Torus knots in Lens spaces, open Gromov-Witten invariants, and topological recursion

Starting from a torus knot $\mathcal{K}$ in the lens space $L(p,-1)$, we construct a Lagrangian sub-manifold $L_{\mathcal{K}}$ in $\mathcal{X}=\big(\mathcal{O}_{\mathbb{P}^1}(-1)\oplus \mathcal{O}_{\mathbb{P}^1}(-1)\big)/\mathbb{Z}_p$ under the conifold transition. We prove a mirror theorem which relates the all genus open-closed Gromov-Witten invariants of $(\mathcal{X},L_{\mathcal{K}})$ 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

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

The Gerby Gopakumar-Mari\~no-Vafa Formula

We prove a formula for certain cubic $\Z_n$-Hodge integrals in terms of loop Schur functions. We use this identity to prove the Gromov-Witten/Donaldson-Thomas correspondence for local $\Z_n$-gerbes over \proj^1.Comment: 43 pages, Published Versio