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 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. Ordinary GV invariantsLet X be a Calabi-Yau threefold (CY3), by which we mean a smooth quasi-projective variety over C of dimension 3 with K X ∼ = O X .In 1998 GV98, Gopakumar and Vafa (GV) defined via physics integer invariants n g (β) which give a virtual count of curves C ⊂ X of genus g and classMathematically, there are two conjecturally equivalent sheaf-theoretic approaches to defining n g (β), one by Pandharipande and Thomas (PT) via their stable pair invariants PT10, and one more recently given by Maulik and Toda (MT) using perverse sheaves MTo18.We begin by reviewing ordinary GV theory, and then we develop in a parallel fashion a theory of GV invariants for CY3s with an involution. GV invariants via PT theoryLet PT β,n (X) be the moduli space of PT pairs PT09: PT β,n (X) = (F, s) : s ∈ H 0 (X, F ), supp(F ) = β, χ(F ) = n ,