6. THE MAIN THEOREM 17

scheme is a way to extract the singular part of certain functions of one

variable. We construct a certain subalgebra

P((0, 1)) ⊂

C∞((0,

1))

consisting of functions f(ε) of a “motivic” nature. Functions in P((0, 1))

arise as the periods of families of algebraic varieties over Zariski open subsets

U ⊂ AQ,

1

such that U(R) contains (0, 1). (For more details, see Chapter 2,

Section 9).

Definition 6.0.1. A renormalization scheme is a subspace

P((0, 1))

0

⊂ P((0, 1))

of “purely singular” functions, complementary to the subspace

P((0, 1))≥0 ⊂ P((0, 1))

of functions whose r → ∞ limit exists.

The choice of a renormalization scheme gives us a way to extract the

singular part of functions in P((0, 1)).

The variant theorem is the following.

Theorem B. The choice of a renormalization scheme leads to a bijec-

tion between the space of theories and the space of local action functionals

S :

C∞(M)

→ R[[ ]].

Equivalently, there is a bijection between the space of theories and the

space of Lagrangians up to the addition of a Lagrangian which is a total

derivative.

Theorem B implies theorem A, but theorem A is the more natural for-

mulation.

There are certain caveats:

(1) Like the effective actions

Seff

[Λ], the local action functional S is a

formal power series both in φ ∈

C∞(M)

and in .

(2) Modulo , we require that S is of the form

S(φ) = −

1

2

M

φ D φ + cubic and higher terms.

There is a more general formulation of this theorem, where the space of

fields is allowed to be the space of sections of a graded vector bundle. In

the more general formulation, the action functional S must have a quadratic

term which is elliptic in a certain sense.

6.1. Let me sketch how to prove theorem A. Given the action S, we

construct the low-energy effective action

Seff

[Λ] by renormalizing a certain

functional integral. The formula for the functional integral is

Seff

[Λ](φL) = log

φH ∈C∞(M)(Λ,∞)

eS(φL+φH )/

.