Connection (principal bundle)

Examples

Maurer–Cartan connection

For the trivial principal ${\displaystyle G}$-bundle ${\displaystyle \pi :E\to X}$ where ${\displaystyle E=G\times X}$, there is a canonical connection^{pg 49}

${\displaystyle {\omega }_{MC}\in {\mathrm{\Omega }}^1(E,\mathfrak{g})}$

called the Maurer-Cartan connection. It is defined at a point ${\displaystyle (g,x)\in G\times X}$ by

${\displaystyle ({\omega }_{MC}{)}_{(g,x)}=(L_{g^{-1}}\circ {\pi }_1{)}_{\ast }}$ for ${\displaystyle x\in X,g\in G}$

which is a composition

${\displaystyle T_{(g,x)}E\stackrel{{\pi }_{1\ast }}{\to }{T}_{g}G\stackrel{(L_{g^{-1}}{)}_{\ast }}{\to }{T}_{e}G=\mathfrak{g}}$

defining the 1-form. Note that

${\displaystyle {\omega }_0=(L_{g^{-1}}{)}_{\ast }:T_{g}G\to T_{e}G=\mathfrak{g}}$

is the Maurer–Cartan form on the Lie group ${\displaystyle G}$ and ${\displaystyle {\omega }_{MC}={\pi }_1^{\ast }{\omega }_0}$.

Trivial bundle

For a trivial principal ${\displaystyle G}$-bundle ${\displaystyle \pi :E\to X}$, the identity section ${\displaystyle i:X\to G\times X}$ given by ${\displaystyle i(x)=(e,x)}$ defines a 1-1 correspondence

${\displaystyle i^{\ast }:{\mathrm{\Omega }}^1(E,\mathfrak{g})\to {\mathrm{\Omega }}^1(X,\mathfrak{g})}$

between connections on ${\displaystyle E}$ and ${\displaystyle \mathfrak{g}}$-valued 1-forms on ${\displaystyle X}$^{pg 53}. For a ${\displaystyle \mathfrak{g}}$-valued 1-form ${\displaystyle A}$ on ${\displaystyle X}$, there is a unique 1-form ${\displaystyle \tilde{A}}$ on ${\displaystyle E}$ such that

1. ${\displaystyle \tilde{A}(X)=0}$ for ${\displaystyle X\in T_{x}E}$ a vertical vector
2. ${\displaystyle R_g^{\ast }\tilde{A}=\text{Ad}(g^{-1})\circ \tilde{A}}$ for any ${\displaystyle g\in G}$

Then given this 1-form, a connection on ${\displaystyle E}$ can be constructed by taking the sum

${\displaystyle {\omega }_{MC}+\tilde{A}}$

giving an actual connection on ${\displaystyle E}$. This unique 1-form can be constructed by first looking at it restricted to ${\displaystyle (e,x)}$ for ${\displaystyle x\in X}$. Then, ${\displaystyle {\tilde{A}}_{(e,x)}}$ is determined by ${\displaystyle A}$ because ${\displaystyle T_{(x,e)}E=ker({\pi }_{\ast })\oplus i_{\ast }{T}_{x}X}$ and we can get ${\displaystyle {\tilde{A}}_{(g,x)}}$by taking

${\displaystyle {\tilde{A}}_{(g,x)}=R_g^{\ast }{\tilde{A}}_{(e,x)}=\text{Ad}(g^{-1})\circ {\tilde{A}}_{(e,x)}}$

Similarly, the form

${\displaystyle {\tilde{A}}_{(x,g)}=\text{Ad}(g^{-1})\circ A_x\circ {\pi }_{\ast }:T_{(x,g)}E\to \mathfrak{g}}$

defines a 1-form giving the properties 1 and 2 listed above.

