the double lines to twist, i.e., one going over the other. If we imagine that line-crossing is
forbidden, then the twist can still be achieved by inserting a "worm-hole" in the sheet on
which the diagram is drawn. It is therefore clear that with the Feynman rules according
to the double-line notation, and with line-crossing forbidden, the sum over all diagrams
would have to include the sum over all topologies of the sheet on which the diagrams
are drawn. Notice also that if a single line closes in a contractible loop, the resulting
amplitude, however complicated, will have a factor of i ii = Nc. If the loop, however,
is not contractible, meaning that the surface on which it is drawn is not planar, then the
index sum does not decouple from the expressions to yield the factor of N,. A rigorous
proof that amplitudes drawn on H-hole surfaces will be suppressed from the planar ones
by 1/NH, is given in 't Hooft's paper [4] using Euler's formula: F-P+V 2-2H relating
the number of faces (F), lines (P), vertices (V) and holes (H) of a planar shape. The limit
Nc +oc with the 't Hooft coupling held constant therefore singles out planar diagram.
Furthermore, the topology of the sheets on which the diagrams are drawn constitutes a
means of systematically improving the zeroth order Nc -- +oc limit.
The concept of "the sheet on which Feynman diagrams are drawn" is not perhaps
fully useful until the Lightcone world sheet concept introduced in the next chapter
identifies this sheet with the world sheet itself. Then the topology is not that of "the
surface on which to draw the Feynman diagrams" but rather the topology of a world sheet
which itself directly holds the dynamical variables. The Lightcone parametrization enables
us to identify the p+ component of propagators with a space coordinate on this sheet and
x+ with the time coordinate. This means that first order 1/Nc corrections to the work
presented here this have a prescription in terms of the topology of the Lightcone World
Sheet.
1.3 Work on Planar Diagrams
The approach due to 't Hooft, considering quantum field theories with U(Nc)
symmetry perturbatively in 1/Nc, is particularly intriguing because it organizes the order