160
7 Feynman Transform of a Modular Operad
given for any finite set S, genus g ∈ A, and symbols u, v. These data satisfy the
following axioms.
(i) For arbitrary isomorphisms ρ : S 1 {a} → T 1 , σ : S 2 {b} → T 2 of finite
sets and genera g 1 , g 2 ∈ A, one has the equality
M
ρ| S 1 σ | S 2
a
L
• b = ρ(a)
L
• σ (b)
T (ρ) ⊗ M(σ )
of maps
T
S 1 {a}; g 1
⊗ M
S 2 {b}; g 2
→ M
T 1 T 2 \ {ρ(a), σ (b)}; g 1 + g 2
.
(ii) For any isomorphism ρ : S {u, v} → T of finite sets and a genus g ∈ A, one
has the equality
M
ρ| S
uv = ρ(u)ρ(v) M(ρ)
of maps M
S {u, v}; g
→ M
T \ {ρ(u), ρ(v)}; g + s
.
(iii) 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
L
• d ) = − c
L
• d ( a • b ⊗1)
(7.1)
of maps from T
S 1 {a}; g 1
⊗ T
S 2 {b, c}; g 2
⊗ M
S 3 {d}; g 3
to the
space M
S 1 S 2 S 3 ; g 1 +g 2 +g 3
.
(iv) For S 1 , S 2 , S 3 , a, b, c, d and g 1 , g 2 , g 3 ∈ A as in (iii), one has the equality
a
L
• b (1 ⊗ d
L
• c ) = − d
L
• c (1 ⊗ a
L
• b )(τ ⊗ 1)
of maps from T
S 1 {a}; g 1
⊗ T
S 2 {d}; g 2
⊗ M
S 3 {b, c}; g 3
to the
space M
S 1 S 2 S 3 ; g 1 +g 2 +g 3
.
(v) For a finite set S, symbols a, b, c, d and a genus g ∈ A one has the equality
ab cd = − cd ab
of maps M
S {a, b, c, d}; g
→ M(S; g + 2s).
(vi) For finite sets S 1 , S 2 , symbols a, b, c, d and genera g 1 , g 2 ∈ A, one has the
equality
ab c
L
• d = − cd a
L
• b
of maps T
S 1 {a, c}; g 1
⊗ M
S 2 {b, d}; g 2
→ M(S 1 S 2 ; g 1 + g 2 + s).