Mirror theorems for root stacks and relative pairs

Abstract

Given a smooth projective variety X with a smooth nef divisor D and a positive integer r, we construct an I-function, an explicit slice of Givental's Lagrangian cone, for Gromov-Witten theory of the root stack X D,r . As an application, we also obtain an I-function for relative Gromov-Witten theory following the relation between relative and orbifold Gromov-Witten invariants.

Structures in genus‐zero relative Gromov–Witten theory

“…Following Definition 3.3, the procedures are very straightforward and we believe it is unnecessary to spell it out. When (X, D) is not a toric pair, a mirror theorem for the pair (X, D) has recently been proved in [12] using our formalism.…”

A mirror theorem for multi-root stacks and applications

“…In this paper, we study genus zero orbifold Gromov-Witten theory of multi-root stacks. We generalize the main theorem of [14] to normal crossing divisors. In other words, we prove a mirror theorem for multi-root stacks by constructing the I-function which lies in Givental's Lagrangian cone.…”

“…In Section 5.1, we use the mirror theorem for relative Gromov-Witten theory of (X, D), derived in [14], to calculate the relative Gromov-Witten invariant on the right-hand side of (2). The local Gromov-Witten invariant on the left-hand side of (2) is calculated using a well-known mirror theorem, see, for example, [20].…”

“…Under suitable assumptions, it means that the extended I-function can be used to compute orbifold invariants with orbifold markings that map to these twisted sectors. An example of such an extended I-function is written in [14,Section 3.4] for root stacks where the root construction is along a smooth divisor.…”

Log BPS numbers of log Calabi-Yau surfaces

“…Stating equation (1.1) only for β-primitive points leads to the following conjecture. It is a BPS version of the log-local principle put forward and proved in some cases in [30], and further developed in [13,14,71], as well as in [25,84] in relation to orbifold Gromov-Witten theory.…”