162
7 Feynman Transform of a Modular Operad
(iii) For finite sets S 1 , S 2 , symbols a, b, u, v, and genera g 1 , g 2 ∈ A, one has the
equality
a
R
• b ( uv ⊗ 1) = − uv a
R
• b
of maps M
S 1 {a, u, v}; g 1
⊗ T
S 2 {b}; g 2
→ M(S 1 S 2 ; g 1 + g 2 + s).
(iv) For finite sets S 1 , S 2 , symbols a, b, u, v, and genera g 1 , g 2 ∈ A, one has the
equality
a
R
• b (1 ⊗ • uv ) = − uv a
R
• b
of maps M
S 1 {a}; g 1
⊗ T
S 2 {b, u, v}; g 2
→ M(S 1 S 2 ; g 1 + g 2 + s).
(v) For mutually disjoint sets S 1 , S 2 , S 3 , symbols a, b, c, d and genera g 1 , g 2 , g 3 ∈
A, one has the equality
a
L
• b (1 ⊗ c
R
• d ) = − c
R
• d ( a
L
• b ⊗1)
(7.3)
of maps from T
S 1 {a}; g 1
⊗ M
S 2 {b, c}; g 2
⊗ T
S 3 {d}; g 3
to the
space M
S 1 S 2 S 3 ; g 1 +g 2 +g 3
.
(vi) For mutually disjoint sets S 1 , S 2 , S 3 , symbols a, b, c, d and genera g 1 , g 2 , g 3 ∈
A, one has the equality
a
R
• b (1 ⊗ c • d ) = − c
R
• d ( a
R
• b ⊗1)
(7.4)
of maps from M
S 1 {a}; g 1
⊗ T
S 2 {b, c}; g 2
⊗ T
S 3 {d}; g 3
to the
space M
S 1 S 2 S 3 ; g 1 +g 2 +g 3
.
Since the pasting schemes for modular operads and their modules are abstract
graphs, there is no concept of “left” and “right” compositions so, unlike, e.g. left
versus right modules over associative algebras, a
L
• b and a
R
• b are materializations of
the same operation which differ only by the way they are written on paper.
Example 7.1 Any odd modular operad T with the structure operations a • b and • uv
is a module over itself, with
a
L
• b := a • b and uv := • uv .
All axioms of Definition 7.1 clearly follow from Definition 6.24 of an odd operad.
Slightly less straightforward is only (iv) obtained by applying (1 ⊗ τ ) from the right
on (6.80), and (vii) obtained by applying τ from the right on (6.83).
7 Feynman Transform of a Modular Operad
(iii) For finite sets S 1 , S 2 , symbols a, b, u, v, and genera g 1 , g 2 ∈ A, one has the
equality
a
R
• b ( uv ⊗ 1) = − uv a
R
• b
of maps M
S 1 {a, u, v}; g 1
⊗ T
S 2 {b}; g 2
→ M(S 1 S 2 ; g 1 + g 2 + s).
(iv) For finite sets S 1 , S 2 , symbols a, b, u, v, and genera g 1 , g 2 ∈ A, one has the
equality
a
R
• b (1 ⊗ • uv ) = − uv a
R
• b
of maps M
S 1 {a}; g 1
⊗ T
S 2 {b, u, v}; g 2
→ M(S 1 S 2 ; g 1 + g 2 + s).
(v) For mutually disjoint sets S 1 , S 2 , S 3 , symbols a, b, c, d and genera g 1 , g 2 , g 3 ∈
A, one has the equality
a
L
• b (1 ⊗ c
R
• d ) = − c
R
• d ( a
L
• b ⊗1)
(7.3)
of maps from T
S 1 {a}; g 1
⊗ M
S 2 {b, c}; g 2
⊗ T
S 3 {d}; g 3
to the
space M
S 1 S 2 S 3 ; g 1 +g 2 +g 3
.
(vi) For mutually disjoint sets S 1 , S 2 , S 3 , symbols a, b, c, d and genera g 1 , g 2 , g 3 ∈
A, one has the equality
a
R
• b (1 ⊗ c • d ) = − c
R
• d ( a
R
• b ⊗1)
(7.4)
of maps from M
S 1 {a}; g 1
⊗ T
S 2 {b, c}; g 2
⊗ T
S 3 {d}; g 3
to the
space M
S 1 S 2 S 3 ; g 1 +g 2 +g 3
.
Since the pasting schemes for modular operads and their modules are abstract
graphs, there is no concept of “left” and “right” compositions so, unlike, e.g. left
versus right modules over associative algebras, a
L
• b and a
R
• b are materializations of
the same operation which differ only by the way they are written on paper.
Example 7.1 Any odd modular operad T with the structure operations a • b and • uv
is a module over itself, with
a
L
• b := a • b and uv := • uv .
All axioms of Definition 7.1 clearly follow from Definition 6.24 of an odd operad.
Slightly less straightforward is only (iv) obtained by applying (1 ⊗ τ ) from the right
on (6.80), and (vii) obtained by applying τ from the right on (6.83).