nLab
reduction and lift of structure groups

Idea

A special case of G-structure.

For $G\to K$ a monomorphism of groups, a $G$-structure on a $K$-principal bundle is a reduction of the structure group from $K$ to $G$.

Alternatively, for $G\to K$ an epimorphism of groups, a $G$-structure on a $K$-principal bundle is a lift of the structure group from $K$ to $G$.

Remark

The distinction between epimorphism and monomorphism, hence between reduction and lift of structure groups is, however, not quite “homotopy correct”. For instance in the first steps of the Whitehead tower of the orthogonal group

$$\mathrm{Spin}(n)\to \mathrm{SO}(n)\hookrightarrow O(n)(\hookrightarrow \mathrm{GL}(n))$$Spin(n) \to SO(n) \hookrightarrow O(n) (\hookrightarrow GL(n))

the morphism $\mathrm{Spin}(n)\to \mathrm{SO}(n)$ is epi, while $\mathrm{SO}(n)\to O(n)$ is mono.

See (NSS, section 4.3).

Definition

We spell out three equivalent definitions.

Let $H$ be the ambient (∞,1)-topos, let $G,K\in \mathrm{Grp}(H$ be two ∞-groups and let $\varphi :G\to K$ be a homomorphism, hence $B\varphi :BG\to BK$ the morphism in $H$ between their deloopings. Write

$$\begin{array}{ccc}K⫽G& \to & BG\\ & & {\downarrow }^{B\varphi }\\ & & BK\end{array}$$\array{ K\sslash G &\to& \mathbf{B}G \\ && \downarrow^{\mathrlap{\mathbf{B}\phi}} \\ && \mathbf{B}K }

for the corresponding fiber sequence, with $K⫽G$ the homotopy fiber of the given morphism. By the discussion at ∞-action this exhibits the canonical $K$-∞-action on the coset object $K⫽G$.

Let furthermore $P\to X$ be a $K$-principal ∞-bundle in $H$. By the discussion there this is modulated essentially uniquely by a cocycle morphism $k:X\to BK$ such that there is a fiber sequence

$$\begin{array}{ccc}P& \to & X\\ & & \downarrow \\ & & BK\end{array}.$$\array{ P &\to& X \\ && \downarrow \\ && \mathbf{B}K } \,.

Reduction of the cocycle

The reduction of the structure of the cocycle $k$ is a diagram

$$\begin{array}{ccccc}X& & \stackrel{\sigma }{\to }& & BG\\ & {}_k\searrow & {⇙}_{\tilde{\sigma }}& \swarrow \\ & & BK\end{array}$$\array{ X &&\stackrel{\sigma}{\to}&& \mathbf{B}G \\ & {}_{\mathllap{k}}\searrow &\swArrow_{\tilde\sigma}& \swarrow \\ && \mathbf{B}K }

in $H$, hence a morphism

$$\sigma :k\to B\varphi$$\sigma : k \to \mathbf{B}\phi

in the slice (∞,1)-topos $B_{/BK}$.

Section of the associated coset-bundle

By the discussion at associated ∞-bundle such a diagram is equivalently a section

$$\sigma \in {\Gamma }_X(P{\times }_{K}K⫽G)$$\sigma \in \Gamma_X(P \times_{K} K\sslash G)

of the associated $K⫽G$ fiber ∞-bundle.

Equivariant map to the coset

The above is the categorical semantics of what in the homotopy type theory internal language of $H$ is given by the syntax

$$⊢(\prod _{x:BK}P\to K⫽G):\mathrm{Type}.$$\vdash (\prod_{x : \mathbf{B}K} P \to K\sslash G) : Type \,.

See the discussion at ∞-action.

This expresses the fact that the reduction of the structure group along $\varphi$ is equivalently a $K$-equivariant map $P\to K⫽G$.

Examples

• reduction of tangent bundle along orthogonal group inclusion $O(n)\hookrightarrow \mathrm{GL}(n)$: vielbein, orthogonal structure,

• reduction of tangent bundle along symplectic group inclusion $\mathrm{Sp}(2n)\to \mathrm{GL}(2n)$: almost symplectic structure;

  subsequent lift to the metaplectic group $\mathrm{Mp}(2n)\to \mathrm{Sp}(2n)$: metaplectic structure

  induced lift over Lagrangian submanifolds to the metalinear group $\mathrm{Ml}(n)\to \mathrm{GL}(n)$: metalinear structure;

• reduction of tangent bundle along inclusion of complex general linear group $\mathrm{GL}(n,ℂ)\hookrightarrow \mathrm{GL}(2n,ℝ)$: almost complex structure;

  further reduction to the unitary group $U(n)\hookrightarrow \mathrm{GL}(n,ℂ)$: almost Hermitian structure;

• reduction of generalized tangent bundle along $U(n,n)\hookrightarrow O(2n,2n)$: generalized complex geometry,

  • further reduction along $\mathrm{SU}(n,n)\hookrightarrow O(2n,2n)$: generalized Calabi-Yau manifold ;

    • for $n=3$: further reduction along $\mathrm{SU}(3)\times \mathrm{SU}(3)\hookrightarrow \mathrm{SU}(3,3)$, N=2 type II sugra compactification

• reduction of generalized tangent bundle along $G_2\times G_2\hookrightarrow \mathrm{SO}(7,7)$: G2-structure;

• reduction of generalized tangent bundle along $O(n)\times O(n)\hookrightarrow O(n,n)$: generalized vielbein, type II geometry;

• reduction of exceptional tangent bundle along maximal compact subgroup of exceptional Lie group $H_n\hookrightarrow E_{n(n)}$: exceptional generalized geometry

• reduction of exceptional tangent bundle along $\mathrm{SU}(7)\hookrightarrow E_{7(7)}$: N=1 11d sugra compactification on

References

In the generality of principal infinity-bundles, reductions/lifts of structure groups are discused in section 4.3 of

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal ∞-bundles -- General theory (arXiv:1207.0248)