Counting curves on surfaces in Calabi–Yau 3-folds
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Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of two-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 one-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 \(X\).
KeywordsModulus Space Line Bundle Modular Form Chern Character Ideal Sheaf
We thank Kai Behrend, Vincent Bouchard, Jim Bryan, Tudor Dimofte, Dagan Karp, Jan Manschot, Davesh Maulik and Yukinobu Toda for useful discussions. The second author would like to thank the University of British Columbia, the Max Planck Institute, Bonn and the Isaac Newton Institute for Mathematical Sciences, Cambridge for their hospitality. The third author is supported by the EPSRC programme Grant EP/G06170X/1.
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