Bifurcations of embedded curves and towards an extension of Taubes' Gromov invariant to Calabi-Yau 3-folds

We define an integer-valued virtual count of embedded pseudo-holomorphic curves of two times a primitive homology class and arbitrary genus in symplectic Calabi–Yau $3$-folds, which can be viewed as an extension of Taubes’ Gromov invariant. The construction depends on a detailed study of bifurcations of moduli spaces of embedded pseudo-holomorphic curves which is partially motivated by Wendl’s recent solution of Bryan–Pandharipande’s super-rigidity conjecture.