Weyl symmetry for curve counting invariants via spherical twists

TL;DR

This paper explores how Weyl symmetry, via spherical twists, influences curve counting invariants in Calabi--Yau 3-folds, leading to new rationality results and functional equations for Pandharipande--Thomas invariants.

Contribution

It introduces a novel approach linking Weyl symmetry and derived auto-equivalences to curve counting invariants, with new rationality and functional equations.

Findings

Derived auto-equivalence group acts on generating functions

Established rationality of Pandharipande--Thomas invariants

Matched results with Calabi--Yau orbifold cases

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 $A_1$-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 DT invariants for surface-like objects.