Counting Invariant Curves: a theory of Gopakumar-Vafa invariants for Calabi-Yau threefolds with an involution

Abstract

We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau threefold (CY3) X which is equipped with an involution ı preserving the holomorphic volume form. We define integers n g,h (β) which give a virtual count of the number of genus g curves C on X, in the class β ∈ H 2 (X), which are invariant under ı, and whose quotient C/ı has genus h. We give two definitions of n g,h (β) 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 × C where S is an Abelian surface with ı(a) = −a or a K3 surface with a symplectic involution (a Nikulin K3 surface). For these cases, we give formulas for our invariants in terms of Jacobi modular forms. For the Abelian surface case, the specialization of our invariants n g,h (β) to h = 0 recovers the count of hyperelliptic curves on an Abelian surface first computed in [9].