Remodeling Conjecture with Descendants

We formulate and prove the Remodeling Conjecture with descendants, which is a version of all-genus equivariant descendant mirror symmetry for semi-projective toric Calabi-Yau 3-orbifolds. We consider the $K$-group of equivariant coherent sheaves on the toric Calabi-Yau 3-orbifold with support bounded in a direction, and prove that it is isomorphic to a certain integral relative first homology group of the equivariant mirror curve. We establish a correspondence between all-genus equivariant descendant Gromov-Witten invariants with $K$-theoretic framings and oscillatory integrals (Laplace transforms) of the Chekhov-Eynard-Orantin topological recursion invariants along relative 1-cycles on the equivariant mirror curve. Our genus-zero correspondence is an equivariant Hodge-theoretic mirror symmetry with integral structures. In the non-equivariant setting, we prove a conjecture of Hosono which equates central charges of compactly supported coherent sheaves with period integrals of integral 3-cycles on the Hori-Vafa mirror 3-fold.