as the density. But ρ\rho itself is more fundamental in the geometry of physics.

Wave functions and canonical Hilbert spaces

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

line bundlesquare rootchoice corresponds to
canonical bundleTheta characteristicover Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundlehalf-density bundle
canonical bundle of Lagrangian submanifoldmetalinear structuremetaplectic correction
determinant line bundlePfaffian line bundle
quadratic secondary intersection pairingpartition function of self-dual higher gauge theoryintegral Wu structure


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 June 21, 2014 at 02:27:01. See the history of this page for a list of all contributions to it.