Vanishing and Symmetries of BPS Invariants for CDV Singularities

Abstract: This paper shows that the motivic BPS invariants associated to a noncommutative crepant resolution of a compound Du-Val singularity are controlled by the labelled Dynkin combinatorics appearing in the work of Iyama--Wemyss. In particular, we show that the invariants vanish for dimension vectors which are not a multiple of a restricted root obtained from the affine root system under a natural quotient map. An immediate corollary is a description of the curve classes for which the Gopakumar--Vafa invariants of geometric crepant resolutions vanish, generalising a recent result of Nabijou--Wemyss to the nonisolated setting. We furthermore formulate a method for finding symmetries among the non-vanishing invariants using derived equivalences, and show how this can be applied to line bundle twists and mutation functors in some settings. In particular, we find new wall-crossing relations among the Gopakumar-Vafa invariants of the different crepant resolutions of a cDV singularity.