The quantum orbifold cohomology of toric stack bundles

Abstract

Abstract. We study Givental's Lagrangian cone for the quantum orbifold cohomology of toric stack bundles. Using Gromov-Witten invariants of the base and combinatorics of the toric stack fibers, we construct an explicit slice of the Lagrangian cone defined by the genus 0 Gromov-Witten theory of a toric stack bundle.

Mirror theorems for root stacks and relative pairs

“…We also refer to [15] and [12] for the S-extended I-function for toric stacks. As mentioned in [26], [15], [30], the non-extended Ifunction only determines the restriction of the J-function to the small parameter space H 2 (X D,r , C) ⊂ H 2 CR (X D,r , C). Taking the S-extended I-function allows one to determine the J-function along twisted sectors.…”

“…Using the S-extended I-function for toric stack bundles in [30], we also state the mirror theorem for root stacks in terms of S-extended I-function. Theorem 1.2 (=Theorem 3.11).…”

“…Y X∞,r is a toric stack bundle over X with the fibers isomorphic to the weighted projective line P 1 1,r . Therefore, by the mirror theorem for toric stack bundles [30], [41], we can write the I-function of Y X∞,r as the hypergeometric modification of the J-function of X:…”

“…There are two reasons that we need to assume X is a smooth projective variety instead of a smooth proper Deligne-Mumford stack. First, the mirror theorem for toric stack bundles in [30] requires the base to be a smooth variety. Whether it holds for smooth Deligne-Mumford stack is currently not yet studied.…”

“…The following mirror theorem again follows from the mirror theorem for toric stack bundles [30] and orbifold quantum Lefschetz [36], [14]. Theorem 3.11 allows one to compute orbifold Gromov-Witten invariants of root stacks when there are orbifold marked points with ages a i r, for 1 ≤ i ≤ m.…”