On the operad structure of admissible G-covers

TL;DR

This paper develops a new operad structure on moduli spaces of pointed stable curves with admissible G-covers, introducing a category-colored operad concept that links to G-equivariant CohFTs and orbifold Gromov--Witten invariants.

Contribution

It introduces a novel operad structure colored by categories, connecting admissible G-covers with G-equivariant CohFTs and orbifold Gromov--Witten invariants.

Findings

Operad structure on moduli spaces of G-covers described.

Category-colored operad concept introduced.

Orbifold Gromov--Witten invariants provide examples of G-CohFTs.

Abstract

We describe the modular operad structure on the moduli spaces of pointed stable curves equipped with an admissible $G$-cover. To do this we are forced to introduce the notion of an operad colored not by a set but by the objects of a category. This construction interpolates in a sense between `framed' and `colored' versions of operads; we hope that it will be of independent interest. An algebra over this operad is the same thing as a $G$-equivariant CohFT, as defined by Jarvis, Kaufmann and Kimura. We prove that the (orbifold) Gromov--Witten invariants of global quotients $[X/G]$ give examples of $G$-CohFTs.