Connection (principal bundle)

Most known non-trivial computations of principal G-connections are done with homogeneous spaces because of the triviality of the (co)tangent bundle. (For example, let ${\displaystyle G\to H\to H/G}$, be a principal G-bundle over ${\displaystyle H/G}$) This means that 1-forms on the total space are canonically isomorphic to ${\displaystyle C^{\infty }(H,{\mathfrak {g}}^{*})}$, where ${\displaystyle {\mathfrak {g}}^{*}}$ is the dual lie algebra, hence G-connections are in bijection with ${\displaystyle C^{\infty }(H,{\mathfrak {g}}^{*}\otimes {\mathfrak {g}})^{G}}$.

A principal G-connection ${\displaystyle \omega }$ on ${\displaystyle P}$ determines an Ehresmann connection on ${\displaystyle P}$ in the following way. First note that the fundamental vector fields generating the ${\displaystyle G}$ action on ${\displaystyle P}$ provide a bundle isomorphism (covering the identity of ${\displaystyle P}$) from the bundle ${\displaystyle VP}$ to ${\displaystyle P\times {\mathfrak {g}}}$, where ${\displaystyle VP=\ker(d\pi )}$ is the kernel of the tangent mapping ${\displaystyle {\mathrm {d} }\pi \colon TP\to TM}$ which is called the vertical bundle of ${\displaystyle P}$. It follows that ${\displaystyle \omega }$ determines uniquely a bundle map ${\displaystyle v:TP\rightarrow V}$ which is the identity on ${\displaystyle V}$. Such a projection ${\displaystyle v}$ is uniquely determined by its kernel, which is a smooth subbundle ${\displaystyle H}$ of ${\displaystyle TP}$ (called the horizontal bundle) such that ${\displaystyle TP=V\oplus H}$. This is an Ehresmann connection.

Conversely, an Ehresmann connection ${\displaystyle H\subset TP}$ (or ${\displaystyle v:TP\rightarrow V}$) on ${\displaystyle P}$ defines a principal ${\displaystyle G}$-connection ${\displaystyle \omega }$ if and only if it is ${\displaystyle G}$-equivariant in the sense that ${\displaystyle H_{pg}=\mathrm {d} (R_{g})_{p}(H_{p})}$.

A trivializing section of a principal bundle P is given by a section s of P over an open subset U of M. Then the pullback s^*ω of a principal connection is a 1-form on U with values in ${\displaystyle {\mathfrak {g}}}$. If the section s is replaced by a new section sg, defined by (sg)(x) = s(x)g(x), where g:M→G is a smooth map, then ${\displaystyle (sg)^{*}\omega =\operatorname {Ad} (g)^{-1}s^{*}\omega +g^{-1}dg}$. The principal connection is uniquely determined by this family of ${\displaystyle {\mathfrak {g}}}$-valued 1-forms, and these 1-forms are also called connection forms or connection 1-forms, particularly in older or more physics-oriented literature.

The group G acts on the tangent bundle TP by right translation. The quotient space TP/G is also a manifold, and inherits the structure of a fibre bundle over TM which shall be denoted dπ:TP/G→TM. Let ρ:TP/G→M be the projection onto M. The fibres of the bundle TP/G under the projection ρ carry an additive structure.

The bundle TP/G is called the bundle of principal connections (Kobayashi 1957). A section Γ of dπ:TP/G→TM such that Γ : TM → TP/G is a linear morphism of vector bundles over M, can be identified with a principal connection in P. Conversely, a principal connection as defined above gives rise to such a section Γ of TP/G.

Finally, let Γ be a principal connection in this sense. Let q:TP→TP/G be the quotient map. The horizontal distribution of the connection is the bundle

${\displaystyle H=q^{-1}\Gamma (TM)\subset TP.}$ We see again the link to the horizontal bundle and thus Ehresmann connection.

If ω and ω′ are principal connections on a principal bundle P, then the difference ω′ − ω is a ${\displaystyle {\mathfrak {g}}}$-valued 1-form on P which is not only G-equivariant, but horizontal in the sense that it vanishes on any section of the vertical bundle V of P. Hence it is basic and so is determined by a 1-form on M with values in the adjoint bundle

${\displaystyle {\mathfrak {g}}_{P}:=P\times ^{G}{\mathfrak {g}}.}$

Conversely, any such one form defines (via pullback) a G-equivariant horizontal 1-form on P, and the space of principal G-connections is an affine space for this space of 1-forms.

