Framed motivic Donaldson-Thomas invariants of small crepant resolutions

TL;DR

This paper computes higher rank framed motivic Donaldson-Thomas invariants for small crepant resolutions of toric Calabi-Yau 3-folds, extending wall-crossing formulas and connecting with refined DT invariants.

Contribution

It introduces a higher rank version of motivic DT/PT invariants and establishes their wall-crossing behavior for small crepant resolutions.

Findings

Derived explicit formulas for r-framed motivic PT and DT invariants.

Extended the motivic DT/PT wall-crossing to higher ranks.

Connected the results with the current development of refined DT invariants.

Abstract

For an arbitrary integer $r\geq 1$, we compute $r$-framed motivic PT and DT invariants of small crepant resolutions of toric Calabi-Yau $3$-folds, establishing a "higher rank" version of the motivic DT/PT wall-crossing formula. This generalises the work of Morrison and Nagao. Our formulae, in particular their relationship with the $r=1$ theory, fit nicely in the current development of higher rank refined DT invariants.