- 25 Downloads
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 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.