Extending this to non-trivial bundles

This statement can be refined^{pg 55} even further for non-trivial bundles ${\displaystyle E\to X}$ by considering an open covering ${\displaystyle \mathcal{U}=\{U_a{\}}_{a\in I}}$ of ${\displaystyle X}$ with trivializations ${\displaystyle \{{\varphi }_a{\}}_{a\in I}}$ and transition functions ${\displaystyle \{g_{ab}{\}}_{a,b\in I}}$. Then, there is a 1-1 correspondence between connections on ${\displaystyle E}$ and collections of 1-forms

${\displaystyle \{A_a\in {\mathrm{\Omega }}_1(U_a,\mathfrak{g}){\}}_{a\in I}}$

which satisfy

${\displaystyle A_b=Ad(g_{ab}^{-1})\circ A_a+g_{ab}^{\ast }{\omega }_0}$

on the intersections ${\displaystyle U_{ab}}$ for ${\displaystyle {\omega }_0}$ the Maurer–Cartan form on ${\displaystyle G}$, ${\displaystyle {\omega }_0=g^{-1}dg}$ in matrix form.

Global reformulation of space of connections

For a principal ${\displaystyle G}$ bundle ${\displaystyle \pi :E\to M}$ the set of connections in ${\displaystyle E}$ is an affine space^{pg 57} for the vector space ${\displaystyle {\mathrm{\Omega }}^1(M,E_{\mathfrak{g}})}$ where ${\displaystyle E_{\mathfrak{g}}}$ is the associated adjoint vector bundle. This implies for any two connections ${\displaystyle {\omega }_0,{\omega }_1}$ there exists a form ${\displaystyle A\in {\mathrm{\Omega }}^1(M,E_{\mathfrak{g}})}$ such that

${\displaystyle {\omega }_0={\omega }_1+A}$

We denote the set of connections as ${\displaystyle \mathcal{A}(E)}$, or just ${\displaystyle \mathcal{A}}$ if the context is clear.

Connection on the complex Hopf-bundle

We^{pg 94} can construct ${\displaystyle {\mathbb{C}\mathbb{P}}^n}$ as a principal ${\displaystyle {\mathbb{C}}^{\ast }}$-bundle ${\displaystyle \gamma :H_{\mathbb{C}}\to {\mathbb{C}\mathbb{P}}^n}$ where ${\displaystyle H_{\mathbb{C}}={\mathbb{C}}^{n+1}-\{0\}}$ and ${\displaystyle \gamma }$ is the projection map

${\displaystyle \gamma (z_0,\dots ,z_n)=[z_0,\dots ,z_n]}$

Note the Lie algebra of ${\displaystyle {\mathbb{C}}^{\ast }=GL(1,\mathbb{C})}$ is just the complex plane. The 1-form ${\displaystyle \omega \in {\mathrm{\Omega }}^1(H_{\mathbb{C}},\mathbb{C})}$ defined as

${\displaystyle \begin{array}{cc}\omega & =\frac{{\overline{z}}^{t}dz}{|z{|}^2}\\ & =\sum _{i=0}^n\frac{{\overline{z}}_i}{|z{|}^2}d{z}_i\end{array}}$

forms a connection, which can be checked by verifying the definition. For any fixed ${\displaystyle \lambda \in {\mathbb{C}}^{\ast }}$ we have

${\displaystyle \begin{array}{cc}{R}_{\lambda }^{\ast }\omega & =\frac{{\overline{(z\lambda )}}^{t}d(z\lambda )}{|z\lambda {|}^2}\\ & =\frac{\overline{\lambda }\lambda {\overline{z}}^{t}dz}{|\lambda {|}^2\cdot |z{|}^2}\end{array}}$

and since ${\displaystyle |\lambda {|}^2=\overline{\lambda }\lambda }$, we have ${\displaystyle {\mathbb{C}}^{\ast }}$-invariance. This is because the adjoint action is trivial since the Lie algebra is Abelian. For constructing the splitting, note for any ${\displaystyle z\in H_{\mathbb{C}}}$ we have a short exact sequence

${\displaystyle 0\to \mathbb{C}\stackrel{v_z}{\to }{T}_z{H}_{\mathbb{C}}\stackrel{{\gamma }_{\ast }}{\to }{T}_{[z]}{\mathbb{C}\mathbb{P}}^n\to 0}$

where ${\displaystyle v_z}$ is defined as

${\displaystyle v_z(\lambda )=z\cdot \lambda }$

so it acts as scaling in the fiber (which restricts to the corresponding ${\displaystyle {\mathbb{C}}^{\ast }}$-action). Taking ${\displaystyle {\omega }_z\circ v_z(\lambda )}$ we get

