Reduction of the structure group

In mathematics, in particular the theory of principal bundles, one can ask if a principal $G$ -bundle over a group $G$ "comes from" a subgroup $H$ of $G$ . This is called reduction of the structure group (to $H$ ), and makes sense for any map $H \to G$ , which need not be an inclusion map (despite the terminology).

Definition

Formally, given a G-bundle B and a map H → G (which need not be an inclusion), a reduction of the structure group (from G to H) is an H-bundle $B_H$ such that the pushout $B_H \times_H G$ is isomorphic to B.

Note that these do not always exist, nor if they exist are they unique.

As a concrete example, every even-dimensional real vector space is the underlying real space of a complex vector space: it admits a linear complex structure. A real vector bundle admits an almost complex structure if and only if it is the underlying real bundle of a complex vector bundle. This is a reduction along the inclusion GL(n,C) → GL(2n,R)

In terms of transition maps, a G-bundle can be reduced if and only if the transition maps can be taken to have values in H. Note that the term reduction is misleading: it suggests that H is a subgroup of G, which is often the case, but need not be (for example for spin structures): it's properly called a lifting.

More abstractly, "G-bundles over X" is a functor^[1] in G: given a map H → G, one gets a map from H-bundles to G-bundles by inducing (as above). Reduction of the structure group of a G-bundle B is choosing an H-bundle whose image is B.

The inducing map from H-bundles to G-bundles is in general neither onto nor one-to-one, so the structure group cannot always be reduced, and when it can, this reduction need not be unique. For example, not every manifold is orientable, and those that are orientable admit exactly two orientations.

If H is a Lie subgroup of G, then there is a natural one-to-one correspondence between reductions of a G-bundle B to H and global sections of the fiber bundle B/H obtained by quotienting B by the right action of H. Specifically, the fibration B → B/H is a principal H-bundle over B/H. If σ : X → B/H is a section, then the pullback bundle B_H = σ⁻¹B is a reduction of B.^[2]

Examples

Examples for vector bundles, particularly the tangent bundle of a manifold:

$GL^+ < GL$ is an orientation, and this is possible if and only if the bundle is orientable
$SL < GL$ is a volume form; since $SL \to GL^+$ is a deformation retract, a volume form exists if and only if a bundle is orientable
$SL^{\pm} < GL$ is a pseudo-volume form, and this is always possible
$O(n) < GL(n)$ is a Riemannian metric; as $O(n)$ is the maximal compact subgroup (so the inclusion is a deformation retract), this is always possible
$O(1,n-1) < GL(n)$ is a pseudo-Riemannian metric;^[3] there is the topological obstruction to this reduction
$GL(n,\mathbf{C}) < GL(2n,\mathbf{R})$ is an almost complex structure
$GL(n,\mathbf{H})\cdot Sp(1) < GL(4n,\mathbf{R})$ (where $GL(n,\mathbf{H})$ is the group of n×n invertible quaternionic matrices acting on $\mathbf{H}^n \cong \mathbf{R}^{4n}$ on the left and Sp(1)=Spin(3) the group of unit quaternions acting on $\mathbf{H}^n$ from the right) is an almost quaternionic structure^[4]
$\mbox{Spin}(n) \to \mbox{SO}(n)$ (which is not an inclusion: it's a 2-fold covering space) is a spin structure.
$GL(k) \times GL(n-k) < GL(n)$ decomposes a vector bundle as a Whitney sum (direct sum) of sub-bundles of rank k and n − k.

Integrability

Many geometric structures are stronger than G-structures; they are G-structures with an integrability condition. Thus such a structure requires a reduction of the structure group (and can be obstructed, as below), but this is not sufficient. Examples include complex structure, symplectic structure (as opposed to almost complex structures and almost symplectic structures).

Another example is for a foliation, which requires a reduction of the tangent bundle to a block matrix subgroup, together with an integrability condition so that the Frobenius theorem applies.

Notes

↑ Indeed, it is a bifunctor in G and X.
↑ In classical field theory, such a section $\sigma$ describes a classical Higgs field (arXiv: hep-th/0510158).
↑ It is a gravitational field in gauge gravitation theory (arXiv: gr-qc/0512115)
↑ Besse 1987, §14.61

References