For the trivial principal ${\displaystyle G}$-bundle ${\displaystyle \pi :E\to X}$ where ${\displaystyle E=G\times X}$, there is a canonical connection[1]^{pg 49}

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

called the Maurer-Cartan connection. It is defined as follows: for a point ${\displaystyle (g,x)\in G\times X}$ define

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

which is a composition

${\displaystyle T_{(g,x)}E\xrightarrow {\pi _{1*}} T_{g}G\xrightarrow {(L_{g^{-1}})_{*}} T_{e}G={\mathfrak {g}}}$

defining the 1-form. Note that

${\displaystyle \omega _{0}=(L_{g^{-1}})_{*}:T_{g}G\to T_{e}G={\mathfrak {g}}}$

is the Maurer-Cartan form on the Lie group ${\displaystyle G}$ and ${\displaystyle \omega _{MC}=\pi _{1}^{*}\omega _{0}}$.

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

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

between connections on ${\displaystyle E}$ and ${\displaystyle {\mathfrak {g}}}$-valued 1-forms on ${\displaystyle X}$[1]^{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. ${\displaystyle R_{g}^{*}{\tilde {A}}={\text{Ad}}(g^{-1})\circ {\tilde {A}}}$ 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 ${\displaystyle T_{(x,e)}E=ker(\pi _{*})\oplus i_{*}T_{x}X}$ and we can get ${\displaystyle {\tilde {A}}_{(g,x)}}$by taking

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

Similarly, the form

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

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

This statement can be refined[1]^{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 ${\displaystyle \{\phi _{a}\}_{a\in I}}$ 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

${\displaystyle \{A_{a}\in \Omega _{1}(U_{a},{\mathfrak {g}})\}_{a\in I}}$

which satisfy

${\displaystyle A_{b}=Ad(g_{ab}^{-1})\circ A_{a}+g_{ab}^{*}\omega _{0}}$

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.

For a principal ${\displaystyle G}$ bundle ${\displaystyle \pi :E\to M}$ the set of connections in ${\displaystyle E}$ is an affine space[1]^{pg 57} for the vector space ${\displaystyle \Omega ^{1}(M,E_{\mathfrak {g}})}$ 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 ${\displaystyle A\in \Omega ^{1}(M,E_{\mathfrak {g}})}$ 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.

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

${\displaystyle \gamma (z_{0},\ldots ,z_{n})=[z_{0},\ldots ,z_{n}]}$

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

${\displaystyle {\begin{aligned}\omega &={\frac {{\overline {z}}^{t}dz}{|z|^{2}}}\\&=\sum _{i=0}^{n}{\frac {{\overline {z}}_{i}}{|z|^{2}}}dz_{i}\end{aligned}}}$

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

${\displaystyle {\begin{aligned}R_{\lambda }^{*}\omega &={\frac {{\overline {(z\lambda )}}^{t}d(z\lambda )}{|z\lambda |^{2}}}\\&={\frac {{\overline {\lambda }}\lambda {\overline {z}}^{t}dz}{|\lambda |^{2}\cdot |z|^{2}}}\end{aligned}}}$

and since ${\displaystyle |\lambda |^{2}={\overline {\lambda }}{\lambda }}$, we have ${\displaystyle \mathbb {C} ^{*}}$-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

${\displaystyle 0\to \mathbb {C} \xrightarrow {v_{z}} T_{z}H_{\mathbb {Z} }\xrightarrow {\gamma _{*}} T_{[z]}\mathbb {CP} ^{n}\to 0}$

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 ${\displaystyle \mathbb {C} ^{*}}$-action). Taking ${\displaystyle \omega _{z}\circ v_{z}(\lambda )}$ we get

${\displaystyle {\begin{aligned}\omega _{z}\circ v_{z}(\lambda )&={\frac {{\overline {z}}dz}{|z|^{2}}}(z\lambda )\\&={\frac {{\overline {z}}z\lambda }{|z|^{2}}}\\&=\lambda \end{aligned}}}$

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

${\displaystyle {\begin{aligned}dz&=dz_{0}+\cdots +dz_{n}\\z&=a_{0}z_{0}+\cdots +a_{n}z_{n}\\dz(z)&=a_{0}+\cdots +a_{n}\\dz(\lambda z)&=\lambda \cdot (a_{0}+\cdots +a_{n})\\{\overline {z}}&={\overline {a_{0}}}+\cdots +{\overline {a_{n}}}\end{aligned}}}$

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

We say that a connection ${\displaystyle \omega }$ is flat if its curvature form ${\displaystyle \Omega =0}$. 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[1]^{pg 68} if and only if there exists an open covering ${\displaystyle \{U_{a}\}_{a\in I}}$ with trivializations ${\displaystyle \left\{\phi _{a}\right\}_{a\in I}}$ 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.