Large N expansion of superconformal index of k=1 ABJM theory and semiclassical M5 brane partition function

It was shown in arXiv:2309.10786 that the leading non-perturbative contribution to the large $N$ expansion of superconformal index of (2,0) 6d theory (which describes low-energy dynamics of $N$ coincident M5 branes) is reproduced by the semiclassical partition function of quantum M2 brane wrapped on $S^1\times S^2$ in a twisted version of AdS${}_7\times S^4$ background. Here we demonstrate an analogous relation for the leading non-perturbative contribution to the large $N$ expansion of the superconformal index of the $n=8$ supersymmetric level-one $U(N)\times U(N)$ ABJM theory (which describes low-energy dynamics of $N$ coincident M2 branes). The roles of M2 and M5 branes get effectively interchanged. Namely, the large $N$ correction to the ABJM index is found to be given by the semiclassical partition function of quantum M5 brane wrapped on $S^1\times S^5$ in a twisted version of AdS${}_4\times S^7$ background. This effectively confirms the suggestion for the “M5 brane index” made in arXiv:2007.05213 on the basis of indirect superconformal algebra considerations.