7
Feynman Transform of a Modular Operad
The aim of this chapter is to recall an analog of the bar construction for modular
operads, called in this context the Feynman transform and introduced in [2], see
also [3, Section II.5.3].
7.1
Modules and Derivations
We define modules over odd modular operads and derivations with values in these
modules. We then introduce trivial extensions of odd operads by these modules
and prove that derivations can be encoded by morphisms of these extensions. This
will imply that a derivation whose source is a free odd modular operad is uniquely
determined by its restriction to the modular module of generators. This scheme is
standard, but some care is needed to get signs right, as the operations of odd operads
and their modules have degree +1.
Definition 7.1 Let T be an odd modular operad with compositions a • b and
contractions • uv as in Definition 6.24. A T -module, or a module over T is a
modular module
M =
M(S; g) ∈ Chain; (S; g) ∈ Cor × A
together with degree +1 morphisms
a
L
• b : T
S 1 {a}; g 1
⊗ M
S 2 {b}; g 2
→ M(S 1 S 2 ; g 1 + g 2 )
defined for arbitrary disjoint finite sets S 1 , S 2 , symbols a, b, and genera g 1 , g 2 ∈ A,
and degree +1 maps
uv = vu : M
S {u, v}; g
→ M(S; g + s)
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_7
159
Feynman Transform of a Modular Operad
The aim of this chapter is to recall an analog of the bar construction for modular
operads, called in this context the Feynman transform and introduced in [2], see
also [3, Section II.5.3].
7.1
Modules and Derivations
We define modules over odd modular operads and derivations with values in these
modules. We then introduce trivial extensions of odd operads by these modules
and prove that derivations can be encoded by morphisms of these extensions. This
will imply that a derivation whose source is a free odd modular operad is uniquely
determined by its restriction to the modular module of generators. This scheme is
standard, but some care is needed to get signs right, as the operations of odd operads
and their modules have degree +1.
Definition 7.1 Let T be an odd modular operad with compositions a • b and
contractions • uv as in Definition 6.24. A T -module, or a module over T is a
modular module
M =
M(S; g) ∈ Chain; (S; g) ∈ Cor × A
together with degree +1 morphisms
a
L
• b : T
S 1 {a}; g 1
⊗ M
S 2 {b}; g 2
→ M(S 1 S 2 ; g 1 + g 2 )
defined for arbitrary disjoint finite sets S 1 , S 2 , symbols a, b, and genera g 1 , g 2 ∈ A,
and degree +1 maps
uv = vu : M
S {u, v}; g
→ M(S; g + s)
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_7
159