The quantum orbifold cohomology of weighted projective spaces

Abstract

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental's heuristic argument, which relates small quantum cohomology to S 1 -equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov-Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.

Computing genus-zero twisted Gromov-Witten invariants

“…We also prove a quantum Lefschetz theorem for orbifolds, Corollary 5.1 below, which directly generalizes [24,Theorem 2] and [58,Theorem 5.15]. (This suffices, for example, to determine the even-degree part of the small quantum orbifold cohomology algebra of any of the 181 Fano 3-fold weighted projective complete intersections with terminal singularities: see [23,Proposition 1.10].) In Section 6 we consider the situation of Example B, computing in Section 6.2 the genus-zero Gromov-Witten potential of the type A surface singularity C 2 /Z n .…”

“…This approach is used in [23] to compute genuszero invariants of weighted projective complete intersections. If we assume more -that H 1 (C, f ⋆ F ) = 0 for all topological types θ which contribute non-trivially to I tw (t ′ , z) -then exactly the same argument allows us to determine those genuszero (n + 1)-point Gromov-Witten invariants of Y which involve n classes coming from (18).…”

Orbifold quasimap theory

“…For this purpose, we construct another quasimap graph space. When X is a weighted projective space, this construction already appeared in [15].…”

Quantum Cohomology and Periods

“…The following will be shown in joint work with Coates, Corti and Tseng [15] (see [18] for the case of weighted projective spaces): …”