3.5.SU(N)REPRESENTATION INVARIANCE 159
x^1x^2 N2
R^2 N2SU(N)xkATASU(N)
τAγ(t)Figure 3.2: Tangent spaceTASU(N)atA∈SU(N).For infinitesimalt, the solution of (3.5.12) is
γ(t) =A+tτA⊂SU(N).It follows that
(3.5.13) (A+tτA)†(A+tτA) =I.
AsA†A=Iandtis infinitesimal, we deduce from (3.5.13) that
(3.5.14) A†τA+τA†A= 0.
Hence, (3.5.14) is the condition for anN-th order complex matrixτ∈TASU(N):A†τis
anti-Hermitian. Namely,
(3.5.15) TASU(N) ={τ|τsatisfies( 3. 5. 14 )}.
Note that
(3.5.16) A+tτA=A(I+tA†τA) =AetA
†τA
,for infinitesimalt. If we replaceτbyiτ, then (3.5.15) can be expressed as
(3.5.17) TASU(N) ={iτ|A†τ=τ†A,Tr(A†τ) = 0 }.
By (3.5.17),∀A∈SU(N)there is a neighborhoodUA⊂SU(N)such that for anyΩ∈UA,Ω
can be written as
(3.5.18) Ω=AeiA
†τ
for someiτ∈TASU(N).Here the traceless condition in (3.5.17)
Tr(A†τ) = 0