Integrating massive modes for supergraviton (D0 brane) scattering

From this paper, I want to integrate out the massive degrees of freedom. When we expand the action about a classical background, diagonalizing the mass matrix gives us particles with the following masses squared:

1. 16 bosons with $m^2 = r^{2}$
2. 2 bosons with $m^2 = r^2 + 2v$
3. 2 bosons with $m^2 = r^2 – 2v$
4. 10 massless modes
5. 8 real fermions with $m^2 = r^2 + v$
6. 8 real fermions with $m^2 = r^2 – v$
7. 2 ghost bosons with $m^2 = r^2$
8. 1 massless ghost boson

For each bosonic field, integrating out yields an inverse square root of the determinant. For each fermionic field, integration gives the determinant. I’m not sure how to integrate a ghost particle. From integrating out the bosons of mass squared $r^{2} pm 2v$, I get the correct determinants $det^{-1}(-partial^{2}_{tau} + r^{2} – 2v) det^{-1}(-partial^{2}_{tau} + r^{2} + 2v)$. However, I am unable to work out the following factors: $det^{-6}(-partial^{2}_{tau} + r^{2}) det^{4}(-partial^{2}_{tau} + r^{2} + v) det^{4}(-partial^{2}_{tau} + r^{2} – v)$. I’m not sure what I’m missing here.

Furthermore, how does fact that the sum of the powers is zero follow from
super symmetry?