Counting curves on surfaces in Calabi-Yau 3-folds

TL;DR

This paper introduces new invariants for counting specific 2-dimensional torsion sheaves on Calabi-Yau threefolds, linking them to existing MNOP invariants and motivated by string theory dualities.

Contribution

It defines novel invariants for pairs of subschemes in Calabi-Yau threefolds and relates them to established enumerative invariants, expanding the understanding of curve counting in this context.

Findings

New invariants for 2D torsion sheaves are defined.

Invariants are expressed in terms of MNOP invariants.

Results are motivated by string theory S-duality conjectures.

Abstract

Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of 2-dimensional torsion sheaves, enumerating pairs $Z\subset H$ in a Calabi-Yau threefold X. Here H is a member of a sufficiently positive linear system and Z is a 1-dimensional subscheme of it. The associated sheaf is the ideal sheaf of $Z\subset H$, pushed forward to X and considered as a certain Joyce-Song pair in the derived category of X. We express these invariants in terms of the MNOP invariants of X.