Open-closed Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric DM stacks

We study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth tonic Deligne-Mumford stacks (with possibly nontrivial generic stabilisers K and semi-projective coarse moduli spaces) relative to Lagrangian branes of Aganagic-Vafa type. An Aganagic-Vafa brane in this paper is a possibly ineffective C-infinity orbifold that admits a presentation [(S-1 x R-2)/G(tau)], where G(tau) is a finite abelian group containing K and G(tau)/K congruent to mu(m) is cyclic of some order m E Z(>0). 1. We present foundational materials of enumerative geometry of stable holomorphic maps from bordered orbifold Riemann surfaces to a 3-dimensional Calabi-Yau smooth toric DM stack X with boundaries mapped into an Aganagic-Vafa brane L. All genus open-closed Gromov-Witten invariants of X relative to L are defined by torus localisation and depend on the choice of a framing f is an element of Z of L. 2. We provide another definition of all genus open-closed Gromov-Witten invariants in (1) based on algebraic relative orbifold Gromov-Witten theory, which agrees with the definition in (1) up to a sign depending on the choice of orientation on moduli of maps in (1). This generalises the definition in [57] for smooth toric Calabi-Yau 3-folds and specifies an orientation on moduli of maps in (1) compatible with the canonical orientation on moduli of relative stable maps determined by the complex structure. 3. When X is a toric Calabi-Yau 3-orbifold (i.e., when the generic stabiliser K is trivial), so that G(tau) = mu(m), we define generating functions F-g,h(X,) (L,f) of open-closed Gromov-Witten invariants of arbitrary genus g and number h of boundary circles; it takes values in H-CR* (B mu(m);C)(circle times h), where H-CR* (B-mu m; C) congruent to C-m is the Chen-Ruan orbifold cohomology of the classifying space B-mu m of mu(m). 4. We prove an open mirror theorem that relates the generating function F-0,1(X,(L,f)) of orbifold disk invariants to Abel-Jacobi maps of the mirror curve of X. This generalises a conjecture by Aganagic-Vafa [6] and Aganagic-Klemm-Vafa [5] (proved in full generality by the first and the second authors in [33]) on the disk potential of a smooth semi-projective toric Calabi-Yau 3-fold.