The operad structure of admissible G-covers

Petersen, Dan

doi:10.2140/ant.2013.7.1953

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The operad structure of admissible $G$-covers

Dan Petersen

Vol. 7 (2013), No. 8, 1953–1975

DOI: 10.2140/ant.2013.7.1953

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 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 $[X ∕ G]$ give examples of $G$ -CohFTs.

Keywords

modular operad, operad colored by groupoid, orbifold Gromov–Witten theory, cohomological field theory

Mathematical Subject Classification 2010

Primary: 18D50

Secondary: 14H10, 14D21

Milestones

Received: 4 June 2012

Revised: 18 January 2013

Accepted: 16 March 2013

Published: 24 November 2013

Authors

Dan Petersen
Department of Mathematics KTH Royal Institute of Technology 100 44 Stockholm Sweden

Vol. 7, No. 8, 2013