Mirror Symmetry for Stable Quotients Invariants

Double and triple Givental’s J-functions for stable quotients invariants

“…is a times the coefficient J n;a (q) d of q d in J n;a (q); see [5,Theorem 1]. The contribution of Z 1,1 0 (d) to SQ d n;a (2, 0) is the same; this explains the first term on the right-hand side of (1.10).…”

“…Using (1.9), the genus 0 two-and three-marked SQ-invariants of a Calabi-Yau complete intersection threefold X n;a can be expressed in terms of the BPS counts of GW-theory. For example, by the m = 2 case of (1.9), SQ 1,1 n;a (q) = a J n;a (q) − ∞ d=1 BPS d n;a (1, 1) ln 1 − q d e dJn;a(q) , (1.10) Table 1: Some genus 0 GW-and SQ-invariants of a quintic threefold X 5; (5) where BPS d n;a (1, 1) are the genus 0 two-marked BPS counts for X n;a defined by…”

“…ν n (a) = 0; thus, a different argument is needed in these cases. We employ the same kind of trick as used in [5] to confirm mirror symmetry for the stable quotients analogue of Givental's J-function and essentially deduce the Calabi-Yau cases from the Fano cases. Specifically, we show that the equivariant three-point mirror formula of Proposition 3.1 is equivalent to the closed formula for twisted three-point Hurwitz numbers of Proposition 4.1, whenever |a| ≤ n. In Section 9, we show that the validity of the latter does not depend n; since it holds whenever |a| < n, it follows that it holds for all a, and so the equivariant three-point mirror formula of Proposition 3.1 holds whenever |a| ≤ n. Along with [21], Proposition 3.1 finally leads to the mirror formula for the stable-quotients analogue of the triple Givental's J-function in Theorem 3; see Section 10.…”

“…We begin this section by reviewing the equivariant setup used in [20,16,5] and of the hypergeometric series Ḟn;a and Fn;a , we state an equivariant version of Theorem 2; see Theorem 3 below. This theorem immediately implies Theorem 2.…”

“…The genus 0 mirror formula in Gromov-Witten theory extends to the twisted Gromov-Witten invariants associated with direct sums of line bundles over projective spaces; see [7,8,11]. By [5], the analogue of Givental's J-function for the twisted stable quotients invariants defined in [12] satisfies a simpler version of the mirror formula from Gromov-Witten theory. In this paper, we obtain mirror formulas for the stable quotients analogues of the double and triple Givental's J-functions for direct sums of line bundles.…”