172
7 Feynman Transform of a Modular Operad
together with degree 0 morphisms (cocompositions)
a
S 1
◦
b
S 2
=
a
S 1 ; g 1
◦
b
S 2 ;g 2
: C (S 1 2 ; g) → C (S 1 g 1 )⊗C (S 2 g 2 ) (7.16)
defined for arbitrary disjoint finite sets S 1 , S 2 , symbols a, b, and genera g 1 , g 2 ∈ A
such that g 1 + g 2 = g. There are, moreover, degree 0 morphisms (cocontractions)
◦
uv
g = ◦
vu
g = ◦
uv
: C (S; g + s) → C (S {u, v}; g)
(7.17)
given for any finite set S, genus g ∈ A, and symbols u, v. These data must satisfy
the following axioms:
(i) For arbitrary isomorphisms ρ : S 1 {a} → T 1 and σ : S 2 {b} → T 2 of finite
sets and genera g 1 , g 2 ∈ A, one has the equality
C (ρ) ⊗ C (σ )
ρ(a)
T 1 \ ρ(a)
◦
ρ(b)
T 2 \σ (a) =
a
S 1
◦
b
S 2
C (ρ| S 1 σ | S 2 )
of maps
C
T 1 T 2 \ {ρ(a), σ (b)}; g 1 + g 2
→ C
S 1 {a}; g 1
⊗ C
S 2 {b}; g 2
.
(ii) For each isomorphism ρ : S {u, v} → T of finite sets and a genus g ∈ A,
one has the equality
C (ρ) ◦
ρ(a)ρ(b)
= ◦
ab
C (ρ| S )
(7.18)
of maps C
T \ {ρ(u), ρ(v)}; g + s
→ C
S {u, v}; g
.
(iii) For S 1 , S 2 , a, b and g 1 , g 2 as in (7.16), one has the equality
τ
S 2
b
◦
S 1
a =
a
S 1
◦
b
S 2
(7.19)
of maps C
S 1 S 2 ; g 1 + g 2
→ C (S 1 {a}; g 1 ) ⊗ C (S 2 {b}; g 2 ).
(iv) 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
(1 ⊗
c
S 2 {b}
◦
d
S 3
)
a
S 1
◦
b
S 2 3
= (
a
S 1
◦
b
S 2 ⊗ 1)
c
S 1 S 2
◦
d
S 3
(7.20)
of maps from C
S 1 S 2 S 3 ; g 1 +g 2 +g 3
to the space
C
S 1 {a}; g 1
⊗ C
S 2 {b, c}; g 2
⊗ C
S 3 {d}; g 3
.