Weyl symmetry for curve counting invariants via spherical twists

Abstract

We study the curve counting invariants of Calabi–Yau 3-folds via the Weyl reflection along a ruled divisor. We obtain a new rationality result and functional equation for the generating functions of Pandharipande–Thomas invariants. When the divisor arises as resolution of a curve of A 1 A_1 -singularities, our results match the rationality of the associated Calabi–Yau orbifold. The symmetry on generating functions descends from the action of an infinite dihedral group of derived auto-equivalences, which is generated by the derived dual and a spherical twist. Our techniques involve wall-crossing formulas and generalized Donaldson–Thomas invariants for surface-like objects.