Vanishing and Symmetries of BPS Invariants for CDV Singularities

TL;DR

This paper investigates the vanishing and symmetry properties of BPS invariants in the context of CDV singularities, revealing their dependence on Dynkin combinatorics and deriving new wall-crossing relations among Gopakumar-Vafa invariants.

Contribution

It establishes a link between motivic BPS invariants and Dynkin combinatorics for CDV singularities, and introduces a method to find symmetries among invariants via derived equivalences.

Findings

BPS invariants vanish for certain dimension vectors not aligned with restricted roots.

Describes when Gopakumar--Vafa invariants vanish in geometric crepant resolutions.

Identifies new wall-crossing relations among invariants of different crepant resolutions.

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.