176
7 Feynman Transform of a Modular Operad
Let us try to define the Feynman transform of a modular cooperad by mimicking
the definition of the cobar construction of a coassociative coalgebra recalled at the
beginning of this section. One is tempted to take the free modular operad F(↑ C )
on the component-wise suspension of a modular cooperad C and equip it with a
differential that extends the structure operations of C into a degree +1 derivation
of F(↑ C ). Quite surprisingly, this would not work. The reason is that, while
the cocompositions (7.16) define a degree +1 operations on this suspensions, the
cocontractions (7.17) induce operations of degree 0! Fortunately, the analog of the
alternative approach to the definition of the cobar construction using odd modular
operads works.
In the following definition, F( ˚
C ) is the free odd modular operad generated by the
modular module ˚
C associated with the underlying modular comodule of C in the
correspondence of Definition 7.6, and ι : ˚
C → F( ˚
C ) the canonical inclusion. Recall
that each derivation θ : F( ˚
C ) → F( ˚
C ) is uniquely determined by its restriction
θι : ˚
C → F( ˚
C ) along the natural inclusion ι : ˚
C → F( ˚
C ) by Theorem 7.1.
Definition 7.9 Let C be a modular cooperad with the structure operations (7.16)
and (7.17), and the internal differential d C . Let ∂, d : F( ˚
C ) → F( ˚
C ) be degree +1
derivations defined, for x ∈ C (S; g) = ˚
C (S; g), by the finite sum 2
∂ι(x) :=
g +s=g
• uv ι ◦
uv
g (x) +
1
2
A
g 1 +g 2 =g
a • b (ι ⊗ ι)
a
A; g 1
◦
b
B;g 2
(x) (7.28)
where a • b and • uv are the degree +1 structure operations of F( ˚
C ), and u, v, a, b
are independent symbols, respectively, by
dι(x) := ιd C (x).
The odd dg-modular operad F (C ) :=
F( ˚
C ), ∂ + d
is called the Feynman
transform of the modular cooperad C . If we need to distinguish between ∂ and d,
we will call ∂ the external and d the internal differential.
Remark 7.4 Let us choose an element s ∈ S. It is clear that the rightmost term
of (7.28) splits into the sum of two terms:
A
s∈A
g 1 +g 2 =g
a • b (ι ⊗ ι)
a
A; g 1
◦
b
B;g 2
(x) +
A
s∈B
g 1 +g 2 =g
a • b (ι ⊗ ι)
a
A; g 1
◦
b
B;g 2
(x).
2 In the first term in the right-hand side, the summation is not performed over the repeated indexes.
All summations are finite by (7.24).
7 Feynman Transform of a Modular Operad
Let us try to define the Feynman transform of a modular cooperad by mimicking
the definition of the cobar construction of a coassociative coalgebra recalled at the
beginning of this section. One is tempted to take the free modular operad F(↑ C )
on the component-wise suspension of a modular cooperad C and equip it with a
differential that extends the structure operations of C into a degree +1 derivation
of F(↑ C ). Quite surprisingly, this would not work. The reason is that, while
the cocompositions (7.16) define a degree +1 operations on this suspensions, the
cocontractions (7.17) induce operations of degree 0! Fortunately, the analog of the
alternative approach to the definition of the cobar construction using odd modular
operads works.
In the following definition, F( ˚
C ) is the free odd modular operad generated by the
modular module ˚
C associated with the underlying modular comodule of C in the
correspondence of Definition 7.6, and ι : ˚
C → F( ˚
C ) the canonical inclusion. Recall
that each derivation θ : F( ˚
C ) → F( ˚
C ) is uniquely determined by its restriction
θι : ˚
C → F( ˚
C ) along the natural inclusion ι : ˚
C → F( ˚
C ) by Theorem 7.1.
Definition 7.9 Let C be a modular cooperad with the structure operations (7.16)
and (7.17), and the internal differential d C . Let ∂, d : F( ˚
C ) → F( ˚
C ) be degree +1
derivations defined, for x ∈ C (S; g) = ˚
C (S; g), by the finite sum 2
∂ι(x) :=
g +s=g
• uv ι ◦
uv
g (x) +
1
2
A
g 1 +g 2 =g
a • b (ι ⊗ ι)
a
A; g 1
◦
b
B;g 2
(x) (7.28)
where a • b and • uv are the degree +1 structure operations of F( ˚
C ), and u, v, a, b
are independent symbols, respectively, by
dι(x) := ιd C (x).
The odd dg-modular operad F (C ) :=
F( ˚
C ), ∂ + d
is called the Feynman
transform of the modular cooperad C . If we need to distinguish between ∂ and d,
we will call ∂ the external and d the internal differential.
Remark 7.4 Let us choose an element s ∈ S. It is clear that the rightmost term
of (7.28) splits into the sum of two terms:
A
s∈A
g 1 +g 2 =g
a • b (ι ⊗ ι)
a
A; g 1
◦
b
B;g 2
(x) +
A
s∈B
g 1 +g 2 =g
a • b (ι ⊗ ι)
a
A; g 1
◦
b
B;g 2
(x).
2 In the first term in the right-hand side, the summation is not performed over the repeated indexes.
All summations are finite by (7.24).