Vol. 7, No. 8, 2013

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The operad structure of admissible $G$-covers

Dan Petersen

Vol. 7 (2013), No. 8, 1953–1975

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 the cohomology of 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 [XG] give examples of G-CohFTs.

modular operad, operad colored by groupoid, orbifold Gromov–Witten theory, cohomological field theory
Mathematical Subject Classification 2010
Primary: 18D50
Secondary: 14H10, 14D21
Received: 4 June 2012
Revised: 18 January 2013
Accepted: 16 March 2013
Published: 24 November 2013
Dan Petersen
Department of Mathematics
KTH Royal Institute of Technology
100 44 Stockholm