9. EXTRACTING THE SINGULAR PART 59

Definition 9.2.2. Let PQ((0, 1)) ⊂

C∞((0,

1)) be the subalgebra of

functions of this form. Elements of this subalgebra will be called rational

periods.

Note that PQ((0, 1)) is, indeed, a subalgebra; the sum of functions cor-

responds to the disjoint union of the pairs (X, D) of algebraic varieties, and

the product of functions corresponds to fibre product of the algebraic va-

rieties (X, D) over U. Further, PQ((0, 1)) is of countable dimension over

Q.

Note that, if f is a rational period, then, for every rational number

t ∈ Q ∩ (0, 1), f(t) is a period in the sense of Kontsevich and Zagier.

Definition 9.2.3. Let

P((0, 1)) = PQ((0, 1)) ⊗ R ⊂

C∞((0,

1))

be the real vector space spanned by the space of rational periods. Elements

of P((0, 1)) will be called periods.

9.3. Now we are ready to state the theorem on the small ε asymptotic

expansions of the functions wγ(P (ε, L),I)(a).

We will regard the functional wγ(P (ε, L),I)(a) as a function of the three

variables ε, L and a ∈

C∞(M).

It is an element of the space of functionals

O(C∞(M ),C∞((0,

1)ε) ⊗

C∞((0,

∞)L))).

The subscripts ε and L indicate the coordinates on the intervales (0, 1) and

(0, ∞). If we fix ε but allow L and a to vary, we get a functional

wγ(P (ε, L),I) ∈

O(C∞(M ),C∞((0,

∞)L)).

This is a topological vector space; we are interested in the behaviour of

wγ(P (ε, L),I) as ε → 0.

The following theorem describes the small ε behaviour of wγ(P (ε, L),I).

Theorem 9.3.1. Let I ∈

Oloc(C∞(M

))[[ ]] be a local functional, and let

γ be a connected stable graph.

(1) There exists a small ε asymptotic expansion

wγ(P (ε, L),I)

∞

i=0

gi(ε)Ψi

where the

gi ∈ P((0, 1)ε)

are periods, and Ψi ∈

O(C∞(M ),C∞((0,

∞)L)).

The precise meaning of “asymptotic expansion” is as follows:

there is a non-decreasing sequence dR ∈ Z, indexed by R ∈ Z 0,

such that dR → ∞ as R → ∞, and such that for all R,

lim

ε→0

ε−dR

wγ(P (ε, L),I) −

R

i=0

gi(ε)Ψi = 0.