On 2-loop AdS$_2$ superstring partition function in AdS$_5 \times S^5$ and AdS$_4\times CP^3$

Abstract

We revisit the 2-loop computation of the partition function of the GS superstring expanded near the AdS$_2$ surface in AdS$_5\times S^5$ and AdS$_4\times {\rm CP}^3$, dual to $\frac{1}{2}$ BPS circular Wilson loop in ${\cal N}=4$ SYM and ABJM theories. In arXiv:2601.08809 it was done in AdS$_5\times S^5$ 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 ${\rm Str}\,E$ 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, ${\rm Str}\,E=0$ and the 2-loop coefficient $f_2$ vanishes, in tension with $f_2=\frac{3}{32π^2}$ in SYM theory. In the AdS$_4\times{\rm CP}^3$ 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, $f_2=\frac{3}{32π^2}$ instead of $f_2=0$ required by localization. This term is the level-zero (string) part of the result of arXiv:2609.14497 for the AdS$_2\times S^1$ M2 brane in AdS$_4\times S^7/\mathbb Z_k$, where it is cancelled by the higher $S^1$ modes, which also remove a finite mass shift of the massless fermion in conflict with the AdS$_2$ 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 AdS$_5\times S^5$ case it is still to be identified and may resolve the remaining 2-loop mismatch with localization.