Pointed Admissible G-Covers and G-equivariant Cohomological Field Theories

Details Pointed Admissible G-Covers and G-equivariant Cohomological Field Theories Pointed Admissible G-Covers and G-equivariant Cohomological Field Theories

2003-02-25

arxiv.org/abs/math/0302316v5

Compositio Mathematica 141 (2005) 926-978

Mathematics

Algebraic Geometry

Corrected proof of the trace axiom and made minor typo corrections. 13 figures. Uses Paul Taylor's diagrams package

Scientific paper

10.1112/S0010437X05001284

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, reduce to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that by taking the "quotient" by G, a G-CohFT reduces 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 also 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 the global quotient [X/G] as well as the ring H*(X,G) of Fantechi and Goettsche.

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