A Planar Diagram Theory for Strong Interactions

This problem aims to go through the basics of Gerard 't Hooft's Planar Diagram Theory for Strong Interactions paper. This was a seminal paper in the development of string theory, particularly the AdS/CFT correspondance.

To begin, look up the Feynman rules for a Yang-Mills Lagrangian and answer the following questions:

1. In terms of just the coupling strength $g$, what contribution does a 4-point vertex produce? Your answer should be in the form of $g^N$ where $N$ is some number.

2. In terms of just the coupling strength $g$, what contribution does a 3-point vertex produce?

Now consider a scenario where we are computing a process with a given Feynman diagram. Then, suppose we make faces out of loops by treating vertices of our diagrams as vertices, propagators as edges, and loops as faces of our solid figure. Suppose there are $V_4$ number of 4-point vertices and $V_3$ number of 3-point vertices and $F$ number of loops in our diagram.

As a function of $g$, we get the following factor from our Feynman diagram. $$r = (g^2)^{A} \, g^B \, N_c^C$$

3. What is A?

4. What is B?

5. What is C?

Now consider making $N_c$ very large and $g^2$ very small i.e. consider the limit $g^2 \to 0$ and $N_c \to \infty$ such that $g^2 N_c$ goes to some finite value $\lambda$. This is called the t'Hooft limit.

Convince yourself that in our solid figure made from Feynman diagrams, we can write the following relation between $E$ and $V_n$: $$ E = \frac{1}{2} \sum_n n \;V_n $$

Then, $r$ can be written entirely in terms of $\lambda$ and $N_c$ as following: $$ \lambda^{f(E, V, F)} N_c^{g (E, V, F)}$$

6. What is f(E, V, F)? It should be a simple algebraic expression. Write your terms in the order- E, V, F.

7. What is g(E, V, F)? It should be a simple algebraic expression. Write your terms in the order- E, V, F.

But, notice that $g(E, V, F)$ is nothing but Euler's characteristic, $\chi$ of our polyhedron.

8. What is $\chi$ for a sphere?

9. What is $\chi$ for a torus?

For a closed orientable surface, Euler's characteristic is related to its genus by $$\chi = 2 - 2g$$ The genus of a surface is the number of handles or holes on it. It's a simple guess to find out the genus of a sphere and a torus.

However, we have shown an important result here. Because we could write the contribution from each Feynman diagram in terms of $\chi$ and by extension, in terms of $g$, there's a natural way to topologically sort our Feynman diagrams. In particular, if we label the contribution by the genus, we will get a sum $$\sim f_0(\lambda) N^2 + f_1(\lambda) N^0 + f_2(\lambda) N^{-2} + \ldots +f_g(\lambda)N^\chi$$

In the t'Hooft limit, where $N\to \infty$, diagrams with genus $0$ dominate. These are called planar diagrams. A planar diagram is a diagram that you can draw on the surface of a sphere. This gives a hint that the partition function of a large-N_C theory is given by a summation over the topologies of a two-dimensional surfaces. This was really one of the first realizations that eventually led to the development of the holography principle.