Establish nonzero-mode determinant cancellation on Eguchi–Hanson space

Calculate the regulated relative nonzero-mode determinants, equivalently the relevant heat kernels or scattering phase shifts, to determine whether supersymmetric pairing yields a unit determinant ratio for the Eguchi–Hanson instanton backgrounds.

Background

The $SU(2)$ and two-block $SU(4)$ calculations assume cancellation between paired nonzero bosonic and fermionic fluctuations. Because Eguchi–Hanson space is noncompact, the fluctuation operators have continuous spectra, and formal supersymmetric pairing does not guarantee equality of their regulated spectral densities or scattering data.

An explicit determinant computation is therefore needed to justify the assumed cancellation and to determine whether finite or logarithmic contributions remain. This issue is especially important for the higher-rank single-root saddle, where continuum fluctuations are required by renormalization-group consistency.