34 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

This functional integral is ill-defined.

3. Generalities on Feynman graphs

In this section, we will describe the Feynman graph expansion for func-

tional integrals of the form appearing in the renormalization group equation.

This section will only deal with finite dimensional vector spaces, as a toy

model for the infinite dimensional functional integrals we will be concerned

with for most of this book. For another mathematical description of the

Feynman diagram expansion in finite dimensions, one can consult, for ex-

ample, (Man99).

3.1.

Definition 3.1.1. A

stable2

graph is a graph γ, possibly with external

edges (or tails); and for each vertex v of γ an element g(v) ∈ Z≥0, called

the genus of the vertex v; with the property that every vertex of genus 0 is

at least trivalent, and every vertex of genus 1 is at least 1-valent (0-valent

vertices are allowed, provided they are of genus 1).

If γ is a stable graph, the genus g(γ) of γ is defined by

g(γ) = b1(γ) +

v∈V (γ)

g(v)

where b1(γ) is the first Betti number of γ.

More formally, a stable graph γ is determined by the following data.

(1) A finite set H(γ) of half-edges of γ.

(2) A finite set V (γ) of vertices of γ.

(3) An involution σ : H(γ) → H(γ). The set of fixed points of this

involution is denoted T (γ), and is called the set of tails of γ. The

set of two-element orbits is denoted E(γ), and is called the set of

edges.

(4) A map π : H(γ) → V (γ), which sends a half-edge to the vertex to

which it is attached.

(5) A map g : V (γ → Z≥0.

From this data we construct a topological space |γ| which is the quotient of

V (γ)

(

H(γ) × [0,

1

2

]

)

by the relation which identifies (h, 0) ∈ H(γ) × [0,

1

2

] with π(h) ∈ V (γ); and

identifies (h,

1

2

) with (σ(h),

1

2

). We say γ is connected if |γ| is. A graph γ is

stable, as above, if every vertex v of genus 0 is at least trivalent, and every

vertex of genus 1 is at least univalent.

We are also interested in automorphisms of stable graphs. It is helpful

to give a formal definition. An element of g ∈ Aut(γ) of the group Aut(γ) is

2The

term “stable” comes from algebraic geometry, where such graphs are used to

label the strata of the Deligne-Mumford moduli space of stable curves.