Stable pairs and BPS invariants

By R. Pandharipande, Richard P. Thomas. 2007

203 citations recorded by the source corpus.

We define the BPS invariants of Gopakumar-Vafa in the case of irreducible curve classes on Calabi-Yau 3-folds. The main tools are the theory of stable pairs in the derived category and Behrend's constructible function approach to the virtual class. We prove that for irreducible classes the stable pairs generating function satisfies the strong BPS rationality conjectures. We define the contribution of each curve to the BPS invariants. A curve $C$ only contributes to the BPS invariants in genera lying between the geometric genus and arithmetic genus of $C$. Complete formulae are derived for nonsingular and nodal curves. A discussion of primitive classes on K3 surfaces from the point of view of stable pairs is given in the Appendix via calculations of Kawai-Yoshioka. A proof of the Yau-Zaslow formula for rational curve counts is obtained. A connection is made to the Katz-Klemm-Vafa formula for BPS counts in all genera.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/acd40488893b7a51b9680bd6912c01100e497e69

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Citations in the corpus

Refined curve counting with tropical geometry (2014, methodology): “This is motivated by the proposed definition the Gopakumar Vafa (BPS) invariants in terms of Pandharipande-Thomas invariants in [21].”
DAHA and iterated torus knots (2014, background): “Versal deformations Last, but not least, we refer the reader to [24] for some relatively recent developments concerning the Hilbert schemes of (locally) plane curves, including their role in the theory of the Gopakumar–Vafa BPS invariants (see [35]) and various generalizations.”
The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link (2012, methodology): “In other words, it satisﬁes the hypothesis of the following lemma, whose statement and proof are modeled on Lemma 3.13 of [58].”
Gromov-Witten theory and Noether-Lefschetz theory (2007, background): “Conjecture 2 for primitive classes β is connected to Euler characteristics of the moduli spaces of stable pairs on K3 by the correspondence of [42, 43].”
Dirichlet Branes and Mirror Symmetry (2009, background): “Without making further technical points, let us just mention that the theory also has connections to topological strings [381, 354, 386, 437], modular forms [111], Ringel-Hall algebras [269, 270, 271, 273, 59] as well as the combinatorics of mutations on quivers [313].”
Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics (2019, methodology): “The Kawai-Yoshioka formula has played a central role in the Gromov-Witten and the stable pairs theory of K3 surfaces [40, 45, 46].”
Gromov-Witten/pairs correspondence for the quintic 3-fold (2012, background): “The second input is the foundation of the theory of stable pairs developed in [34, 35, 36].”
Refined curve counting on complex surfaces (2012, methodology): “One can avoid this unpleasantness by indexing as in Pandharipande and Thomas [62] and elsewhere by the Euler characteristic of the ideal sheaf on the left-hand side and by the genus rather than cogenus on the right; however, for our purposes indexing by the cogenus is far more convenient.”
Mirror symmetry and projective geometry of Reye congruences I (2011, methodology): “les 3,4,5 in Appendix B), are obtained by using mirror symmetry, but should be useful as a testing ground for the recent developments in the study of integral invariants of Calabi-Yau threefolds (see [PT] and references therein). Acknowledgments: This paper is supported in part by Grant-in Aid Scien”
Curves on K3 surfaces and modular forms (2010, background): “) A basic result proved in [18, 47] is the following.”