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We construct stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties. In particular, this gives an elliptic generalization of the results of arXiv:1211.1287. We apply them to the computation of the monodromy of $q$-difference equations arising the enumerative K-theory of rational curves in Nakajima varieties, including the quantum Knizhnik-Zamolodchikov equations.
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This paper relates the elliptic stable envelopes of a hypertoric variety $X$ with the K-theoretic stable envelopes of the loop hypertoric space, $widetilde{mathscr{L}}X$. It thus points to a possible categorification of elliptic stable envelopes.
We revisit the construction of stable envelopes in equivariant elliptic cohomology [arXiv:1604.00423] and give a direct inductive proof of their existence and uniqueness in a rather general situation. We also discuss the specialization of this construction to equivariant K-theory.
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 Behrends 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.
On a Weierstra{ss} elliptic surface $X$, we define a `limit of Bridgeland stability conditions, denoted as $Z^l$-stability, by moving the polarisation towards the fiber direction in the ample cone while keeping the volume of the polarisation fixed. We describe conditions under which a slope stable torsion-free sheaf is taken by a Fourier-Mukai transform to a $Z^l$-stable object, and describe a modification upon which a $Z^l$-semistable object is taken by the inverse Fourier-Mukai transform to a slope semistable torsion-free sheaf. We also study wall-crossing for Bridgeland stability, and show that 1-dimensional twisted Gieseker semistable sheaves are taken by a Fourier-Mukai transform to Bridgeland semistable objects.
The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3-folds. We evaluate the equivariant vertex for stable pairs on toric 3-folds in terms of weighted box counting. In the toric Calabi-Yau case, the result simplifies to a new form of pure box counting. The conjectural equivalence with the DT vertex predicts remarkable identities. The equivariant vertex governs primary insertions in the theory of stable pairs for toric varieties. We consider also the descendent vertex and conjecture the complete rationality of the descendent theory for stable pairs.
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