On the Remodeling Conjecture for Toric Calabi-Yau 3-Orbifolds

Stable quotients and the holomorphic anomaly equation

“…The holomorphic anomaly equation can be proven for toric Calabi-Yau 3-folds by the topological recursion of Eynard and Orantin [16] together with the remodelling conjecture [5] proven recently in [17]. The path of [5,16,17] to the holomorphic anomaly equation for toric Calabi-Yau 3-folds involves explicit manipulation of the Gromov-Witten partition function (which can be computed by now in several different ways in toric Calabi-Yau cases [1,30]). Our approach involves the geometry of the moduli space of stable quotients -the derivation of the holomorphic anomaly equation takes place on the stable quotient side.…”

Modularity of open-closed GW invariants of $K_{\mathbb{P}^2/μ_3}$

“…Among them, an attractive class of examples are toric Calabi-Yau 3-folds where an explicit B-model is available via Eynard-Orantin topological recursion on mirror curve. This mirror symmetry is remodeling conjecture [EO07, BKMP09,BKMP10] proved in [FLZ16]. There are 16 local toric surfaces where the mirror curve is genus one.…”

“…Remodeling conjecture, proposed in [BKMP09, BKMP10], relates A-model open-closed Gromov-Witten invariants with B-model Eynar-Orantin topological recursion invariants under mirror map proved in [FLZ16].…”

$\mathcal{N}=1$ Super Topological Recursion

“…The Eynard-Orantin topological recursion introduced in [17,28,29] can be used to compute various kinds of enumerative invariants, such as Gromov-Witten invariants, Hurwitz numbers, knot invariants, and more (see [11,12,15,25,31,30,32,33,37,41] and references therein). Starting with a spectral curve, the Eynard-Orantin topological recursion provides an infinite sequence of multilinear differentials (known as correlation functions) which are generating functions for those enumerative invariants.…”