1
Introduction
Loosely speaking, a modular operad can be pictured as a set of vertices, or corollas,
labeled by the number of legs n and a further integer g interpreted as the genus. Two
corollas can be composed. For each pair of legs (one from the first corolla, another
from the second one), the composition is done by joining the legs and contracting
the resulting edge. This composition decreases the total number of legs by two and
is additive with respect to the genera. Similarly, for each pair of legs of the same
corolla, there is a self-composition consisting of joining the legs and contracting
the resulting edge. Such a self-composition reduces the number of legs by two and
increases the genus by one. Further, corollas admit an action of the symmetric group,
by permuting the legs of a corolla. To define the structure of a modular operad (in
the category of differential graded vector spaces) we need a prescription associating
to each corolla a dg (differential graded) vector space. Also, we need a prescription
transferring the compositions and the action of the symmetric group from corollas
to the associated vector spaces in a natural way. These prescriptions determine the
particular kind of a modular operad we are dealing with, e.g., modular commutative
operad, modular associative and so forth.
The Feynman transform associates to a modular operad an odd modular operad in
a way reminiscent of constructing Feynman graphs in quantum field theory. Here,
“odd” refers to the operadic compositions, which have now degree one. Roughly
speaking, Feynman transform is a free operad generated by a properly suspended
dual and a differential formed using the duals of the operadic compositions of the
original modular operad. Since as an odd operad the Feynman transform is generated
by corollas, it can be described in terms of graphs with external legs and any
numbers of loops resulting from concatenation of corollas, similarly as Feynman
graphs are composed from the interaction vertices.
An important example of an odd modular operad is the endomorphism operad
associated with a differential graded vector space equipped with an odd symplectic
form. An algebra over an odd modular operad is a morphism from it to the
endomorphism operad, i.e. its representation. The algebra resulting in this way
© 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_1
1
Introduction
Loosely speaking, a modular operad can be pictured as a set of vertices, or corollas,
labeled by the number of legs n and a further integer g interpreted as the genus. Two
corollas can be composed. For each pair of legs (one from the first corolla, another
from the second one), the composition is done by joining the legs and contracting
the resulting edge. This composition decreases the total number of legs by two and
is additive with respect to the genera. Similarly, for each pair of legs of the same
corolla, there is a self-composition consisting of joining the legs and contracting
the resulting edge. Such a self-composition reduces the number of legs by two and
increases the genus by one. Further, corollas admit an action of the symmetric group,
by permuting the legs of a corolla. To define the structure of a modular operad (in
the category of differential graded vector spaces) we need a prescription associating
to each corolla a dg (differential graded) vector space. Also, we need a prescription
transferring the compositions and the action of the symmetric group from corollas
to the associated vector spaces in a natural way. These prescriptions determine the
particular kind of a modular operad we are dealing with, e.g., modular commutative
operad, modular associative and so forth.
The Feynman transform associates to a modular operad an odd modular operad in
a way reminiscent of constructing Feynman graphs in quantum field theory. Here,
“odd” refers to the operadic compositions, which have now degree one. Roughly
speaking, Feynman transform is a free operad generated by a properly suspended
dual and a differential formed using the duals of the operadic compositions of the
original modular operad. Since as an odd operad the Feynman transform is generated
by corollas, it can be described in terms of graphs with external legs and any
numbers of loops resulting from concatenation of corollas, similarly as Feynman
graphs are composed from the interaction vertices.
An important example of an odd modular operad is the endomorphism operad
associated with a differential graded vector space equipped with an odd symplectic
form. An algebra over an odd modular operad is a morphism from it to the
endomorphism operad, i.e. its representation. The algebra resulting in this way
© 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_1
1