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 ,