Curve counting via stable pairs in the derived category
R. Pandharipande, R. P. Thomas
TL;DR
This work develops a comprehensive stable-pairs theory for nonsingular projective 3-folds by modeling pairs as objects in the derived category and constructing a virtual class via a fixed-determinant obstruction theory. It proves that the resulting integer invariants P_{n,β} should encode GW/DT data and BPS counts, and formulates precise conjectures equating stable-pairs, Gromov–Witten, and Donaldson–Thomas theories under canonical transformations. The paper provides explicit local and toric calculations, including a complete local P^1 example and a local-curve contribution, and introduces a novel stable-pairs vertex that parallels the DT vertex in the toric Calabi–Yau setting. Collectively, it ties stable-pairs invariants to BPS integrality, wall-crossing in the derived category, and the topological-vertex machinery, offering a unified, integer-valued perspective on curve counting in 3-folds with broad consequences for enumerative geometry and string theory.
Abstract
For a nonsingular projective 3-fold $X$, we define integer invariants virtually enumerating pairs $(C,D)$ where $C\subset X$ is an embedded curve and $D\subset C$ is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of $X$. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of $X$. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.