38 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

Proof. We will prove this by first verifying the result for P = 0, and

then checking that both sides satisfy the same differential equation as a

function of P . When P = 0, we have seen that

W (0,I) = I

which of course is the same as log exp( ∂P ) exp(I/ ).

Now let us turn to proving the general case. It is easier to consider the

exponentiated version: so we will actually verify that

exp

(

−1W

(P, I)

)

= exp( ∂P ) exp(I/ ).

We will do this by verifying that, if ε is a parameter of square zero, and

P ∈

Sym2

U,

exp

(

−1W

(

P + εP , I

))

= (1 + ε∂P ) exp

(

−1W

(P, I)

)

.

Let a1,...,ak ∈ U, and let us consider

∂

∂a1

· · ·

∂

∂ak

exp

(

−1W

(P, I)

)

(0).

It will suﬃce to prove a similar differential equation for this expression.

It follows immediately from the definition of the weight function wγ(P, I)

of a graph that

∂

∂a1

· · ·

∂

∂ak

eW (P,I)/

(0) =

γ,φ

g(γ)

|Aut(γ, φ)|

wγ,φ(P, I)(a1,...,ak)

where the sum is over all possibly disconnected stable graphs γ with an

isomorphism φ : {1,...,k}

∼

= T (γ). The automorphism group Aut(γ, φ)

consists of those automorphisms preserving the ordering φ of the set of tails

of γ.

Let ε be a parameter of square zero, and let P ∈

Sym2

U. Let us

consider varying P to P + εP . We find that

d

dε

∂

∂a1

· · ·

∂

∂ak

exp

(

−1W

(

P + εP , I

))

(0)

=

γ,e,φ

g(γ)

|Aut(γ, e, φ)|

wγ,e,φ(P, I)(a1,...,ak).

Here, the sum is over possibly disconnected stable graphs γ with a distin-

guished edge e ∈ E(γ). The weight wγ,e,φ is defined in the same way as

wγ,φ except that the distinguished edge e is labelled by P , whereas all other

edges are labelled by P . The automorphism group considered must preserve

the edge e as well as φ.

Given any graph γ with a distinguished edge e, we can cut along this

edge to get another graph γ with two more tails. These tails can be ordered