On Some Computations of Higher Rank Refined Donaldson-Thomas Invariants

TL;DR

This paper computes higher rank refined Donaldson-Thomas invariants for local curve geometries, deriving and verifying a refined wall-crossing formula, and applying it to specific cases with higher D6 or D4 ranks.

Contribution

It introduces a refined wall-crossing formula for higher D6 and D4 ranks and applies it to compute invariants on local rational curves.

Findings

Refined wall-crossing formula derived and verified.

Computed invariants for specific local curve geometries.

Confirmed agreement with existing unrefined formulas.

Abstract

We present some computations of higher rank refined Donaldson-Thomas invariants on local curve geometries, corresponding to local D6-D2-D0 or D4-D2-D0 configurations. A refined wall-crossing formula for invariants with higher D6 or D4 ranks is derived and verified to agree with the existing formulas under the unrefined limit. Using the formula, refined invariants on the $(-1,-1)$ and $(-2,0)$ local rational curve with higher D6 or D4 ranks are computed.