Quantum Cohomology of Stacks

From tomorrow onwards, we will be working with toric stacks, but today we’ll be more general. Let f : Γ → X be a stable representable morphism from an orbi-curve to a stack. An orbi-curve is a proper, projective, algebraic curve Γ with some marked points xi. Each of the marked points has a chart of the form [∆/μri ] (where ∆ is a disk and μr is the group of r-th roots of unity). An orbi-curve has a fundamental group π 1 = π1(Γ r {xi})/〈γ ri i 〉, where the γi is a small loop around xi. Orbi-curves have line bundles on them, which are line bundles L, together with an action of μri on Lxi , given by v 7→ ζ iv. Riemann-Roch tells you that