Justification of the wall fixed-point prescription by an equivariant family index theorem

Prove, using the equivariant families APS theorem and an appropriate adiabatic-limit argument, that the equivariant family eta invariant of the Lens-space fibration over the nine-dimensional M9-wall boundary is represented by the non-identity fixed-point sum used to define the wall anomaly prescription.

Background

The paper computes the non-identity ALE fixed-point contribution on the M9 wall by using a degree-eight characteristic-class expression involving the normal factor K_j(r) and the equivariant Chern character of the Hořava–Witten virtual fermion bundle. This expression reproduces the Kac-label-dependent remainder of the unflavored tensor-branch anomaly polynomial for A-type orbi-instanton theories.

The authors explicitly distinguish this successful algebraic match from a derivation of the physical boundary spectral problem. The actual APS boundary operator acts on the full nine-dimensional space W_6 × L(k,1), whereas the calculation uses the vertical three-dimensional Lens-space operator. Establishing the correspondence requires the equivariant families APS theorem and an adiabatic-limit analysis, which the paper leaves unresolved.