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

  • Arkady A. Tseytlin
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Abstract

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 cal 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.

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Cite

BibTeXRIS
BibTeX
@misc{tseytlin20262loop,
  title = {{On 2-loop AdS$_2$ superstring partition function in AdS$_5 \times S^5$ and AdS$_4\times CP^3$}},
  author = {Arkady A. Tseytlin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37096},
  eprint = {2610.37096},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37096},
}
RIS
TY  - UNPB
TI  - On 2-loop AdS_2 superstring partition function in AdS_5 × S^5 and AdS_4× CP^3
AU  - Arkady A. Tseytlin
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37096
UR  - https://arxiv.org/abs/2610.37096
AB  - We revisit the 2-loop computation of the partition function of the GS superstring expanded near the AdS_2 surface in AdS_5× S^5 and AdS_4× CP^3, dual to 1/2 BPS circular Wilson loop in cal N=4 SYM and ABJM theories. In arXiv:2601.08809 it was done in AdS_5× 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 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 f_2 vanishes, in tension with f_2=3/32π^2 in SYM theory. In the AdS_4×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=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× S^1 M2 brane in AdS_4× S^7/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× S^5 case it is still to be identified and may resolve the remaining 2-loop mismatch with localization.
ER  -