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Pointed admissible G-covers and G-equivariant cohomological field theories

Published online by Cambridge University Press:  21 June 2005

Tyler J. Jarvis
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, USAjarvis@math.byu.edu
Ralph Kaufmann
Affiliation:
Department of Mathematics, University of Connecticut, 196 Auditorium Road, Storrs, CT 06269-3009, USAkaufmann@math.uconn.edu
Takashi Kimura
Affiliation:
Department of Mathematics and Statistics, 111 Cummington Street, Boston University, Boston, MA 02215, USA and School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA kimura@math.bu.edu
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Abstract

For any finite group G we define the moduli space of pointed admissible G-covers and the concept of a G-equivariant cohomological field theory (G-CohFT), which, when G is the trivial group, reduces to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that taking the ‘quotient’ by G reduces a G-CohFT to a CohFT. We also prove that a G-CohFT contains a G-Frobenius algebra, a G-equivariant generalization of a Frobenius algebra, and that the ‘quotient’ by G agrees with the obvious Frobenius algebra structure on the space of G-invariants, after rescaling the metric. We then introduce the moduli space of G-stable maps into a smooth, projective variety X with G action. Gromov–Witten-like invariants of these spaces provide the primary source of examples of G-CohFTs. Finally, we explain how these constructions generalize (and unify) the Chen–Ruan orbifold Gromov–Witten invariants of $[X/G]$ as well as the ring $H^{\bullet}(X,G)$ of Fantechi and Göttsche.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2005

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