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7 Feynman Transform of a Modular Operad
The main source of examples of modular cooperads are the piecewise linear duals
of modular operads satisfying suitable finiteness conditions.
Definition 7.8 A Z-graded vector space V is of finite type if each component V k ,
k ∈ Z, is finite-dimensional. It is finite-dimensional if the associated total space
V =
k∈Z
V
k
is finite-dimensional. A modular operad M is of finite type (resp. finitedimensional) if the graded vector space M (S; g) is such for each (S, g) ∈ Cor× A.
Proposition 7.1 Let M be a modular operad of finite type. For k 1 , k 2 ∈ Z, g 1 , g 2 ∈
A and S 1 , S 2 ∈ Cor denote by
g 1 ,k 1
a ◦
g 2 ,k 2
b
: M
S 1 {a}; g 1
k 1 ⊗ M
S 2 {b}; g 2
k 2 → M (S 1 S 2 ; g 1 + g 2 )
k 1 +k 2
the restriction of the structure operation (6.51) to the indicated components. Assume
that the set
(k 1 , k 2 ) ∈ Z
×2 , k 1 + k 2 = k |
k 1 ,g 1
a ◦
k 2 ,g 2
b
= 0
(7.25)
is finite for each S 1 , S 2 , g 1 , g 2 ∈ A and k ∈ Z. Then the modular module
M
#
=
M (S; g)
#
∈ Chain
(S, g) ∈ Cor × A
of piecewise linear duals has a natural modular cooperad structure induced from
the modular operad structure of M .
Remark 7.3 The finiteness of (7.25) is always satisfied when M is non-negatively
or non-positively graded, or finite-dimensional,
Proof (of Proposition 7.1) To shorten the notation, we denote for k ∈ Z, g ∈ A
and S ∈ Cor by C (S; g) k the linear dual
M (S; g) k # of the degree-k graded
component of M (S; g). It is clear that the collection
M
#
= C =
C (S; g) ∈ Chain
(S, g) ∈ Cor × A
with
C (S; g) = M (S; g)
#
=
k∈Z
C (S; g)
k