Counting curves on surfaces in Calabi–Yau 3-folds

Abstract

Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of two-dimensional torsion sheaves, enumerating pairs (formula presented). Here (formula presented) is a member of a sufficiently positive linear system and (formula presented) is a one-dimensional subscheme of it. The associated sheaf is the ideal sheaf of (formula presented), pushed forward to (formula presented) and considered as a certain Joyce–Song pair in the derived category of (formula presented). We express these invariants in terms of the MNOP invariants of (formula presented).