156 CHAPTER 3. MATHEMATICAL FOUNDATIONS
It follows that the covector fieldsψj( 1 ≤j≤N)in (3.4.52) satisfy
(3.4.55) ∆ψj= 0 for 1≤j≤N,
where∆=dδ+δdis the Laplace-Beltrami operator as defined in (3.2.31). Hence, the equa-
tions in (3.4.55) can be equivalently rewritten in the form
DkDkψlj=−Rklψkj for 1≤j≤N, 1 ≤l≤n,
which are exactly the equations in (3.4.51). The proof is complete.
3.5 SU(N)Representation Invariance
3.5.1 SU(N)gauge representation
We briefly recapitulate theSU(N)gauge theory. In the general case, a set ofSU(N)gauge
fields consists ofK=N^2 −1 vector fieldsAaμandNspinor fieldsψj:
(3.5.1) A^1 μ,···,AKμ, Ψ= (ψ^1 ,···,ψN)T.
For the fields (3.5.1), a gauge invariant functional is the Yang-Mills action:
(3.5.2) LYM=
∫
[LG+LD]dx,
whereLGandLDare the gauge field section and Dirac spinor section, and are written as
(3.5.3)
LG=−
1
4
Fμ νaFμ νa,
Fμ νa =∂μAνa−∂νAaμ+gλbcaAbμAcμ,
and
(3.5.4)
LD=Ψ(iγμDμ−m)Ψ,
Dμ=∂μ+igAaμτa,
whereτa( 1 ≤a≤K=N^2 − 1 )are the generators ofSU(N).
The actions (3.5.2)-(3.5.4) are invariant under both the Lorentz transformation and the
SU(N)gauge transformation as follows
(3.5.5)
Ψ ̃=ΩΨ,
A ̃aμτa=ΩAaμτaΩ−^1 +i
g
(∂μΩ)Ω−^1 ,
andΩ⊂SU(N)can be expressed as
Ω=eiθ
aτa
, τais in( 3. 5. 4 ).