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
