25 Summary: Large N and topology 
	Statistical Mechanics: Entropy, Order Parameters, and Complexity
		James P. Sethna, Cornell University, Spring 2020
				Lecture 23

One obscure fact. The projective plane, RP2, that you studied today in
class, has Euler number chi = 1. 

Suppose we have a surface, which we have sewn together from a bunch of 
polygons. The surface then has faces (the polygons), edges, and vertices.
The The Euler number of the surface is
    chi = #Faces - #Edges + #Vertices. 
The Euler number only depends on the topology of the surface (not the 
choice of polygons. The sphere S^2 has Euler number 2; the torus has chi=0.
The projective plane RP2 has Euler number one.

Now let me indulge you in another story about grad school. David Gross
(before his Nobel Prize) was teaching QFT at Princeton, and was telling
us about the large-N expansion. The idea is that three colors of quarks
is too hard: take the limit of N colors as N goes to infinity. The 
Feynman diagrams then have vertices on which you need to sum over N colors,
edges which suppress the contribution by a factor of N, and an integral over
each face that multiplies by N. So each diagram contributes N^chi, where
chi is the Euler number of the surface formed by sewing the faces together.

David explained that the lowest order theory was given by summing over
diagrams like this one that could be drawn on the sphere (so-called planar
diagrams), and the first correction (down by 1/N^2) were diagrams that 
could be drawn on the torus. I, the smart aleck in the audience,
immediately piped up, asking about RP2... Two days later in class, he 
explained that for the theories he was working on, the diagrams only formed
orientable surfaces (but that other theories might have 1/N corrections
from "projective-planar diagrams".

Have a good weekend. Hope problem set 9 goes well...