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7 Feynman Transform of a Modular Operad
On the other hand, one has
t 1 a • b (t 2 c • d t 3 ) = −(t 1 a • b t 2 ) c • d t 3
by (6.80),
t 1 a
L
• b (t 2 c
L
• d m 3 ) = −(t 1 a • b t 2 ) c
L
• d m 3
by (7.1),
t 1 a
L
• b (m 2 c
R
• d t 3 ) = −(t 1 a
L
• b m 2 ) c
R
• d t 3
by (7.3), and
m 1 a
R
• b (t 2 c • d t 3 ) = −(m 1 a
R
• b t 2 ) c
R
• d t 3
by (7.4). The requisite equality (6.80) follows immediately.
Definition 7.4 Let T be an odd modular operad and M a T -module with the
structure operations a
L
• b and ab . We define the modular module sM by
sM(S; g) := ↑ M(S; g), (S; g) ∈ Cor × A,
with the actions (6.48) given as sM(σ ) := ↑ M(σ ) ↓. For disjoint finite sets S 1 , S 2 ,
symbols a, b and genera g 1 , g 2 ∈ A we define degree +1 morphisms
a
L
• b : T
S 1 {a}; g 1
⊗ sM
S 2 {b}; g 2
→ sM(S 1 S 2 ; g 1 + g 2 )
by
a
L
• b := −↑ a
L
• b (1 ⊗ ↓).
(7.7)
For a finite set S, genus g ∈ A, and symbols u, v we also define degree +1 maps
uv = vu : sM
S {u, v}; g
→ sM(S; g + s)
by the formula
uv := −↑ uv ↓ .
(7.8)
The auxiliary “right” actions a
R
• b : sM ⊗ T → sM of the suspension sM are given
by a
R
• b := −↑ a
R
• b (↓ ⊗ 1).