148
6 Operads
followed by
(s ⊗ 1) : k ⊗ V 1 ⊗ V 2 ⊗ V 3 → V u ⊗ V v ⊗ V 1 ⊗ V 2 ⊗ V 3
composed with the permutation
ρ : V u ⊗ V v ⊗ V 1 ⊗ V 2 ⊗ V 3
∼ =
−→ V 1 ⊗ V u ⊗ V 2 ⊗ V v ⊗ V 3
and finally followed by
f : V 1 ⊗ V u ⊗ V 2 ⊗ V v ⊗ V 3 −→ k.
Let us apply this composition on the element
v 1 ⊗ · · · ⊗ v n = ω 1 ⊗ ω 2 ⊗ ω 3 ∈ V
⊗(n) .
Isomorphism (6.75) brings ω 1 ⊗ ω 2 ⊗ ω 3 into 1 ⊗ ω 1 ⊗ ω 2 ⊗ ω 3 . One then has
(s ⊗ 1)(1 ⊗ ω 1 ⊗ ω 2 ⊗ ω 3 ) =
s
i ⊗ s
i ⊗ ω 1 ⊗ ω 2 ⊗ ω 3 ,
while
ρ(s
i ⊗ s
i ⊗ ω 1 ⊗ ω 2 ⊗ ω 3 ) = (−1)
|s
i ||ω 2 | ω 1 ⊗ s
i ⊗ ω 2 ⊗ s
i ⊗ ω 3 ,
(6.76)
so the result is
(−1)
|s
i ||ω 2 | f (ω 1 ⊗ s
i ⊗ ω 2 ⊗ s
i ⊗ ω 3 ) as claimed.
6.5
Odd Modular Operads
One of our fundamental constructions used in this work is the Feynman transform
of a modular operads recalled below in Sect. 7.2. Quite surprisingly, the Feynman
transform is not an ordinary modular operad, but its odd version.
Definition 6.24 An odd modular operad 12 with step s is a modular module
T =
T (S; g) ∈ Chain; (S; g) ∈ Cor × A
(6.77)
together with degree +1 morphisms (compositions)
a • b : T
S 1 {a}; g 1
⊗ T
S 2 {b}; g 2
→ T (S 1 S 2 ; g 1 + g 2 )
(6.78)
12 This terminology was introduced by Ralph Kaufmann; the name “twisted modular operad” is
sometimes used, too.