Connection (principal bundle)

In differential geometry , the relationship is a concept that can be used to explain parallel transport between the fibers of a principal bundle . In physics, such relationships are used to describe fields in the Yang-Mills theories .

definition

Be a principal bundle with the structure group . The group works through ${\ displaystyle \ pi \ colon P \ rightarrow M}$ ${\ displaystyle G}$

{\ displaystyle R \ colon P \ times G \ rightarrow P}

.

Also denote the Lie algebra of the Lie group . ${\ displaystyle {\ mathfrak {g}}}$ ${\ displaystyle G}$

A relationship is then a -value 1-form , which is -equivariant and whose restriction to the fibers corresponds to the Maurer-Cartan form . So the following two conditions should be met: ${\ displaystyle {\ mathfrak {g}}}$ ${\ displaystyle \ omega \ in \ Omega ^ {1} (P, {\ mathfrak {g}})}$ ${\ displaystyle G}$

{\ displaystyle D (R_ {g}) \ omega = \ operatorname {Ad} (g ^ {- 1}) (\ omega)}

for all

{\ displaystyle g \ in G}

and

{\ displaystyle \ omega (X ^ {\ sharp}) = X}

for everyone .

{\ displaystyle X \ in {\ mathfrak {g}}}

Here is defined by . denotes the differential of . is the adjoint effect and is the so-called fundamental vector field . It will go through ${\ displaystyle R_ {g} \ colon P \ to P}$ ${\ displaystyle R_ {g} (p) = R (p, g)}$ ${\ displaystyle D (R_ {g})}$ ${\ displaystyle R_ {g}}$ ${\ displaystyle \ operatorname {Ad} \ colon G \ to GL ({\ mathfrak {g}})}$ ${\ displaystyle X ^ {\ sharp}}$

{\ displaystyle X ^ {\ sharp} (p): = \ left. {\ frac {\ mathrm {d}} {\ mathrm {d} t}} \ right | _ {t = 0} R _ {\ exp ( tX)} (p)}

For

{\ displaystyle p \ in P}

on defined. ${\ displaystyle P}$

curvature

The curvature of a connected form is defined by

{\ displaystyle \ Omega = d \ omega + {\ tfrac {1} {2}} [\ omega \ wedge \ omega].}

Here the commutator is Lie algebra valued differential forms by

{\ displaystyle [\ omega \ wedge \ eta] (v_ {1}, v_ {2}) = [\ omega (v_ {1}), \ eta (v_ {2})] - [\ omega (v_ {2 }), \ eta (v_ {1})]}

and the outer derivative through ${\ displaystyle d \ omega}$

{\ displaystyle d \ omega (X, Y) = X (\ omega (Y)) - Y (\ omega (X)) - \ omega ([X, Y])}

Are defined.

The curve form is -invariant and therefore defines a 2-form on . ${\ displaystyle G}$ ${\ displaystyle \ Omega \ in \ Omega ^ {2} (M, {\ mathfrak {g}})}$ ${\ displaystyle M}$

Bianchi identity

Relationship and curvature form satisfy the equation

{\ displaystyle d \ Omega = \ left [\ Omega, \ omega \ right]}

.

Horizontal subspaces

For a form of connection on a principal bundle , the horizontal subspaces are defined by ${\ displaystyle \ omega \ in \ Omega ^ {1} (P, {\ mathfrak {g}})}$ ${\ displaystyle G}$ ${\ displaystyle \ pi: P \ rightarrow M}$ ${\ displaystyle H_ {p}, p \ in P}$

{\ displaystyle H_ {p}: = ker (\ omega: T_ {p} P \ rightarrow {\ mathfrak {g}})}

.

The horizontal subspaces are transverse to the tangent spaces of the fibers of , and they are -invariant, i.e. H. for everyone . ${\ displaystyle \ pi}$ ${\ displaystyle G}$ ${\ displaystyle H_ {gp} = DR_ {g} (H_ {p})}$ ${\ displaystyle g \ in G, p \ in P}$

From the horizontal subspaces can form the connection recover (for identification of the tangent of the fiber ) by projection of along the tangent of the fiber. ${\ displaystyle {\ mathfrak {g}}}$ ${\ displaystyle T_ {p} P}$ ${\ displaystyle H_ {p}}$

Parallel transport

For every path and every there is a path with and . (This follows from the existence and uniqueness theorem for ordinary differential equations .) ${\ displaystyle \ gamma: \ left [0.1 \ right] \ rightarrow M}$ ${\ displaystyle x \ in \ pi ^ {- 1} (\ gamma (0))}$ ${\ displaystyle {\ tilde {\ gamma}}: \ left [0,1 \ right] \ rightarrow P}$ ${\ displaystyle {\ tilde {\ gamma}} _ {x} (0) = x}$ ${\ displaystyle \ pi ({\ tilde {\ gamma}} _ {x}) = \ gamma}$

In particular, you have one through each way ${\ displaystyle \ gamma: \ left [0.1 \ right] \ rightarrow M}$

{\ displaystyle P _ {\ gamma} (x) = {\ tilde {\ gamma}} _ {x} (1)}

defined figure

{\ displaystyle P _ {\ gamma}: \ pi ^ {- 1} (\ gamma (0)) \ rightarrow \ pi ^ {- 1} (\ gamma (1))}

,

the so-called parallel transport along the way . ${\ displaystyle \ gamma}$

At one point , the holonomy group is defined as a subset of the diffeomorphisms of the fiber as follows. To a closed path with and one there is a clear elevation with and we define . The group that is for everyone is the holonomy group. ${\ displaystyle b \ in B}$ ${\ displaystyle F_ {b}: = \ pi ^ {- 1} (b)}$ ${\ displaystyle \ gamma: \ left [0,1 \ right] \ rightarrow B}$ ${\ displaystyle \ gamma (0) = \ gamma (1) = b}$ ${\ displaystyle x \ in F_ {b}}$ ${\ displaystyle {\ tilde {\ gamma}}}$ ${\ displaystyle {\ tilde {\ gamma}} (0) = x}$ ${\ displaystyle f _ {\ gamma} (x): = {\ tilde {\ gamma}} (1)}$ ${\ displaystyle f _ {\ gamma}}$ ${\ displaystyle \ gamma}$

Riemannian connection

For a Riemannian manifold , the frame bundle is a principal bundle with the linear group . ${\ displaystyle M}$ ${\ displaystyle GL (n, \ mathbb {R})}$