Lie algebra U(NA) which is determined by the structure constants fabc by the relation:
[r, b] I fabcTc (1-9)
Since the matrices Ta form a basis for the U(Nc) algebra the indices must run from 1 to
NA the dimension of the algebra. A representation of U(Nc) is a set of matrices which
satisfy relation (1-9), and the so called fundamental representation is the set of Nc x Nc
unitary matrices (hence the label U(Nc)). Another such set (which satisfies (1-9)) are
the structure constants themselves, namely fabc thought of as a matrix, with the index a
labelling which generator and indices b and c labelling the rows and columns of the matrix.
The structure constants are in this way said to form the adjoint representation of the
algebra U(Nc). Notice that in this representation the matrices are Nc x Nc but there are
still just N) degrees of freedom, i.e. U(Nc) is found here as a small subalgebra among all
matrices of this size.
1.2 't Hooft's Large Nc Limit
In 1974 a paper [4] was published by 't Hooft explaining how a systematic expansion
in 1/Nc for field theories with U(Nc) symmetries could be achieved. The sheets on which
Feynman diagrams are drawn were classified into planar, one-hole, two-hole, etc. surfaces
and it was shown that this topological classification corresponds to the order of 1/Nc
in the expansion. In his article 't Hooft pointed out that this was also the topology of
the classes of string diagrams in the quantized dual string models with quarks at its
ends. He further pursued the string analogy by going to Lightcone frame and proposed
a world-sheet non-local Hamiltonian theory which would sum all Feynman diagrams
perturbatively with Nc -- +oc and g2Nc fixed. The coupling of g2Nc has since been
called the 'tHooft-coupling and his world-sheet model was the beginning of extensive
considerations along the same lines by theoretical physicists. Since this work by 't Hooft
most certainly was the foundation on which the Lightcone World-Sheet picture later built,