${\displaystyle \begin{array}{cc}{\omega }_z\circ v_z(\lambda )& =\frac{\overline{z}dz}{|z{|}^2}(z\lambda )\\ & =\frac{\overline{z}z\lambda }{|z{|}^2}\\ & =\lambda \end{array}}$

where the second equality follows because we are considering ${\displaystyle z\lambda }$ a vertical tangent vector, and ${\displaystyle dz(z\lambda )=z\lambda }$. The notation is somewhat confusing, but if we expand out each term

${\displaystyle \begin{array}{cc}dz& =d{z}_0+\cdots +d{z}_n\\ z& =a_0{z}_0+\cdots +a_n{z}_n\\ dz(z)& =a_0+\cdots +a_n\\ dz(\lambda z)& =\lambda \cdot (a_0+\cdots +a_n)\\ \overline{z}& =\overline{a_0}+\cdots +\overline{a_n}\end{array}}$

it becomes more clear (where ${\displaystyle a_i\in \mathbb{C}}$).

Induced covariant and exterior derivatives

For any linear representation W of G there is an associated vector bundle ${\displaystyle P{\times }^{G}W}$ over M, and a principal connection induces a covariant derivative on any such vector bundle. This covariant derivative can be defined using the fact that the space of sections of ${\displaystyle P{\times }^{G}W}$ over M is isomorphic to the space of G-equivariant W-valued functions on P. More generally, the space of k-forms with values in ${\displaystyle P{\times }^{G}W}$ is identified with the space of G-equivariant and horizontal W-valued k-forms on P. If α is such a k-form, then its exterior derivative dα, although G-equivariant, is no longer horizontal. However, the combination dα+ωΛα is. This defines an exterior covariant derivative d^ω from ${\displaystyle P{\times }^{G}W}$-valued k-forms on M to ${\displaystyle P{\times }^{G}W}$-valued (k+1)-forms on M. In particular, when k=0, we obtain a covariant derivative on ${\displaystyle P{\times }^{G}W}$.

Curvature form

The curvature form of a principal G-connection ω is the ${\displaystyle \mathfrak{g}}$-valued 2-form Ω defined by

${\displaystyle \mathrm{\Omega }=d\omega +\frac{1}{2}[\omega \wedge \omega ].}$

It is G-equivariant and horizontal, hence corresponds to a 2-form on M with values in ${\displaystyle {\mathfrak{g}}_P}$. The identification of the curvature with this quantity is sometimes called the (Cartan's) second structure equation. Historically, the emergence of the structure equations are found in the development of the Cartan connection. When transposed into the context of Lie groups, the structure equations are known as the Maurer–Cartan equations: they are the same equations, but in a different setting and notation.

Flat connections and characterization of bundles with flat connections

We say that a connection ${\displaystyle \omega }$ is flat if its curvature form ${\displaystyle \mathrm{\Omega }=0}$. There is a useful characterization of principal bundles with flat connections; that is, a principal ${\displaystyle G}$-bundle ${\displaystyle \pi :E\to X}$ has a flat connection^{pg 68} if and only if there exists an open covering ${\displaystyle \{U_a{\}}_{a\in I}}$ with trivializations ${\displaystyle {\left\{{\varphi }_a\right\}}_{a\in I}}$ such that all transition functions

${\displaystyle g_{ab}:U_a\cap U_b\to G}$

are constant. This is useful because it gives a recipe for constructing flat principal ${\displaystyle G}$-bundles over smooth manifolds; namely taking an open cover and defining trivializations with constant transition functions.

Connections on frame bundles and torsion

If the principal bundle P is the frame bundle, or (more generally) if it has a solder form, then the connection is an example of an affine connection, and the curvature is not the only invariant, since the additional structure of the solder form θ, which is an equivariant Rⁿ-valued 1-form on P, should be taken into account. In particular, the torsion form on P, is an Rⁿ-valued 2-form Θ defined by

${\displaystyle \mathrm{\Theta }=\mathrm{d}\theta +\omega \wedge \theta .}$

Θ is G-equivariant and horizontal, and so it descends to a tangent-valued 2-form on M, called the torsion. This equation is sometimes called the (Cartan's) first structure equation.

Definition in algebraic geometry

If X is a scheme (or more generally, stack, derived stack, or even prestack), we can associate to it its so-called de Rham stack, denoted X_dR. This has the property that a principal G bundle over X_dR is the same thing as a G bundle with *flat* connection over X.