nLab reduction and lift of structure groups

Idea

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 $G$.

A $G$-reduction of the frame bundle of a smooth manifold is called a G-structure.

Remark

As one passes to higher differential geometry, the (epi, mono) factorization system dissolves into the infinite tower of (n-epi, n-mono) factorization systems, and hence the distinction between reduction and lift of structure groups blurs. One may just consider generally for $G\to K$ a homomorphism of ∞-groups the problem of factoring a modulating morphism $X\to \mathbf{B}K$ through this morphism, up to a chosen homotopy.

Definition

We spell out three equivalent definitions.

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

$$\array{ K\sslash G &\to& \mathbf{B}G \\ && \downarrow^{\mathrlap{\mathbf{B}\phi}} \\ && \mathbf{B}K }$$

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

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

$$\array{ P &\to& X \\ && \downarrow \\ && \mathbf{B}K } \,.$$

Reduction of the cocycle

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

$$\array{ X &&\stackrel{\sigma}{\to}&& \mathbf{B}G \\ & {}_{\mathllap{k}}\searrow &\swArrow_{\tilde\sigma}& \swarrow \\ && \mathbf{B}K }$$

in $\mathbf{H}$, hence a morphism

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

in the slice (∞,1)-topos $\mathbf{B}_{/\mathbf{B}K}$.

Section of the associated coset-bundle

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

$$\sigma \in \Gamma_X(P \times_{K} K\sslash G)$$

of the associated $K \sslash G$ fiber ∞-bundle.

Equivariant map to the coset

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

$$\vdash (\prod_{x : \mathbf{B}K} P \to K\sslash G) : Type \,.$$

See the discussion at ∞-action.

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

Examples

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

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

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

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

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

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

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

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

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

• reduction of generalized tangent bundle along $G_2 \times G_2 \hookrightarrow SO(7,7)$: 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 $SU(7) \hookrightarrow E_{7(7)}$: 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)