On Some Computations of Higher Rank Refined Donaldson-Thomas Invariants

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.