Counting invariant curves: A theory of Gopakumar-Vafa invariants for Calabi-Yau threefolds with an involution

We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau threefold (CY3) X which is equipped with an involution & imath; preserving the holomorphic volume form. We define integers n g,h ( beta ) which give a virtual count of the number of genus g curves C on X , in the class beta E H 2 ( X ), which are invariant under & imath; , and whose quotient C/& imath; has genus h . We give two definitions of n g,h ( beta ) which we conjecture to be equivalent: one in terms of a version of Pandharipande-Thomas theory and one in terms of a version of Maulik-Toda theory. We compute our invariants and give evidence for our conjecture in several cases. In particular, we compute our invariants when X = S x C , where S is an Abelian surface with & imath; ( a ) = - a or a K 3 surface with a symplectic involution (a Nikulin K 3 surface). For these cases, we give formulas for our invariants in terms of Jacobi modular forms.