Figure 1.2: Two examples of contributions to the vacuum amplitude in dou- ble line notation. The diagram on the left is planar and hence contributes N $^{2}$. The diagram on the right is of genus one and hence contributes N 0 indices in the loop. Hence we have So a diagram with V vertices, E propagators and F loops will include a factor where χ is the Euler characteristic of the diagram. It is a topological invariant, depending only on the genus (number of holes), g , of the surface: Therefore the perturbative expansion of any diagram in field theory can be written as where f$_{g}$ is some polynomial in λ . In the large N limit, we see that any computation will be dominated by those diagrams which are topologically 14 ∞ ∑ g =0 N 2 − 2 g ∞ ∑ i =0 c$_{g,i}$λ i = ∞ ∑ g =0 N 2 − 2 $^{g}$f$_{g}$ ( λ ) (1.21) χ = 2 − 2 g (1.20) N N − $^{E+F}$λ E − V = N $^{χ}$λE − V (1.19) Vertex : N$_{c}$ λ (1.16) Propagator : λ N$_{c}$ (1.17) Loop : N$_{c}$ (1.18)