Geometry and quantum field theory - 04 Matrix integrals.pdf

MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 19
4. Matrix integrals
Let hN be the space of Hermitian matrices of size N . The inner product on hN is given by (A, B)=
Tr(AB). In this section we will consider integrals of the form

2
−N /2 −S(A)/

ZN =
e dA,
hN
2
where the Lebesgue measure dA is normalized by the condition e−Tr(A )/2dA =1,andS(A)=

2 − m 3
Tr(A )/2 m≥0 gmTr(A )/m is the action functional. We will be interested the behavior of the
coefficients of the expansion of ZN in gi for large N . The study of this behavior will lead us to considering
not simply Feynman graphs, but actually fat (or ribbon) graphs, which are in fact 2-dimensional surfaces.
Thus, before we proceed further, we need to do some 2-binatorial topology.
. Fat from the proof of Feynman’s theorem that given a finite collection of flowers
and a pairing σ on the set T of endpoints of their edges, we can obtain a graph Γσ by connecting (or
gluing) the points which fall in the same pair.
Now, given an i-flower, let us inscribe it in a closed disk D (so that the ends of the edges are on
the boundary) and take its small tubular neighborhood in D. This produces a region with piecewise
smooth boundary. We will equip this region with an orientation, and call it a fat i-valent
boundary of a fat i-valent flower has the