One-loop divergences for KK theories on $\mathrm{AdS}\times S$ spaces; a reanalysis of $\mathrm{AdS}_4 \times S^7\,\big/$ ABJM precision holography

Abstract

We provide a systematic framework for computing the logarithmically divergent part of one-loop partition functions on product spaces $\mathrm{AdS}_{d_A} \times S^{d_S}$ of arbitrary dimension. By expanding the higher-dimensional kinetic operators in spherical harmonics, we reduce the ($d_A+d_S$)-dimensional spectral problem to an infinite tower of $d_A$-dimensional determinants, which are then represented via spectral $ζ$-function methods. We isolate the logarithmic divergences arising from the interplay between the individual AdS determinants and the infinite Kaluza-Klein sum, carefully accounting for the contributions of zero modes on the sphere that produce additional AdS determinants. We test this framework on different fields and apply it to the complete multiplet of 11-dimensional supergravity on $\mathrm{AdS}_4 \times S^7$. We recover in a 4d language the result of arXiv:1210.6057, namely that the only non-vanishing logarithmic divergence originates entirely from the 2-form AdS mode in the ghost sector, reproducing the well-known $\frac{1}{4}\log N$ correction to the ABJM free energy predicted by supersymmetric localization.