Related papers: A note on BPS structures and Gopakumar-Vafa invari…
We study a class of flat bundles, of finite rank $N$, which arise naturally from the Donaldson-Thomas theory of a Calabi-Yau threefold $X$ via the notion of a variation of BPS structure. We prove that in a large $N$ limit their flat…
In this paper, we generalize a mathematical definition of Gopakumar-Vafa (GV) invariants on Calabi-Yau 3-folds introduced by Maulik and the author, using an analogue of BPS sheaves introduced by Davison-Meinhardt on the coarse moduli spaces…
The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove…
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…
We prove a closed formula for leading Gopakumar- Vafa BPS invariants of local Calabi-Yau geometries given by the canonical line bundles of toric Fano surfaces. It shares some similar features with Goettsche-Yau-Zaslow formula: Connection…
Using the interpretation of certain generalised Donaldson-Thomas invariants, including stable pairs curve counts, as the monodromy of a flat connection on a formal principal bundle, we show that the conjectural Gopakumar-Vafa contributions…
In 2008, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold $X$ using Gromov-Witten theory. Recently, Cao-Maulik-Toda proposed a conjectural description of these invariants in terms of stable pair theory. When…
In analogy with the Gopakumar-Vafa (GV) conjecture on Calabi-Yau (CY) 3-folds, Klemm and Pandharipande defined GV type invariants on Calabi-Yau 4-folds using Gromov-Witten theory and conjectured their integrality. In a joint work with…
Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds,…
The Gopakumar-Vafa conjecture predicts that the BPS invariants of a symplectic 6-manifold, defined in terms of the Gromov-Witten invariants, are integers and all but finitely many vanish in every homology class. The integrality part of this…
We study the equivariant version of the genus zero BPS invariants of the total space of a rank 2 bundle on P^1 whose determinant is O(-2). We define the equivariant genus zero BPS invariants by the residue integrals on the moduli space of…
The Gopakumar-Vafa type invariants on Calabi-Yau 4-folds (which are non-trivial only for genus zero and one) are defined by Klemm-Pandharipande from Gromov-Witten theory, and their integrality is conjectured. In a previous work of…
Supersymmetric D-branes supported on the complex two-dimensional base $S$ of the local Calabi-Yau threefold $K_S$ are described by semi-stable coherent sheaves on $S$. Under suitable conditions, the BPS indices counting these objects (known…
The Gopakumar-Vafa conjecture is defined and studied for the local geometry of a curve in a Calabi-Yau 3-fold. The integrality predicted in Gromov-Witten theory by the Gopakumar-Vafa BPS count is verified in a natural series of cases in…
We prove the integrality and finiteness of open BPS invariants of toric Calabi-Yau 3-folds relative to Aganagic-Vafa outer branes, defined from open Gromov-Witten invariants by the Labastida-Mari\~no-Ooguri-Vafa formula. Specializing to…
We define $p$-adic BPS or $p$BPS-invariants for moduli spaces $M_{\beta,\chi}$ of 1-dimensional sheaves on del Pezzo surfaces by means of integration over a non-archimedean local field $F$ . Our definition relies on a canonical measure…
We study properties of the recently established refined topological recursion for some simple spectral curves associated to quadratic differentials. We prove explicit formulas for the free energy and Voros coefficients of the corresponding…
Kim, Kresch and Oh defined unramified Gromov-Witten invariants. For a threefold, Pandharipande conjectured that they are equal to Gopakumar-Vafa invariants (BPS invariants) in the case of Fano classes and primitive Calabi-Yau classes. We…
As an analogy to Gopakumar-Vafa conjecture on Calabi-Yau 3-folds, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold $X$ using Gromov-Witten theory. When $X$ is holomorphic symplectic, Gromov-Witten invariants…
In analogy with the Gopakumar-Vafa conjecture on CY 3-folds, Klemm and Pandharipande defined GV type invariants on Calabi-Yau 4-folds using Gromov-Witten theory and conjectured their integrality. In this paper, we propose a sheaf-theoretic…