3.1. BASIC CONCEPTS 121
By (3.1.55),A=I. The connection ofSU(N)group is the gauge fieldsGaμ:
Γμ={
igGaμτa| {τa}N(^2) − 1
a= 1 is a set of generators ofSU(N)
}
.
Hence, relations (3.1.56) for theSU(N)gauge fieldsGaμare written as
D ̃Ψ ̃=ΩΨ, Ψ:M→M⊗p(C^4 )N,G ̃aμτa=ΩGaμτaΩ−^1 −^1
ig∂ΩΩ−^1 , Ω∈SU(N).
3.Connections for general linear group GL(n).For(k,r)-tensors:(3.1.57) F:M→TrkM,
theGL(n)group action
(3.1.58) Ap:TM→TM, Ap∈GL(n),
induces a linear transformation:
(3.1.59) Tp=Ap⊗ ··· ⊗Ap
︸ ︷︷ ︸
k
⊗A−p^1 ⊗ ··· ⊗A−p^1
︸ ︷︷ ︸
r:TrkM→TrkM,which can be equivalently expressed aK×Kmatrix withK=nk+r, and⊗is the tensor
product of matrices defined by (3.1.67); see Remark3.5. In this case, the matrixAof (3.1.54)-
(3.1.55) is precisely theApin (3.1.58). Hence, by (3.1.56) we have
(3.1.60) D ̃F ̃= (A⊗T)DF,
whereFis as in (3.1.57),Tis as in (3.1.59), andAis as in (3.1.58).
The covariant derivative operatorDdepends on the indiceskandrof bundle spacesTrkM,
and are derived by induction.
4.Derivative on TM.ForF= (F^1 ,···,Fn),(3.1.61) DjFi=∂jFi+ΓijlFl,
whereΓijlare connections defined onTM. AsMis a Riemannian manifold,{Γijk}are the
Levi-Civita connection as given by (2.3.25). It follows from (3.1.56) that the connection of
(3.1.61) transforms as
(3.1.62) Γ ̃=A⊗A⊗(AT)−^1 Γ−A⊗∂A⊗A−^1 ,
which is the the same as those of (2.3.23).
5.Derivative on T∗M.ConsiderF= (F 1 ,···,Fn).The derivative operators satisfy that