On 2-loop AdS₂ superstring partition function in AdS₅ × S⁵ and AdS₄× CP³

We revisit the 2-loop computation of the partition function of the GS superstring expanded near the AdS₂ surface in AdS₅× S⁵ and AdS₄× CP³, dual to 1/2 BPS circular Wilson loop in N=4 SYM and ABJM theories. In arXiv:2601.08809 it was done in AdS₅× S⁵ using dimensional regularization and a UV pole was found. Since dimensional regularization is problematic for the GS string, here we use a heat-kernel cutoff instead. Then the contribution of the local path integral measure, proportional to $δ(0)$, can no longer be ignored. We show that it cancels the $δ(0)$ terms coming from the quartic interactions in the action, leaving the square of the difference Str\,E between the regularized numbers of bosonic and fermionic modes. The result is thus UV finite, the pole of arXiv:2601.08809 being an artefact of dimensional regularization. If all fields are regularized with the same smearing kernel, Str\,E=0 and the 2-loop coefficient f₂ vanishes, in tension with f₂=3/32π² in SYM theory. In the AdS₄× CP³ case the same cancellation leaves a finite term proportional to the square of the coincident value of the massless 2d fermion Green's function, giving, ironically, f₂=3/32π² instead of f₂=0 required by localization. This term is the level-zero (string) part of the result of arXiv:2609.14497 for the AdS₂× S¹ M2 brane in AdS₄× S⁷/ Zₖ, where it is cancelled by the higher S¹ modes, which also remove a finite mass shift of the massless fermion in conflict with the AdS₂ supersymmetry. This suggests that the type II string path integral measure should contain an extra contribution, supplied in the IIA case by the parent M2 brane; in the AdS₅× S⁵ case it is still to be identified and may resolve the remaining 2-loop mismatch with localization.