SUNT[a] is the SU(N)
T^a generator in the fundamental
representation. The fundamental indices are implicit.
Overview, CA, CF, SUND, SUNDelta, SUNF, SUNSimplify.
SUNT[a]T^a
Since T^a is a noncommutative
object, products have to separated by a Dot
(.).
SUNT[a] . SUNT[b] . SUNT[c]T^a.T^b.T^c
SUNT[a, b, c, d]T^a.T^b.T^c.T^d
SUNSimplify[SUNT[a, b, a], SUNNToCACF -> False]-\frac{T^b}{2 N}
SUNSimplify[SUNT[a, b, b, a]]C_F^2
SUNSimplify[SUNT[a, b, a]]-\frac{1}{2} T^b \left(C_A-2 C_F\right)
SUNSimplify[SUNT[a, b, a], SUNNToCACF -> False]-\frac{T^b}{2 N}
The normalization of the generators is chosen in the standard way, therefore \textrm{Tr}(T^aT^b) = \frac{1}{2} \delta _{ab}
SUNTrace[SUNT[a, b]]\frac{\delta ^{ab}}{2}
In case you want T_f, you need to
include a factor 2*Tfinside the trace.
SUNTrace[2 Tf SUNT[a, b]]T_f \delta ^{ab}
SUNTrace[SUNT[a, b]] // StandardForm\frac{1}{2} \;\text{SUNDelta}[\text{SUNIndex}[a],\text{SUNIndex}[b]]
SUNT[a] // FCI // StandardForm
(*SUNT[SUNIndex[a]]*)SUNT[a] // FCI // FCE // StandardForm
(*SUNT[a]*)