18 research outputs found
The SYZ mirror symmetry and the BKMP remodeling conjecture
The Remodeling Conjecture proposed by Bouchard-Klemm-Mari\~{n}o-Pasquetti
(BKMP) relates the A-model open and closed topological string amplitudes (open
and closed Gromov-Witten invariants) of a symplectic toric Calabi-Yau 3-fold to
Eynard-Orantin invariants of its mirror curve. The Remodeling Conjecture can be
viewed as a version of all genus open-closed mirror symmetry. The SYZ
conjecture explains mirror symmetry as -duality. After a brief review on SYZ
mirror symmetry and mirrors of symplectic toric Calabi-Yau 3-orbifolds, we give
a non-technical exposition of our results on the Remodeling Conjecture for
symplectic toric Calabi-Yau 3-orbifolds. In the end, we apply SYZ mirror
symmetry to obtain the descendent version of the all genus mirror symmetry for
toric Calabi-Yau 3-orbifolds.Comment: 18 pages. Exposition of arXiv:1604.0712
Torus knots in Lens spaces, open Gromov-Witten invariants, and topological recursion
Starting from a torus knot in the lens space , we
construct a Lagrangian sub-manifold in
under the conifold
transition. We prove a mirror theorem which relates the all genus open-closed
Gromov-Witten invariants of to the topological
recursion on the B-model spectral curve. This verifies a conjecture in
\cite{Bor-Bri} in the case of lens space.Comment: 43 pages, 6 figure
Twisted Equivariant Gromov-Witten Theory of the Classifying Space of a Finite Group
For any finite group , the equivariant Gromov-Witten invariants of
can be viewed as a certain twisted Gromov-Witten invariants
of the classifying stack . In this paper, we use Tseng's
orbifold quantum Riemann-Roch theorem to express the equivariant Gromov-Witten
invariants of as a sum over Feynman graphs, where the weight
of each graph is expressed in terms of descendant integrals over moduli spaces
of stable curves and representations of .Comment: This paper is a non-abelian generalization of arXiv:1310.481
Equivariant Gromov-Witten Theory of GKM Orbifolds
In this paper, we study the all genus Gromov-Witten theory for any GKM orbifold X. We generalize the Givental formula which is studied in the smooth case in [41] [42] [43] to the orbifold case. Specifically, we recover the higher genus Gromov-Witten invariants of a GKM orbifold X by its genus zero data. When X is toric, the genus zero Gromov-Witten invariants of X can be explicitly computed by the mirror theorem studied in [22] and our main theorem gives a closed formula for the all genus Gromov-Witten invariants of X. When X is a toric Calabi-Yau 3-orbifold, our formula leads to a proof of the remodeling conjecture in [38]. The remodeling conjecture can be viewed as an all genus mirror symmetry for toric Calabi-Yau 3-orbifolds. In this case, we apply our formula to the A-model higher genus potential and prove the remodeling conjecture by matching it to the B-model higher genus potential