Holographic Tests of the $μ$ Ensemble

We study in detail a recent proposal stating that M-theory observables arising from the quantisation of M2-branes are naturally computed in a fixed $A_3$ ensemble. In a holographic setting this implies that in a semiclassical approximation 11d bulk observables in an asymptotically AdS$_4$ background are computed in a grand canonical ensemble of fixed chemical potential $\mu$ conjugate to the integer $N$ determining the rank of the gauge group in the dual CFT. We provide detailed precision tests of holography in the fixed $\mu$ ensemble by focusing on the two leading terms in the derivative expansion of the 11d theory. In addition, we study one-loop logarithmic corrections which we compute in detail. For asymptotically AdS$_4\times S^7/\mathbf{Z}_k$ 11d backgrounds our results are in agreement with supersymmetric localisation calculations in the dual ABJM SCFT. Moreover, we employ the logarithmic corrections in the $\mu$ ensemble to determine the measure of the integral Laplace transform and use this to compute dual SCFT supersymmetric partition functions, like the superconformal index and the squashed $S^3$ partition function, to all orders in the perturbative $1/N$ expansion in terms of Airy functions in harmony with results and conjectures in the dual SCFT.