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7 Feynman Transform of a Modular Operad
Proof. For homogeneous x ∈ T
S 1 {a}; g 1
and y ∈ T
S 2 {a}; g 2
, one has
αβ(x a • b y) = (−1)
l α
β(x) a • b y + (−1)
l|x| x a • b βy
= (−1)
k+l
αβ(x) a • b y + (−1)
(l+|x|)k β(x) a • b α(y)
+ (−1)
l|x| α(x) a • b β(y) + (−1)
(k+l)|x| x a • b αβ(y)
.
Likewise
−(−1)
kl βα(x a • b y) = − (−1)
kl (−1)
k β
α(x) a • b y + (−1)
k|x| x a • b αy
= (−1)
k+l
− (−1)
kl βα(x) a • b y − (−1)
l|x| α(x) a • b β(y)
− (−1)
(k+l)|x| β(x) a • b α(y) − (−1)
kl+(k+l)|x| x a • b βα(y)
.
Summing the above two equations we get that
[α, β](x a • b y) = (−1)
(k+l)
[α, β](x) a • b y + (−1)
(k+l)|x| x a • b [α, β](y)
,
which is (7.5) for θ = [α, β] evaluated at x ⊗ y. In the same vein, one obtains for
z ∈ T
S {u, v}
that
αβ(• uv z) = (−1)
l α • uv β(z) = (−1)
k+l
• uv αβ(z)
while
−(−1)
kl βα(• uv z) = −(−1)
kl (−1)
k β • uv α(z) = (−1)
k+l
− (−1)
kl
• uv βα(z)
therefore
[α, β](• uv z) = (−1)
k+l
• uv
[α, β](z)
,
which is (7.6) for θ = [α, β] evaluated at z. To prove the last sentence of the lemma,
one needs to observe that if deg(α) = deg(β) = 1, then α 2 = 2[α, α], β 2 = 2[β, β]
and αβ + βα = [α, β].
Definition 7.3 The trivial extension of an odd modular operad T by a T -module
M is an odd modular operad T ⊕M whose underlying modular module is defined by
(T ⊕ M)(S; g) := T (S; g) ⊕ M(S; g), (S; g) ∈ Cor × A,