Mirror symmetry for stable quotients invariants

Abstract

The moduli space of stable quotients introduced by Marian-Oprea-Pandharipande provides a natural compactification of the space of morphisms from nonsingular curves to a nonsingular projective variety and carries a natural virtual class. We show that the analogue of Givental's J-function for the resulting twisted projective invariants is described by the same mirror hypergeometric series as the corresponding Gromov-Witten invariants (which arise from the moduli space of stable maps), but without the mirror transform (in the Calabi-Yau case). This implies that the stable quotients and Gromov-Witten twisted invariants agree if there is enough "positivity", but not in all cases. As a corollary of the proof, we show that certain twisted Hurwitz numbers arising in the stable quotients theory are also described by a fundamental object associated with this hypergeometric series. We thus completely answer some of the questions posed by Marian-Oprea-Pandharipande concerning their invariants. Our results suggest a deep connection between the stable quotients invariants of complete intersections and the geometry of the mirror families. As in Gromov-Witten theory, computing Givental's J-function (essentially a generating function for genus 0 invariants with 1 marked point) is key to computing stable quotients invariants of higher genus and with more marked points; we exploit this in forthcoming papers.

The equivariant A-twist and gauged linear sigma models on the two-sphere

“…30 In particular, the contour integral formula for correlators in the geometric phase presented in [15] can be obtained by summing the instantons in the naïve version of our formula (8.4). 31 The statement extends to all F a,b correlators of positive axial R-charge (a + b > 3). Those which violate the m = 0 selection rule for the axial R-symmetry vanish in the m → 0 limit.…”

“…These recursion relations simplify many explicit computations, and have deep relationships to enumerative geometry. 5 It is interesting to observe that the one-loop determinant Z 1-loop we have computed can be identified with the densities computed in [19,20,21,31] that are integrated over moduli spaces of curves to obtain certain geometric invariants, once Ω is identified with the "equivariant parameter" of these works.…”

“…30 For n < 0 we find that our formula (8.4) gives the same result: limit and residue commute. 31 It was also argued in [15] that their naive formula is incorrect for the n = 0 correlators F 3−b,b . Note that the formula follows from n < 0 correlators and quantum cohomology relations at vanishing twisted masses.…”

Energy bounds and vanishing results for the Gromov–Witten invariants of the projective space

“…As demonstrated in [6,18,20], equivariant localization computations in GW-theory can sometimes be carried out by working with the residues of the equivariant mirror B-side functions and by extracting the non-equivariant terms at the end. In such situations, precise knowledge of the equivariant coefficients C…”

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

“…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.…”